Gamma-ray compactness problem

Rapidly varying high-energy sources can appear too dense in target photons to transmit their observed gamma rays. For an approximately isotropic stationary region with target-band luminosity

L_{2}

, radius

R

and target photon energy

ϵ_{2}

, the target number density is

n \sim L_{2} / (4 π R^{2} c ϵ_{2})

. photon-photon pair production then gives optical depth

τ \sim nσ R

. A luminosity variability size bound can make this very large. relativistic beaming both increases the inferred rest-frame size and reduces the inferred rest-frame radiation density, allowing escape. This radiation-opacity argument is distinct from gravitational stellar compactness and topological compactness.

Table of contents
- Beamed gamma-ray optical depth Gamma-ray compactness problem
- Internal pair-production target energy in a relativistic blob Gamma-ray compactness problem

Beamed gamma-ray optical depth ( $τ^{'} \propto D^{- (4 + 2 α)}$ )

 0  0

For an isotropically emitting rest-frame blob moving with Doppler factor

D

, take the observed target-band spectral flux density to satisfy

F_{ν} \propto ν^{- α}

over the relevant frequencies. The sign convention for

α

matters. At fixed cosmological redshift, absorb fixed distance and redshift factors into a reference optical depth

τ_{0}

A light-crossing estimate gives

R^{'} \propto D

for fixed observed variability time, while a corresponding logarithmic frequency bin has observed-to-rest-frame luminosity boost

D^{4}

. At a fixed observed high-energy photon energy, the internal pair-production target energy in a relativistic blob shifts the observed target energy by

D^{2}

. Combining these factors gives

τ^{'} (D) \sim τ_{0} D^{- 6} \frac{F _{l o g} ( D ^{2} ϵ _{20} )}{F _{l o g} ( ϵ _{20} )} = τ_{0} D^{- 6 + 2 (1 - α)} = τ_{0} D^{- (4 + 2 α)},

(1)

where

F_{l o g} = ν F_{ν}

and

ϵ_{20}

is the reference target energy at

D = 1

. For example,

α = 1

gives

D^{- 6}

, whereas a locally flat

F_{ν}

gives

D^{- 4}

. The power-law assumption must cover the target energies swept by changing

D

. Spectral breaks, different emission geometries and external stationary illumination require a different calculation;

D^{- 6}

is not a universal pair-opacity law.

Internal pair-production target energy in a relativistic blob ( $ϵ_{2, obs} = D^{2} K / ϵ_{1, obs}$ )

 0  0

Let the relevant rest-frame photon-photon pair production energy product be

K = ϵ_{1}^{'} ϵ_{2}^{'}

for a specified collision-angle prescription. For fixed observed high-energy photon energy,

ϵ_{1}^{'} = ϵ_{1, obs} / D

, so the target has

ϵ_{2}^{'} = DK / ϵ_{1, obs}

. Observing the same internally produced target population supplies a second Doppler factor, giving the displayed result. An external stationary radiation field requires its own angular transformation instead.