## FANDOM

157 Pages

Consider a relativistically expanding shell $\Gamma \gg 1$. The equal arrival time surface (EATS) is the physical surface from which emitted photons arrive at the same time to the observer. A photon emitted at time $t$ will reach the observer at a time:

$T=t+\frac{D-R(t)cos \theta}{c}$

where $D$ is the distance between the shell's centre of expansion and the observer , $R(t)$ is the shell's radius at a time $t$ and $\theta$ is the angle between the emission site and the line of sight to the observer. Plugging in $R(t)=\beta c t$, and shifting the arrival time by a constant amount: $T_0=T-\frac{D}{c}$, we have:

$T_0=\frac{R}{\beta c}-\frac{R cos \theta}{c}$.

It follows that: $\beta c T=R(1-\beta cos \theta) \rightarrow R=\frac{\beta c T}{1-\beta cos \theta}$. Recalling that an ellipse is the shape defined by: $R=\frac{a(1-\epsilon^2)}{1-\epsilon cos \theta}$, we see that the EATS is an ellipsoid with ellipticity: $\epsilon=\beta$, a semi-major axis: $a=\gamma^2 \beta c T_0$ and semi-minor axis: $b=\beta c T_0$.