The Colorado College
Supported by the NSF-REU and Michigan Space Grant Consortium
Numerical Integration of Relativistic, Magnetic Compton Scattering
This summer I went to NASAs Goddard Space Flight Center with Professor Peter Gonthier and worked with a computer code that dealt with magnetic Compton scattering. The results from our program will be applied to a pulsar model simulating the emission of gamma-rays from pulsars.
Pulsars, which are very rapidly rotating neutron stars, are observed to be emitting blackbody radiation with an average energy of about 0.1 keV, in the x-ray part of the spectrum. They are also found to have extremely strong magnetic fields, on the order of 1011 or 1012 Gauss, where the "critical field" is about 4.414 x 1013 Gauss. There is growing evidence of other, more exotic pulsars known as magnetars, with super-strong fields, exceeding the critical magnetic field. In the polar cap theory, electrons near the surface of the star are accelerated by a gap in the magnetosphere, above the magnetic poles, over which there is a large electric potential difference. Being accelerated across this potential, the electrons come off the star at relativistic speeds.
In inverse Compton scattering, the electron is in motion in the presence of a magnetic field and collides with a photon, slowing down and giving some of its kinetic energy to the photon, thus increasing the energy of the photon. Magnetic Compton scattering is an important contributing factor to the seed photons that initiate the cascade above the polar cap. In order to see how magnetic Compton scattering might contribute, we were interested in looking at the behavior of the cross section. We used the exact QED differential cross section in the rest frame of the electron.
Klein and Nishina had previously gotten a cross section that was fully relativistic and QED, but had no magnetic field. We are testing to see if the exact QED magnetic cross section tends toward the Klein-Nishina limit, since under ultra relativistic conditions, the magnetic fields contribution should decrease to the point where it does not have an effect. To do this, we have to add the contributions of the many Landau states that the electron can be in, because of the strong magnetic field. We take the differential cross section, d2s/dq(w, q), and integrate it over q, and sum the Landau states, to get a cross section as a function of energy, s(w).
This is done numerically by a computer program, written in C, using different numerical integration routines. A lot of the time was spent wrestling with the numerical integration part of the program, as they were originally more generic routines not specifically meant for our task, and so ran into various problems. For instance, as the Landau states increase, their contribution becomes smaller and smaller, and eventually the numbers were too tiny for the computer to handle within its limits of precision. After considerate effort, we were able to observe that the exact QED cross section tends toward the Klein-Nishina limit at large energies.
I think that this summer has been an interesting and useful experience for me. I learned a lot about computer programming, some astrophysics, and most of all, what theoretical physicists spend their days doing. It was fun and valuable to see ahead of college to the possible things I will do during and after graduate school. I would like to thank Hope College, Professor Gonthier, Michelle Ouellette, Alice Harding, and NASA for having me this summer and arranging all of this!