### 7 Mathematics

The optional globular cluster is generated using a King (1962)
star density law

D(r)=k((1+(r/r_{c})^{2})^{-1/2}-(1+(r_{t}/r_{c})^{2})^{-1/2})^{2}

where $D(r)$ is the star surface density a projected distance $r$ from the
centre of the cluster.
$k$, $r_{c}$ and $r_{t}$ are constants, $k$ being a scale factor and $r_{c}$
and $r_{t}$ the core and tidal radii respectively.

Moffat’s formula gives the intensity I(r) at a radial distance r from the
centre of the star image as

I(r)=I_{0}/(1+(\frac{r}{R})^{2})^{\beta}

where $I_{0}$, R and $\beta$ are all constants.
The total luminosity, Lt, of such a profile is

Lt=\frac{\pi R^{2}I_{0}}{\beta -1}

so

I(r)=\frac{Lt(\beta -1)}{\pi^{2}R^{2}}/(1+(\frac{r}{R})^{2})^{\beta}

If the intensity threshold beyond which the profile is truncated is $I_{th}$,
then the corresponding radius $r_{th}$ is

r_{th}=((\frac{Lt(\beta -1)}{I_{th}\pi R^{2}})^{1/\beta}-1)^{1/2}R

from this, the fraction of the total light emitted beyond this boundary, f, may
be calculated

f=(1+(\frac{r_{th}}{R})^{2})^{1-\beta}

\begin{displaymath}
B=\frac {\pi( (X-(\frac{IXEXT}{2}-1))^{2} +
(Y-\frac{IYEXT}{2})^{2} )^{\frac{1}{2}}} {A}
\end{displaymath}

LATEX OUTPUT

The optional globular cluster is generated using a King (1962) star density law

 $D\left(r\right)=k{\left({\left(1+{\left(r/{r}_{c}\right)}^{2}\right)}^{-1/2}-{\left(1+{\left({r}_{t}/{r}_{c}\right)}^{2}\right)}^{-1/2}\right)}^{2}$ (1)

where $D\left(r\right)$ is the star surface density a projected distance $r$ from the centre of the cluster. $k$, ${r}_{c}$ and ${r}_{t}$ are constants, $k$ being a scale factor and ${r}_{c}$ and ${r}_{t}$ the core and tidal radii respectively.

Moffat’s formula gives the intensity I(r) at a radial distance r from the centre of the star image as

 $I\left(r\right)={I}_{0}/{\left(1+{\left(\frac{r}{R}\right)}^{2}\right)}^{\beta }$ (2)

where ${I}_{0}$, R and $\beta$ are all constants. The total luminosity, Lt, of such a profile is

 $Lt=\frac{\pi {R}^{2}{I}_{0}}{\beta -1}$ (3)

so

 $I\left(r\right)=\frac{Lt\left(\beta -1\right)}{{\pi }^{2}{R}^{2}}/{\left(1+{\left(\frac{r}{R}\right)}^{2}\right)}^{\beta }$ (4)

If the intensity threshold beyond which the profile is truncated is ${I}_{th}$, then the corresponding radius ${r}_{th}$ is

 ${r}_{th}={\left({\left(\frac{Lt\left(\beta -1\right)}{{I}_{th}\pi {R}^{2}}\right)}^{1/\beta }-1\right)}^{1/2}R$ (5)

from this, the fraction of the total light emitted beyond this boundary, f, may be calculated

 $f={\left(1+{\left(\frac{{r}_{th}}{R}\right)}^{2}\right)}^{1-\beta }$ (6)
$B=\frac{\pi {\left({\left(X-\left(\frac{IXEXT}{2}-1\right)\right)}^{2}+{\left(Y-\frac{IYEXT}{2}\right)}^{2}\right)}^{\frac{1}{2}}}{A}$

An AR model is of the form

X_{i}=\sum_{j=1}^{M} A_{j}X_{i-j}+E_{i}

for equally spaced observations $X_{i}$ and for a set of constants $A_{j}$.
$E_{i}$ is the error in using this model.
The method involves choosing the $A_{j}$ to minimize the $E_{i}$.

The Q, U and E frames can now be calculated as follows:
$Q_{ij}=\frac{A_{ij}-B_{ij}}{A_{ij}+B_{ij}}$
$U_{ij}=\frac{C_{ij}-D_{ij}}{C_{ij}+D_{ij}}$
$E_{ij}=\frac{2}{A_{ij}+B_{ij}+C_{ij}+D_{ij}}$
The total polarization frame, P, and polarization angle frame, T, are generated
from Q, U and (if required) E as follows:
$P_{ij}=\sqrt{{Q_{ij}}^{2}+{U_{ij}}^{2}}-E_{ij}$
$T_{ij}=0.5\arctan \frac{U_{ij}}{Q_{ij}}$
The values of Tij are restricted to the range 0 to +$\pi$ radians.

Every pixel (X,Y) in the image frame is multiplied by SIN(B)/B where
\begin{displaymath}
B=\frac {\pi( (X-(\frac{IXEXT}{2}-1))^{2} +
(Y-\frac{IYEXT}{2})^{2} )^{\frac{1}{2}}} {A}
\end{displaymath}

LATEX OUTPUT

An AR model is of the form

 ${X}_{i}=\sum _{j=1}^{M}{A}_{j}{X}_{i-j}+{E}_{i}$ (7)

for equally spaced observations ${X}_{i}$ and for a set of constants ${A}_{j}$. ${E}_{i}$ is the error in using this model. The method involves choosing the ${A}_{j}$ to minimize the ${E}_{i}$.

The Q, U and E frames can now be calculated as follows:

${Q}_{ij}=\frac{{A}_{ij}-{B}_{ij}}{{A}_{ij}+{B}_{ij}}$

${U}_{ij}=\frac{{C}_{ij}-{D}_{ij}}{{C}_{ij}+{D}_{ij}}$

${E}_{ij}=\frac{2}{{A}_{ij}+{B}_{ij}+{C}_{ij}+{D}_{ij}}$

The total polarization frame, P, and polarization angle frame, T, are generated from Q, U and (if required) E as follows:

${P}_{ij}=\sqrt{{{Q}_{ij}}^{2}+{{U}_{ij}}^{2}}-{E}_{ij}$

${T}_{ij}=0.5arctan\frac{{U}_{ij}}{{Q}_{ij}}$

The values of Tij are restricted to the range 0 to +$\pi$ radians.

Every pixel (X,Y) in the image frame is multiplied by SIN(B)/B where

$B=\frac{\pi {\left({\left(X-\left(\frac{IXEXT}{2}-1\right)\right)}^{2}+{\left(Y-\frac{IYEXT}{2}\right)}^{2}\right)}^{\frac{1}{2}}}{A}$