Source Models

Source models¶

CosmoSim implements spherical and ellipsoid sources with sersic light profiles Graham & Driver, 2005. This is the most important source model for realistic scenarios.

The Sersic index¶

The parameter n is the key to the profile’s flexibility — since it controls the shape of the light distribution:

• n = 1: reduces to an exponential profile, typical of galactic disks (like spiral galaxy disks)

• n = 4: reduces to the de Vaucouleurs profile, which closely matches the light distribution of elliptical galaxies and bulges

• Higher n: profile has a steep central peak (cuspy core) but very extended, faint outer wings

• Lower n: profile is flatter and more uniform, dropping off more sharply at large radii.

General Ellipsoid Model¶

We will define the formulæ for a general ellipsoid source. The spherical source is obtained by setting $\sigma_1=\sigma_2$. We describe the ellipse centred at the origin in a Cartesian coordinate system, with the major axis along the $y$-axis and the minor axis along the $x$-axis. This image is then rotated for the desired orientation $\lambda_S$.

• Parameters (as used in the implementation)

  • Sersic index $n_s$ or n_sersic

  • luminosity $l$ or luminosity

  • major half-light radius $\sigma_1$ (sigma)

  • minor half-light radius $\sigma_2$ (sigma2)

• Coefficients (currently hard-coded)

  • $b_1 = 1.992$

  • $b_0 = - 0.3271$

• Derived quantities

  • Elliptic ratio $q = \sigma_2/\sigma_1$

  • $b = b_0 + b_1\cdot n$ (bn)

  • Polar radius $r = \sqrt{(x/q)^2+y^2}$

  • Effective radius $r_{\textrm{eff}} = \sigma_1$ (re)

The pixel value $(x,y)$ is then given as

where

Discussion and proof¶

Ellipsoid Source with Sersic profile¶

We have hardcoded an implementation of an elliptic profile, namely the de Vaucouleurs’ profile. In our implementation, this individual source is defined by two size parameters, $\sigma_1$ and $\sigma_2$, for the minor and major axes of the ellipsoid.

Additionally some parameters are currently hard-coded.

• The Sersic index $n=4$

• Coefficients $b_1 = 1.992$, $b_0 = - 0.3271$

We define

• elliptic ratio $q = \frac{\sigma_2}{\sigma_1}$

• effective radius $r_{\textrm{eff}} = 10\sigma_1$ (see CHECK in the code below)

• $r = \sqrt{(x/q)^2+y^2}$

• $b = b_0 + b_1\cdot n$

The pixel value is given as

where, as in the spherical case but with $\sigma^2\to\sigma_1\sigma_2$,

See General elliptical Sersic profile for the general recipe (arbitrary $n$ and position angle), including the proof that $r_{\textrm{eff}}$ is the half-light radius for any axis ratio $q$ -- this holds for whatever value $r_{\textrm{eff}}$ is actually given, including the current $r_{\textrm{eff}}=10\sigma_1$; it just means the half-light ellipse currently has semi-axes $(10\sigma_2,10\sigma_1)$ rather than $(\sigma_1,\sigma_2)$.

The code is given as

auto q = sigma2/sigma1;
int n = 4;  // Sersic index
auto re = 10*sigma1; // effective radius
// CHECK: this factor of 10 means the half-light ellipse (where r==re)
// actually has semi-axes (10*sigma2, 10*sigma1), not (sigma1, sigma2).
// If sigma1/sigma2 are meant to *be* the half-light semi-axes, this
// looks like a bug or a leftover unit conversion -- kept as-is for now
// since it matches the current implementation; revisit once confirmed.
auto r = std::sqrt(std::pow(x/q, 2)+std::pow(y, 2));
auto bn = 1.992*n - 0.3271;
auto F = luminosity * std::pow(10, 3);
auto pi = 3.141592;
auto I_eff = F*std::pow(bn, 2.0*n)
           / (2*pi*sigma1*sigma2*n*std::exp(bn)*std::tgamma(2.0*n));
auto value = round(I_eff*std::exp(-bn*((std::pow(r/re, 1.0/n))-1.0)));
if (value > 255) { value = 255; }
dst.at<uchar>(row, col) = (uchar)value;

General elliptical Sersic profile¶

The hard-coded implementation above fixes $n=4$ and only handles ellipses aligned with the $(x,y)$ axes. This section gives the general recipe: arbitrary Sersic index $n_s$, an arbitrary position angle, and a luminosity normalisation consistent with the spherical case.

Elliptical radius¶

Replace the circular radius $r=\sqrt{x'^2+y'^2}$ with an elliptical radius, using the axis ratio $q=\sigma_2/\sigma_1\le 1$:

This is exactly what the hard-coded code above already does for the axis-aligned case (with $y'=y$ playing the role of the unscaled axis). The pixel value is then given by the same 1D formula as the spherical case, with $r\to\tilde r$ and general $n_s$:

Luminosity normalisation¶

An ellipse with semi-axes $\sigma_1,\sigma_2$ has area $\pi\sigma_1\sigma_2$, in place of the circle’s $\pi\sigma^2$. The $I_{\mathrm{eff}}$ normalisation therefore becomes

i.e. literally the spherical-case formula with $\sigma^2\to \sigma_1\sigma_2 = r_{\textrm{eff,circ}}^2$. This is the I_eff used in the code above.