This app uses an emulator to interpolate the results for the gravothermal evolution of a fiducial NFW halo computed using a gravothermal fluid (GTF) code for 29 dimensionless collisional cross sections spanning approximately . Here, , where with and .
It interpolates the dimensionless evolution and rescales radius, density, mass, velocity and time using the requested halo mass, concentration and redshift following scaling relations that follow from matching the dimensionless cross section. This works because the dimensionless evolution of an initial NFW halo only depends on , subject to the inherited truncation and interpolation limits (see below).
If the properties of the user-defined halo require a dimensionless cross section outside the tabulated range (approximately ), the code notifies the user and no new evolution is computed. The app does not allow extrapolation outside the GTF results; it checks the full-precision tabulated endpoints.
If density needs to be evaluated at radii inside the innermost data point from the GTF calculations, the inner profile is assumed to have constant density equal to that of the innermost data point.
The default source is a circular projected Plummer with radial surface brightness profile and total unlensed flux . The ray-traced image is convolved with a normalized Gaussian PSF with a user-specified FWHM. The image uses a fixed brightness stretch, applied identically in all frames, with an ASINH-dependent brightness-to-color mapping.
Sérsic options use I(R) = I₀ exp[−bₙ(R/R½)^(1/n)], with bₙ chosen so R½ encloses half the total light. Uploaded images use bilinear surface-brightness interpolation inside the supplied angular rectangle and zero brightness outside. PNG is converted to linear-sRGB luminance times transparency; FITS uses nonnegative BSCALE/BZERO-scaled values. Uncompressed 2D FITS primary arrays and IMAGE extensions are supported; FITS requires an undistorted RA/Dec TAN WCS with CD or PC/CDELT (legacy CROTA2 also supported). It is reprojected to east/right (RA offset) and north/up (Dec offset), centered at Source offset x. Axes show local angular offsets, not absolute sky coordinates. Other projections and distortion conventions must be reprojected to TAN before upload. All sources are peak-normalized for the fixed display stretch. Magnification uses each source’s integrated unlensed flux. Sérsic and uploaded sources can take longer to compute.
The fiducial halo on which the gravothermal simulations entering the emulator are based has a virial mass , a concentration parameter , and is truncated in energy space such that its total mass, integrated to infinity, is equal to twice the virial mass. This truncation is taken to be weak enough not to affect the gravothermal evolution.
The virial mass is defined as the mass inside the virial radius, which encloses an average density equal to 200 times the critical density at the redshift of the halo (the lens), assuming a flat ΛCDM cosmology with and . This cosmology is also used to compute all angular diameter distances for the lensing calculations.
Total source magnification is defined as , where is the positive total lensed flux, computed by adaptive integration independently of the displayed pixels, before PSF convolution. A normalized PSF preserves total flux on the full image plane. Automatic display framing targets at least 99.5% of the integrated lensed flux and adds PSF padding; flux outside the displayed frame is not excluded from the reported magnification.
The quantity is the area enclosed by the radial caustics in the source plane and measures how likely a source is to be multiply imaged. The tangential caustic area At is shown in rose alongside Ar in blue. It is zero for zero external shear. With shear, both areas are computed by mapping the critical curves into the source plane.
External shear adds a constant tidal deflection Γθ, with Γ = [[γ cos(2φ), γ sin(2φ)], [γ sin(2φ), −γ cos(2φ)]], without external convergence or changes to the halo evolution. The angle φ is measured counterclockwise from +x toward +y. This version supports 0 ≤ γ ≤ 0.3. The effective angular Einstein radius is √(Acrit/π), using the area enclosed by the tangential (Einstein) critical curve in the image plane. The Einstein critical curve is always overlaid on the lensed image. The Show caustics & critical curves button adds the radial critical curve and both source-plane caustics at the same angular scale and origin. At zero shear the tangential caustic is a point. In the unlensed source view only the source-plane caustics are shown. Curves outside the displayed field are clipped; subcritical snapshots may have no critical curves or caustics. The profile arrows mark the corresponding effective physical radius. At γ = 0, the original spherical computations are used. Nonzero-shear magnification uses exact radial integration for Plummer and bilinear uploaded sources, adaptive radial integration for Sérsic sources, and converged angular quadrature, independently of display pixels.
Notes for the displayed evolution