autogalaxy.profiles.mass.dPIEMassLenstool#

class dPIEMassLenstool[source]#

Bases: dPIEMass

The dual Pseudo Isothermal Elliptical Mass Distribution(dPIEMass) profiles, which is a two component PIEMass with both a core radius and a truncation radius, see Eliasdottir (2007): https://arxiv.org/abs/0710.5636 This profile is ported from Lenstool’s C code, which has the same formulation.

This proflie describes an elliptic isothermal mass distribution with a finite core, rho r^{-2} while in the transition region (ra<=R<=rs), and rho r^{-4} in the outer parts: rho propto [(ra^2 + R^2) (rs^2 + R^2)]^{-1}

The convergence is given by two PIEMass with core radius ra and rs: kappa(r_{em}) = rs / (rs - ra) * (kappa_{PIEMass,ra} - kappa_{PIEMass,rs})

= b_0 / 2 * rs / (rs - ra) * ( frac{1}{sqrt{ ra^2 + r_{em}^2}} - frac{1}{sqrt{ rs^2 + r_{em}^2}} )

where r_{em}^2 = x^2 / (1 + epsilon)^2 + y^2 / (1 - epsilon)^2. Note in Eliasdottir (2007), E0 = 6pi * sigma_{dPIEPotential}^2 / c^2 * (D_{LS} / D_{S}). Eliasdottir’s E0 is not the same as E0 in Kassiola & Kovner(1993) which is also b0. There is frac{sigma_{dPIEPotential}^2}{sigma_0^2} = frac{2}{3}

rac{rs^2}{rs^2-ra^2},

thus E0(Kassiola & Kovner(1993)) = b0 = E0(Eliasdottir (2007)) * (rs^2 - ra^2) / rs^2. So when s->infty and a->0, they are equivalent.

In this implementation: - ra is the core radius in unit of arcseconds. - rs is the truncation 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

Parameters:

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

Methods

`analytical_hessian_2d_from`	Calculate the hessian matrix on a grid of (y,x) arc-second coordinates.
`analytical_magnification_2d_from`
`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.
`from_lenstool`	Construct a `dPIEMass` from Lenstool's native dPIE / PIEMD parameterization, as read directly out of a Lenstool `.par` file (`potentiel` profil 81) or the parameter tables of Lenstool-based papers.
`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 two dimensional projected lensing 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.
`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.