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 $\deg$ | $0 \leq \theta < 180$ |