Gaussian

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Usage

gaussian  
  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; peak surface brightness)  
  sigma     # width of the Gaussian (standard deviation, in arcsec)  

Surface Brightness Profile

After rotating by $-\text{pa}$ into the frame aligned with the light profile (major axis along $x_1$, minor along $x_2$), define:

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

The 2D elliptical Gaussian intensity is then:

$I(r) = A \cdot \exp\left(-\frac{1}{2} \cdot \frac{r^2}{\sigma^2}\right)$

where:

• $A$ is the amplitude (Amp), the peak intensity at the centre
• $q$ is the axis ratio
• $\sigma$ is the Gaussian width (standard deviation)