(Was an edit, moved to new post due to eran_rathan's post):
What we really want, ideally, is a three-dimensional contour plot showing the photon flux at every point in the two-dimensional ecliptic plane. To get it accurately, we'd need to give that plane some unit thickness and quantize it to a series of cubes. Then, we'd need to produce a list of black-hole-affected trajectories for a random series of rays coming out of the supernova origin (being careful not to bias this with clustering around the poles). Finally, we'd want each cube to run a check for intersection with the trajectory of every ray in our list and assign the number of intersections to its xy coordinates to give us the flux density plot.
Then condense that whole process into a program which will produce the plot for any arbitrarily-sized and arbitrarily-located black hole.
Arguably, there's radial symmetry around the line connecting the black hole and supernova (if we ignore the effects of mutual orbit and so forth). The null geodesic trajectory of a given ray depends only on the azimuthal angle of the initial ray vector. But this only helps us in producing the initial list of trajectory equations; I think we'll still need to use a three-dimensional treatment in order to get a good idea of flux via the intersection approach.
Finally, this whole thing assumes a time-independent flux, which may not be the case. Is the peak power output of a supernova constant for long enough to produce a time-independent flux for this system?
davidstarlingm wrote:Anyone know the equation for the trajectory of a photon with arbitrary direction, given a black hole of specified size and distance from the primary? That would be the first step in setting up a proper Monte Carlo.
that level of math is, unfortunately, well outside my pay grade.
Surely the equation of the null geodesic trajectory around a black hole has been derived in plenty of places, no? Photons are only following the shape of space, after all; it's not they have rest mass to deal with.
EDIT: This assumes a Schwarzchild black hole, of course; a Kerr black hole is going to introduce all manner of nasty interactions from frame-dragging, particularly if the damn thing has a rotational axis not
normal to the ecliptic. Then again, a proper rotation could theoretically cause constructive interference within the spherical orbit itself like the pumping mechanism
in lasing fluorescent microspheres. Possibly a workable supernova laser?
ANOTHER EDIT: I'm not running a Mac, but apparently this dude
developed a calculator to determine the trajectory of photons around a black hole last month. Any possible use?