Regimes of Particulate and Wave-like Behavior

[Vrbic]

Homework Statement

A gamma-ray burster is an astrophysical object (probably a fireball of hot gas exploding outward from the vicinity of a newborn black hole or colliding neutron stars or colliding neutron star and black hole) at a cosmological distance from Earth (∼ $10^{10}$ light years). The fireball emits gamma rays with individual photon energies as measured at Earth E ∼ 100 keV. These photons arive at Earth in a burst whose total energy per unit area is roughly e=$10^{−6} ergs/cm^2$ and that lasts about one second. Assume the diameter of the emitting surface as seen from Earth is D∼ 1000 km and there is no absorption along the route to earth. Make a rough estimate of the mean occupation number of the burst’s photon states. Your answer should be in the region η << 1, so the photons behave like classical, distinguishable particles. Will the occupation number change as the photons propagate from the source to earth?

Homework Equations

$n=\frac{c^2 I_{\nu}}{h^4 \nu^3}$...distribution function
$E=h\nu$...energy of one photon
$I_{\nu}=\frac{dE}{dAdtd\nu d\Omega}$ intesity = energy per unit area unit time unit frequency unit solid angle
$\eta=\frac{h^3}{g_s}n$ ...mean occupation number, where $g_s=1$ for photons

The Attempt at a Solution

I believe it is easy, I have only one uncertainty, I suppose that $I_{\nu}\sim e/(D/2)^2=10^{−6} ergs/cm^2/(D/2)^2$. Is it good approximation?

 

[jbsteele]

Assuming that, I get:$n=\frac{e}{h^4 \nu^3 D^2}$ $\eta=\frac{h^3}{g_s}n=\frac{eh^2}{g_s \nu^3 D^2}$ $\eta=\frac{10^{−6} ergs/cm^2 h^2}{100 keV^3 1000 km^2}≈2×10^{−20}$No, the occupation number will not change as the photons propagate from the source to Earth.