TIME-INDEPENDENT INVERSE COMPTON SPECTRUM FOR PHOTONS FROM A PLASMA CLOUD OF FINITE EXTENT

The general theoretical aspects of inverse Compton scattering was investigated and an equation for the timeindependent inverse Compton spectrum for photons from a plasma cloud of finite extent was derived. This was done by convolving the Kompaneets equation used for describing the evolution of the photon spectrum from a plasma cloud of infinite extent and the distribution of photons over escape time. By looking for a power law solution to the equation so derived, was revealed that the spectrum produced by inverse Compton scattering depend only on the electron temperature and optical depth.


FINITE EXTENT CLOUD SOLUTION TO THE KOMPANEETS EQUATION
The Kompaneets equation (Kompaneets, 1957;Birkinshaw, 1999) which is a parabolic partial differential equation is given by is the dimensionless electron temperature, with e T the electron temperature andτ is the optical depth of the cloud.For a stationary case the y-parameter is defined as the product τ Θ , and is a measure of the total amount of energy transferred from the electrons to the photons.The electron density e n is assumed to be constant.
A more general solution of the equation is given as use the idea that if one knew how many of the photons had escaped from a slab between time t and t + dt, that is, the distribution of photons over the escape time from the slab, one could convolve this with eqn. 2 and determine the spectrum of escaping photons in a cloud with slab-like geometry.By considering the one-dimensional diffusion equation  3) is separable, so if one looks for solutions of the form ( ) ( ) In general, the boundary condition of the problem is used to determine the separation constant n λ , while the initial distribution ( ) x n is used to determine the coefficient n c .For an arbitrary injection of photons, there may be infinitely many terms in the series solution given by eqn. 4.However, because n λ increases with n, after some time the higher terms or modes will have died away compared to the first term or 'fundamental' mode.Therefore, the spatial distribution of photons approaches the distribution given by the first eigenfunction The distribution of photons over the escape time is simply the intensity at the boundary of the slab as a function of time u, and this can be given by eqn. 5.A normalization condition is also required for the fraction of photons that have escaped.Using eqn. 5 we write The function P(u) is the required distribution of photons over the escape time.As discussed above, if the function P(u) is known it can be convolved with the solution to the Kompaneets' equation to give the spectrum from a plasma cloud of finite extent, i.e.
Therefore the parameter β must be determined and this is done by appealing to the boundary condition of the problem.
For the spatial part of the problem one has the equation If one appeals to symmetry about the mid-plane of the disk and discards the sine solution, we get [ ] This equation is then used with the boundary condition to give n λ (or equivalently β ).The boundary condition from diffusion theory is (Zweifel, 1973) where the diffusion constant D in this case equals 1/3, from eqn. 3.This boundary condition implies that the number of escaping photons (the right hand side) is equal to half the number of photons at the boundary (the left hand side).
Presumably the other half of the photons is directed back into the cloud due to the assumed isotropy of the photon distribution.In addition, this condition holds that no photons are incident on the cloud from the outside.The boundary condition given in eqn.11 presents difficulties such as, the extent to which the intensity goes to zero and as such an easier approach was formulated by considering the linear extrapolation Substituting this into the boundary condition (eqn.11) and solving for k gives k = 3/2.So the flux outside the surface of the cloud can be taken to obey This implies that the intensity goes to zero at a distance τ = 2/3 outside the boundary.This gives a new and simple boundary condition.Evaluating the intensity at the point outside the boundary (the so-called extrapolated length) and equating to zero, one obtains Now that β is known, everything about the distribution of photons over the escape time P(u) is known.It is also seen that the spatial transport depends only on the optical depth τ o .This analysis can be repeated for spherical and cylindrical geometries, the principal difference being that value of β (obtained from the boundary condition) is different.

THE STATIONARY EQUATION
Knowing the distribution of photons over the escape time P(u), and given eqn.7, the spectrum from a finite cloud can be found by performing the integration.However, more insight can be gained by using the same equations to derive another equation for the spectrum of a finite cloud; one that is stationary in time.To do this, we construct an equation for the time-independent spectrum N(x) such that This is done by substituting both the Kompaneets expression for n(x,u) and the equation for P(u) (eqn.6).Dropping the ( ) 2 ,u x n term in Kompaneets equation, multiplying each side by ( ) The left hand side equals [ ] where ƒ(x) is a source that is equal to ( ) 3 0 , x x n . So the equation becomes The left hand side is identical to the Kompaneets equation.The first term on the right hands side describes the escape of photons from the plasma cloud, while the second term describes the injection of photons due to a source, ƒ(x).The value of γ is given by the spatial diffusion problem.
By looking for a power law solution to the stationary equation of the form for a slab geometry (using the value of β found earlier).Solving this quadratic equation gives This equation relates the energy spectral index α to the temperature and optical depth of the cloud.From the equation, we see that the spectrum produced by inverse Compton scattering depends only on the electron temperature and optical depth.It is also seen that the spectral slope is inversely proportional to Θ and τ o .

CONCLUSION
The Comptonization process for photons undergoing repeated scattering in a cloud of finite slab-like geometry was investigated and an expression for the time-independent spectrum N(x) consequently derived.It was shown, after appealing to a power law solution, that the spectrum produced by inverse Compton scattering depends only on the electron temperature and optical depth.This analysis can be extended to spherical and cylindrical geometries; with the principal difference being that the value of β (obtained from the boundary condition) is different.
electron number density, T σ is the Thompson cross section, c is the speed of light;

.
r measures the distance from the slab mid-plane.I is the angle-averaged intensity, where the photon occupation number is related to the intensity I It is assumed that one face of the slab is along the plane half-thickness of the slab.Equation (