Is the gamma-ray burst a spherical field, or is it a narrow, directed cone? If the latter, I wonder how the burst spreads/diffuses as it travels and what sort of equation governs the energy flux with respect to distance and initial conditions.
Great question! Since they're emanated in jets, the GRBs travel in a conical shape that spreads out as it travels. Check out page 19 of this paper, it has a histogram of the jet opening angle;
http://arxiv.org/pdf/1101.2458.pdf
Then you can figure out the flux from the area of a circle on a sphere, which is something like
2πr^2(1−cosθ)
Where theta is the opening angle and r is the distance from point of emanation of the GRB.
Thanks for the paper link! I'll check it out. (The histogram legend seems to indicate two distributions of observed GRBs, the larger centered around 8 degree cones.)
I think there are multiple types of GRBs and some can be spherical. Two that I have read about are neutron stars colliding and large solar mass > 100 exploding and leaving no remnant. I've read that these types of bursts can blow the atmosphere off of a planet. http://en.wikipedia.org/wiki/Hypernova
