# Sources

The sources in Lensed are two-dimensional surface brightness distributions $I_S(x, y) = S(R(x, y)) \;,$ where $S(r)$ is one of the one-dimensional profiles below, ellipticised by the radius function $R(x, y) = \sqrt{\tilde{x}^2 + q^2 \tilde{y}^2}$ for the central coordinates $\tilde{x} = (x - x_S) \cos(\theta) + (y - y_S) \sin(\theta) \;,$ $\tilde{y} = (y - y_S) \cos(\theta) - (x - x_S) \sin(\theta) \;,$ using source position $(x_S, y_S)$, axis ratio $q$ and position angle $\theta$.

Sources are usually normalised by their total luminosity $L_{\text{tot}}$, which is given as the magnitude $m$ above the zero-point of the image, i.e. $L_{\text{tot}} = 10^{-0.4 m} \;.$ This means that the magnitude $m$ of a source with more than 1 count per second must be less than $0$, and more negative values result in brighter sources.

## Gaussian

The gauss profile is a Gaussian blob given by $S(r) = \frac{L_{\text{tot}}}{q \, \pi \sigma^2} \, \frac{1}{2} \, e^{ -\frac{1}{2} r^2/\sigma^2 } \;,$ where $\sigma^2$ is the variance of the profile.

### Parameters

Name Description Range
x source position $x_S$ image pixels
y source position $y_S$ image pixels
mag total magnitude $m$
sigma standard deviation $\sigma$ $\sigma > 0$
q axis ratio $q$ $0 < q < 1$
pa position angle $\theta$ in $\deg$ $0 \leq \theta < 180$

## Exponential

The exp profile is an exponential disk given by $S(r) = \frac{L_{\text{tot}}}{q \, \pi R_s^2} \, \frac{1}{2} \, e^{ - r/R_s } \;,$ where $R_s$ is the scale length of the profile.

### Parameters

Name Description Range
x source position $x_S$ image pixels
y source position $y_S$ image pixels
mag total magnitude $m$
rs scale length $R_s$ $R_s > 0$
q axis ratio $q$ $0 < q < 1$
pa position angle $\theta$ in $\deg$ $0 \leq \theta < 180$

## De Vaucouleurs

The devauc profile is De Vaucouleurs law, given by $S(r) = \frac{L_{\text{tot}}}{q \, \pi R_{\text{eff}}^2} \, \frac{b^8}{8!} \, \exp\left( -b \, (r/R_{\text{eff}})^{1/4} \right) \;,$ where $b = 7.6692494425008039044$ and $R_{\text{eff}}$ is the effective radius containing half the total luminosity.

### Parameters

Name Description Range
x source position $x_S$ image pixels
y source position $y_S$ image pixels
mag total magnitude $m$
r effective radius $R_{\text{eff}}$ $R_{\text{eff}} > 0$
q axis ratio $q$ $0 < q < 1$
pa position angle $\theta$ in $\deg$ $0 \leq \theta < 180$

## Sérsic

The sersic profile is an extension of De Vaucouleurs law to the general Sérsic $1/n$ law given by12 $S(r) = \frac{L_{\text{tot}}}{q \, \pi R_{\text{eff}}^2} \, \frac{b_n^{2n}}{\Gamma(2n+1)} \, \exp\left( -b_n \, (r/R_{\text{eff}})^{1/n} \right) \;,$ where $R_{\text{eff}}$ is the effective radius containing half the total luminosity, and the coefficient $b_n$ is the solution of $\frac{\Gamma(2n, b_n)}{\Gamma(2n)} = \frac{1}{2} \;,$ which can be approximated by the minimax polynomial $b_n \approx 1.9992n - 0.3271$ in the range $0.5 < n < 8$.

The Sérsic profile with $n = 0.5$, $n = 1$ and $n = 4$ is equivalent to the Gaussian, exponential and De Vaucouleurs profile, respectively, with a different radius definition.

### Parameters

Name Description Range
x source position $x_S$ image pixels
y source position $y_S$ image pixels
mag total magnitude $m$
r effective radius $R_{\text{eff}}$ $R_{\text{eff}} > 0$
n Sérsic index $n$ $0.5 < n < 8$
q axis ratio $q$ $0 < q < 1$
pa position angle $\theta$ in $\deg$ $0 \leq \theta < 180$

1. J. L. Sérsic, (1968).

2. A. W. Graham and S. P. Driver, Publ. Astron. Soc. Aust 22, 118 (2005).