lenstronomy.LensModel.Profiles package

Submodules

lenstronomy.LensModel.Profiles.arc_perturbations module

class lenstronomy.LensModel.Profiles.arc_perturbations.ArcPerturbations[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

uses radial and tangential fourier modes within a specific range in both directions to perturb a lensing potential

derivatives(x, y, coeff, d_r, d_phi, center_x, center_y)[source]
Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • coeff – float, amplitude of basis
  • d_r – period of radial sinusoidal in units of angle
  • d_phi – period of tangential sinusoidal in radian
  • center_x – center of rotation for tangential basis
  • center_y – center of rotation for tangential basis
Returns:

f_x, f_y

function(x, y, coeff, d_r, d_phi, center_x, center_y)[source]
Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • coeff – float, amplitude of basis
  • d_r – period of radial sinusoidal in units of angle
  • d_phi – period of tangential sinusoidal in radian
  • center_x – center of rotation for tangential basis
  • center_y – center of rotation for tangential basis
Returns:

hessian(x, y, coeff, d_r, d_phi, center_x, center_y)[source]
Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • coeff – float, amplitude of basis
  • d_r – period of radial sinusoidal in units of angle
  • d_phi – period of tangential sinusoidal in radian
  • center_x – center of rotation for tangential basis
  • center_y – center of rotation for tangential basis
Returns:

f_xx, f_yy, f_xy

lenstronomy.LensModel.Profiles.base_profile module

class lenstronomy.LensModel.Profiles.base_profile.LensProfileBase(*args, **kwargs)[source]

Bases: object

this class acts as the base class of all lens model functions and indicates raise statements and default outputs if these functions are not defined in the specific lens model class

density_lens(*args, **kwargs)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:kwargs – keywords of the profile
Returns:raise as definition is not defined
derivatives(*args, **kwargs)[source]

deflection angles

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

function(*args, **kwargs)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(*args, **kwargs)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

mass_3d_lens(*args, **kwargs)[source]

mass enclosed a 3d sphere or radius r given a lens parameterization with angular units For this profile those are identical.

Parameters:kwargs – keywords of the profile
Returns:raise as definition is not defined
set_dynamic()[source]
Returns:no return, deletes pre-computed variables for certain lens models
set_static(**kwargs)[source]

pre-computes certain computations that do only relate to the lens model parameters and not to the specific position where to evaluate the lens model

Parameters:kwargs – lens model parameters
Returns:no return, for certain lens model some private self variables are initiated

lenstronomy.LensModel.Profiles.chameleon module

class lenstronomy.LensModel.Profiles.chameleon.Chameleon(static=False)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class of the Chameleon model (See Suyu+2014) an elliptical truncated double isothermal profile

density_lens(r, alpha_1, w_c, w_t, e1=0, e2=0, center_x=0, center_y=0)[source]

spherical average density as a function of 3d radius

Parameters:
  • r – 3d radius
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • w_c – see Suyu+2014
  • w_t – see Suyu+2014
  • e1 – ellipticity parameter
  • e2 – ellipticity parameter
  • center_x – ra center
  • center_y – dec center
Returns:

matter density at 3d radius r

derivatives(x, y, alpha_1, w_c, w_t, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – ra-coordinate
  • y – dec-coordinate
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • w_c – see Suyu+2014
  • w_t – see Suyu+2014
  • e1 – ellipticity parameter
  • e2 – ellipticity parameter
  • center_x – ra center
  • center_y – dec center
Returns:

deflection angles (RA, DEC)

function(x, y, alpha_1, w_c, w_t, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – ra-coordinate
  • y – dec-coordinate
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • w_c – see Suyu+2014
  • w_t – see Suyu+2014
  • e1 – ellipticity parameter
  • e2 – ellipticity parameter
  • center_x – ra center
  • center_y – dec center
Returns:

lensing potential

hessian(x, y, alpha_1, w_c, w_t, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – ra-coordinate
  • y – dec-coordinate
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • w_c – see Suyu+2014
  • w_t – see Suyu+2014
  • e1 – ellipticity parameter
  • e2 – ellipticity parameter
  • center_x – ra center
  • center_y – dec center
Returns:

second derivatives of the lensing potential (Hessian: f_xx, f_xy, f_yx, f_yy)

lower_limit_default = {'alpha_1': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.8, 'e2': -0.8, 'w_c': 0, 'w_t': 0}
mass_3d_lens(r, alpha_1, w_c, w_t, e1=0, e2=0, center_x=0, center_y=0)[source]

mass enclosed 3d radius

Parameters:
  • r – 3d radius
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • w_c – see Suyu+2014
  • w_t – see Suyu+2014
  • e1 – ellipticity parameter
  • e2 – ellipticity parameter
  • center_x – ra center
  • center_y – dec center
Returns:

mass enclosed 3d radius r

param_convert(alpha_1, w_c, w_t, e1, e2)[source]

convert the parameter alpha_1 (deflection angle one arcsecond from the center) into the “Einstein radius” scale parameter of the two NIE profiles

Parameters:
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • w_c – see Suyu+2014
  • w_t – see Suyu+2014
  • e1 – eccentricity modulus
  • ee – eccentricity modulus
Returns:

param_names = ['alpha_1', 'w_c', 'w_t', 'e1', 'e2', 'center_x', 'center_y']
set_dynamic()[source]
Returns:
set_static(alpha_1, w_c, w_t, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • alpha_1
  • w_c
  • w_t
  • e1
  • e2
  • center_x
  • center_y
Returns:

upper_limit_default = {'alpha_1': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.8, 'e2': 0.8, 'w_c': 100, 'w_t': 100}
class lenstronomy.LensModel.Profiles.chameleon.DoubleChameleon[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class of the Chameleon model (See Suyu+2014) an elliptical truncated double isothermal profile

density_lens(r, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]
Parameters:
  • r – 3d radius
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
  • w_c1 – Suyu+2014 for first profile
  • w_t1 – Suyu+2014 for first profile
  • e11 – ellipticity parameter for first profile
  • e21 – ellipticity parameter for first profile
  • w_c2 – Suyu+2014 for second profile
  • w_t2 – Suyu+2014 for second profile
  • e12 – ellipticity parameter for second profile
  • e22 – ellipticity parameter for second profile
  • center_x – ra center
  • center_y – dec center
Returns:

3d density at radius r

derivatives(x, y, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]
Parameters:
  • x – ra-coordinate
  • y – dec-coordinate
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
  • w_c1 – Suyu+2014 for first profile
  • w_t1 – Suyu+2014 for first profile
  • e11 – ellipticity parameter for first profile
  • e21 – ellipticity parameter for first profile
  • w_c2 – Suyu+2014 for second profile
  • w_t2 – Suyu+2014 for second profile
  • e12 – ellipticity parameter for second profile
  • e22 – ellipticity parameter for second profile^V
  • center_x – ra center
  • center_y – dec center
Returns:

deflection angles (RA, DEC)

function(x, y, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]
Parameters:
  • x – ra-coordinate
  • y – dec-coordinate
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
  • w_c1 – Suyu+2014 for first profile
  • w_t1 – Suyu+2014 for first profile
  • e11 – ellipticity parameter for first profile
  • e21 – ellipticity parameter for first profile
  • w_c2 – Suyu+2014 for second profile
  • w_t2 – Suyu+2014 for second profile
  • e12 – ellipticity parameter for second profile
  • e22 – ellipticity parameter for second profile
  • center_x – ra center
  • center_y – dec center
Returns:

lensing potential

hessian(x, y, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]
Parameters:
  • x – ra-coordinate
  • y – dec-coordinate
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
  • w_c1 – Suyu+2014 for first profile
  • w_t1 – Suyu+2014 for first profile
  • e11 – ellipticity parameter for first profile
  • e21 – ellipticity parameter for first profile
  • w_c2 – Suyu+2014 for second profile
  • w_t2 – Suyu+2014 for second profile
  • e12 – ellipticity parameter for second profile
  • e22 – ellipticity parameter for second profile
  • center_x – ra center
  • center_y – dec center
Returns:

second derivatives of the lensing potential (Hessian: f_xx, f_yy, f_xy)

lower_limit_default = {'alpha_1': 0, 'center_x': -100, 'center_y': -100, 'e11': -0.8, 'e12': -0.8, 'e21': -0.8, 'e22': -0.8, 'ratio': 0, 'w_c1': 0, 'w_c2': 0, 'w_t1': 0, 'w_t2': 0}
mass_3d_lens(r, alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]
Parameters:
  • r – 3d radius
  • alpha_1 – deflection angle at 1 (arcseconds) from the center
  • ratio – ratio of deflection amplitude at radius = 1 of the first to second Chameleon profile
  • w_c1 – Suyu+2014 for first profile
  • w_t1 – Suyu+2014 for first profile
  • e11 – ellipticity parameter for first profile
  • e21 – ellipticity parameter for first profile
  • w_c2 – Suyu+2014 for second profile
  • w_t2 – Suyu+2014 for second profile
  • e12 – ellipticity parameter for second profile
  • e22 – ellipticity parameter for second profile
  • center_x – ra center
  • center_y – dec center
Returns:

mass enclosed 3d radius

param_names = ['alpha_1', 'ratio', 'w_c1', 'w_t1', 'e11', 'e21', 'w_c2', 'w_t2', 'e12', 'e22', 'center_x', 'center_y']
set_dynamic()[source]
Returns:no return, deletes pre-computed variables for certain lens models
set_static(alpha_1, ratio, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]

pre-computes certain computations that do only relate to the lens model parameters and not to the specific position where to evaluate the lens model

Parameters:kwargs – lens model parameters
Returns:no return, for certain lens model some private self variables are initiated
upper_limit_default = {'alpha_1': 100, 'center_x': 100, 'center_y': 100, 'e11': 0.8, 'e12': 0.8, 'e21': 0.8, 'e22': 0.8, 'ratio': 100, 'w_c1': 100, 'w_c2': 100, 'w_t1': 100, 'w_t2': 100}
class lenstronomy.LensModel.Profiles.chameleon.TripleChameleon[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class of the Chameleon model (See Suyu+2014) an elliptical truncated double isothermal profile

density_lens(r, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]
Parameters:
  • r – 3d radius
  • alpha_1
  • ratio12 – ratio of first to second amplitude
  • ratio13 – ratio of first to third amplitude
  • w_c1
  • w_t1
  • e11
  • e21
  • w_c2
  • w_t2
  • e12
  • e22
  • center_x
  • center_y
Returns:

density at radius r (spherical average)

derivatives(x, y, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]
Parameters:
  • alpha_1
  • ratio12 – ratio of first to second amplitude
  • ratio13 – ratio of first to third amplidute
  • w_c1
  • w_t1
  • e11
  • e21
  • w_c2
  • w_t2
  • e12
  • e22
  • center_x
  • center_y
Returns:

function(x, y, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]
Parameters:
  • alpha_1
  • ratio12 – ratio of first to second amplitude
  • ratio13 – ratio of first to third amplitude
  • w_c1
  • w_t1
  • e11
  • e21
  • w_c2
  • w_t2
  • e12
  • e22
  • center_x
  • center_y
Returns:

hessian(x, y, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]
Parameters:
  • alpha_1
  • ratio12 – ratio of first to second amplitude
  • ratio13 – ratio of first to third amplidute
  • w_c1
  • w_t1
  • e11
  • e21
  • w_c2
  • w_t2
  • e12
  • e22
  • center_x
  • center_y
Returns:

lower_limit_default = {'alpha_1': 0, 'center_x': -100, 'center_y': -100, 'e11': -0.8, 'e12': -0.8, 'e13': -0.8, 'e21': -0.8, 'e22': -0.8, 'e23': -0.8, 'ratio12': 0, 'ratio13': 0, 'w_c1': 0, 'w_c2': 0, 'w_c3': 0, 'w_t1': 0, 'w_t2': 0, 'w_t3': 0}
mass_3d_lens(r, alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]
Parameters:
  • r – 3d radius
  • alpha_1
  • ratio12 – ratio of first to second amplitude
  • ratio13 – ratio of first to third amplitude
  • w_c1
  • w_t1
  • e11
  • e21
  • w_c2
  • w_t2
  • e12
  • e22
  • center_x
  • center_y
Returns:

mass enclosed 3d radius

param_names = ['alpha_1', 'ratio12', 'ratio13', 'w_c1', 'w_t1', 'e11', 'e21', 'w_c2', 'w_t2', 'e12', 'e22', 'w_c3', 'w_t3', 'e13', 'e23', 'center_x', 'center_y']
set_dynamic()[source]
Returns:no return, deletes pre-computed variables for certain lens models
set_static(alpha_1, ratio12, ratio13, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, w_c3, w_t3, e13, e23, center_x=0, center_y=0)[source]

pre-computes certain computations that do only relate to the lens model parameters and not to the specific position where to evaluate the lens model

Parameters:kwargs – lens model parameters
Returns:no return, for certain lens model some private self variables are initiated
upper_limit_default = {'alpha_1': 100, 'center_x': 100, 'center_y': 100, 'e11': 0.8, 'e12': 0.8, 'e13': 0.8, 'e21': 0.8, 'e22': 0.8, 'e23': 0.8, 'ratio12': 100, 'ratio13': 100, 'w_c1': 100, 'w_c2': 100, 'w_c3': 100, 'w_t1': 100, 'w_t2': 100, 'w_t3': 100}
class lenstronomy.LensModel.Profiles.chameleon.DoubleChameleonPointMass[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class of the Chameleon model (See Suyu+2014) an elliptical truncated double isothermal profile

derivatives(x, y, alpha_1, ratio_pointmass, ratio_chameleon, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]
Parameters:
  • x
  • y
  • alpha_1
  • ratio_pointmass – ratio of point source Einstein radius to combined Chameleon deflection angle at r=1
  • ratio_chameleon – ratio in deflection angles at r=1 for the two Chameleon profiles
  • w_c1 – Suyu+2014 for first profile
  • w_t1 – Suyu+2014 for first profile
  • e11 – ellipticity parameter for first profile
  • e21 – ellipticity parameter for first profile
  • w_c2 – Suyu+2014 for second profile
  • w_t2 – Suyu+2014 for second profile
  • e12 – ellipticity parameter for second profile
  • e22 – ellipticity parameter for second profile
  • center_x – ra center
  • center_y – dec center
Returns:

function(x, y, alpha_1, ratio_pointmass, ratio_chameleon, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]

#TODO chose better parameterization for combining point mass and Chameleon profiles :param x: ra-coordinate :param y: dec-coordinate :param alpha_1: deflection angle at 1 (arcseconds) from the center :param ratio_pointmass: ratio of point source Einstein radius to combined Chameleon deflection angle at r=1 :param ratio_chameleon: ratio in deflection angles at r=1 for the two Chameleon profiles :param w_c1: Suyu+2014 for first profile :param w_t1: Suyu+2014 for first profile :param e11: ellipticity parameter for first profile :param e21: ellipticity parameter for first profile :param w_c2: Suyu+2014 for second profile :param w_t2: Suyu+2014 for second profile :param e12: ellipticity parameter for second profile :param e22: ellipticity parameter for second profile :param center_x: ra center :param center_y: dec center :return:

hessian(x, y, alpha_1, ratio_pointmass, ratio_chameleon, w_c1, w_t1, e11, e21, w_c2, w_t2, e12, e22, center_x=0, center_y=0)[source]
Parameters:
  • x
  • y
  • alpha_1
  • ratio_pointmass – ratio of point source Einstein radius to combined Chameleon deflection angle at r=1
  • ratio_chameleon – ratio in deflection angles at r=1 for the two Chameleon profiles
  • w_c1 – Suyu+2014 for first profile
  • w_t1 – Suyu+2014 for first profile
  • e11 – ellipticity parameter for first profile
  • e21 – ellipticity parameter for first profile
  • w_c2 – Suyu+2014 for second profile
  • w_t2 – Suyu+2014 for second profile
  • e12 – ellipticity parameter for second profile
  • e22 – ellipticity parameter for second profile
  • center_x – ra center
  • center_y – dec center
Returns:

lower_limit_default = {'alpha_1': 0, 'center_x': -100, 'center_y': -100, 'e11': -0.8, 'e12': -0.8, 'e21': -0.8, 'e22': -0.8, 'ratio_chameleon': 0, 'ratio_pointmass': 0, 'w_c1': 0, 'w_c2': 0, 'w_t1': 0, 'w_t2': 0}
param_names = ['alpha_1', 'ratio_chameleon', 'ratio_pointmass', 'w_c1', 'w_t1', 'e11', 'e21', 'w_c2', 'w_t2', 'e12', 'e22', 'center_x', 'center_y']
upper_limit_default = {'alpha_1': 100, 'center_x': 100, 'center_y': 100, 'e11': 0.8, 'e12': 0.8, 'e21': 0.8, 'e22': 0.8, 'ratio_chameleon': 100, 'ratio_pointmass': 100, 'w_c1': 100, 'w_c2': 100, 'w_t1': 100, 'w_t2': 100}

lenstronomy.LensModel.Profiles.cnfw module

class lenstronomy.LensModel.Profiles.cnfw.CNFW[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class computes the lensing quantities of a cored NFW profile: rho = rho0 * (r + r_core)^-1 * (r + rs)^-2 alpha_Rs is the normalization equivalent to the deflection angle at rs in the absence of a core

alpha_r(R, Rs, rho0, r_core)[source]

deflection angel of NFW profile along the radial direction

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
Returns:

Epsilon(R) projected density at radius R

cnfwGamma(R, Rs, rho0, r_core, ax_x, ax_y)[source]

shear gamma of NFW profile (times Sigma_crit) along the projection to coordinate ‘axis’

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r200 (float>0) – radius of (sub)halo
  • axis (same as R) – projection to either x- or y-axis
Returns:

Epsilon(R) projected density at radius R

density(R, Rs, rho0, r_core)[source]

three dimensional truncated NFW profile

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (central core density)
Returns:

rho(R) density

density_2d(x, y, Rs, rho0, r_core, center_x=0, center_y=0)[source]

projected two dimenstional NFW profile (kappa*Sigma_crit)

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r200 (float>0) – radius of (sub)halo
Returns:

Epsilon(R) projected density at radius R

density_lens(R, Rs, alpha_Rs, r_core)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

derivatives(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]

deflection angles

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

function(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]
Parameters:
  • x – angular position
  • y – angular position
  • Rs – angular turn over point
  • alpha_Rs – deflection at Rs (in the absence of a core
  • r_core – core radius
  • center_x – center of halo
  • center_y – center of halo
Returns:

hessian(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy

lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'r_core': 0}
mass_2d(R, Rs, rho0, r_core)[source]

analytic solution of the projection integral (convergence)

Parameters:x (float >0) – R/Rs
mass_3d(R, Rs, rho0, r_core)[source]

mass enclosed a 3d sphere or radius r

Parameters:
  • r
  • Ra
  • Rs
Returns:

mass_3d_lens(R, Rs, alpha_Rs, r_core)[source]

mass enclosed a 3d sphere or radius r given a lens parameterization with angular units

Returns:
model_name = 'CNFW'
param_names = ['Rs', 'alpha_Rs', 'r_core', 'center_x', 'center_y']
upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'r_core': 100}

lenstronomy.LensModel.Profiles.cnfw_ellipse module

class lenstronomy.LensModel.Profiles.cnfw_ellipse.CNFW_ELLIPSE[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions concerning the NFW profile

relation are: R_200 = c * Rs

density_lens(R, Rs, alpha_Rs, r_core, e1=0, e2=0)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

derivatives(x, y, Rs, alpha_Rs, r_core, e1, e2, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function (integral of NFW)

function(x, y, Rs, alpha_Rs, r_core, e1, e2, center_x=0, center_y=0)[source]

returns double integral of NFW profile

hessian(x, y, Rs, alpha_Rs, r_core, e1, e2, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy

lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'r_core': 0}
mass_3d_lens(R, Rs, alpha_Rs, r_core, e1=0, e2=0)[source]

mass enclosed a 3d sphere or radius r given a lens parameterization with angular units

Returns:
param_names = ['Rs', 'alpha_Rs', 'r_core', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'r_core': 100}

lenstronomy.LensModel.Profiles.const_mag module

class lenstronomy.LensModel.Profiles.const_mag.ConstMag(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class implements the macromodel potential of Diego et al. Convergence and shear are computed according to Diego2018

derivatives(x, y, mu_r, mu_t, parity, phi_G, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • mu_r – radial magnification
  • mu_t – tangential magnification
  • parity – parity of the side of the macromodel. Either +1 (positive parity) or -1 (negative parity)
  • phi_G – shear orientation angle (relative to the x-axis)
Returns:

deflection angle (in angles)

function(x, y, mu_r, mu_t, parity, phi_G, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • mu_r – radial magnification
  • mu_t – tangential magnification
  • parity – parity side of the macromodel. Either +1 (positive parity) or -1 (negative parity)
  • phi_G – shear orientation angle (relative to the x-axis)
Returns:

lensing potential

hessian(x, y, mu_r, mu_t, parity, phi_G, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • mu_r – radial magnification
  • mu_t – tangential magnification
  • parity – parity of the side of the macromodel. Either +1 (positive parity) or -1 (negative parity)
  • phi_G – shear orientation angle (relative to the x-axis)
Returns:

hessian matrix (in angles)

lower_limit_default = {'center_x': -100, 'center_y': -100, 'mu_r': 1, 'mu_t': 1000, 'parity': -1, 'phi_G': 0.0}
param_names = ['center_x', 'center_y', 'mu_r', 'mu_t', 'parity', 'phi_G']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'mu_r': 1, 'mu_t': 1000, 'parity': 1, 'phi_G': 3.141592653589793}

lenstronomy.LensModel.Profiles.constant_shift module

class lenstronomy.LensModel.Profiles.constant_shift.Shift(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

Lens model with a constant shift of the deflection field

derivatives(x, y, alpha_x, alpha_y)[source]
Parameters:
  • x – coordinate in image plane (angle)
  • y – coordinate in image plane (angle)
  • alpha_x – shift in x-direction (angle)
  • alpha_y – shift in y-direction (angle)
Returns:

deflection in x- and y-direction

function(x, y, alpha_x, alpha_y)[source]
Parameters:
  • x – coordinate in image plane (angle)
  • y – coordinate in image plane (angle)
  • alpha_x – shift in x-direction (angle)
  • alpha_y – shift in y-direction (angle)
Returns:

lensing potential

hessian(x, y, alpha_x, alpha_y)[source]
Parameters:
  • x – coordinate in image plane (angle)
  • y – coordinate in image plane (angle)
  • alpha_x – shift in x-direction (angle)
  • alpha_y – shift in y-direction (angle)
Returns:

hessian elements f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'alpha_x': -1000, 'alpha_y': -1000}
param_names = ['alpha_x', 'alpha_y']
upper_limit_default = {'alpha_x': 1000, 'alpha_y': 1000}

lenstronomy.LensModel.Profiles.convergence module

class lenstronomy.LensModel.Profiles.convergence.Convergence(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

a single mass sheet (external convergence)

derivatives(x, y, kappa, ra_0=0, dec_0=0)[source]

deflection angle

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kappa – (external) convergence
Returns:

deflection angles (first order derivatives)

function(x, y, kappa, ra_0=0, dec_0=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kappa – (external) convergence
Returns:

lensing potential

hessian(x, y, kappa, ra_0=0, dec_0=0)[source]

Hessian matrix

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kappa – external convergence
  • ra_0 – zero point of polynomial expansion (no deflection added)
  • dec_0 – zero point of polynomial expansion (no deflection added)
Returns:

second order derivatives f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'dec_0': -100, 'kappa': -10, 'ra_0': -100}
model_name = 'CONVERGENCE'
param_names = ['kappa', 'ra_0', 'dec_0']
upper_limit_default = {'dec_0': 100, 'kappa': 10, 'ra_0': 100}

lenstronomy.LensModel.Profiles.coreBurkert module

class lenstronomy.LensModel.Profiles.coreBurkert.CoreBurkert(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

lensing properties of a modified Burkert profile with variable core size normalized by rho0, the central core density

cBurkGamma(R, Rs, rho0, r_core, ax_x, ax_y)[source]
Parameters:
  • R – projected distance
  • Rs – scale radius
  • rho0 – central core density
  • r_core – core radius
  • ax_x – x coordinate
  • ax_y – y coordinate
Returns:

cBurkPot(R, Rs, rho0, r_core)[source]
Parameters:
  • R – projected distance
  • Rs – scale radius
  • rho0 – central core density
  • r_core – core radius
coreBurkAlpha(R, Rs, rho0, r_core, ax_x, ax_y)[source]

deflection angle

Parameters:
  • R
  • Rs
  • rho0
  • r_core
  • ax_x
  • ax_y
Returns:

density(R, Rs, rho0, r_core)[source]

three dimensional cored Burkert profile

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – characteristic density
Returns:

rho(R) density

density_2d(x, y, Rs, rho0, r_core, center_x=0, center_y=0)[source]

projected two dimenstional core Burkert profile (kappa*Sigma_crit)

Parameters:
  • x – x coordinate
  • y – y coordinate
  • Rs – scale radius
  • rho0 – central core density
  • r_core – core radius
derivatives(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]

deflection angles :param x: x coordinate :param y: y coordinate :param Rs: scale radius :param rho0: central core density :param r_core: core radius :param center_x: :param center_y: :return:

function(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]
Parameters:
  • x – angular position
  • y – angular position
  • Rs – angular turn over point
  • alpha_Rs – deflection angle at Rs
  • center_x – center of halo
  • center_y – center of halo
Returns:

hessian(x, y, Rs, alpha_Rs, r_core, center_x=0, center_y=0)[source]
Parameters:
  • x – x coordinate
  • y – y coordinate
  • Rs – scale radius
  • rho0 – central core density
  • r_core – core radius
  • center_x
  • center_y
Returns:

lower_limit_default = {'Rs': 1, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'r_core': 0.5}
mass_2d(R, Rs, rho0, r_core)[source]

analytic solution of the projection integral (convergence)

Parameters:
  • R – projected distance
  • Rs – scale radius
  • rho0 – central core density
  • r_core – core radius
mass_3d(R, Rs, rho0, r_core)[source]
Parameters:
  • R – projected distance
  • Rs – scale radius
  • rho0 – central core density
  • r_core – core radius
param_names = ['Rs', 'alpha_Rs', 'r_core', 'center_x', 'center_y']
upper_limit_default = {'Rs': 100, 'alpha_Rs': 100, 'center_x': 100, 'center_y': 100, 'r_core': 50}

lenstronomy.LensModel.Profiles.cored_density module

class lenstronomy.LensModel.Profiles.cored_density.CoredDensity(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for a uniform cored density dropping steep in the outskirts This profile is e.g. featured in Blum et al. 2020 https://arxiv.org/abs/2001.07182v1 3d rho(r) = 2/pi * Sigma_crit R_c**3 * (R_c**2 + r**2)**(-2)

static alpha_r(r, sigma0, r_core)[source]

radial deflection angle of the cored density profile

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

deflection angle

static d_alpha_dr(r, sigma0, r_core)[source]

radial derivatives of the radial deflection angle

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

dalpha/dr

static density(r, sigma0, r_core)[source]

rho(r) = 2/pi * Sigma_crit R_c**3 * (R_c**2 + r**2)**(-2)

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

density at radius r

density_2d(x, y, sigma0, r_core, center_x=0, center_y=0)[source]

projected density at projected radius r

Parameters:
  • x – x-coordinate in angular units
  • y – y-coordinate in angular units
  • sigma0 – convergence in the core
  • r_core – core radius
  • center_x – center of the profile
  • center_y – center of the profile
Returns:

projected density

density_lens(r, sigma0, r_core)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

desnity at radius r

derivatives(x, y, sigma0, r_core, center_x=0, center_y=0)[source]

deflection angle of cored density profile

Parameters:
  • x – x-coordinate in angular units
  • y – y-coordinate in angular units
  • sigma0 – convergence in the core
  • r_core – core radius
  • center_x – center of the profile
  • center_y – center of the profile
Returns:

alpha_x, alpha_y at (x, y)

function(x, y, sigma0, r_core, center_x=0, center_y=0)[source]

potential of cored density profile

Parameters:
  • x – x-coordinate in angular units
  • y – y-coordinate in angular units
  • sigma0 – convergence in the core
  • r_core – core radius
  • center_x – center of the profile
  • center_y – center of the profile
Returns:

lensing potential at (x, y)

hessian(x, y, sigma0, r_core, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in angular units
  • y – y-coordinate in angular units
  • sigma0 – convergence in the core
  • r_core – core radius
  • center_x – center of the profile
  • center_y – center of the profile
Returns:

Hessian df/dxdx, df/dxdy, df/dydx, df/dydy at position (x, y)

static kappa_r(r, sigma0, r_core)[source]

convergence of the cored density profile. This routine is also for testing

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

convergence at r

lower_limit_default = {'center_x': -100, 'center_y': -100, 'r_core': 0, 'sigma0': -1}
mass_2d(r, sigma0, r_core)[source]

mass enclosed in cylinder of radius r

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

mass enclosed in cylinder of radius r

static mass_3d(r, sigma0, r_core)[source]

mass enclosed 3d radius

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

mass enclosed 3d radius

mass_3d_lens(r, sigma0, r_core)[source]

mass enclosed a 3d sphere or radius r given a lens parameterization with angular units For this profile those are identical.

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

mass enclosed 3d radius

param_names = ['sigma0', 'r_core', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'r_core': 100, 'sigma0': 10}

lenstronomy.LensModel.Profiles.cored_density_2 module

class lenstronomy.LensModel.Profiles.cored_density_2.CoredDensity2(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for a uniform cored density dropping steep in the outskirts credits for suggesting this profile goes to Kfir Blum

\[\rho(r) = 2/\pi * \Sigma_{\rm crit} R_c^2 * (R_c^2 + r^2)^{-3/2}\]

This profile drops like an NFW profile as math:rho(r)^{-3}.

static alpha_r(r, sigma0, r_core)[source]

radial deflection angle of the cored density profile

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

deflection angle

static d_alpha_dr(r, sigma0, r_core)[source]

radial derivatives of the radial deflection angle

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

dalpha/dr

static density(r, sigma0, r_core)[source]

rho(r) = 2/pi * Sigma_crit R_c**3 * (R_c**2 + r**2)**(-3/2)

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

density at radius r

density_2d(x, y, sigma0, r_core, center_x=0, center_y=0)[source]

projected density at projected radius r

Parameters:
  • x – x-coordinate in angular units
  • y – y-coordinate in angular units
  • sigma0 – convergence in the core
  • r_core – core radius
  • center_x – center of the profile
  • center_y – center of the profile
Returns:

projected density

density_lens(r, sigma0, r_core)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

density at radius r

derivatives(x, y, sigma0, r_core, center_x=0, center_y=0)[source]

deflection angle of cored density profile

Parameters:
  • x – x-coordinate in angular units
  • y – y-coordinate in angular units
  • sigma0 – convergence in the core
  • r_core – core radius
  • center_x – center of the profile
  • center_y – center of the profile
Returns:

alpha_x, alpha_y at (x, y)

function(x, y, sigma0, r_core, center_x=0, center_y=0)[source]

potential of cored density profile

Parameters:
  • x – x-coordinate in angular units
  • y – y-coordinate in angular units
  • sigma0 – convergence in the core
  • r_core – core radius
  • center_x – center of the profile
  • center_y – center of the profile
Returns:

lensing potential at (x, y)

hessian(x, y, sigma0, r_core, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in angular units
  • y – y-coordinate in angular units
  • sigma0 – convergence in the core
  • r_core – core radius
  • center_x – center of the profile
  • center_y – center of the profile
Returns:

Hessian df/dxdx, df/dxdy, df/dydx, df/dydy at position (x, y)

static kappa_r(r, sigma0, r_core)[source]

convergence of the cored density profile. This routine is also for testing

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

convergence at r

lower_limit_default = {'center_x': -100, 'center_y': -100, 'r_core': 0, 'sigma0': -1}
mass_2d(r, sigma0, r_core)[source]

mass enclosed in cylinder of radius r

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

mass enclosed in cylinder of radius r

static mass_3d(r, sigma0, r_core)[source]

mass enclosed 3d radius

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

mass enclosed 3d radius

mass_3d_lens(r, sigma0, r_core)[source]

mass enclosed a 3d sphere or radius r given a lens parameterization with angular units For this profile those are identical.

Parameters:
  • r – radius (angular scale)
  • sigma0 – convergence in the core
  • r_core – core radius
Returns:

mass enclosed 3d radius

model_name = 'CORED_DENSITY_2'
param_names = ['sigma0', 'r_core', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'r_core': 100, 'sigma0': 10}

lenstronomy.LensModel.Profiles.cored_density_exp module

class lenstronomy.LensModel.Profiles.cored_density_exp.CoredDensityExp(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions concerning an exponential cored density profile, namely

..math::
rho(r) = rho_0 exp(- (theta / theta_c)^2)
static alpha_radial(r, kappa_0, theta_c)[source]

returns the radial part of the deflection angle :param x: angular position (normally in units of arc seconds) :param y: angular position (normally in units of arc seconds) :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :return: radial deflection angle

density(R, kappa_0, theta_c)[source]

three dimensional density profile in angular units (rho0_physical = rho0_angular Sigma_crit / D_lens)

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
Returns:

rho(R) density

density_2d(x, y, kappa_0, theta_c, center_x=0, center_y=0)[source]

projected two dimensional ULDM profile (convergence * Sigma_crit), but given our units convention for rho0, it is basically the convergence

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
Returns:

Epsilon(R) projected density at radius R

density_lens(r, kappa_0, theta_c)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
Returns:

density rho(r)

derivatives(x, y, kappa_0, theta_c, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function (lensing potential), which are the deflection angles

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

deflection angle in x, deflection angle in y

function(x, y, kappa_0, theta_c, center_x=0, center_y=0)[source]
Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

lensing potential (in arcsec^2)

hessian(x, y, kappa_0, theta_c, center_x=0, center_y=0)[source]
Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

kappa_r(R, kappa_0, theta_c)[source]

convergence of the cored density profile. This routine is also for testing

Parameters:
  • R – radius (angular scale)
  • kappa_0 – convergence in the core
  • theta_c – core radius
Returns:

convergence at r

lower_limit_default = {'center_x': -100, 'center_y': -100, 'kappa_0': 0, 'theta_c': 0}
mass_2d(R, kappa_0, theta_c)[source]

mass enclosed a 2d sphere of radius r returns

\[M_{2D} = 2 \pi \int_0^r dr' r' \int dz \rho(\sqrt(r'^2 + z^2))\]
Parameters:
  • kappa_0 – central convergence of soliton
  • theta_c – core radius (in arcsec)
Returns:

M_2D (ULDM only)

static mass_3d(R, kappa_0, theta_c)[source]

mass enclosed a 3d sphere or radius r :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param R: radius in arcseconds :return: mass of soliton in angular units

mass_3d_lens(r, kappa_0, theta_c)[source]

mass enclosed a 3d sphere or radius r :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param center_x: center of halo (in angular units) :param center_y: center of halo (in angular units) :return: mass

param_names = ['kappa_0', 'theta_c', 'center_x', 'center_y']
rhotilde(kappa_0, theta_c)[source]

Computes the central density in angular units :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :return: central density in 1/arcsec

upper_limit_default = {'center_x': 100, 'center_y': 100, 'kappa_0': 10, 'theta_c': 100}

lenstronomy.LensModel.Profiles.cored_density_mst module

class lenstronomy.LensModel.Profiles.cored_density_mst.CoredDensityMST(profile_type='CORED_DENSITY')[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

approximate mass-sheet transform of a density core. This routine takes the parameters of the density core and subtracts a mass-sheet that approximates the cored profile in it’s center to counter-act (in approximation) this model. This allows for better sampling of the mass-sheet transformed quantities that do not have strong covariances. The subtraction of the mass-sheet is done such that the sampler returns the real central convergence of the original model (but be careful, the output of quantities like the Einstein angle of the main deflector are still the not-scaled one). Attention!!! The interpretation of the result is that the mass sheet as ‘CONVERGENCE’ that is present needs to be subtracted in post-processing.

derivatives(x, y, lambda_approx, r_core, center_x=0, center_y=0)[source]

deflection angles of approximate mass-sheet correction

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • lambda_approx – approximate mass sheet transform
  • r_core – core radius of the cored density profile
  • center_x – x-center of the profile
  • center_y – y-center of the profile
Returns:

alpha_x, alpha_y

function(x, y, lambda_approx, r_core, center_x=0, center_y=0)[source]

lensing potential of approximate mass-sheet correction

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • lambda_approx – approximate mass sheet transform
  • r_core – core radius of the cored density profile
  • center_x – x-center of the profile
  • center_y – y-center of the profile
Returns:

lensing potential correction

hessian(x, y, lambda_approx, r_core, center_x=0, center_y=0)[source]

Hessian terms of approximate mass-sheet correction

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • lambda_approx – approximate mass sheet transform
  • r_core – core radius of the cored density profile
  • center_x – x-center of the profile
  • center_y – y-center of the profile
Returns:

df/dxx, df/dxy, df/dyx, df/dyy

lower_limit_default = {'center_x': -100, 'center_y': -100, 'lambda_approx': -1, 'r_core': 0}
param_names = ['lambda_approx', 'r_core', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'lambda_approx': 10, 'r_core': 100}

lenstronomy.LensModel.Profiles.curved_arc_const module

class lenstronomy.LensModel.Profiles.curved_arc_const.CurvedArcConstMST[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

lens model that describes a section of a highly magnified deflector region. The parameterization is chosen to describe local observables efficient.

Observables are: - curvature radius (basically bending relative to the center of the profile) - radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope) - tangential stretch (plus sign). Infinity means at critical curve - direction of curvature - position of arc

Requirements: - Should work with other perturbative models without breaking its meaning (say when adding additional shear terms) - Must best reflect the observables in lensing - minimal covariances between the parameters, intuitive parameterization.

derivatives(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

function(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]

ATTENTION: there may not be a global lensing potential!

Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

hessian(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'radial_stretch': -5, 'tangential_stretch': -100}
param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'direction', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'radial_stretch': 5, 'tangential_stretch': 100}

lenstronomy.LensModel.Profiles.curved_arc_sis_mst module

class lenstronomy.LensModel.Profiles.curved_arc_sis_mst.CurvedArcSISMST[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

lens model that describes a section of a highly magnified deflector region. The parameterization is chosen to describe local observables efficient.

Observables are: - curvature radius (basically bending relative to the center of the profile) - radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope) - tangential stretch (plus sign). Infinity means at critical curve - direction of curvature - position of arc

Requirements: - Should work with other perturbative models without breaking its meaning (say when adding additional shear terms) - Must best reflect the observables in lensing - minimal covariances between the parameters, intuitive parameterization.

derivatives(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

function(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]

ATTENTION: there may not be a global lensing potential!

Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

hessian(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'radial_stretch': -5, 'tangential_stretch': -100}
param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'direction', 'center_x', 'center_y']
static sis_mst2stretch(theta_E, kappa_ext, center_x_sis, center_y_sis, center_x, center_y)[source]

turn Singular power-law lens model into stretch parameterization at position (center_x, center_y) This is the inverse function of stretch2spp()

Parameters:
  • theta_E – Einstein radius of SIS profile
  • kappa_ext – external convergence (MST factor 1 - kappa_ext)
  • center_x_sis – center of SPP model
  • center_y_sis – center of SPP model
  • center_x – center of curved model definition
  • center_y – center of curved model definition
Returns:

tangential_stretch, radial_stretch, curvature, direction

Returns:

static stretch2sis_mst(tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]
Parameters:
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

parameters in terms of a spherical SIS + MST resulting in the same observables

upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'radial_stretch': 5, 'tangential_stretch': 100}

lenstronomy.LensModel.Profiles.curved_arc_spp module

class lenstronomy.LensModel.Profiles.curved_arc_spp.CurvedArcSPP[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

lens model that describes a section of a highly magnified deflector region. The parameterization is chosen to describe local observables efficient.

Observables are: - curvature radius (basically bending relative to the center of the profile) - radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope) - tangential stretch (plus sign). Infinity means at critical curve - direction of curvature - position of arc

Requirements: - Should work with other perturbative models without breaking its meaning (say when adding additional shear terms) - Must best reflect the observables in lensing - minimal covariances between the parameters, intuitive parameterization.

derivatives(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

function(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]

ATTENTION: there may not be a global lensing potential!

Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

hessian(x, y, tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'radial_stretch': -5, 'tangential_stretch': -100}
param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'direction', 'center_x', 'center_y']
static spp2stretch(theta_E, gamma, center_x_spp, center_y_spp, center_x, center_y)[source]

turn Singular power-law lens model into stretch parameterization at position (center_x, center_y) This is the inverse function of stretch2spp()

Parameters:
  • theta_E – Einstein radius of SPP model
  • gamma – power-law slope
  • center_x_spp – center of SPP model
  • center_y_spp – center of SPP model
  • center_x – center of curved model definition
  • center_y – center of curved model definition
Returns:

tangential_stretch, radial_stretch, curvature, direction

static stretch2spp(tangential_stretch, radial_stretch, curvature, direction, center_x, center_y)[source]
Parameters:
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

parameters in terms of a spherical power-law profile resulting in the same observables

upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'radial_stretch': 5, 'tangential_stretch': 100}

lenstronomy.LensModel.Profiles.curved_arc_spt module

class lenstronomy.LensModel.Profiles.curved_arc_spt.CurvedArcSPT[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

Curved arc model based on SIS+MST with an additional non-linear shear distortions applied on the source coordinates around the center. This profile is effectively a Source Position Transform of a curved arc and a shear distortion.

derivatives(x, y, tangential_stretch, radial_stretch, curvature, direction, gamma1, gamma2, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • gamma1 – non-linear reduced shear distortion in the source plane
  • gamma2 – non-linear reduced shear distortion in the source plane
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

function(x, y, tangential_stretch, radial_stretch, curvature, direction, gamma1, gamma2, center_x, center_y)[source]

ATTENTION: there may not be a global lensing potential!

Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • gamma1 – non-linear reduced shear distortion in the source plane
  • gamma2 – non-linear reduced shear distortion in the source plane
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

hessian(x, y, tangential_stretch, radial_stretch, curvature, direction, gamma1, gamma2, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • gamma1 – non-linear reduced shear distortion in the source plane
  • gamma2 – non-linear reduced shear distortion in the source plane
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'gamma1': -0.5, 'gamma2': -0.5, 'radial_stretch': -5, 'tangential_stretch': -100}
param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'direction', 'gamma1', 'gamma2', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'gamma1': 0.5, 'gamma2': 0.5, 'radial_stretch': 5, 'tangential_stretch': 100}

lenstronomy.LensModel.Profiles.curved_arc_tan_diff module

class lenstronomy.LensModel.Profiles.curved_arc_tan_diff.CurvedArcTanDiff[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

Curved arc model with an additional non-zero tangential stretch differential in tangential direction component

Observables are: - curvature radius (basically bending relative to the center of the profile) - radial stretch (plus sign) thickness of arc with parity (more generalized than the power-law slope) - tangential stretch (plus sign). Infinity means at critical curve - direction of curvature - position of arc

Requirements: - Should work with other perturbative models without breaking its meaning (say when adding additional shear terms) - Must best reflect the observables in lensing - minimal covariances between the parameters, intuitive parameterization.

derivatives(x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • dtan_dtan – d(tangential_stretch) / d(tangential direction) / tangential stretch
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

function(x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)[source]

ATTENTION: there may not be a global lensing potential!

Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • dtan_dtan – d(tangential_stretch) / d(tangential direction) / tangential stretch
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

hessian(x, y, tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)[source]
Parameters:
  • x
  • y
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • direction – float, angle in radian
  • dtan_dtan – d(tangential_stretch) / d(tangential direction) / tangential stretch
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

lower_limit_default = {'center_x': -100, 'center_y': -100, 'curvature': 1e-06, 'direction': -3.141592653589793, 'dtan_dtan': -10, 'radial_stretch': -5, 'tangential_stretch': -100}
param_names = ['tangential_stretch', 'radial_stretch', 'curvature', 'dtan_dtan', 'direction', 'center_x', 'center_y']
static stretch2sie_mst(tangential_stretch, radial_stretch, curvature, dtan_dtan, direction, center_x, center_y)[source]
Parameters:
  • tangential_stretch – float, stretch of intrinsic source in tangential direction
  • radial_stretch – float, stretch of intrinsic source in radial direction
  • curvature – 1/curvature radius
  • dtan_dtan – d(tangential_stretch) / d(tangential direction) / tangential stretch
  • direction – float, angle in radian
  • center_x – center of source in image plane
  • center_y – center of source in image plane
Returns:

parameters in terms of a spherical SIS + MST resulting in the same observables

upper_limit_default = {'center_x': 100, 'center_y': 100, 'curvature': 100, 'direction': 3.141592653589793, 'dtan_dtab': 10, 'radial_stretch': 5, 'tangential_stretch': 100}

lenstronomy.LensModel.Profiles.dipole module

class lenstronomy.LensModel.Profiles.dipole.Dipole(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for dipole response of two massive bodies (experimental)

derivatives(x, y, com_x, com_y, phi_dipole, coupling)[source]

deflection angles

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

function(x, y, com_x, com_y, phi_dipole, coupling)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, com_x, com_y, phi_dipole, coupling)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

lower_limit_default = {'com_x': -100, 'com_y': -100, 'coupling': -10, 'phi_dipole': -10}
param_names = ['com_x', 'com_y', 'phi_dipole', 'coupling']
upper_limit_default = {'com_x': 100, 'com_y': 100, 'coupling': 10, 'phi_dipole': 10}
class lenstronomy.LensModel.Profiles.dipole.DipoleUtil[source]

Bases: object

pre-calculation of dipole properties

angle(center1_x, center1_y, center2_x, center2_y)[source]

compute the rotation angle of the dipole :return:

com(center1_x, center1_y, center2_x, center2_y, Fm)[source]
Returns:center of mass
mass_ratio(theta_E, theta_E_sub)[source]

computes mass ration of the two clumps with given Einstein radius and power law slope (clump1/sub-clump) :param theta_E: :param theta_E_sub: :return:

lenstronomy.LensModel.Profiles.elliptical_density_slice module

class lenstronomy.LensModel.Profiles.elliptical_density_slice.ElliSLICE(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This class computes the lensing quantities for an elliptical slice of constant density. Based on Schramm 1994 https://ui.adsabs.harvard.edu/abs/1994A%26A…284…44S/abstract

Computes the lensing quantities of an elliptical slice with semi major axis ‘a’ and semi minor axis ‘b’, centered on ‘center_x’ and ‘center_y’, oriented with an angle ‘psi’ in radian, and with constant surface mass density ‘sigma_0’. In other words, this lens model is characterized by the surface mass density :

..math::
kappa(x,y) = left{
begin{array}{ll}
sigma_0 & mbox{if } frac{x_{rot}^2}{a^2} + frac{y_{rot}^2}{b^2} leq 1 0 & mbox{else}

end{array}

right}.

with

..math::
x_{rot} = x_c cos psi + y_c sin psi y_{rot} = - x_c sin psi + y_c cos psi x_c = x - center_x y_c = y - center_y
alpha_ext(x, y, kwargs_slice)[source]

deflection angle for (x,y) outside the elliptical slice

Parameters:kwargs_slice – dict, dictionary with the slice definition (a,b,psi,sigma_0)
alpha_in(x, y, kwargs_slice)[source]

deflection angle for (x,y) inside the elliptical slice

Parameters:kwargs_slice – dict, dictionary with the slice definition (a,b,psi,sigma_0)
derivatives(x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0)[source]

lensing deflection angle

Parameters:
  • a – float, semi-major axis, must be positive
  • b – float, semi-minor axis, must be positive
  • psi – float, orientation in radian
  • sigma_0 – float, surface mass density, must be positive
  • center_x – float, center on the x axis
  • center_y – float, center on the y axis
function(x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0)[source]

lensing potential

Parameters:
  • a – float, semi-major axis, must be positive
  • b – float, semi-minor axis, must be positive
  • psi – float, orientation in radian
  • sigma_0 – float, surface mass density, must be positive
  • center_x – float, center on the x axis
  • center_y – float, center on the y axis
hessian(x, y, a, b, psi, sigma_0, center_x=0.0, center_y=0.0)[source]

lensing second derivatives

Parameters:
  • a – float, semi-major axis, must be positive
  • b – float, semi-minor axis, must be positive
  • psi – float, orientation in radian
  • sigma_0 – float, surface mass density, must be positive
  • center_x – float, center on the x axis
  • center_y – float, center on the y axis
lower_limit_default = {'a': 0.0, 'b': 0.0, 'center_x': -100.0, 'center_y': -100.0, 'psi': -1.5707963267948966}
param_names = ['a', 'b', 'psi', 'sigma_0', 'center_x', 'center_y']
pot_ext(x, y, kwargs_slice)[source]

lensing potential for (x,y) outside the elliptical slice

Parameters:kwargs_slice – dict, dictionary with the slice definition (a,b,psi,sigma_0)
pot_in(x, y, kwargs_slice)[source]

lensing potential for (x,y) inside the elliptical slice

Parameters:kwargs_slice – dict, dictionary with the slice definition (a,b,psi,sigma_0)
sign(z)[source]

sign function

Parameters:z – complex
upper_limit_default = {'a': 100.0, 'b': 100.0, 'center_x': 100.0, 'center_y': 100.0, 'psi': 1.5707963267948966}

lenstronomy.LensModel.Profiles.epl module

class lenstronomy.LensModel.Profiles.epl.EPL[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

” Elliptical Power Law mass profile

\[\kappa(x, y) = \frac{3-\gamma}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q}} \right)^{\gamma-1}\]

with \(\theta_{E}\) is the (circularized) Einstein radius, \(\gamma\) is the negative power-law slope of the 3D mass distributions, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate sys- tem aligned with the major and minor axis of the lens.

In terms of eccentricities, this profile is defined as

\[\kappa(r) = \frac{3-\gamma}{2} \left(\frac{\theta'_{E}}{r \sqrt{1 − e*\cos(2*\phi)}} \right)^{\gamma-1}\]

with \(\epsilon\) is the ellipticity defined as

\[\epsilon = \frac{1-q^2}{1+q^2}\]

And an Einstein radius \(\theta'_{\rm E}\) related to the definition used is

\[\left(\frac{\theta'_{\rm E}}{\theta_{\rm E}}\right)^{2} = \frac{2q}{1+q^2}.\]

The mathematical form of the calculation is presented by Tessore & Metcalf (2015), https://arxiv.org/abs/1507.01819. The current implementation is using hyperbolic functions. The paper presents an iterative calculation scheme, converging in few iterations to high precision and accuracy.

A (faster) implementation of the same model using numba is accessible as ‘EPL_NUMBA’ with the iterative calculation scheme.

derivatives(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • gamma – power law slope
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – profile center
  • center_y – profile center
Returns:

alpha_x, alpha_y

function(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • gamma – power law slope
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – profile center
  • center_y – profile center
Returns:

lensing potential

hessian(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • gamma – power law slope
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – profile center
  • center_y – profile center
Returns:

f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 1.5, 'theta_E': 0}
param_conv(theta_E, gamma, e1, e2)[source]

converts parameters as defined in this class to the parameters used in the EPLMajorAxis() class

Parameters:
  • theta_E – Einstein radius as defined in the profile class
  • gamma – negative power-law slope
  • e1 – eccentricity modulus
  • e2 – eccentricity modulus
Returns:

b, t, q, phi_G

param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']
set_dynamic()[source]
Returns:
set_static(theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • gamma – power law slope
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – profile center
  • center_y – profile center
Returns:

self variables set

upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 2.5, 'theta_E': 100}
class lenstronomy.LensModel.Profiles.epl.EPLMajorAxis[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This class contains the function and the derivatives of the elliptical power law.

\[\kappa = (2-t)/2 * \left[\frac{b}{\sqrt{q^2 x^2 + y^2}}\right]^t\]

where with \(t = \gamma - 1\) (from EPL class) being the projected power-law slope of the convergence profile, critical radius b, axis ratio q.

Tessore & Metcalf (2015), https://arxiv.org/abs/1507.01819

derivatives(x, y, b, t, q)[source]

returns the deflection angles

Parameters:
  • x – x-coordinate in image plane relative to center (major axis)
  • y – y-coordinate in image plane relative to center (minor axis)
  • b – critical radius
  • t – projected power-law slope
  • q – axis ratio
Returns:

f_x, f_y

function(x, y, b, t, q)[source]

returns the lensing potential

Parameters:
  • x – x-coordinate in image plane relative to center (major axis)
  • y – y-coordinate in image plane relative to center (minor axis)
  • b – critical radius
  • t – projected power-law slope
  • q – axis ratio
Returns:

lensing potential

hessian(x, y, b, t, q)[source]

Hessian matrix of the lensing potential

Parameters:
  • x – x-coordinate in image plane relative to center (major axis)
  • y – y-coordinate in image plane relative to center (minor axis)
  • b – critical radius
  • t – projected power-law slope
  • q – axis ratio
Returns:

f_xx, f_yy, f_xy

param_names = ['b', 't', 'q', 'center_x', 'center_y']

lenstronomy.LensModel.Profiles.epl_numba module

class lenstronomy.LensModel.Profiles.epl_numba.EPL_numba[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

” Elliptical Power Law mass profile - computation accelerated with numba

\[\kappa(x, y) = \frac{3-\gamma}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q}} \right)^{\gamma-1}\]

with \(\theta_{E}\) is the (circularized) Einstein radius, \(\gamma\) is the negative power-law slope of the 3D mass distributions, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate system aligned with the major and minor axis of the lens.

In terms of eccentricities, this profile is defined as

\[\kappa(r) = \frac{3-\gamma}{2} \left(\frac{\theta'_{E}}{r \sqrt{1 − e*\cos(2*\phi)}} \right)^{\gamma-1}\]

with \(\epsilon\) is the ellipticity defined as

\[\epsilon = \frac{1-q^2}{1+q^2}\]

And an Einstein radius \(\theta'_{\rm E}\) related to the definition used is

\[\left(\frac{\theta'_{\rm E}}{\theta_{\rm E}}\right)^{2} = \frac{2q}{1+q^2}.\]

The mathematical form of the calculation is presented by Tessore & Metcalf (2015), https://arxiv.org/abs/1507.01819. The current implementation is using hyperbolic functions. The paper presents an iterative calculation scheme, converging in few iterations to high precision and accuracy.

A (slower) implementation of the same model using hyperbolic functions without the iterative calculation is accessible as ‘EPL’ not requiring numba.

derivatives[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

deflection angles alpha_x, alpha_y

function[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

lensing potential

hessian[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

Hessian components f_xx, f_yy, f_xy

lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 1.5, 'theta_E': 0}
param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 2.5, 'theta_E': 100}

lenstronomy.LensModel.Profiles.flexion module

class lenstronomy.LensModel.Profiles.flexion.Flexion(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for flexion

derivatives(x, y, g1, g2, g3, g4, ra_0=0, dec_0=0)[source]

deflection angles

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

function(x, y, g1, g2, g3, g4, ra_0=0, dec_0=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, g1, g2, g3, g4, ra_0=0, dec_0=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

lower_limit_default = {'dec_0': -100, 'g1': -0.1, 'g2': -0.1, 'g3': -0.1, 'g4': -0.1, 'ra_0': -100}
param_names = ['g1', 'g2', 'g3', 'g4', 'ra_0', 'dec_0']
upper_limit_default = {'dec_0': 100, 'g1': 0.1, 'g2': 0.1, 'g3': 0.1, 'g4': 0.1, 'ra_0': 100}

lenstronomy.LensModel.Profiles.flexionfg module

class lenstronomy.LensModel.Profiles.flexionfg.Flexionfg[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

Flexion consist of basis F flexion and G flexion (F1,F2,G1,G2), see formulas 2.54, 2.55 in Massimo Meneghetti 2017 - “Introduction to Gravitational Lensing”.

derivatives(x, y, F1, F2, G1, G2, ra_0=0, dec_0=0)[source]

deflection angle :param x: x-coordinate :param y: y-coordinate :param F1: F1 flexion, derivative of kappa in x direction :param F2: F2 flexion, derivative of kappa in y direction :param G1: G1 flexion :param G2: G2 flexion :param ra_0: center x-coordinate :param dec_0: center x-coordinate :return: deflection angle

function(x, y, F1, F2, G1, G2, ra_0=0, dec_0=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • F1 – F1 flexion, derivative of kappa in x direction
  • F2 – F2 flexion, derivative of kappa in y direction
  • G1 – G1 flexion
  • G2 – G2 flexion
  • ra_0 – center x-coordinate
  • dec_0 – center y-coordinate
Returns:

lensing potential

hessian(x, y, F1, F2, G1, G2, ra_0=0, dec_0=0)[source]

Hessian matrix :param x: x-coordinate :param y: y-coordinate :param F1: F1 flexion, derivative of kappa in x direction :param F2: F2 flexion, derivative of kappa in y direction :param G1: G1 flexion :param G2: G2 flexion :param ra_0: center x-coordinate :param dec_0: center y-coordinate :return: second order derivatives f_xx, f_yy, f_xy

lower_limit_default = {'F1': -0.1, 'F2': -0.1, 'G1': -0.1, 'G2': -0.1, 'dec_0': -100, 'ra_0': -100}
param_names = ['F1', 'F2', 'G1', 'G2', 'ra_0', 'dec_0']
static transform_fg(F1, F2, G1, G2)[source]

basis transform from (F1,F2,G1,G2) to (g1,g2,g3,g4) :param f1: F1 flexion, derivative of kappa in x direction :param f2: F2 flexion, derivative of kappa in y direction :param g1: G1 flexion :param g2: G2 flexion :return: g1,g2,g3,g4 (phi_xxx, phi_xxy, phi_xyy, phi_yyy)

upper_limit_default = {'F1': 0.1, 'F2': 0.1, 'G1': 0.1, 'G2': 0.1, 'dec_0': 100, 'ra_0': 100}

lenstronomy.LensModel.Profiles.gauss_decomposition module

This module contains the class to compute lensing properties of any elliptical profile using Shajib (2019)’s Gauss decomposition.

class lenstronomy.LensModel.Profiles.gauss_decomposition.GaussianEllipseKappaSet(use_scipy_wofz=True, min_ellipticity=1e-05)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This class computes the lensing properties of a set of concentric elliptical Gaussian convergences.

density_2d(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

Compute the density of a set of concentric elliptical Gaussian convergence profiles \(\sum A/(2\pi \sigma^2) \exp(-( x^2+y^2/q^2)/2\sigma^2)\).

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • amp (numpy.array with dtype=float) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)
  • sigma (numpy.array with dtype=float) – Standard deviation of Gaussian
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordianate of centroid
Returns:

Density \(\kappa\) for elliptical Gaussian convergence

Return type:

float, or numpy.array with shape equal to x.shape

derivatives(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

Compute the derivatives of function angles \(\partial f/\partial x\), \(\partial f/\partial y\) at \(x,\ y\) for a set of concentric elliptic Gaussian convergence profiles.

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • amp (numpy.array with dtype=float) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)
  • sigma (numpy.array with dtype=float) – Standard deviation of Gaussian
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordianate of centroid
Returns:

Deflection angle \(\partial f/\partial x\), \(\partial f/\partial y\) for elliptical Gaussian convergence

Return type:

tuple (float, float) or (numpy.array, numpy.array) with each numpy array’s shape equal to x.shape

function(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

Compute the potential function for a set of concentric elliptical Gaussian convergence profiles.

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • amp (numpy.array with dtype=float) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)
  • sigma (numpy.array with dtype=float) – Standard deviation of Gaussian
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordianate of centroid
Returns:

Potential for elliptical Gaussian convergence

Return type:

float, or numpy.array with shape = x.shape

hessian(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

Compute Hessian matrix of function \(\partial^2f/\partial x^2\), \(\partial^2 f/\partial y^2\), \(\partial^2 f/\partial x\partial y\) for a set of concentric elliptic Gaussian convergence profiles.

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • amp (numpy.array with dtype=float) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)
  • sigma (numpy.array with dtype=float) – Standard deviation of Gaussian
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordianate of centroid
Returns:

Hessian \(\partial^2f/\partial x^2\), \(\partial^2/\partial x\partial y\), \(\partial^2/\partial y\partial x\), \(\partial^2 f/\partial y^2\) for elliptical Gaussian convergence.

Return type:

tuple (float, float, float) , or (numpy.array, numpy.array, numpy.array) with each numpy array’s shape equal to x.shape

lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma': 0}
param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma': 100}
class lenstronomy.LensModel.Profiles.gauss_decomposition.GaussDecompositionAbstract(n_sigma=15, sigma_start_mult=0.02, sigma_end_mult=15.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]

Bases: object

This abstract class sets up a template for computing lensing properties of an elliptical convergence through Shajib (2019)’s Gauss decomposition.

density_2d(x, y, e1=0.0, e2=0.0, center_x=0.0, center_y=0.0, **kwargs)[source]

Compute the convergence profile for Gauss-decomposed elliptic Sersic profile.

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordinate of centroid
  • **kwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile in the child class.
Returns:

Convergence profile

Return type:

type(x)

derivatives(x, y, e1=0.0, e2=0.0, center_x=0.0, center_y=0.0, **kwargs)[source]

Compute the derivatives of the deflection potential \(\partial f/\partial x\), \(\partial f/\partial y\) for a Gauss-decomposed elliptic convergence.

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordinate of centroid
  • **kwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile
Returns:

Derivatives of deflection potential

Return type:

tuple (type(x), type(x))

function(x, y, e1=0.0, e2=0.0, center_x=0.0, center_y=0.0, **kwargs)[source]

Compute the deflection potential of a Gauss-decomposed elliptic convergence.

Parameters:
  • x (float) – x coordinate
  • y (float) – y coordinate
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordinate of centroid
  • **kwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile
Returns:

Deflection potential

Return type:

float

gauss_decompose(**kwargs)[source]

Compute the amplitudes and sigmas of Gaussian components using the integral transform with Gaussian kernel from Shajib (2019). The returned values are in the convention of eq. (2.13).

Parameters:
  • func (function) – The function to decompose
  • **kwargs – Keyword arguments to send to func
Returns:

Amplitudes and standard deviations of the Gaussian components

Return type:

tuple (numpy.array, numpy.array)

get_kappa_1d(y, **kwargs)[source]

Abstract method to compute the spherical Sersic profile at y. The concrete method has to defined by the child class.

Parameters:
  • y (float or numpy.array) – y coordinate
  • **kwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile
get_scale(**kwargs)[source]

Abstract method to identify the keyword argument for the scale size among the profile parameters of the child class’ convergence profile.

Parameters:**kwargs – Keyword arguments
Returns:Scale size
Return type:float
hessian(x, y, e1=0.0, e2=0.0, center_x=0.0, center_y=0.0, **kwargs)[source]

Compute the Hessian of the deflection potential \(\partial^2f/\partial x^2\), \(\partial^2 f/ \partial y^2\), \(\partial^2 f/\partial x\partial y\) of a Gauss-decomposed elliptic Sersic convergence.

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordinate of centroid
  • **kwargs – Keyword arguments that are defined by the child class that are particular for the convergence profile
Returns:

Hessian of deflection potential

Return type:

tuple (type(x), type(x), type(x))

class lenstronomy.LensModel.Profiles.gauss_decomposition.SersicEllipseGaussDec(n_sigma=15, sigma_start_mult=0.02, sigma_end_mult=15.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]

Bases: lenstronomy.LensModel.Profiles.gauss_decomposition.GaussDecompositionAbstract

This class computes the lensing properties of an elliptical Sersic profile using the Shajib (2019)’s Gauss decomposition method.

get_kappa_1d(y, **kwargs)[source]

Compute the spherical Sersic profile at y.

Parameters:
  • y (float) – y coordinate
  • **kwargs – Keyword arguments
Keyword Arguments:
 
  • n_sersic (float) – Sersic index
  • R_sersic (float) – Sersic scale radius
  • k_eff (float) – Sersic convergence at R_sersic
Returns:

Sersic function at y

Return type:

type(y)

get_scale(**kwargs)[source]

Identify the scale size from the keyword arguments.

Parameters:

**kwargs – Keyword arguments

Keyword Arguments:
 
  • n_sersic (float) – Sersic index
  • R_sersic (float) – Sersic scale radius
  • k_eff (float) – Sersic convergence at R_sersic
Returns:

Sersic radius

Return type:

float

lower_limit_default = {'R_sersic': 0.0, 'center_x': -100.0, 'center_y': -100.0, 'e1': -0.5, 'e2': -0.5, 'k_eff': 0.0, 'n_sersic': 0.5}
param_names = ['k_eff', 'R_sersic', 'n_sersic', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'R_sersic': 100.0, 'center_x': 100.0, 'center_y': 100.0, 'e1': 0.5, 'e2': 0.5, 'k_eff': 100.0, 'n_sersic': 8.0}
class lenstronomy.LensModel.Profiles.gauss_decomposition.NFWEllipseGaussDec(n_sigma=15, sigma_start_mult=0.005, sigma_end_mult=50.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]

Bases: lenstronomy.LensModel.Profiles.gauss_decomposition.GaussDecompositionAbstract

This class computes the lensing properties of an elliptical, projected NFW profile using Shajib (2019)’s Gauss decomposition method.

get_kappa_1d(y, **kwargs)[source]

Compute the spherical projected NFW profile at y.

Parameters:
  • y (float) – y coordinate
  • **kwargs – Keyword arguments
Keyword Arguments:
 
  • alpha_Rs (float) – Deflection angle at Rs
  • R_s (float) – NFW scale radius
Returns:

projected NFW profile at y

Return type:

type(y)

get_scale(**kwargs)[source]

Identify the scale size from the keyword arguments.

Parameters:

**kwargs – Keyword arguments

Keyword Arguments:
 
  • alpha_Rs (float) – Deflection angle at Rs
  • R_s (float) – NFW scale radius
Returns:

NFW scale radius

Return type:

float

lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5}
param_names = ['Rs', 'alpha_Rs', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5}
class lenstronomy.LensModel.Profiles.gauss_decomposition.GaussDecompositionAbstract3D(n_sigma=15, sigma_start_mult=0.02, sigma_end_mult=15.0, precision=10, use_scipy_wofz=True, min_ellipticity=1e-05)[source]

Bases: lenstronomy.LensModel.Profiles.gauss_decomposition.GaussDecompositionAbstract

This abstract class sets up a template for computing lensing properties of a convergence from 3D spherical profile through Shajib (2019)’s Gauss decomposition.

gauss_decompose(**kwargs)[source]

Compute the amplitudes and sigmas of Gaussian components using the integral transform with Gaussian kernel from Shajib (2019). The returned values are in the convention of eq. (2.13).

Parameters:
  • func (function) – The function to decompose
  • **kwargs – Keyword arguments to send to func
Returns:

Amplitudes and standard deviations of the Gaussian components

Return type:

tuple (numpy.array, numpy.array)

class lenstronomy.LensModel.Profiles.gauss_decomposition.CTNFWGaussDec(n_sigma=15, sigma_start_mult=0.01, sigma_end_mult=20.0, precision=10, use_scipy_wofz=True)[source]

Bases: lenstronomy.LensModel.Profiles.gauss_decomposition.GaussDecompositionAbstract3D

This class computes the lensing properties of an projection from a spherical cored-truncated NFW profile using Shajib (2019)’s Gauss decomposition method.

get_kappa_1d(y, **kwargs)[source]

Compute the spherical cored-truncated NFW profile at y.

Parameters:
  • y (float) – y coordinate
  • **kwargs – Keyword arguments
Keyword Arguments:
 
  • r_s (float) – Scale radius
  • r_trunc (float) – Truncation radius
  • r_core (float) – Core radius
  • rho_s (float) – Density normalization
  • a (float) – Core regularization parameter
Returns:

projected NFW profile at y

Return type:

type(y)

get_scale(**kwargs)[source]

Identify the scale size from the keyword arguments.

Parameters:

**kwargs – Keyword arguments

Keyword Arguments:
 
  • r_s (float) – Scale radius
  • r_trunc (float) – Truncation radius
  • r_core (float) – Core radius
  • rho_s (float) – Density normalization
  • a (float) – Core regularization parameter
Returns:

NFW scale radius

Return type:

float

lower_limit_default = {'a': 0.0, 'center_x': -100, 'center_y': -100, 'r_core': 0, 'r_s': 0, 'r_trunc': 0, 'rho_s': 0}
param_names = ['r_s', 'r_core', 'r_trunc', 'a', 'rho_s', 'center_xcenter_y']
upper_limit_default = {'a': 10.0, 'center_x': 100, 'center_y': 100, 'r_core': 100, 'r_s': 100, 'r_trunc': 100, 'rho_s': 1000}

lenstronomy.LensModel.Profiles.gaussian_ellipse_kappa module

This module defines class GaussianEllipseKappa to compute the lensing properties of an elliptical Gaussian profile with ellipticity in the convergence using the formulae from Shajib (2019).

class lenstronomy.LensModel.Profiles.gaussian_ellipse_kappa.GaussianEllipseKappa(use_scipy_wofz=True, min_ellipticity=1e-05)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This class contains functions to evaluate the derivative and hessian matrix of the deflection potential for an elliptical Gaussian convergence.

The formulae are from Shajib (2019).

density_2d(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

Compute the density of elliptical Gaussian \(A/(2 \pi \sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\).

Parameters:
  • x (float or numpy.array) – x coordinate.
  • y (float or numpy.array) – y coordinate.
  • amp (float) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)
  • sigma (float) – Standard deviation of Gaussian.
  • e1 (float) – Ellipticity parameter 1.
  • e2 (float) – Ellipticity parameter 2.
  • center_x (float) – x coordinate of centroid.
  • center_y (float) – y coordianate of centroid.
Returns:

Density \(\kappa\) for elliptical Gaussian convergence.

Return type:

float, or numpy.array with shape = x.shape.

derivatives(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

Compute the derivatives of function angles \(\partial f/\partial x\), \(\partial f/\partial y\) at \(x,\ y\).

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • amp (float) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)
  • sigma (float) – Standard deviation of Gaussian
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordianate of centroid
Returns:

Deflection angle \(\partial f/\partial x\), \(\partial f/\partial y\) for elliptical Gaussian convergence.

Return type:

tuple (float, float) or (numpy.array, numpy.array) with each numpy.array’s shape equal to x.shape.

function(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

Compute the potential function for elliptical Gaussian convergence.

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • amp (float) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)
  • sigma (float) – Standard deviation of Gaussian
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordianate of centroid
Returns:

Potential for elliptical Gaussian convergence

Return type:

float, or numpy.array with shape equal to x.shape

hessian(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

Compute Hessian matrix of function \(\partial^2f/\partial x^2\), \(\partial^2 f/\partial y^2\), \(\partial^2/\partial x\partial y\).

Parameters:
  • x (float or numpy.array) – x coordinate
  • y (float or numpy.array) – y coordinate
  • amp (float) – Amplitude of Gaussian, convention: \(A/(2 \pi\sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\)
  • sigma (float) – Standard deviation of Gaussian
  • e1 (float) – Ellipticity parameter 1
  • e2 (float) – Ellipticity parameter 2
  • center_x (float) – x coordinate of centroid
  • center_y (float) – y coordianate of centroid
Returns:

Hessian \(A/(2 \pi \sigma^2) \exp(-(x^2+y^2/q^2)/2\sigma^2)\) for elliptical Gaussian convergence.

Return type:

tuple (float, float, float) , or (numpy.array, numpy.array, numpy.array) with each numpy.array’s shape equal to x.shape.

lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma': 0}
param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']
static sgn(z)[source]

Compute the sign function \(\mathrm{sgn}(z)\) factor for deflection as sugggested by Bray (1984). For current implementation, returning 1 is sufficient.

Parameters:z (complex) – Complex variable \(z = x + \mathrm{i}y\)
Returns:\(\mathrm{sgn}(z)\)
Return type:float
sigma_function(x, y, q)[source]

Compute the function \(\varsigma (z; q)\) from equation (4.12) of Shajib (2019).

Parameters:
  • x (float or numpy.array) – Real part of complex variable, \(x = \mathrm{Re}(z)\)
  • y (float or numpy.array) – Imaginary part of complex variable, \(y = \mathrm{Im}(z)\)
  • q (float) – Axis ratio
Returns:

real and imaginary part of \(\varsigma(z; q)\) function

Return type:

tuple (type(x), type(x))

upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma': 100}
static w_f_approx(z)[source]

Compute the Faddeeva function \(w_{\mathrm F}(z)\) using the approximation given in Zaghloul (2017).

Parameters:z (complex or numpy.array(dtype=complex)) – complex number
Returns:\(w_\mathrm{F}(z)\)
Return type:complex

lenstronomy.LensModel.Profiles.gaussian_ellipse_potential module

class lenstronomy.LensModel.Profiles.gaussian_ellipse_potential.GaussianEllipsePotential[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix with ellipticity in the convergence

the calculation follows Glenn van de Ven et al. 2009

density(r, amp, sigma, e1, e2)[source]
Parameters:
  • r
  • amp
  • sigma
Returns:

density_2d(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • R
  • am
  • sigma_x
  • sigma_y
Returns:

derivatives(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

returns Gaussian

hessian(x, y, amp, sigma, e1, e2, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma': 0}
mass_2d(R, amp, sigma, e1, e2)[source]
Parameters:
  • R
  • amp
  • sigma_x
  • sigma_y
Returns:

mass_2d_lens(R, amp, sigma, e1, e2)[source]
Parameters:
  • R
  • amp
  • sigma_x
  • sigma_y
Returns:

mass_3d(R, amp, sigma, e1, e2)[source]
Parameters:
  • R
  • amp
  • sigma
  • e1
  • e2
Returns:

mass_3d_lens(R, amp, sigma, e1, e2)[source]
Parameters:
  • R
  • amp
  • sigma
  • e1
  • e2
Returns:

param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma': 100}

lenstronomy.LensModel.Profiles.gaussian_kappa module

class lenstronomy.LensModel.Profiles.gaussian_kappa.GaussianKappa[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix

alpha_abs(R, amp, sigma)[source]

absolute value of the deflection :param R: :param amp: :param sigma_x: :param sigma_y: :return:

d_alpha_dr(R, amp, sigma_x, sigma_y)[source]
Parameters:
  • R
  • amp
  • sigma_x
  • sigma_y
Returns:

density(r, amp, sigma)[source]
Parameters:
  • r
  • amp
  • sigma
Returns:

density_2d(x, y, amp, sigma, center_x=0, center_y=0)[source]
Parameters:
  • R
  • am
  • sigma_x
  • sigma_y
Returns:

derivatives(x, y, amp, sigma, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, amp, sigma, center_x=0, center_y=0)[source]

returns Gaussian

hessian(x, y, amp, sigma, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'sigma': 0}
mass_2d(R, amp, sigma)[source]
Parameters:
  • R
  • amp
  • sigma_x
  • sigma_y
Returns:

mass_2d_lens(R, amp, sigma)[source]
Parameters:
  • R
  • amp
  • sigma_x
  • sigma_y
Returns:

mass_3d(R, amp, sigma)[source]
Parameters:
  • R
  • amp
  • sigma_x
  • sigma_y
Returns:

mass_3d_lens(R, amp, sigma)[source]
Parameters:
  • R
  • amp
  • sigma_x
  • sigma_y
Returns:

param_names = ['amp', 'sigma', 'center_x', 'center_y']
upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'sigma': 100}

lenstronomy.LensModel.Profiles.gaussian_potential module

class lenstronomy.LensModel.Profiles.gaussian_potential.Gaussian(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions to evaluate a Gaussian function and calculates its derivative and hessian matrix

derivatives(x, y, amp, sigma_x, sigma_y, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, amp, sigma_x, sigma_y, center_x=0, center_y=0)[source]

returns Gaussian

hessian(x, y, amp, sigma_x, sigma_y, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'sigma': 0}
param_names = ['amp', 'sigma_x', 'sigma_y', 'center_x', 'center_y']
upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'sigma': 100}

lenstronomy.LensModel.Profiles.hernquist module

class lenstronomy.LensModel.Profiles.hernquist.Hernquist(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class to compute the Hernquist 1990 model, which is in 3d: rho(r) = rho0 / (r/Rs * (1 + (r/Rs))**3)

in lensing terms, the normalization parameter ‘sigma0’ is defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0

density(r, rho0, Rs)[source]

computes the 3-d density

Parameters:
  • r – 3-d radius
  • rho0 – density normalization
  • Rs – Hernquist radius
Returns:

density at radius r

density_2d(x, y, rho0, Rs, center_x=0, center_y=0)[source]

projected density along the line of sight at coordinate (x, y)

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • rho0 – density normalization
  • Rs – Hernquist radius
  • center_x – x-center of the profile
  • center_y – y-center of the profile
Returns:

projected density

density_lens(r, sigma0, Rs)[source]

Density as a function of 3d radius in lensing parameters This function converts the lensing definition sigma0 into the 3d density

Parameters:
  • r – 3d radius
  • sigma0 – rho0 * Rs (units of projected density)
  • Rs – Hernquist radius
Returns:

enclosed mass in 3d

derivatives(x, y, sigma0, Rs, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate position (units of angle)
  • y – y-coordinate position (units of angle)
  • sigma0 – normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
  • Rs – Hernquist radius in units of angle
  • center_x – x-center of the profile (units of angle)
  • center_y – y-center of the profile (units of angle)
Returns:

derivative of function (deflection angles in x- and y-direction)

function(x, y, sigma0, Rs, center_x=0, center_y=0)[source]

lensing potential

Parameters:
  • x – x-coordinate position (units of angle)
  • y – y-coordinate position (units of angle)
  • sigma0 – normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
  • Rs – Hernquist radius in units of angle
  • center_x – x-center of the profile (units of angle)
  • center_y – y-center of the profile (units of angle)
Returns:

lensing potential at (x,y)

grav_pot(x, y, rho0, Rs, center_x=0, center_y=0)[source]

#TODO decide whether these functions are needed or not

gravitational potential (modulo 4 pi G and rho0 in appropriate units) :param x: x-coordinate position (units of angle) :param y: y-coordinate position (units of angle) :param rho0: density normalization parameter of Hernquist profile :param Rs: Hernquist radius in units of angle :param center_x: x-center of the profile (units of angle) :param center_y: y-center of the profile (units of angle) :return: gravitational potential at projected radius

hessian(x, y, sigma0, Rs, center_x=0, center_y=0)[source]

Hessian terms of the function

Parameters:
  • x – x-coordinate position (units of angle)
  • y – y-coordinate position (units of angle)
  • sigma0 – normalization parameter defined such that the deflection at projected RS leads to alpha = 2./3 * Rs * sigma0
  • Rs – Hernquist radius in units of angle
  • center_x – x-center of the profile (units of angle)
  • center_y – y-center of the profile (units of angle)
Returns:

df/dxdx, df/dxdy, df/dydx, df/dydy

lower_limit_default = {'Rs': 0, 'center_x': -100, 'center_y': -100, 'sigma0': 0}
mass_2d(r, rho0, Rs)[source]

mass enclosed projected 2d sphere of radius r

Parameters:
  • r – projected radius
  • rho0 – density normalization
  • Rs – Hernquist radius
Returns:

mass enclosed 2d projected radius

mass_2d_lens(r, sigma0, Rs)[source]

mass enclosed projected 2d sphere of radius r Same as mass_2d but with input normalization in units of projected density :param r: projected radius :param sigma0: rho0 * Rs (units of projected density) :param Rs: Hernquist radius :return: mass enclosed 2d projected radius

mass_3d(r, rho0, Rs)[source]

mass enclosed a 3d sphere or radius r

Parameters:
  • r – 3-d radius within the mass is integrated (same distance units as density definition)
  • rho0 – density normalization
  • Rs – Hernquist radius
Returns:

enclosed mass

mass_3d_lens(r, sigma0, Rs)[source]

mass enclosed a 3d sphere or radius r for lens parameterisation This function converts the lensing definition sigma0 into the 3d density

Parameters:
  • sigma0 – rho0 * Rs (units of projected density)
  • Rs – Hernquist radius
Returns:

enclosed mass in 3d

mass_tot(rho0, Rs)[source]

total mass within the profile :param rho0: density normalization :param Rs: Hernquist radius :return: total mass within profile

param_names = ['sigma0', 'Rs', 'center_x', 'center_y']
rho2sigma(rho0, Rs)[source]

converts 3d density into 2d projected density parameter :param rho0: 3d density normalization of Hernquist model :param Rs: Hernquist radius :return: sigma0 defined quantity in projected units

sigma2rho(sigma0, Rs)[source]

converts projected density parameter (in units of deflection) into 3d density parameter :param sigma0: density defined quantity in projected units :param Rs: Hernquist radius :return: rho0 the 3d density normalization of Hernquist model

upper_limit_default = {'Rs': 100, 'center_x': 100, 'center_y': 100, 'sigma0': 100}

lenstronomy.LensModel.Profiles.hernquist_ellipse module

class lenstronomy.LensModel.Profiles.hernquist_ellipse.Hernquist_Ellipse[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions for the elliptical Hernquist profile. Ellipticity is defined in the potential.

density(r, rho0, Rs, e1=0, e2=0)[source]

computes the 3-d density

Parameters:
  • r – 3-d radius
  • rho0 – density normalization
  • Rs – Hernquist radius
Returns:

density at radius r

density_2d(x, y, rho0, Rs, e1=0, e2=0, center_x=0, center_y=0)[source]

projected density along the line of sight at coordinate (x, y)

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • rho0 – density normalization
  • Rs – Hernquist radius
  • center_x – x-center of the profile
  • center_y – y-center of the profile
Returns:

projected density

density_lens(r, sigma0, Rs, e1=0, e2=0)[source]

Density as a function of 3d radius in lensing parameters This function converts the lensing definition sigma0 into the 3d density

Parameters:
  • r – 3d radius
  • sigma0 – rho0 * Rs (units of projected density)
  • Rs – Hernquist radius
Returns:

enclosed mass in 3d

derivatives(x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function (integral of NFW)

function(x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0)[source]

returns double integral of NFW profile

hessian(x, y, sigma0, Rs, e1, e2, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma0': 0}
mass_2d(r, rho0, Rs, e1=0, e2=0)[source]

mass enclosed projected 2d sphere of radius r

Parameters:
  • r – projected radius
  • rho0 – density normalization
  • Rs – Hernquist radius
Returns:

mass enclosed 2d projected radius

mass_2d_lens(r, sigma0, Rs, e1=0, e2=0)[source]

mass enclosed projected 2d sphere of radius r Same as mass_2d but with input normalization in units of projected density :param r: projected radius :param sigma0: rho0 * Rs (units of projected density) :param Rs: Hernquist radius :return: mass enclosed 2d projected radius

mass_3d(r, rho0, Rs, e1=0, e2=0)[source]

mass enclosed a 3d sphere or radius r

Parameters:
  • r – 3-d radius within the mass is integrated (same distance units as density definition)
  • rho0 – density normalization
  • Rs – Hernquist radius
Returns:

enclosed mass

mass_3d_lens(r, sigma0, Rs, e1=0, e2=0)[source]

mass enclosed a 3d sphere or radius r in lensing parameterization

Parameters:
  • r – 3-d radius within the mass is integrated (same distance units as density definition)
  • sigma0 – rho0 * Rs (units of projected density)
  • Rs – Hernquist radius
Returns:

enclosed mass

param_names = ['sigma0', 'Rs', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'Rs': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma0': 100}

lenstronomy.LensModel.Profiles.hessian module

class lenstronomy.LensModel.Profiles.hessian.Hessian(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for constant Hessian distortion (second order) The input is in the same convention as the LensModel.hessian() output.

derivatives(x, y, f_xx, f_yy, f_xy, f_yx, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • f_xx – dalpha_x/dx
  • f_yy – dalpha_y/dy
  • f_xy – dalpha_x/dy
  • f_yx – dalpha_y/dx
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

deflection angles

function(x, y, f_xx, f_yy, f_xy, f_yx, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • f_xx – dalpha_x/dx
  • f_yy – dalpha_y/dy
  • f_xy – dalpha_x/dy
  • f_yx – dalpha_y/dx
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

lensing potential

hessian(x, y, f_xx, f_yy, f_xy, f_yx, ra_0=0, dec_0=0)[source]

Hessian. Attention: If f_xy != f_yx then this function is not accurate!

Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • f_xx – dalpha_x/dx
  • f_yy – dalpha_y/dy
  • f_xy – dalpha_x/dy
  • f_yx – dalpha_y/dx
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

f_xx, f_yy, f_xy

lower_limit_default = {'dec_0': -100, 'f_xx': -100, 'f_xy': -100, 'f_yx': -100, 'f_yy': -100, 'ra_0': -100}
param_names = ['f_xx', 'f_yy', 'f_xy', 'f_yx', 'ra_0', 'dec_0']
upper_limit_default = {'dec_0': 100, 'f_xx': 100, 'f_xy': 100, 'f_yx': 100, 'f_yy': 100, 'ra_0': 100}

lenstronomy.LensModel.Profiles.interpol module

class lenstronomy.LensModel.Profiles.interpol.Interpol(grid=False, min_grid_number=100, kwargs_spline=None)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class which uses an interpolation of a lens model and its first and second order derivatives

See also the tests in lenstronomy.test.test_LensModel.test_Profiles.test_interpol.py for example use cases as checks against known analytic models.

The deflection angle is in the same convention as the one in the LensModel module, meaning that: source position = image position - deflection angle

derivatives(x, y, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]

returns df/dx and df/dy of the function

Parameters:
  • x – x-coordinate (angular position), float or numpy array
  • y – y-coordinate (angular position), float or numpy array
  • grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
  • grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
  • f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
  • f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
  • f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
  • f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
Returns:

f_x, f_y at interpolated positions (x, y)

do_interp(x_grid, y_grid, f_, f_x, f_y, f_xx=None, f_yy=None, f_xy=None)[source]
f_interp(x, y, x_grid=None, y_grid=None, f_=None, grid=False)[source]
f_x_interp(x, y, x_grid=None, y_grid=None, f_x=None, grid=False)[source]
f_xx_interp(x, y, x_grid=None, y_grid=None, f_xx=None, grid=False)[source]
f_xy_interp(x, y, x_grid=None, y_grid=None, f_xy=None, grid=False)[source]
f_y_interp(x, y, x_grid=None, y_grid=None, f_y=None, grid=False)[source]
f_yy_interp(x, y, x_grid=None, y_grid=None, f_yy=None, grid=False)[source]
function(x, y, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]
Parameters:
  • x – x-coordinate (angular position), float or numpy array
  • y – y-coordinate (angular position), float or numpy array
  • grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
  • grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
  • f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
  • f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
  • f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
  • f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
Returns:

potential at interpolated positions (x, y)

hessian(x, y, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

Parameters:
  • x – x-coordinate (angular position), float or numpy array
  • y – y-coordinate (angular position), float or numpy array
  • grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
  • grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
  • f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
  • f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
  • f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
  • f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
Returns:

f_xx, f_xy, f_yx, f_yy at interpolated positions (x, y)

lower_limit_default = {}
param_names = ['grid_interp_x', 'grid_interp_y', 'f_', 'f_x', 'f_y', 'f_xx', 'f_yy', 'f_xy']
upper_limit_default = {}
class lenstronomy.LensModel.Profiles.interpol.InterpolScaled(grid=True, min_grid_number=100, kwargs_spline=None)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for handling an interpolated lensing map and has the freedom to scale its lensing effect. Applications are e.g. mass to light ratio.

derivatives(x, y, scale_factor=1, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]
Parameters:
  • x – x-coordinate (angular position), float or numpy array
  • y – y-coordinate (angular position), float or numpy array
  • scale_factor – float, overall scaling of the lens model relative to the input interpolation grid
  • grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
  • grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
  • f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
  • f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
  • f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
  • f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
Returns:

deflection angles in x- and y-direction at position (x, y)

function(x, y, scale_factor=1, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]
Parameters:
  • x – x-coordinate (angular position), float or numpy array
  • y – y-coordinate (angular position), float or numpy array
  • scale_factor – float, overall scaling of the lens model relative to the input interpolation grid
  • grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
  • grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
  • f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
  • f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
  • f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
  • f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
Returns:

potential at interpolated positions (x, y)

hessian(x, y, scale_factor=1, grid_interp_x=None, grid_interp_y=None, f_=None, f_x=None, f_y=None, f_xx=None, f_yy=None, f_xy=None)[source]
Parameters:
  • x – x-coordinate (angular position), float or numpy array
  • y – y-coordinate (angular position), float or numpy array
  • scale_factor – float, overall scaling of the lens model relative to the input interpolation grid
  • grid_interp_x – numpy array (ascending) to mark the x-direction of the interpolation grid
  • grid_interp_y – numpy array (ascending) to mark the y-direction of the interpolation grid
  • f – 2d numpy array of lensing potential, matching the grids in grid_interp_x and grid_interp_y
  • f_x – 2d numpy array of deflection in x-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_y – 2d numpy array of deflection in y-direction, matching the grids in grid_interp_x and grid_interp_y
  • f_xx – 2d numpy array of df/dxx, matching the grids in grid_interp_x and grid_interp_y
  • f_yy – 2d numpy array of df/dyy, matching the grids in grid_interp_x and grid_interp_y
  • f_xy – 2d numpy array of df/dxy, matching the grids in grid_interp_x and grid_interp_y
Returns:

second derivatives of the lensing potential f_xx, f_yy, f_xy at position (x, y)

lower_limit_default = {'scale_factor': 0}
param_names = ['scale_factor', 'grid_interp_x', 'grid_interp_y', 'f_', 'f_x', 'f_y', 'f_xx', 'f_yy', 'f_xy']
upper_limit_default = {'scale_factor': 100}

lenstronomy.LensModel.Profiles.multi_gaussian_kappa module

class lenstronomy.LensModel.Profiles.multi_gaussian_kappa.MultiGaussianKappa[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

density(r, amp, sigma, scale_factor=1)[source]
Parameters:
  • r
  • amp
  • sigma
Returns:

density_2d(x, y, amp, sigma, center_x=0, center_y=0, scale_factor=1)[source]
Parameters:
  • R
  • am
  • sigma_x
  • sigma_y
Returns:

derivatives(x, y, amp, sigma, center_x=0, center_y=0, scale_factor=1)[source]
Parameters:
  • x
  • y
  • amp
  • sigma
  • center_x
  • center_y
Returns:

function(x, y, amp, sigma, center_x=0, center_y=0, scale_factor=1)[source]
Parameters:
  • x
  • y
  • amp
  • sigma
  • center_x
  • center_y
Returns:

hessian(x, y, amp, sigma, center_x=0, center_y=0, scale_factor=1)[source]
Parameters:
  • x
  • y
  • amp
  • sigma
  • center_x
  • center_y
Returns:

lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'sigma': 0}
mass_3d_lens(R, amp, sigma, scale_factor=1)[source]
Parameters:
  • R
  • amp
  • sigma
Returns:

param_names = ['amp', 'sigma', 'center_x', 'center_y']
upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'sigma': 100}
class lenstronomy.LensModel.Profiles.multi_gaussian_kappa.MultiGaussianKappaEllipse[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

density(r, amp, sigma, e1, e2, scale_factor=1)[source]
Parameters:
  • r
  • amp
  • sigma
Returns:

density_2d(x, y, amp, sigma, e1, e2, center_x=0, center_y=0, scale_factor=1)[source]
Parameters:
  • R
  • am
  • sigma_x
  • sigma_y
Returns:

derivatives(x, y, amp, sigma, e1, e2, center_x=0, center_y=0, scale_factor=1)[source]
Parameters:
  • x
  • y
  • amp
  • sigma
  • center_x
  • center_y
Returns:

function(x, y, amp, sigma, e1, e2, center_x=0, center_y=0, scale_factor=1)[source]
Parameters:
  • x
  • y
  • amp
  • sigma
  • center_x
  • center_y
Returns:

hessian(x, y, amp, sigma, e1, e2, center_x=0, center_y=0, scale_factor=1)[source]
Parameters:
  • x
  • y
  • amp
  • sigma
  • center_x
  • center_y
Returns:

lower_limit_default = {'amp': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma': 0}
mass_3d_lens(R, amp, sigma, e1, e2, scale_factor=1)[source]
Parameters:
  • R
  • amp
  • sigma
Returns:

param_names = ['amp', 'sigma', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'amp': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma': 100}

lenstronomy.LensModel.Profiles.multipole module

class lenstronomy.LensModel.Profiles.multipole.Multipole(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This class contains a multipole contribution (for 1 component with m>=2) This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf m : int, multipole order, m>=2 a_m : float, multipole strength phi_m : float, multipole orientation in radian

derivatives(x, y, m, a_m, phi_m, center_x=0, center_y=0)[source]

Deflection of a multipole contribution (for 1 component with m>=2) This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf

Parameters:
  • m – int, multipole order, m>=2
  • a_m – float, multipole strength
  • phi_m – float, multipole orientation in radian
  • center_x – x-position
  • center_y – x-position
Returns:

deflection angles alpha_x, alpha_y

function(x, y, m, a_m, phi_m, center_x=0, center_y=0)[source]

Lensing potential of multipole contribution (for 1 component with m>=2) This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf

Parameters:
  • m – int, multipole order, m>=2
  • a_m – float, multipole strength
  • phi_m – float, multipole orientation in radian
  • center_x – x-position
  • center_y – x-position
Returns:

lensing potential

hessian(x, y, m, a_m, phi_m, center_x=0, center_y=0)[source]

Hessian of a multipole contribution (for 1 component with m>=2) This uses the same definitions as Xu et al.(2013) in Appendix B3 https://arxiv.org/pdf/1307.4220.pdf

Parameters:
  • m – int, multipole order, m>=2
  • a_m – float, multipole strength
  • phi_m – float, multipole orientation in radian
  • center_x – x-position
  • center_y – x-position
Returns:

f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'a_m': 0, 'center_x': -100, 'center_y': -100, 'm': 2, 'phi_m': -3.141592653589793}
param_names = ['m', 'a_m', 'phi_m', 'center_x', 'center_y']
upper_limit_default = {'a_m': 100, 'center_x': 100, 'center_y': 100, 'm': 100, 'phi_m': 3.141592653589793}

lenstronomy.LensModel.Profiles.nfw module

class lenstronomy.LensModel.Profiles.nfw.NFW(interpol=False, num_interp_X=1000, max_interp_X=10)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions concerning the NFW profile

relation are: R_200 = c * Rs The definition of ‘Rs’ is in angular (arc second) units and the normalization is put in in regards to a deflection angle at ‘Rs’ - ‘alpha_Rs’. To convert a physical mass and concentration definition into those lensing quantities for a specific redshift configuration and cosmological model, you can find routines in lenstronomy.Cosmo.lens_cosmo.py

>>> from lenstronomy.Cosmo.lens_cosmo import LensCosmo
>>> from astropy.cosmology import FlatLambdaCDM
>>> cosmo = FlatLambdaCDM(H0=70, Om0=0.3, Ob0=0.05)
>>> lens_cosmo = LensCosmo(z_lens=0.5, z_source=1.5, cosmo=cosmo)

Here we compute the angular scale of Rs on the sky (in arc seconds) and the deflection angle at Rs (in arc seconds):

>>> Rs_angle, alpha_Rs = lens_cosmo.nfw_physical2angle(M=10**13, c=6)

And here we perform the inverse calculation given Rs_angle and alpha_Rs to return the physical halo properties.

>>> rho0, Rs, c, r200, M200 = lens_cosmo.nfw_angle2physical(Rs_angle=Rs_angle, alpha_Rs=alpha_Rs)

The lens model calculation uses angular units as arguments! So to execute a deflection angle calculation one uses

>>> from lenstronomy.LensModel.Profiles.nfw import NFW
>>> nfw = NFW()
>>> alpha_x, alpha_y = nfw.derivatives(x=1, y=1, Rs=Rs_angle, alpha_Rs=alpha_Rs, center_x=0, center_y=0)
F_(X)[source]

computes h()

Parameters:X
Returns:
static alpha2rho0(alpha_Rs, Rs)[source]

convert angle at Rs into rho0

Parameters:
  • alpha_Rs – deflection angle at RS
  • Rs – scale radius
Returns:

density normalization (characteristic density)

density(R, Rs, rho0)[source]

three dimensional NFW profile

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
Returns:

rho(R) density

density_2d(x, y, Rs, rho0, center_x=0, center_y=0)[source]

projected two dimensional NFW profile (kappa*Sigma_crit)

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r200 (float>0) – radius of (sub)halo
Returns:

Epsilon(R) projected density at radius R

density_lens(r, Rs, alpha_Rs)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:
  • r – 3d radios
  • Rs – turn-over radius of NFW profile
  • alpha_Rs – deflection at Rs
Returns:

density rho(r)

derivatives(x, y, Rs, alpha_Rs, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function (integral of NFW), which are the deflection angles

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • Rs – turn over point in the slope of the NFW profile in angular unit
  • alpha_Rs – deflection (angular units) at projected Rs
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

deflection angle in x, deflection angle in y

function(x, y, Rs, alpha_Rs, center_x=0, center_y=0)[source]
Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • Rs – turn over point in the slope of the NFW profile in angular unit
  • alpha_Rs – deflection (angular units) at projected Rs
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

lensing potential

g_(X)[source]

computes h()

Parameters:X (float >0) – R/Rs
Returns:
h_(X)[source]

computes h()

Parameters:X (float >0) – R/Rs
Returns:h(X)
hessian(x, y, Rs, alpha_Rs, center_x=0, center_y=0)[source]
Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • Rs – turn over point in the slope of the NFW profile in angular unit
  • alpha_Rs – deflection (angular units) at projected Rs
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100}
mass_2d(R, Rs, rho0)[source]

mass enclosed a 2d cylinder or projected radius R :param R: projected radius :param Rs: scale radius :param rho0: density normalization (characteristic density) :return: mass in cylinder

mass_3d(r, Rs, rho0)[source]

mass enclosed a 3d sphere or radius r

Parameters:
  • r – 3d radius
  • Rs – scale radius
  • rho0 – density normalization (characteristic density)
Returns:

M(<r)

mass_3d_lens(r, Rs, alpha_Rs)[source]

mass enclosed a 3d sphere or radius r. This function takes as input the lensing parameterization.

Parameters:
  • r – 3d radius
  • Rs – scale radius
  • alpha_Rs – deflection (angular units) at projected Rs
Returns:

M(<r)

nfwAlpha(R, Rs, rho0, ax_x, ax_y)[source]

deflection angel of NFW profile (times Sigma_crit D_OL) along the projection to coordinate ‘axis’

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r200 (float>0) – radius of (sub)halo
  • axis (same as R) – projection to either x- or y-axis
Returns:

Epsilon(R) projected density at radius R

nfwGamma(R, Rs, rho0, ax_x, ax_y)[source]

shear gamma of NFW profile (times Sigma_crit) along the projection to coordinate ‘axis’

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r200 (float>0) – radius of (sub)halo
  • axis (same as R) – projection to either x- or y-axis
Returns:

Epsilon(R) projected density at radius R

nfwPot(R, Rs, rho0)[source]

lensing potential of NFW profile (Sigma_crit D_OL**2)

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r200 (float>0) – radius of (sub)halo
Returns:

Epsilon(R) projected density at radius R

param_names = ['Rs', 'alpha_Rs', 'center_x', 'center_y']
profile_name = 'NFW'
static rho02alpha(rho0, Rs)[source]

convert rho0 to angle at Rs

Parameters:
  • rho0 – density normalization (characteristic density)
  • Rs – scale radius
Returns:

deflection angle at RS

upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100}

lenstronomy.LensModel.Profiles.nfw_ellipse module

class lenstronomy.LensModel.Profiles.nfw_ellipse.NFW_ELLIPSE(interpol=False, num_interp_X=1000, max_interp_X=10)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions concerning the NFW profile with an ellipticity defined in the potential parameterization of alpha_Rs and Rs is the same as for the spherical NFW profile

from Glose & Kneib: https://cds.cern.ch/record/529584/files/0112138.pdf

relation are: R_200 = c * Rs

density_lens(r, Rs, alpha_Rs, e1=1, e2=0)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:
  • r – 3d radios
  • Rs – turn-over radius of NFW profile
  • alpha_Rs – deflection at Rs
Returns:

density rho(r)

derivatives(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function, calculated as an elliptically distorted deflection angle of the spherical NFW profile

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • Rs – turn over point in the slope of the NFW profile in angular unit
  • alpha_Rs – deflection (angular units) at projected Rs
  • e1 – eccentricity component in x-direction
  • e2 – eccentricity component in y-direction
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

deflection in x-direction, deflection in y-direction

function(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]

returns elliptically distorted NFW lensing potential

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • Rs – turn over point in the slope of the NFW profile in angular unit
  • alpha_Rs – deflection (angular units) at projected Rs
  • e1 – eccentricity component in x-direction
  • e2 – eccentricity component in y-direction
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

lensing potential

hessian(x, y, Rs, alpha_Rs, e1, e2, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy the calculation is performed as a numerical differential from the deflection field. Analytical relations are possible

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • Rs – turn over point in the slope of the NFW profile in angular unit
  • alpha_Rs – deflection (angular units) at projected Rs
  • e1 – eccentricity component in x-direction
  • e2 – eccentricity component in y-direction
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5}
mass_3d_lens(R, Rs, alpha_Rs, e1=1, e2=0)[source]
Parameters:
  • R – radius (in angular units)
  • Rs
  • alpha_Rs
  • e1
  • e2
Returns:

param_names = ['Rs', 'alpha_Rs', 'e1', 'e2', 'center_x', 'center_y']
profile_name = 'NFW_ELLIPSE'
upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5}

lenstronomy.LensModel.Profiles.nfw_mass_concentration module

class lenstronomy.LensModel.Profiles.nfw_mass_concentration.NFWMC(z_lens, z_source, cosmo=None, static=False)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions parameterises the NFW profile with log10 M200 and the concentration rs/r200 relation are: R_200 = c * Rs

ATTENTION: the parameterization is cosmology and redshift dependent! The cosmology to connect mass and deflection relations is fixed to default H0=70km/s Omega_m=0.3 flat LCDM. It is recommended to keep a given cosmology definition in the lens modeling as the observable reduced deflection angles are sensitive in this parameterization. If you do not want to impose a mass-concentration relation, it is recommended to use the default NFW lensing profile parameterized in reduced deflection angles.

derivatives(x, y, logM, concentration, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function (integral of NFW)

function(x, y, logM, concentration, center_x=0, center_y=0)[source]
Parameters:
  • x – angular position
  • y – angular position
  • Rs – angular turn over point
  • alpha_Rs – deflection at Rs
  • center_x – center of halo
  • center_y – center of halo
Returns:

hessian(x, y, logM, concentration, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'center_x': -100, 'center_y': -100, 'concentration': 0.01, 'logM': 0}
param_names = ['logM', 'concentration', 'center_x', 'center_y']
set_dynamic()[source]
Returns:
set_static(logM, concentration, center_x=0, center_y=0)[source]
Parameters:
  • logM
  • concentration
  • center_x
  • center_y
Returns:

upper_limit_default = {'center_x': 100, 'center_y': 100, 'concentration': 1000, 'logM': 16}

lenstronomy.LensModel.Profiles.nfw_vir_trunc module

class lenstronomy.LensModel.Profiles.nfw_vir_trunc.NFWVirTrunc(z_lens, z_source, cosmo=None)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions concerning the NFW profile that is sharply truncated at the virial radius https://arxiv.org/pdf/astro-ph/0304034.pdf

relation are: R_200 = c * Rs

kappa(theta, logM, c)[source]

projected surface brightness

Parameters:
  • theta – radial angle from the center of the profile
  • logM – log_10 halo mass in physical units of M_sun
  • c – concentration of the halo; r_vir = c * r_s
Returns:

convergence at theta

lenstronomy.LensModel.Profiles.nie module

class lenstronomy.LensModel.Profiles.nie.NIE[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

Non-singular isothermal ellipsoid (NIE)

\[\kappa = \theta_E/2 \left[s_{scale} + qx^2 + y^2/q]−1/2\]
density_lens(r, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]

3d mass density at 3d radius r. This function assumes spherical symmetry/ignoring the eccentricity.

Parameters:
  • r – 3d radius
  • theta_E – Einstein radius
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale
  • center_x – profile center
  • center_y – profile center
Returns:

3d mass density at 3d radius r

derivatives(x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale
  • center_x – profile center
  • center_y – profile center
Returns:

alpha_x, alpha_y

function(x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale
  • center_x – profile center
  • center_y – profile center
Returns:

lensing potential

hessian(x, y, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale
  • center_x – profile center
  • center_y – profile center
Returns:

f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 's_scale': 0, 'theta_E': 0}
mass_3d_lens(r, theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]

mass enclosed a 3d radius r. This function assumes spherical symmetry/ignoring the eccentricity.

Parameters:
  • r – 3d radius
  • theta_E – Einstein radius
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale
  • center_x – profile center
  • center_y – profile center
Returns:

3d mass density at 3d radius r

param_conv(theta_E, e1, e2, s_scale)[source]
param_names = ['theta_E', 'e1', 'e2', 's_scale', 'center_x', 'center_y']
set_dynamic()[source]
Returns:
set_static(theta_E, e1, e2, s_scale, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale
  • center_x – profile center
  • center_y – profile center
Returns:

self variables set

upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 's_scale': 100, 'theta_E': 10}
class lenstronomy.LensModel.Profiles.nie.NIEMajorAxis(diff=1e-10)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This class contains the function and the derivatives of the non-singular isothermal ellipse. See Keeton and Kochanek 1998, https://arxiv.org/pdf/astro-ph/9705194.pdf

\[\kappa = b * (q2(s2 + x2) + y2􏰉)^{−1/2}`\]
derivatives(x, y, b, s, q)[source]

returns df/dx and df/dy of the function

function(x, y, b, s, q)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, b, s, q)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

static kappa(x, y, b, s, q)[source]

convergence

Parameters:
  • x – major axis coordinate
  • y – minor axis coordinate
  • b – normalization
  • s – smoothing scale
  • q – axis ratio
Returns:

convergence

param_names = ['b', 's', 'q', 'center_x', 'center_y']

lenstronomy.LensModel.Profiles.nie_potential module

class lenstronomy.LensModel.Profiles.nie_potential.NIE_POTENTIAL[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class implements the elliptical potential of Eq. (67) of LECTURES ON GRAVITATIONAL LENSING and Eq. (1) of Blandford & Kochanek 1987, mapped to Eq. (8) of Barnaka1998 to find the ellipticity bounds

derivatives(x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • theta_E – Einstein radius (in angles)
  • theta_c – core radius (in angles)
  • e1 – eccentricity component, x direction(dimensionless)
  • e2 – eccentricity component, y direction (dimensionless)
Returns:

deflection angle (in angles)

function(x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • theta_E – Einstein radius (in angles)
  • theta_c – core radius (in angles)
  • e1 – eccentricity component, x direction(dimensionless)
  • e2 – eccentricity component, y direction (dimensionless)
Returns:

lensing potential

hessian(x, y, theta_E, theta_c, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • theta_E – Einstein radius (in angles)
  • theta_c – core radius (in angles)
  • e1 – eccentricity component, x direction(dimensionless)
  • e2 – eccentricity component, y direction (dimensionless)
Returns:

hessian matrix (in angles)

lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': 0, 'e2': 0, 'theta_E': 0, 'theta_c': 0}
param_conv(theta_E, theta_c, e1, e2)[source]
param_names = ['center_x', 'center_y', 'theta_E', 'theta_c', 'e1', 'e2']
set_dynamic()[source]
Returns:
set_static(theta_E, theta_c, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate in image plane
  • y – y-coordinate in image plane
  • theta_E – Einstein radius
  • theta_c – core radius
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – profile center
  • center_y – profile center
Returns:

self variables set

upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.2, 'e2': 0.2, 'theta_E': 10, 'theta_c': 10}
class lenstronomy.LensModel.Profiles.nie_potential.NIEPotentialMajorAxis(diff=1e-10)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class implements the elliptical potential of Eq. (67) of LECTURES ON GRAVITATIONAL LENSING and Eq. (1) of Blandford & Kochanek 1987, mapped to Eq. (8) of Barnaka1998 to find the ellipticity bounds

derivatives(x, y, theta_E, theta_c, eps)[source]

returns df/dx and df/dy of the function

function(x, y, theta_E, theta_c, eps)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, theta_E, theta_c, eps)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

param_names = ['theta_E', 'theta_c', 'eps', 'center_x', 'center_y']

lenstronomy.LensModel.Profiles.numerical_deflections module

class lenstronomy.LensModel.Profiles.numerical_deflections.NumericalAlpha(custom_class)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This class allows one to incorporate any lens profile into the usage framework of lenstronomy. When creating the instance of LensModel with this lens profile, you must pass in numerical_alpha_class = CustomClass(), where CustomClass is a class with a call method that returns the x/y deflection angles. This allows one to numerically compute and interpolate deflection angles for potentially very complex mass profiles, and then use the results with lenstronomy without having to heavily modify the existing structure of the software.

derivatives(x, y, center_x=0, center_y=0, **kwargs)[source]
Parameters:
  • x – x coordinate [arcsec]
  • y – x coordinate [arcsec]
  • center_x – deflector x center [arcsec]
  • center_y – deflector y center [arcsec]
  • kwargs – keyword arguments for the custom profile
Returns:

function(x, y, center_x=0, center_y=0, **kwargs)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, center_x=0, center_y=0, **kwargs)[source]

Returns the components of the hessian matrix :param x: x coordinate [arcsec] :param y: y coordinate [arcsec] :param center_x: the deflector x coordinate :param center_y: the deflector y coordinate :param kwargs: keyword arguments for the profile :return: the derivatives of the deflection angles that make up the hessian matrix

lenstronomy.LensModel.Profiles.p_jaffe module

class lenstronomy.LensModel.Profiles.p_jaffe.PJaffe[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class to compute the DUAL PSEUDO ISOTHERMAL ELLIPTICAL MASS DISTRIBUTION based on Eliasdottir (2007) https://arxiv.org/pdf/0710.5636.pdf Appendix A

Module name: ‘PJAFFE’;

An alternative name is dPIED.

The 3D density distribution is

\[\rho(r) = \frac{\rho_0}{(1+r^2/Ra^2)(1+r^2/Rs^2)}\]

with \(Rs > Ra\).

The projected density is

\[\Sigma(R) = \Sigma_0 \frac{Ra Rs}{Rs-Ra}\left(\frac{1}{\sqrt{Ra^2+R^2}} - \frac{1}{\sqrt{Rs^2+R^2}} \right)\]

with

\[\Sigma_0 = \pi \rho_0 \frac{Ra Rs}{Rs + Ra}\]

In the lensing parameterization,

\[\sigma_0 = \frac{\Sigma_0}{\Sigma_{\rm crit}}\]
density(r, rho0, Ra, Rs)[source]

computes the density

Parameters:
  • r – radial distance from the center (in 3D)
  • rho0 – density normalization (see class documentation above)
  • Ra – core radius
  • Rs – transition radius from logarithmic slope -2 to -4
Returns:

density at r

density_2d(x, y, rho0, Ra, Rs, center_x=0, center_y=0)[source]

projected density

Parameters:
  • x – projected coordinate on the sky
  • y – projected coordinate on the sky
  • rho0 – density normalization (see class documentation above)
  • Ra – core radius
  • Rs – transition radius from logarithmic slope -2 to -4
  • center_x – center of profile
  • center_y – center of profile
Returns:

projected density

derivatives(x, y, sigma0, Ra, Rs, center_x=0, center_y=0)[source]

deflection angles

Parameters:
  • x – projected coordinate on the sky
  • y – projected coordinate on the sky
  • sigma0 – sigma0/sigma_crit (see class documentation above)
  • Ra – core radius (see class documentation above)
  • Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
  • center_x – center of profile
  • center_y – center of profile
Returns:

f_x, f_y

function(x, y, sigma0, Ra, Rs, center_x=0, center_y=0)[source]

lensing potential

Parameters:
  • x – projected coordinate on the sky
  • y – projected coordinate on the sky
  • sigma0 – sigma0/sigma_crit (see class documentation above)
  • Ra – core radius (see class documentation above)
  • Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
  • center_x – center of profile
  • center_y – center of profile
Returns:

lensing potential

grav_pot(r, rho0, Ra, Rs)[source]

gravitational potential (modulo 4 pi G and rho0 in appropriate units)

Parameters:
  • r – radial distance from the center (in 3D)
  • rho0 – density normalization (see class documentation above)
  • Ra – core radius
  • Rs – transition radius from logarithmic slope -2 to -4
Returns:

gravitational potential (modulo 4 pi G and rho0 in appropriate units)

hessian(x, y, sigma0, Ra, Rs, center_x=0, center_y=0)[source]

Hessian of lensing potential

Parameters:
  • x – projected coordinate on the sky
  • y – projected coordinate on the sky
  • sigma0 – sigma0/sigma_crit (see class documentation above)
  • Ra – core radius (see class documentation above)
  • Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
  • center_x – center of profile
  • center_y – center of profile
Returns:

f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'Ra': 0, 'Rs': 0, 'center_x': -100, 'center_y': -100, 'sigma0': 0}
mass_2d(r, rho0, Ra, Rs)[source]

mass enclosed projected 2d sphere of radius r

Parameters:
  • r – radial distance from the center in projection
  • rho0 – density normalization (see class documentation above)
  • Ra – core radius
  • Rs – transition radius from logarithmic slope -2 to -4
Returns:

Sigma(<r)

mass_3d(r, rho0, Ra, Rs)[source]

mass enclosed a 3d sphere or radius r

Parameters:
  • r – radial distance from the center (in 3D)
  • rho0 – density normalization (see class documentation above)
  • Ra – core radius
  • Rs – transition radius from logarithmic slope -2 to -4
Returns:

M(<r)

mass_3d_lens(r, sigma0, Ra, Rs)[source]

mass enclosed a 3d sphere or radius r given a lens parameterization with angular units

Parameters:
  • r – radial distance from the center (in 3D)
  • rho0 – density normalization (see class documentation above)
  • Ra – core radius
  • Rs – transition radius from logarithmic slope -2 to -4
Returns:

M(<r) in angular units (modulo critical mass density)

mass_tot(rho0, Ra, Rs)[source]

total mass within the profile

Parameters:
  • rho0 – density normalization (see class documentation above)
  • Ra – core radius
  • Rs – transition radius from logarithmic slope -2 to -4
Returns:

total mass

param_names = ['sigma0', 'Ra', 'Rs', 'center_x', 'center_y']
rho2sigma(rho0, Ra, Rs)[source]

converts 3d density into 2d projected density parameter, Equation A4 in Eliasdottir (2007)

Parameters:
  • rho0 – density normalization
  • Ra – core radius (see class documentation above)
  • Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
Returns:

projected density normalization

sigma2rho(sigma0, Ra, Rs)[source]

inverse of rho2sigma()

Parameters:
  • sigma0 – projected density normalization
  • Ra – core radius (see class documentation above)
  • Rs – transition radius from logarithmic slope -2 to -4 (see class documentation above)
Returns:

3D density normalization

upper_limit_default = {'Ra': 100, 'Rs': 100, 'center_x': 100, 'center_y': 100, 'sigma0': 10}

lenstronomy.LensModel.Profiles.p_jaffe_ellipse module

class lenstronomy.LensModel.Profiles.p_jaffe_ellipse.PJaffe_Ellipse[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions concerning the NFW profile

relation are: R_200 = c * Rs

derivatives(x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function (integral of NFW)

function(x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0)[source]

returns double integral of NFW profile

hessian(x, y, sigma0, Ra, Rs, e1, e2, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'Ra': 0, 'Rs': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'sigma0': 0}
mass_3d_lens(r, sigma0, Ra, Rs, e1=0, e2=0)[source]
Parameters:
  • r
  • sigma0
  • Ra
  • Rs
  • q
  • phi_G
Returns:

param_names = ['sigma0', 'Ra', 'Rs', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'Ra': 100, 'Rs': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'sigma0': 10}

lenstronomy.LensModel.Profiles.pemd module

class lenstronomy.LensModel.Profiles.pemd.PEMD(suppress_fastell=False)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for power law ellipse mass density profile. This class effectively calls the class SPEMD_SMOOTH with a fixed and very small central smoothing scale to perform the numerical integral using the FASTELL code by Renan Barkana.

\[\kappa(x, y) = \frac{3-\gamma}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q}} \right)^{\gamma-1}\]

with \(\theta_{E}\) is the (circularized) Einstein radius, \(\gamma\) is the negative power-law slope of the 3D mass distributions, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate system aligned with the major and minor axis of the lens.

In terms of eccentricities, this profile is defined as

\[\kappa(r) = \frac{3-\gamma}{2} \left(\frac{\theta'_{E}}{r \sqrt{1 − e*\cos(2*\phi)}} \right)^{\gamma-1}\]

with \(\epsilon\) is the ellipticity defined as

\[\epsilon = \frac{1-q^2}{1+q^2}\]

And an Einstein radius \(\theta'_{\rm E}\) related to the definition used is

\[\left(\frac{\theta'_{\rm E}}{\theta_{\rm E}}\right)^{2} = \frac{2q}{1+q^2}.\]
density_lens(r, theta_E, gamma, e1=None, e2=None)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:
  • r – radius within the mass is computed
  • theta_E – Einstein radius
  • gamma – power-law slope
  • e1 – eccentricity component (not used)
  • e2 – eccentricity component (not used)
Returns:

mass enclosed a 3D radius r

derivatives(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

deflection angles alpha_x, alpha_y

function(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

lensing potential

hessian(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

Hessian components f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 1.5, 'theta_E': 0}
mass_3d_lens(r, theta_E, gamma, e1=None, e2=None)[source]

computes the spherical power-law mass enclosed (with SPP routine) :param r: radius within the mass is computed :param theta_E: Einstein radius :param gamma: power-law slope :param e1: eccentricity component (not used) :param e2: eccentricity component (not used) :return: mass enclosed a 3D radius r

param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 2.5, 'theta_E': 100}

lenstronomy.LensModel.Profiles.point_mass module

class lenstronomy.LensModel.Profiles.point_mass.PointMass[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class to compute the physical deflection angle of a point mass, given as an Einstein radius

derivatives(x, y, theta_E, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • theta_E – Einstein radius (in angles)
Returns:

deflection angle (in angles)

function(x, y, theta_E, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • theta_E – Einstein radius (in angles)
Returns:

lensing potential

hessian(x, y, theta_E, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coord (in angles)
  • y – y-coord (in angles)
  • theta_E – Einstein radius (in angles)
Returns:

hessian matrix (in angles)

lower_limit_default = {'center_x': -100, 'center_y': -100, 'theta_E': 0}
param_names = ['theta_E', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'theta_E': 100}

lenstronomy.LensModel.Profiles.sersic module

class lenstronomy.LensModel.Profiles.sersic.Sersic(smoothing=1e-05, sersic_major_axis=False)[source]

Bases: lenstronomy.LensModel.Profiles.sersic_utils.SersicUtil, lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions to evaluate a Sersic mass profile: https://arxiv.org/pdf/astro-ph/0311559.pdf

derivatives(x, y, n_sersic, R_sersic, k_eff, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, n_sersic, R_sersic, k_eff, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • n_sersic – Sersic index
  • R_sersic – half light radius
  • k_eff – convergence at half light radius
  • center_x – x-center
  • center_y – y-center
Returns:

hessian(x, y, n_sersic, R_sersic, k_eff, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'R_sersic': 0, 'center_x': -100, 'center_y': -100, 'k_eff': 0, 'n_sersic': 0.5}
param_names = ['k_eff', 'R_sersic', 'n_sersic', 'center_x', 'center_y']
upper_limit_default = {'R_sersic': 100, 'center_x': 100, 'center_y': 100, 'k_eff': 10, 'n_sersic': 8}

lenstronomy.LensModel.Profiles.sersic_ellipse_kappa module

class lenstronomy.LensModel.Profiles.sersic_ellipse_kappa.SersicEllipseKappa[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains the function and the derivatives of an elliptical sersic profile with the ellipticity introduced in the convergence (not the potential).

This requires the use of numerical integrals (Keeton 2004)

derivatives(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]

deflection angles

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

function(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'R_sersic': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'k_eff': 0, 'n_sersic': 0.5}
param_names = ['k_eff', 'R_sersic', 'n_sersic', 'e1', 'e2', 'center_x', 'center_y']
projected_mass(x, y, q, n_sersic, R_sersic, k_eff, u=1, power=1)[source]
upper_limit_default = {'R_sersic': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'k_eff': 10, 'n_sersic': 8}

lenstronomy.LensModel.Profiles.sersic_ellipse_potential module

class lenstronomy.LensModel.Profiles.sersic_ellipse_potential.SersicEllipse[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions to evaluate a Sersic mass profile: https://arxiv.org/pdf/astro-ph/0311559.pdf

derivatives(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]

returns Gaussian

hessian(x, y, n_sersic, R_sersic, k_eff, e1, e2, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'R_sersic': 0, 'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'k_eff': 0, 'n_sersic': 0.5}
param_names = ['k_eff', 'R_sersic', 'n_sersic', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'R_sersic': 100, 'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'k_eff': 10, 'n_sersic': 8}

lenstronomy.LensModel.Profiles.sersic_utils module

class lenstronomy.LensModel.Profiles.sersic_utils.SersicUtil(smoothing=1e-05, sersic_major_axis=False)[source]

Bases: object

alpha_abs(x, y, n_sersic, r_eff, k_eff, center_x=0, center_y=0)[source]
Parameters:
  • x
  • y
  • n_sersic
  • r_eff
  • k_eff
  • center_x
  • center_y
Returns:

static b_n(n)[source]

b(n) computation. This is the approximation of the exact solution to the relation, 2*incomplete_gamma_function(2n; b_n) = Gamma_function(2*n). :param n: the sersic index :return:

d_alpha_dr(x, y, n_sersic, r_eff, k_eff, center_x=0, center_y=0)[source]
Parameters:
  • x
  • y
  • n_sersic
  • r_eff
  • k_eff
  • center_x
  • center_y
Returns:

density(x, y, n_sersic, r_eff, k_eff, center_x=0, center_y=0)[source]

de-projection of the Sersic profile based on Prugniel & Simien (1997) :return:

get_distance_from_center(x, y, e1, e2, center_x, center_y)[source]

Get the distance from the center of Sersic, accounting for orientation and axis ratio :param x: :param y: :param e1: eccentricity :param e2: eccentricity :param center_x: center x of sersic :param center_y: center y of sersic

k_Re(n, k)[source]
k_bn(n, Re)[source]

returns normalisation of the sersic profile such that Re is the half light radius given n_sersic slope

total_flux(amp, R_sersic, n_sersic, e1=0, e2=0, **kwargs)[source]

computes analytical integral to compute total flux of the Sersic profile

Parameters:
  • amp – amplitude parameter in Sersic function (surface brightness at R_sersic
  • R_sersic – half-light radius in semi-major axis
  • n_sersic – Sersic index
  • e1 – eccentricity
  • e2 – eccentricity
Returns:

Analytic integral of the total flux of the Sersic profile

lenstronomy.LensModel.Profiles.shapelet_pot_cartesian module

class lenstronomy.LensModel.Profiles.shapelet_pot_cartesian.CartShapelets(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains the function and the derivatives of the cartesian shapelets

H_n(n, x)[source]

constructs the Hermite polynomial of order n at position x (dimensionless)

Parameters:
  • n – The n’the basis function.
  • x – 1-dim position (dimensionless)
Returns:

array– H_n(x).

Raises:

AttributeError, KeyError

derivatives(x, y, coeffs, beta, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, coeffs, beta, center_x=0, center_y=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, coeffs, beta, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'beta': 0, 'center_x': -100, 'center_y': -100, 'coeffs': [0]}
param_names = ['coeffs', 'beta', 'center_x', 'center_y']
phi_n(n, x)[source]

constructs the 1-dim basis function (formula (1) in Refregier et al. 2001)

Parameters:
  • n – The n’the basis function.
  • x – 1-dim position (dimensionless)
Returns:

array– phi_n(x).

Raises:

AttributeError, KeyError

pre_calc(x, y, beta, n_order, center_x, center_y)[source]

calculates the H_n(x) and H_n(y) for a given x-array and y-array :param x: :param y: :param amp: :param beta: :param n_order: :param center_x: :param center_y: :return: list of H_n(x) and H_n(y)

upper_limit_default = {'beta': 100, 'center_x': 100, 'center_y': 100, 'coeffs': [100]}

lenstronomy.LensModel.Profiles.shapelet_pot_polar module

class lenstronomy.LensModel.Profiles.shapelet_pot_polar.PolarShapelets[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains the function and the derivatives of the Singular Isothermal Sphere

derivatives(x, y, coeffs, beta, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, coeffs, beta, center_x=0, center_y=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, coeffs, beta, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'beta': 0, 'center_x': -100, 'center_y': -100, 'coeffs': [0]}
param_names = ['coeffs', 'beta', 'center_x', 'center_y']
upper_limit_default = {'beta': 100, 'center_x': 100, 'center_y': 100, 'coeffs': [100]}

lenstronomy.LensModel.Profiles.shear module

class lenstronomy.LensModel.Profiles.shear.Shear(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for external shear gamma1, gamma2 expression

derivatives(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • gamma1 – shear component
  • gamma2 – shear component
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

deflection angles

function(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • gamma1 – shear component
  • gamma2 – shear component
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

lensing potential

hessian(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • gamma1 – shear component
  • gamma2 – shear component
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'dec_0': -100, 'gamma1': -0.5, 'gamma2': -0.5, 'ra_0': -100}
param_names = ['gamma1', 'gamma2', 'ra_0', 'dec_0']
upper_limit_default = {'dec_0': 100, 'gamma1': 0.5, 'gamma2': 0.5, 'ra_0': 100}
class lenstronomy.LensModel.Profiles.shear.ShearGammaPsi[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class to model a shear field with shear strength and direction. The translation ot the cartesian shear distortions is as follow:

\[\gamma_1 = \gamma_{ext} \cos(2 \phi_{ext} \gamma_2 = \gamma_{ext} \sin(2 \phi_{ext}\]
derivatives(x, y, gamma_ext, psi_ext, ra_0=0, dec_0=0)[source]

deflection angles

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

static function(x, y, gamma_ext, psi_ext, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • gamma_ext – shear strength
  • psi_ext – shear angle (radian)
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

hessian(x, y, gamma_ext, psi_ext, ra_0=0, dec_0=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

lower_limit_default = {'dec_0': -100, 'gamma_ext': 0, 'psi_ext': -3.141592653589793, 'ra_0': -100}
param_names = ['gamma_ext', 'psi_ext', 'ra_0', 'dec_0']
upper_limit_default = {'dec_0': 100, 'gamma_ext': 1, 'psi_ext': 3.141592653589793, 'ra_0': 100}
class lenstronomy.LensModel.Profiles.shear.ShearReduced[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

reduced shear distortions \(\gamma' = \gamma / (1-\kappa)\). This distortion keeps the magnification as unity and, thus, does not change the size of apparent objects. To keep the magnification at unity, it requires

\[(1-\kappa)^2 - \gamma_1^2 - \gamma_2^ = 1\]

Thus, for given pair of reduced shear \((\gamma'_1, \gamma'_2)\), an additional convergence term is calculated and added to the lensing distortions.

derivatives(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • gamma1 – shear component
  • gamma2 – shear component
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

deflection angles

function(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • gamma1 – shear component
  • gamma2 – shear component
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

lensing potential

hessian(x, y, gamma1, gamma2, ra_0=0, dec_0=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y0-coordinate (angle)
  • gamma1 – shear component
  • gamma2 – shear component
  • ra_0 – x/ra position where shear deflection is 0
  • dec_0 – y/dec position where shear deflection is 0
Returns:

f_xx, f_xy, f_yx, f_yy

lower_limit_default = {'dec_0': -100, 'gamma1': -0.5, 'gamma2': -0.5, 'ra_0': -100}
param_names = ['gamma1', 'gamma2', 'ra_0', 'dec_0']
upper_limit_default = {'dec_0': 100, 'gamma1': 0.5, 'gamma2': 0.5, 'ra_0': 100}

lenstronomy.LensModel.Profiles.sie module

class lenstronomy.LensModel.Profiles.sie.SIE(NIE=True)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for singular isothermal ellipsoid (SIS with ellipticity)

\[\kappa(x, y) = \frac{1}{2} \left(\frac{\theta_{E}}{\sqrt{q x^2 + y^2/q}} \right)\]

with \(\theta_{E}\) is the (circularized) Einstein radius, \(q\) is the minor/major axis ratio, and \(x\) and \(y\) are defined in a coordinate sys- tem aligned with the major and minor axis of the lens.

In terms of eccentricities, this profile is defined as

\[\kappa(r) = \frac{1}{2} \left(\frac{\theta'_{E}}{r \sqrt{1 − e*\cos(2*\phi)}} \right)\]

with \(\epsilon\) is the ellipticity defined as

\[\epsilon = \frac{1-q^2}{1+q^2}\]

And an Einstein radius \(\theta'_{\rm E}\) related to the definition used is

\[\left(\frac{\theta'_{\rm E}}{\theta_{\rm E}}\right)^{2} = \frac{2q}{1+q^2}.\]
static density(r, rho0, e1=0, e2=0)[source]

computes the density :param r: radius in angles :param rho0: density at angle=1 :return: density at r

static density_2d(x, y, rho0, e1=0, e2=0, center_x=0, center_y=0)[source]

projected density :param x: :param y: :param rho0: :param center_x: :param center_y: :return:

density_lens(r, theta_E, e1=0, e2=0)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:
  • r – radius in angles
  • theta_E – Einstein radius
  • e1 – eccentricity component
  • e2 – eccentricity component
Returns:

density

derivatives(x, y, theta_E, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x
  • y
  • theta_E
  • q
  • phi_G
  • center_x
  • center_y
Returns:

function(x, y, theta_E, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x
  • y
  • theta_E
  • q
  • phi_G
  • center_x
  • center_y
Returns:

grav_pot(x, y, rho0, e1=0, e2=0, center_x=0, center_y=0)[source]

gravitational potential (modulo 4 pi G and rho0 in appropriate units) :param x: :param y: :param rho0: :param a: :param s: :param center_x: :param center_y: :return:

hessian(x, y, theta_E, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x
  • y
  • theta_E
  • q
  • phi_G
  • center_x
  • center_y
Returns:

lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'theta_E': 0}
mass_2d(r, rho0, e1=0, e2=0)[source]

mass enclosed projected 2d sphere of radius r :param r: :param rho0: :param a: :param s: :return:

mass_2d_lens(r, theta_E, e1=0, e2=0)[source]
Parameters:
  • r
  • theta_E
Returns:

static mass_3d(r, rho0, e1=0, e2=0)[source]

mass enclosed a 3d sphere or radius r :param r: radius in angular units :param rho0: density at angle=1 :return: mass in angular units

mass_3d_lens(r, theta_E, e1=0, e2=0)[source]

mass enclosed a 3d sphere or radius r given a lens parameterization with angular units

Parameters:
  • r – radius in angular units
  • theta_E – Einstein radius
Returns:

mass in angular units

param_names = ['theta_E', 'e1', 'e2', 'center_x', 'center_y']
static theta2rho(theta_E)[source]

converts projected density parameter (in units of deflection) into 3d density parameter :param theta_E: :return:

upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'theta_E': 100}

lenstronomy.LensModel.Profiles.sis module

class lenstronomy.LensModel.Profiles.sis.SIS(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains the function and the derivatives of the Singular Isothermal Sphere

\[\kappa(x, y) = \frac{1}{2} \left(\frac{\theta_{E}}{\sqrt{x^2 + y^2}} \right)\]

with \(\theta_{E}\) is the Einstein radius,

static density(r, rho0)[source]

computes the density :param r: radius in angles :param rho0: density at angle=1 :return: density at r

static density_2d(x, y, rho0, center_x=0, center_y=0)[source]

projected density :param x: :param y: :param rho0: :param center_x: :param center_y: :return:

density_lens(r, theta_E)[source]

computes the density at 3d radius r given lens model parameterization. The integral in projected in units of angles (i.e. arc seconds) results in the convergence quantity.

Parameters:
  • r – 3d radius
  • theta_E – Einstein radius
Returns:

density(r)

derivatives(x, y, theta_E, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, theta_E, center_x=0, center_y=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

grav_pot(x, y, rho0, center_x=0, center_y=0)[source]

gravitational potential (modulo 4 pi G and rho0 in appropriate units) :param x: :param y: :param rho0: :param a: :param s: :param center_x: :param center_y: :return:

hessian(x, y, theta_E, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'center_x': -100, 'center_y': -100, 'theta_E': 0}
mass_2d(r, rho0)[source]

mass enclosed projected 2d sphere of radius r :param r: :param rho0: :param a: :param s: :return:

mass_2d_lens(r, theta_E)[source]
Parameters:
  • r – radius
  • theta_E – Einstein radius
Returns:

mass within a radius in projection

static mass_3d(r, rho0)[source]

mass enclosed a 3d sphere or radius r :param r: radius in angular units :param rho0: density at angle=1 :return: mass in angular units

mass_3d_lens(r, theta_E)[source]

mass enclosed a 3d sphere or radius r given a lens parameterization with angular units

Parameters:
  • r – radius in angular units
  • theta_E – Einstein radius
Returns:

mass in angular units

param_names = ['theta_E', 'center_x', 'center_y']
static rho2theta(rho0)[source]

converts 3d density into 2d projected density parameter :param rho0: :param gamma: :return:

static theta2rho(theta_E)[source]

converts projected density parameter (in units of deflection) into 3d density parameter :param theta_E: :return:

upper_limit_default = {'center_x': 100, 'center_y': 100, 'theta_E': 100}

lenstronomy.LensModel.Profiles.sis_truncate module

class lenstronomy.LensModel.Profiles.sis_truncate.SIS_truncate(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains the function and the derivatives of the Singular Isothermal Sphere

derivatives(x, y, theta_E, r_trunc, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function

function(x, y, theta_E, r_trunc, center_x=0, center_y=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, theta_E, r_trunc, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

lower_limit_default = {'center_x': -100, 'center_y': -100, 'r_trunc': 0, 'theta_E': 0}
param_names = ['theta_E', 'r_trunc', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'r_trunc': 100, 'theta_E': 100}

lenstronomy.LensModel.Profiles.spemd module

class lenstronomy.LensModel.Profiles.spemd.SPEMD(suppress_fastell=False)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for smooth power law ellipse mass density profile. This class effectively performs the FASTELL calculations by Renan Barkana. The parameters are changed and represent a spherically averaged Einstein radius an a logarithmic 3D mass profile slope.

The Einstein ring parameter converts to the definition used by GRAVLENS as follow: (theta_E / theta_E_gravlens) = sqrt[ (1+q^2) / (2 q) ]

FASTELL has the following defintions: The parameters are position (x1,x2), overall factor (q), power (gam), axis ratio (arat) which is <=1, core radius squared (s2), and the output potential (phi). The projected mass density distribution, in units of the critical density, is kappa(x1,x2)=q [u2+s2]^(-gam), where u2=[x1^2+x2^2/(arat^2)].

static convert_params(theta_E, gamma, q, s_scale)[source]

converts parameter defintions into quantities used by the FASTELL fortran library

Parameters:
  • theta_E – Einstein radius
  • gamma – 3D power-law slope of mass profile
  • q – axis ratio minor/major
  • s_scale – float, smoothing scale in the core
Returns:

pre-factors to SPEMP profile for FASTELL

derivatives(x, y, theta_E, gamma, e1, e2, s_scale, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale in the center of the profile
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

deflection angles alpha_x, alpha_y

function(x, y, theta_E, gamma, e1, e2, s_scale, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale in the center of the profile (angle)
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

lensing potential

hessian(x, y, theta_E, gamma, e1, e2, s_scale, center_x=0, center_y=0)[source]
Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale in the center of the profile
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

Hessian components f_xx, f_xy, f_yx, f_yy

static is_not_empty(x1, x2)[source]

Check if float or not an empty array :return: True if x1 and x2 are either floats/ints or an non-empty array, False if e.g. objects are [] :rtype: bool

lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 0, 's_scale': 0, 'theta_E': 0}
param_names = ['theta_E', 'gamma', 'e1', 'e2', 's_scale', 'center_x', 'center_y']
param_transform(x, y, theta_E, gamma, e1, e2, s_scale, center_x=0, center_y=0)[source]

transforms parameters in the format of fastell4py

Parameters:
  • x – x-coordinate (angle)
  • y – y-coordinate (angle)
  • theta_E – Einstein radius (angle), pay attention to specific definition!
  • gamma – logarithmic slope of the power-law profile. gamma=2 corresponds to isothermal
  • e1 – eccentricity component
  • e2 – eccentricity component
  • s_scale – smoothing scale in the center of the profile
  • center_x – x-position of lens center
  • center_y – y-position of lens center
Returns:

x-rotated, y-rotated, q_fastell, gam, s2, q, phi_G

upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 100, 's_scale': 100, 'theta_E': 100}

lenstronomy.LensModel.Profiles.spep module

class lenstronomy.LensModel.Profiles.spep.SPEP[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for Softened power-law elliptical potential (SPEP)

density_lens(r, theta_E, gamma, e1=0, e2=0)[source]
Parameters:
  • r
  • theta_E
  • gamma
  • e1
  • e2
Returns:

derivatives(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]

deflection angles

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

function(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]
Parameters:
  • x (array of size (n)) – set of x-coordinates
  • theta_E (float.) – Einstein radius of lense
  • gamma (<2 float) – power law slope of mass profifle
  • q (0<phi_G<pi/2) – Axis ratio
  • phi_G – position angel of SES
Returns:

function

Raises:

AttributeError, KeyError

hessian(x, y, theta_E, gamma, e1, e2, center_x=0, center_y=0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

lower_limit_default = {'center_x': -100, 'center_y': -100, 'e1': -0.5, 'e2': -0.5, 'gamma': 0, 'theta_E': 0}
mass_3d_lens(r, theta_E, gamma, e1=0, e2=0)[source]

computes the spherical power-law mass enclosed (with SPP routiune) :param r: :param theta_E: :param gamma: :param q: :param phi_G: :return:

param_names = ['theta_E', 'gamma', 'e1', 'e2', 'center_x', 'center_y']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'e1': 0.5, 'e2': 0.5, 'gamma': 100, 'theta_E': 100}

lenstronomy.LensModel.Profiles.splcore module

class lenstronomy.LensModel.Profiles.splcore.SPLCORE(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This lens profile corresponds to a spherical power law (SPL) mass distribution with logarithmic slope gamma and a 3D core radius r_core

\[\]

holeft(r, ho_0, r_c, gamma ight) = ho_0 rac{{r_c}^gamma}{left(r^2 + r_c^2 ight)^{ rac{gamma}{2}}}

The difference between this and EPL is that this model contains a core radius, is circular, and is also defined for gamma=3.

With respect to SPEMD, this model is different in that it is also defined for gamma = 3, is circular, and is defined in terms of a physical density parameter rho0, or the central density at r=0 divided by the critical density for lensing such that rho0 has units 1/arcsec.

This class is defined for all gamma > 1

alpha(r, sigma0, r_core, gamma)[source]

Returns the deflection angle at r :param r: radius [arcsec] :param sigma0: convergence at r=0 :param r_core: core radius [arcsec] :param gamma: logarithmic slope at r -> infinity :return: deflection angle at r

density(r, rho0, r_core, gamma)[source]

Returns the 3D density at r :param r: radius [arcsec] :param sigma0: convergence at r=0 :param r_core: core radius [arcsec] :param gamma: logarithmic slope at r -> infinity :return: density at r

density_2d(x, y, rho0, r_core, gamma)[source]

Returns the convergence at radius r :param x: x position [arcsec] :param y: y position [arcsec] :param sigma0: convergence at r=0 :param r_core: core radius [arcsec] :param gamma: logarithmic slope at r -> infinity :return: convergence at r

density_lens(r, sigma0, r_core, gamma)[source]

Returns the 3D density at r :param r: radius [arcsec] :param sigma0: convergence at r=0 :param r_core: core radius [arcsec] :param gamma: logarithmic slope at r -> infinity :return: density at r

derivatives(x, y, sigma0, r_core, gamma, center_x=0, center_y=0)[source]
Parameters:
  • x – projected x position at which to evaluate function [arcsec]
  • y – projected y position at which to evaluate function [arcsec]
  • sigma0 – convergence at r = 0
  • r_core – core radius [arcsec]
  • gamma – logarithmic slope at r -> infinity
  • center_x – x coordinate center of lens model [arcsec]
  • center_y – y coordinate center of lens model [arcsec]
Returns:

deflection angle alpha in x and y directions

function(x, y, sigma0, r_core, gamma, center_x=0, center_y=0)[source]

lensing potential

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

hessian(x, y, sigma0, r_core, gamma, center_x=0, center_y=0)[source]
Parameters:
  • x – projected x position at which to evaluate function [arcsec]
  • y – projected y position at which to evaluate function [arcsec]
  • sigma0 – convergence at r = 0
  • r_core – core radius [arcsec]
  • gamma – logarithmic slope at r -> infinity
  • center_x – x coordinate center of lens model [arcsec]
  • center_y – y coordinate center of lens model [arcsec]
Returns:

hessian elements

alpha_(x/y) = alpha_r * cos/sin(x/y / r)

lower_limit_default = {'center_x': -100, 'center_y': -100, 'gamma': 1.000001, 'r_core': 1e-06, 'sigma0': 0}
mass_2d(r, rho0, r_core, gamma)[source]

mass enclosed projected 2d disk of radius r :param r: radius [arcsec] :param rho0: density at r = 0 in units [rho_0_physical / sigma_crit] (which should be equal to [1/arcsec]) where rho_0_physical is a physical density normalization and sigma_crit is the critical density for lensing :param r_core: core radius [arcsec] :param gamma: logarithmic slope at r -> infinity :return: projected mass inside disk of radius r

mass_2d_lens(r, sigma0, r_core, gamma)[source]

mass enclosed projected 2d disk of radius r :param r: radius [arcsec] :param sigma0: convergence at r = 0 where rho_0_physical is a physical density normalization and sigma_crit is the critical density for lensing :param r_core: core radius [arcsec] :param gamma: logarithmic slope at r -> infinity :return: projected mass inside disk of radius r

mass_3d(r, rho0, r_core, gamma)[source]

mass enclosed a 3d sphere or radius r :param r: radius [arcsec] :param rho0: density at r = 0 in units [rho_0_physical / sigma_crit] (which should be equal to [arcsec]) where rho_0_physical is a physical density normalization and sigma_crit is the critical density for lensing :param r_core: core radius [arcsec] :param gamma: logarithmic slope at r -> infinity :return: mass inside radius r

mass_3d_lens(r, sigma0, r_core, gamma)[source]

mass enclosed a 3d sphere or radius r :param r: radius [arcsec] :param sigma0: convergence at r = 0 :param r_core: core radius [arcsec] :param gamma: logarithmic slope at r -> infinity :return: mass inside radius r

param_names = ['sigma0', 'center_x', 'center_y', 'r_core', 'gamma']
upper_limit_default = {'center_x': 100, 'center_y': 100, 'gamma': 5.0, 'r_core': 100, 'sigma0': 1000000000000.0}

lenstronomy.LensModel.Profiles.spp module

class lenstronomy.LensModel.Profiles.spp.SPP(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

class for circular power-law mass distribution

static density(r, rho0, gamma)[source]

computes the density :param x: :param y: :param rho0: :param a: :param s: :return:

static density_2d(x, y, rho0, gamma, center_x=0, center_y=0)[source]

projected density :param x: :param y: :param rho0: :param a: :param s: :param center_x: :param center_y: :return:

density_lens(r, theta_E, gamma)[source]

computes the density at 3d radius r given lens model parameterization. The integral in projected in units of angles (i.e. arc seconds) results in the convergence quantity.

derivatives(x, y, theta_E, gamma, center_x=0.0, center_y=0.0)[source]

deflection angles

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

function(x, y, theta_E, gamma, center_x=0, center_y=0)[source]
Parameters:
  • x (array of size (n)) – set of x-coordinates
  • theta_E (float.) – Einstein radius of lens
  • gamma (<2 float) – power law slope of mass profile
Returns:

function

Raises:

AttributeError, KeyError

grav_pot(x, y, rho0, gamma, center_x=0, center_y=0)[source]

gravitational potential (modulo 4 pi G and rho0 in appropriate units) :param x: :param y: :param rho0: :param a: :param s: :param center_x: :param center_y: :return:

hessian(x, y, theta_E, gamma, center_x=0.0, center_y=0.0)[source]

returns Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

Parameters:
  • x – x-coordinate
  • y – y-coordinate
  • kwargs – keywords of the profile
Returns:

raise as definition is not defined

lower_limit_default = {'center_x': -100, 'center_y': -100, 'gamma': 1.5, 'theta_E': 0}
mass_2d(r, rho0, gamma)[source]

mass enclosed projected 2d sphere of radius r :param r: :param rho0: :param a: :param s: :return:

mass_2d_lens(r, theta_E, gamma)[source]
Parameters:
  • r – projected radius
  • theta_E – Einstein radius
  • gamma – power-law slope
Returns:

2d projected radius enclosed

static mass_3d(r, rho0, gamma)[source]

mass enclosed a 3d sphere or radius r :param r: :param a: :param s: :return:

mass_3d_lens(r, theta_E, gamma)[source]
Parameters:
  • r
  • theta_E
  • gamma
Returns:

param_names = ['theta_E', 'gamma', 'center_x', 'center_y']
static rho2theta(rho0, gamma)[source]

converts 3d density into 2d projected density parameter :param rho0: :param gamma: :return:

static theta2rho(theta_E, gamma)[source]

converts projected density parameter (in units of deflection) into 3d density parameter :param theta_E: :param gamma: :return:

upper_limit_default = {'center_x': 100, 'center_y': 100, 'gamma': 2.5, 'theta_E': 100}

lenstronomy.LensModel.Profiles.tnfw module

class lenstronomy.LensModel.Profiles.tnfw.TNFW[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

this class contains functions concerning the truncated NFW profile with a truncation function (r_trunc^2)*(r^2+r_trunc^2)

relation are: R_200 = c * Rs

F(x)[source]

Classic NFW function in terms of arctanh and arctan :param x: r/Rs :return:

static alpha2rho0(alpha_Rs, Rs)[source]

convert angle at Rs into rho0; neglects the truncation

Parameters:
  • alpha_Rs – deflection angle at RS
  • Rs – scale radius
Returns:

density normalization (characteristic density)

static density(r, Rs, rho0, r_trunc)[source]

three dimensional truncated NFW profile

Parameters:
  • r (float/numpy array) – radius of interest
  • Rs (float > 0) – scale radius
  • r_trunc (float > 0) – truncation radius (angular units)
Returns:

rho(r) density

density_2d(x, y, Rs, rho0, r_trunc, center_x=0, center_y=0)[source]

projected two dimensional NFW profile (kappa*Sigma_crit)

Parameters:
  • R (float/numpy array) – projected radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r_trunc (float > 0) – truncation radius (angular units)
Returns:

Epsilon(R) projected density at radius R

derivatives(x, y, Rs, alpha_Rs, r_trunc, center_x=0, center_y=0)[source]

returns df/dx and df/dy of the function (integral of TNFW), which are the deflection angles

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • Rs – turn over point in the slope of the NFW profile in angular unit
  • alpha_Rs – deflection (angular units) at projected Rs
  • r_trunc – truncation radius (angular units)
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

deflection angle in x, deflection angle in y

function(x, y, Rs, alpha_Rs, r_trunc, center_x=0, center_y=0)[source]
Parameters:
  • x – angular position
  • y – angular position
  • Rs – angular turn over point
  • alpha_Rs – deflection at Rs
  • r_trunc – truncation radius
  • center_x – center of halo
  • center_y – center of halo
Returns:

lensing potential

hessian(x, y, Rs, alpha_Rs, r_trunc, center_x=0, center_y=0)[source]

returns d^2f/dx^2, d^2f/dxdy, d^2f/dydx, d^2f/dy^2 of the TNFW potential f

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • Rs – turn over point in the slope of the NFW profile in angular unit
  • alpha_Rs – deflection (angular units) at projected Rs
  • r_trunc – truncation radius (angular units)
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

Hessian matrix of function d^2f/dx^2, d^f/dy^2, d^2/dxdy

lower_limit_default = {'Rs': 0, 'alpha_Rs': 0, 'center_x': -100, 'center_y': -100, 'r_trunc': 0}
mass_2d(R, Rs, rho0, r_trunc)[source]

analytic solution of the projection integral (convergence)

Parameters:
  • R – projected radius
  • Rs – scale radius
  • rho0 – density normalization (characteristic density)
  • r_trunc – truncation radius (angular units)
Returns:

mass enclosed 2d projected cylinder

mass_3d(r, Rs, rho0, r_trunc)[source]

mass enclosed a 3d sphere or radius r

Parameters:
  • r – 3d radius
  • Rs – scale radius
  • rho0 – density normalization (characteristic density)
  • r_trunc – truncation radius (angular units)
Returns:

M(<r)

nfwAlpha(R, Rs, rho0, r_trunc, ax_x, ax_y)[source]

deflection angel of NFW profile along the projection to coordinate axis

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r_trunc (float > 0) – truncation radius (angular units)
  • axis (same as R) – projection to either x- or y-axis
Returns:

nfwGamma(R, Rs, rho0, r_trunc, ax_x, ax_y)[source]

shear gamma of NFW profile (times Sigma_crit) along the projection to coordinate ‘axis’

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r_trunc (float > 0) – truncation radius (angular units)
  • axis (same as R) – projection to either x- or y-axis
Returns:

nfwPot(R, Rs, rho0, r_trunc)[source]

lensing potential of truncated NFW profile

Parameters:
  • R (float/numpy array) – radius of interest
  • Rs (float) – scale radius
  • rho0 (float) – density normalization (characteristic density)
  • r_trunc (float > 0) – truncation radius (angular units)
Returns:

lensing potential

param_names = ['Rs', 'alpha_Rs', 'r_trunc', 'center_x', 'center_y']
profile_name = 'TNFW'
static rho02alpha(rho0, Rs)[source]

convert rho0 to angle at Rs; neglects the truncation

Parameters:
  • rho0 – density normalization (characteristic density)
  • Rs – scale radius
Returns:

deflection angle at RS

upper_limit_default = {'Rs': 100, 'alpha_Rs': 10, 'center_x': 100, 'center_y': 100, 'r_trunc': 100}

lenstronomy.LensModel.Profiles.uldm module

class lenstronomy.LensModel.Profiles.uldm.Uldm(*args, **kwargs)[source]

Bases: lenstronomy.LensModel.Profiles.base_profile.LensProfileBase

This class contains functions concerning the ULDM soliton density profile, whose good approximation is (see for example https://arxiv.org/pdf/1406.6586.pdf )

\[\rho = \rho_0 (1 + a(\theta/\theta_c)^2)^{-\beta}\]

where \(\theta_c\) is the core radius, corresponding to the radius where the density drops by half its central value, :math: beta is the slope (called just slope in the parameters of this model), :math: rho_0 = kappa_0 Sigma_c/D_lens, and :math: a is a parameter, dependent on :math: beta, chosen such that :math: theta_c indeed corresponds to the radius where the density drops by half (simple math gives :math: a = 0.5^{-1/beta} - 1 ). For an ULDM soliton profile without contributions to background potential, it turns out that :math: beta = 8, a = 0.091. We allow :math: beta to be different from 8 to model solitons which feel the influence of background potential (see 2105.10873) The profile has, as parameters: :param kappa_0: central convergence :param theta_c: core radius (in arcseconds) :param slope: exponent entering the profile, default value is 8

alpha_radial(r, kappa_0, theta_c, slope=8)[source]

returns the radial part of the deflection angle

Parameters:
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • slope – exponent entering the profile
  • r – radius where the deflection angle is computed
Returns:

radial deflection angle

density(R, kappa_0, theta_c, slope=8)[source]

three dimensional ULDM profile in angular units (rho0_physical = rho0_angular Sigma_crit / D_lens) :param R: radius of interest :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: rho(R) density in angular units

density_2d(x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8)[source]

projected two dimensional ULDM profile (convergence * Sigma_crit), but given our units convention for rho0, it is basically the convergence

Parameters:
  • R – radius of interest
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • slope – exponent entering the profile
Returns:

Epsilon(R) projected density at radius R

density_lens(r, kappa_0, theta_c, slope=8)[source]

computes the density at 3d radius r given lens model parameterization. The integral in the LOS projection of this quantity results in the convergence quantity.

Parameters:
  • r – 3d radius
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • slope – exponent entering the profile
Returns:

density rho(r)

derivatives(x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8)[source]

returns df/dx and df/dy of the function (lensing potential), which are the deflection angles

Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • slope – exponent entering the profile
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

deflection angle in x, deflection angle in y

function(x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8)[source]
Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • slope – exponent entering the profile
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

lensing potential (in arcsec^2)

hessian(x, y, kappa_0, theta_c, center_x=0, center_y=0, slope=8)[source]
Parameters:
  • x – angular position (normally in units of arc seconds)
  • y – angular position (normally in units of arc seconds)
  • kappa_0 – central convergence of profile
  • theta_c – core radius (in arcsec)
  • slope – exponent entering the profile
  • center_x – center of halo (in angular units)
  • center_y – center of halo (in angular units)
Returns:

Hessian matrix of function d^2f/dx^2, d^2/dxdy, d^2/dydx, d^f/dy^2

kappa_r(R, kappa_0, theta_c, slope=8)[source]

convergence of the cored density profile. This routine is also for testing

Parameters:
  • R – radius (angular scale)
  • kappa_0 – convergence in the core
  • theta_c – core radius
  • slope – exponent entering the profile
Returns:

convergence at r

lower_limit_default = {'center_x': -100, 'center_y': -100, 'kappa_0': 0, 'slope': 3.5, 'theta_c': 0}
mass_2d(R, kappa_0, theta_c, slope=8)[source]

mass enclosed a 2d sphere or radius r :param R: radius over which the mass is computed :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: mass enclosed in 2d sphere

mass_3d(R, kappa_0, theta_c, slope=8)[source]

mass enclosed a 3d sphere or radius r :param R: radius in arcseconds :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: mass of soliton in angular units

mass_3d_lens(r, kappa_0, theta_c, slope=8)[source]

mass enclosed a 3d sphere or radius r :param r: radius over which the mass is computed :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: mass enclosed in 3D ball

param_names = ['kappa_0', 'theta_c', 'slope', 'center_x', 'center_y']
rhotilde(kappa_0, theta_c, slope=8)[source]

Computes the central density in angular units :param kappa_0: central convergence of profile :param theta_c: core radius (in arcsec) :param slope: exponent entering the profile :return: central density in 1/arcsec

upper_limit_default = {'center_x': 100, 'center_y': 100, 'kappa_0': 1.0, 'slope': 10, 'theta_c': 100}

Module contents