autogalaxy.profiles.mass.Isothermal#

class Isothermal[source]#

Bases: PowerLaw

Represents an elliptical isothermal density distribution, which is equivalent to the elliptical power-law density distribution for the value slope = 2.0.

Parameters:

centre (Tuple[float, float]) – The (y,x) arc-second coordinates of the profile centre.
ell_comps (Tuple[float, float]) – The first and second ellipticity components of the elliptical coordinate system.
einstein_radius (float) – The arc-second Einstein radius.

Methods

`angle`	The position angle in degrees of the major-axis of the ellipse defined by profile, defined counter clockwise from the positive x-axis (0.0 > angle > 180.0).
`angle_radians`	The position angle in radians of the major-axis of the ellipse defined by profile, defined counter clockwise from the positive x-axis (0.0 > angle > 2pi).
`angle_to_profile_grid_from`	The angle between each angle theta on the grid and the profile, in radians.
`axis_ratio`	The ratio of the minor-axis to major-axis (b/a) of the ellipse defined by profile (0.0 > q > 1.0).
`convergence_2d_from`	Returns the two dimensional projected convergence on a grid of (y,x) arc-second coordinates.
`convergence_func`	Returns the convergence of the mass profile as a function of the radial coordinate.
`deflection_func`
`deflections_2d_via_potential_2d_from`	Returns the 2D deflection angles of the mass profile by numerically differentiating the lensing potential on the input grid.
`deflections_yx_2d_from`	Calculate the deflection angles on a grid of (y,x) arc-second coordinates.
`density_between_circular_annuli`	Calculate the mass between two circular annuli and compute the density by dividing by the annuli surface area.
`eccentric_radii_grid_from`	Convert a grid of (y,x) coordinates to an eccentric radius: :math: axis_ratio^0.5 (x^2 + (y^2/q))^0.5
`einstein_radius_rescaled`	Rescale the einstein radius by slope and axis_ratio, to reduce its degeneracy with other mass-profiles parameters.
`elliptical_radii_grid_from`	Convert a grid of (y,x) coordinates to their elliptical radii values: :math: (x^2 + (y^2/q))^0.5
`ellipticity_rescale`	A rescaling factor applied to account for the ellipticity of the mass profile when computing the Einstein radius from the average convergence equals unity criterion.
`extract_attribute`	Returns an attribute of a class and its children profiles in the galaxy as a ValueIrregular or Grid2DIrregular object.
`has`	Returns True if any attribute of this profile is an instance of the input class cls, else False.
`mass_angular_within_circle_from`	Integrate the mass profiles's convergence profile to compute the total mass within a circle of specified radius.
`mass_integral`	Integrand used by mass_angular_within_circle_from to compute the total projected mass within a circle.
`potential_2d_from`	Calculate the potential on a grid of (y,x) arc-second coordinates.
`potential_func`	Returns the integrand of the lensing potential at a single point, used in numerical integration schemes for computing the potential from the mass profile's convergence.
`radial_deflection_from`
`radial_grid_from`	Convert a grid of (y, x) coordinates, to their radial distances from the profile centre (e.g. :math: r = sqrt(x2 + y2)).
`rotated_grid_from_reference_frame_from`	Rotate a grid of (y,x) coordinates which have been transformed to the elliptical reference frame of a profile back to the original unrotated coordinate grid reference frame.
`shear_yx_2d_from`	Returns the analytic 2D weak-lensing shear vector field \((\gamma_2, \gamma_1)\) of the elliptical isothermal mass distribution on a grid of `(y, x)` arc-second coordinates.
`transformed_from_reference_frame_grid_from`	Transform a grid of (y,x) coordinates from the reference frame of the profile to the original observer reference frame.
`transformed_to_reference_frame_grid_from`	Transform a grid of (y,x) coordinates to the reference frame of the profile.
`vmapped_deflections_from`

Attributes

`average_convergence_of_1_radius`	The radius a critical curve forms for this mass profile, e.g. where the mean convergence is equal to 1.0.
`unit_mass`

axis_ratio(xp=<module 'numpy' from '/home/docs/checkouts/readthedocs.org/user_builds/pyautogalaxy/envs/latest/lib/python3.11/site-packages/numpy/__init__.py'>)[source]#: The ratio of the minor-axis to major-axis (b/a) of the ellipse defined by profile (0.0 > q > 1.0).

deflections_yx_2d_from(grid, xp=<module 'numpy' from '/home/docs/checkouts/readthedocs.org/user_builds/pyautogalaxy/envs/latest/lib/python3.11/site-packages/numpy/__init__.py'>, **kwargs)[source]#

Calculate the deflection angles on a grid of (y,x) arc-second coordinates.

Parameters:: grid (Union[ndarray, Grid2D, Grid2DIrregular]) – The grid of (y,x) arc-second coordinates the deflection angles are computed on.

shear_yx_2d_from(grid, xp=<module 'numpy' from '/home/docs/checkouts/readthedocs.org/user_builds/pyautogalaxy/envs/latest/lib/python3.11/site-packages/numpy/__init__.py'>, **kwargs)[source]#

Returns the analytic 2D weak-lensing shear vector field \((\gamma_2, \gamma_1)\) of the elliptical isothermal mass distribution on a grid of (y, x) arc-second coordinates.

For an axis-aligned isothermal profile centred on the origin the shear components reduce to:

\[ \begin{align}\begin{aligned}\gamma_1 = -\kappa(\theta) \, \frac{x^2 - y^2}{x^2 + y^2}\\\gamma_2 = -2 \, \kappa(\theta) \, \frac{x \, y}{x^2 + y^2}\end{aligned}\end{align} \]

where \(\kappa(\theta)\) is the convergence at the rotated grid coordinate. After evaluation in the profile’s reference frame the shear vector field is rotated back into the original frame using the 2 * angle rotation appropriate for a spin-2 quantity (the shear transforms as a spin-2 field, so a coordinate rotation by angle rotates the components by 2 * angle).

This analytic path is mathematically equivalent to LensCalc.shear_yx_2d_via_hessian_from, which derives the same shear from finite-difference (or JAX) derivatives of deflections_yx_2d_from. The cross-check is exercised in test_autogalaxy/profiles/mass/total/test_isothermal.py::test__shear_yx_2d_from__matches_via_hessian.

convergence_func(grid_radius, xp=<module 'numpy' from '/home/docs/checkouts/readthedocs.org/user_builds/pyautogalaxy/envs/latest/lib/python3.11/site-packages/numpy/__init__.py'>)[source]#

Returns the convergence of the mass profile as a function of the radial coordinate.

This is used to integrate the convergence profile to compute enclosed masses and the Einstein radius.

Parameters:: grid_radius (float) – The radial distance from the profile centre at which the convergence is evaluated.
Returns:: The convergence at the input radial distance.
Return type:: float