Sérsic

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Usage

sersic  
  x1c       # x-coordinate of centroid  
  x2c       # y-coordinate of centroid  
  q         # axis ratio (minor/major)  
  pa        # position angle (in radians; counter-clockwise from +x)  
  Amp       # amplitude (normalisation, surface brightness at $r = R_{\text{eff}}$)  
  reff      # effective radius (half-light radius)  
  nsersic   # Sersic index  

Surface Brightness Profile

After rotating by $-\text{pa}$ into the frame of the Sersic light (so the major axis lies along $x_1$ and minor axis along $x_2$), define:

$r = \sqrt{x_1^2 + \frac{x_2^2}{q^2}}$

The Sersic intensity is approximated using the expression from Dutton et al. (2011), based on Ciotti & Bertin (1999). The approximation for $\kappa$ is:

$\kappa = 2n - \frac{1}{3} + \frac{4}{405n} + \frac{46}{25515n^2}$

Then, the surface brightness is:

$I(r) = A \cdot \exp\left[-\kappa \left( \left( \frac{r}{r_{\text{eff}}} \right)^{1/n} - 1 \right) \right]$

where:

• $A$ is the amplitude (Amp), the surface brightness at $r = r_{\text{eff}}$
• $n$ is the Sersic index (nsersic)
• $r_{\text{eff}}$ is the effective (half-light) radius

Notes on Accuracy

• The truncated expansion of $\kappa$ is accurate to within $\sim10^{-3}$ for $n > 0.36$ and $n < 10$
• Intensity $I(r)$ is in surface brightness per arcsec²

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Usage in simulate with sersic_mag

When using sersic_mag (Sersic profile via integrated magnitude), amplitude is calculated as:

Amp = texpo * pow(10, -0.4 * (mag - mzpt)) / 
      (2 * M_PI * exp(kap) * n * kap^{-2n} * r_eff^2 * q * Gamma(2n));
Amp *= dx * dy;  // convert to counts per pixel

Where:

• mag is the total integrated magnitude
• mzpt is the magnitude zeropoint
• texpo is the exposure time
• dx, dy are pixel scales
• Gamma is the gamma function

The approximation for $\kappa$ in this context is:
$\kappa = 2n - \frac{1}{3} + \frac{4}{405n} + \frac{46}{25515n^2}$

This gives a robust conversion from magnitude to amplitude for use in image simulations with GLEE.

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Units of Amplitude

The amplitude parameter (Amp) is in the same units as the science data image:

• If the science image is in ADU (Analog-to-Digital Units), then Amp is in ADU
• If the science image is in e⁻/s (electrons per second), then Amp is in e⁻/s

Predicted Image Generation

The predicted image from the light profile is obtained through the following steps:

1. Profile Evaluation:

   • The profile is evaluated on a pixel grid
   • If subsampling factor = 1: grid matches the science image exactly
   • If subsampling factor > 1: grid is supersampled (e.g., 3×3 subpixels per science pixel)

2. PSF Convolution:

   • The grid from step 1 is convolved with the PSF
   • Uses subsampled PSF if subsampling factor > 1

3. Binning (if subsampled):

   • Convolved image is binned down to science image resolution
   • Each pixel value is divided by (subsampling factor)² to conserve total flux

The resulting image is directly compared to the science image to compute χ² for the lens light (in regions outside the arc mask and inside the lens mask).

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Citations for Profile

@ARTICLE{Sersic63,
       author = {{S{\'e}rsic}, J.~L.},
        title = "{Influencia de la dispersión atmosférica e instrumental en la distribución de brillo de una galaxia [Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy]}",
      journal = {Boletin de la Asociacion Argentina de Astronomia La Plata Argentina},
         year = 1963,
        month = feb,
       volume = {6},
        pages = {41-43},
        url={http://sedici.unlp.edu.ar/handle/10915/91601}
}