Thermal Emission Component in GRB Prompt Emission

II. Theory

1. Photosphere in relativistically expanding plasma
The main concept in the study of thermal photons is that of the Photosphere. The basic idea is that deep enough in the flow, the optical depth is very large, and thus photons cannot escape, but are advected with the flow. Since near the base of the flow the optical depth in GRBs is huge - ~10¹⁵, photons that are produced by any radiative process that occurs near the base of the flow thermalize before escaping.

As the flow propagates, the density of the plasma (electrons, protons, pairs) decreases. Thus, above a certain radius, the optical depth becomes smaller than unity, and the photons escape. The radius at which this happens is, by definition, the photosphere. The first difficulty that arises when trying to calculate the properties of the photosphere, is that in GRBs the flow is highly relativistic, that is it travels at speeds very close to the speed of light, hence special relativity effects take place. As a result, the photospheric radius strongly depends on the angle to the line of sight.

When I carried these calculations, I found that the dependence can be put in a surprisingly simple form:

Here,

is the Lorentz factor of the GRB relativistic outflow, and

is the angle to the line of sight. The angular dependence of the photospheric radius appears in the figure below (note the logarithmic scale)

2. Photon diffusion and probability density functions
The description of the photosphere is important in understanding where the thermal photons decouple from the plasma. Assuming that the photons diffuse below the photosphere, it can also be used to calculate how long it takes the photons to escape, and hence be used to calculate the observed flux.

Propagation of photons is a random process, and therefore not all of the photons decouple from the plasma exactly at the photospheric radius. In fact, photons have a finite probability of being emitted (=last scattering event) from every point in space. While the photosphere gives a first order approximation for the place where this emission takes place, a full calculation (see Pe'er 08) of this probability is required.

Figure 2, to the right, shows the results of a numerical simulation that traces photons from deep in the flow until their escape. Every point represent the last scattering position in the r - plane (same as in figure 1). The photospheric radius, drawn in green, indeed shows a first order approximation to the last scattering event positions, which span the entire space.

In fact, I showed that the probability of a photon to be last scattered at a given point is (r,

) space has an analytical description. Thus, assuming that all the thermal photons are emitted at the center of the explosion at time t=0 (a delta function approximation), and assuming that before they escape the photons effective velocity (=their velocity in the direction of the flow) is equal to the outflow velocity, one can calculate the observed flux at late times, by integrating over the probability of photons to escape, and "counting" all the photons that arrive at the same time.

The results of the calculations is presented in the figure to the left. Both the analytical calculations (red) and the numerical simulation (blue) indicate that at late times, the flux drops as
F(t) ~ t^-2.

The second important ingredient is the temperature. In principle, the calculation of the temperature is similar to that of the flux, with an additional complexity: photons lose their energy by multiple scattering before they escape. Thus, one first needs to calculate the local energy of a photon, before averaging over all the observed photons at any given instance (which gives the temperature).

Photons lose their energy by multiple Compton scattering; i.e., they convert their energy to the kinetic energy of the flow, in a process similar to (but not identical !) to classical adiabatic energy losses. The results of the calculation, presented to the right, shows a comoving energy decrease with the radius, as ~r^-2/3, with a correction at large radii.

In addition to the local (comoving) temperature, the observed temperature of the photons is blue shifted by the relativistic Doppler effect. The Doppler shift by itself depends on the angle to the line of sight, . Therefore, once the local temperature drop and the angle are known, it is possible to calculate the drop in the observed temperature. The results (shown in the left) indicate T(t) ~ t^-1/2 - t^-2/3.

Implications
The results obtained can be useful in understanding late time evolution of thermal photons originating from relativistically expanding plasmas. We believe that this is exactly what happens in the late time prompt emission in GRB's. In fact, in a recent analysis, Felix Ryde and myself found a behavior that is remarkably similar to the prediction given here. You can read more about our findings here.

Since thermal photons originate from the photosphere, the innermost radius from which information can reach the observer, studying it may thus give important information on the progenitor, and the outflow. Thus, we believe that the results here are an important step towards understanding basic properties of GRBs.

Below is a connection to further explanations on the implications of these results. The paper by Pe'er (2008) where the full theory is presented, can be found here.

Study of thermal emission in GRBs:
I- identification

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Study of thermal emission in GRBs:
III- Implications

Asaf Pe'er, Created: 2-Dec-2008