autogalaxy.profiles.mass.dPIEPotentialSph#

class dPIEPotentialSph[source]#

Bases: dPIEPotential

The dual Pseudo-Isothermal mass profile (dPIEPotential) without ellipticity, based on the formulation from Eliasdottir (2007): https://arxiv.org/abs/0710.5636.

This profile describes a circularly symmetric (non-elliptical) projected mass distribution with two scale radii (ra and rs) and a normalization factor kappa_scale. Although originally called the dPIEPotential (Elliptical), this version lacks ellipticity, so the “E” may be a misnomer.

The projected surface mass density is given by:

\[\Sigma(R) = \Sigma_0 (ra * rs) / (rs - ra) * (1 / \sqrt(ra^2 + R^2) - 1 / \sqrt(rs^2 + R^2))\]

(See Eliasdottir 2007, Eq. A3.)

In this implementation: - ra is the core radius in unit of arcseconds. - b0 is the lens strength in unit of arcseconds, when ra->0 & rs->infty & q->1, b0 is the Einstein radius.

b0 is related to the central velocity dispersion sigma_0: b_0 = 4pi * sigma_0^2 / c^2 * (D_{LS} / D_{S}) b0 is not in the Intermediate-Axis-Convention for its r_{em}^2 = x^2 / (1 + epsilon)^2 + y^2 / (1 - epsilon)^2

Credit: Jackson O’Donnell for implementing this profile in PyAutoLens. Note: This dPIEPotentialSph should be the same with dPIEMassSph for their same mathamatical formulations.

Parameters:

centre (Tuple[float, float]) – The (y,x) arc-second coordinates of the profile centre.
ra (float) – The inner core scale radius in arcseconds.
rs (float) – The outer truncation scale radius in arcseconds.
b0 (float) – The lens strength in arcseconds.

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.
`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
`elliptical_radii_grid_from`	Convert a grid of (y,x) coordinates to their elliptical radii values: :math: (x^2 + (y^2/q))^0.5
`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`	Returns the 2D lensing potential of the mass profile from a 2D grid of Cartesian (y,x) 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.
`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.
`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.

deflections_yx_2d_from(grid, xp=<module 'numpy' from '/home/docs/checkouts/readthedocs.org/user_builds/pyautolens/envs/latest/lib/python3.12/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.

convergence_2d_from(grid, xp=<module 'numpy' from '/home/docs/checkouts/readthedocs.org/user_builds/pyautolens/envs/latest/lib/python3.12/site-packages/numpy/__init__.py'>, **kwargs)[source]#

Returns the two dimensional projected convergence on a grid of (y,x) arc-second coordinates.

The grid_2d_to_structure decorator reshapes the ndarrays the convergence is outputted on. See aa.grid_2d_to_structure for a description of the output.

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

potential_2d_from(grid, xp=<module 'numpy' from '/home/docs/checkouts/readthedocs.org/user_builds/pyautolens/envs/latest/lib/python3.12/site-packages/numpy/__init__.py'>, **kwargs)[source]#

Returns the 2D lensing potential of the mass profile from a 2D grid of Cartesian (y,x) coordinates.

The lensing potential ψ(θ) is the gravitational (Shapiro) time-delay term. It quantifies how much the passage of light through the gravitational field delays its arrival relative to a straight-line path in empty space.

The potential enters directly into the Fermat potential:

φ(θ) = ½ |θ − β|² − ψ(θ)

which governs time delays between multiple lensed images of the same source.

Parameters:: grid (Union[ndarray, Grid2D, Grid2DIrregular]) – The 2D (y, x) coordinates where the lensing potential is evaluated.
Returns:: The lensing potential ψ(θ) at every coordinate on the input grid.
Return type:: aa.Array2D