6. Source brightness models
A ray reaching a plane at position $\boldsymbol{\beta}$ (arcsec) picks up the surface brightness $I(\boldsymbol{\beta})$ of every source on that plane. The analytic profiles depend on $\boldsymbol{\beta}$ through the elliptical radius
\[\mathbf{d} = \boldsymbol{\beta} - (c_x, c_y), \qquad \begin{pmatrix}x_r \\ y_r\end{pmatrix} = R(-\varphi)\,\mathbf{d}, \qquad r_\text{ell}^2 = x_r^2 + (y_r/q)^2,\]where $\mathbf{d}$ is the offset from the source centre, $R(-\varphi)$ is the rotation by $-\varphi$, and $(x_r, y_r)$ is the offset along the source’s major and minor axes. This differs from the lens elliptical radius (§5) by a factor of $q$: here $\sigma$ is the semi-major axis of a contour of equal brightness, as in galaxy photometry, while lenses follow Kormann et al. (1994) so that their deflection scale $b$ does not depend on $q$.
| Symbol | Control | Meaning |
|---|---|---|
| $(c_x, c_y)$ | drag the marker | Source centre on its plane (arcsec) |
| $\varphi$ | φ | Position angle of the major axis (radians) |
| $q$ | q | Axis ratio minor/major, 0.05 to 1 |
| $\sigma$ | σ | Scale radius (arcsec), rendered no smaller than 0.01″ |
| $A$ | A | Amplitude, the peak surface brightness |
| Color | Colour the profile is drawn in (white by default) |
The point source, uniform disc and pasted image use some of these with a different meaning; each section below says how.
Gaussian
\[I = A\exp\left(-\frac{r_\text{ell}^2}{2\sigma^2}\right)\]Exponential
\[I = A\exp\left(-\frac{r_\text{ell}}{\sigma}\right)\]This is the Sérsic profile $I \propto \exp\big[-(r_\text{ell}/\sigma)^{1/n}\big]$ with index $n = 1$, more extended than the Gaussian.
Uniform disc
\[I = \begin{cases} A & r_\text{ell} \leq \sigma \\ 0 & \text{otherwise} \end{cases}\]A filled ellipse with a sharp edge. Here $\sigma$ is the semi-major axis, set as r. The disc has no A control and uses $A = 1$ unless a config sets amplitude.
Show shape draws the outline at $r_\text{ell} = 2\sigma$ for the Gaussian and exponential, and at $\sigma$ for the uniform disc.
Point source
A point source models a quasar or other compact object. The code finds every image-plane position $\boldsymbol{\theta}_i$ that maps to the source centre, $\boldsymbol{\beta}(\boldsymbol{\theta}_i) = (c_x, c_y)$, and draws a dot at each. Here $\sigma$ (Size) is the dot radius in arcsec, $A$ its opacity, and Color its colour; there is no $q$ or $\varphi$. The dots are drawn over the image rather than added to the light, so they are not stretched by lensing (like a real quasar image, a point blurred only by the telescope) and do not go through the brightness stretch below. The Unlensed view shows a single dot at $(c_x, c_y)$.
Sign changes of $\boldsymbol{\beta} - (c_x, c_y)$ on a coarse grid of $\boldsymbol{\theta}$ give starting guesses, and Newton’s method (repeated straight-line steps toward the solution, shortened when a step overshoots) refines each until $\lvert\boldsymbol{\beta}(\boldsymbol{\theta}) - (c_x, c_y)\rvert^2 \lt 10^{-14}$ arcsec². Solutions closer than $10^{-5}$″ are merged. Images are searched for within a square $\max(1.1 \times \text{FOV}, 3″)$ across, so images well outside the view are not found. The grid is the Point source setting under Settings: 150, 300 (default), 600 or 1200 points across the field. Denser grids find images near caustics more reliably but are slower.
Heads up. A point source never forms arcs or rings; use a Gaussian or uniform disc for those. Highly demagnified images (such as the central image of an SIE) can be missed by the grid search.
Pasted image
Select the object and paste a picture with Ctrl+V (Cmd+V), or use Load image… in its controls. It is centred on $(c_x, c_y)$, and its height is half the FOV at the moment you load it, times Scale (later zooming does not resize it). Here $\sigma$ is Scale and $A$ is Brightness; there is no $q$, $\varphi$ or Color. The picture is sampled smoothly from a set of pre-shrunk copies (mipmaps), so it stays clean when shrunk. In the light theme, where the lensed image is shown inverted, a pasted picture is inverted with it (it shows as a negative); the background and observation are not. At most four pasted images are drawn at once; any more (the farthest by redshift) are not drawn, and a warning on the image says so.
A pasted image is opaque: its dark pixels hide what is behind them rather than letting the background through (§7). Its coverage $C$ is 1 inside the picture and falls to 0 over a soft border,
\[C = \prod_{i \in \{u,v\}} \operatorname{smoothstep}\big(0,\ w,\ \min(uv_i,\, 1 - uv_i)\big), \qquad w = \min\big(\tfrac{3}{2}\,\text{fwidth}(uv) + w_\text{blend},\ 0.45\big),\]where $uv = (uv_u, uv_v) \in [0,1]^2$ is the position of $\boldsymbol{\beta}$ within the picture, $w$ is the border width in those units, $\operatorname{smoothstep}(0, w, x)$ rises smoothly from 0 at $x = 0$ to 1 at $x = w$, and $\text{fwidth}(uv)$ is how much $uv$ changes between neighbouring screen pixels. The first term keeps the border about 1.5 pixels wide at any magnification. Edge blend (0 to 1) sets $w_\text{blend} = 0.4 \times$ Edge blend, fading the picture into the field. Because the picture is opaque, Brightness darkens it rather than fading it out; use Edge blend to let the field through.
Contrast $k \in [0.3, 2.5]$ and Saturation $s \in [0, 2]$ adjust the picture’s colour $\mathbf{c} = (r, g, b)$ at each point before Brightness:
\[\mathbf{c}' = \operatorname{clamp}\Big(\big(Y + s\,(\mathbf{c} - Y) - \tfrac12\big)\,k + \tfrac12,\ 0,\ 1\Big), \qquad Y = 0.2126\,r + 0.7152\,g + 0.0722\,b,\]where $Y$ is the perceived brightness of $\mathbf{c}$, a weighted sum of its channels with the standard weights for the eye’s sensitivity (most to green, least to blue; ITU-R BT.709), and clamp limits each channel to $[0, 1]$. $s = 0$ gives greyscale and $s = k = 1$ leaves the picture unchanged. A picture spread along an arc tends to look washed out, and $k > 1$ brings back its shadows.
Apply brightness stretch (off by default) sends the picture through the brightness stretch below. A photograph is already display-encoded, so the stretch washes it out. Turn it on if the picture is a surface-brightness map.
Your own pictures are not saved in a config. A config can name one of the site’s images with an image: key on the object (the Realistic preset does this, §11), checked against the same allowlist as bgImage (§7).
Compositing and brightness stretch
Each pixel collects the light of every visible source: Color $\times\, I$ for the analytic profiles and $A\,C\,\mathbf{c}’$ for pasted images. Hidden objects add no light and no lensing. The light is split into $I_\text{analytic}$ (analytic sources), $I_\text{stretched}$ (pasted images with Apply brightness stretch on) and $I_\text{as-is}$ (pasted images with it off), and the displayed source light is
\[I_\text{src}' = \operatorname{clamp}\Big(f\big(\operatorname{clamp}(I_\text{analytic} + I_\text{stretched},\,0,\,1)\big) + I_\text{as-is},\ 0,\ 1\Big),\]applied to each colour channel separately. Pasted images with the stretch off keep their tones, while analytic light still adds on top.
The stretch $f$ is set under Colors → Color map, with the same controls as the quantity maps (§4). Min and Max (default 0 and 1) set the black and white points, $u = \operatorname{clamp}\big((I - \text{Min})/(\text{Max} - \text{Min}),\ 0,\ 1\big)$, and Scale picks the curve:
| Scale | $f$ |
|---|---|
| Linear | $u$ |
| Square root (default) | $\sqrt{u}$ |
| Power law | $u^{\gamma}$, with Power $\gamma \in [0.1, 2]$; lower $\gamma$ lifts faint light more |
| Asinh | $\operatorname{asinh}(a u) / \operatorname{asinh}(a)$, with Softening $a \in [0.5, 20]$; linear for faint light, logarithmic for bright (Lupton et al. 2004) |
| Log | $(\ln I - \ln \text{Min}) / (\ln \text{Max} - \ln \text{Min})$, clamped to $[0,1]$; Min is floored just above zero |
There is no palette menu here, since the lensed image carries its own colour.