Theory of Compton scattering by anisotropic electrons

Theory of Compton scattering by anisotropic electrons

Abstract

Compton scattering plays an important role in various astrophysical objects such as accreting black holes and neutron stars, pulsars, and relativistic jets, clusters of galaxies as well as the early Universe. In most of the calculations it is assumed that the electrons have isotropic angular distribution in some frame. However, there are situations where the anisotropy may be significant due to the bulk motions, or anisotropic cooling by synchrotron radiation, or anisotropic source of seed soft photons. We develop here an analytical theory of Compton scattering by anisotropic distribution of electrons that can simplify significantly the calculations. Assuming that the electron angular distribution can be represented by a second order polynomial over cosine of some angle (dipole and quadrupole anisotropy), we integrate the exact Klein-Nishina cross-section over the angles. Exact analytical and approximate formulae valid for any photon and electron energies are derived for the redistribution functions describing Compton scattering of photons with arbitrary angular distribution by anisotropic electrons. The analytical expressions for the corresponding photon scattering cross-section on such electrons as well as the mean energy of scattered photons, its dispersion and radiation pressure force are also derived. We applied the developed formalism to the accurate calculations of the thermal and kinematic Sunyaev-Zeldovich effects for arbitrary electron distributions.

Subject headings:
accretion, accretion disks – cosmic background radiation – galaxies: jets – methods: analytical – radiation mechanisms: nonthermal – scattering
2

1. Introduction

Compton scattering is one of the most important radiative process that shapes the spectra of various sources: black holes and neutron stars in X-ray binaries, pulsars and pulsar wind nebulae, jets from active galactic nuclei, and the early Universe. Compton scattering kernel takes a simple form if electrons are ultra-relativistic with the Lorentz factor (Blumenthal & Gould, 1970). In a general case, when no restrictions are made on the energies of photon and electrons, Jones (1968) derived the kernel for isotropic electrons and photons. The formulae there contain a few misprints, but even by correcting those (see e.g. Pe’er & Waxman 2005) they cannot be used for calculations because of a number of numerical cancellations (see e.g. Belmont, 2009). An alternative derivation to that kernel was given by Brinkmann (1984) and Nagirner & Poutanen (1994, NP94 hereafter), who showed how to extend numerical scheme to cover all photon and electron energies of interest in astrophysical sources.

In real astrophysical environments, the radiation field does not need to be isotropic and a more general redistribution function is required to describe angle-dependent Compton scattering. NP94 extended previous results to the situation when the photon distribution can be represented as a linear function of some polar angle cosine (Eddington approximation), deriving an analytical formula for the first moment of the kernel. Aharonian & Atoyan (1981) were first to derive a redistribution function for arbitrary photon angular dependence (see also Prasad et al. 1986). Kershaw et al. (1986) and Kershaw (1987) have developed numerical methods to compute the kernel efficiently and with a high accuracy. All these works neglect the effect of photon polarization.

Nagirner & Poutanen (1993) have derived a general Compton scattering redistribution matrix for Stokes parameters assuming an isotropic electron distribution. A general relativistic kinetic equation incorporating the effects of induced scattering and polarization of photons as well as electron polarization and degeneracy has been derived by Nagirner & Poutanen (2001).

In this paper, we propose a method to extend previous results to the case where the electron distribution is no longer isotropic, but can have weak anisotropies which can be represented as a second order polynomial of the cosine of some polar angle. The proposed formalism can find its application in a number of astrophysical problems. The distortion of the cosmic microwave background (CMB) caused by the hot electron gas in clusters of galaxies (i.e. kinematic and thermal Sunyaev-Zeldovich effect) is an obvious application. The electron distribution, isotropic in the cluster frame, can be Lorentz transformed to the CMB frame, resulting in a dipole term linear in cluster velocity and a small quadrupole correction. Compton scattering then can be directly computed in the CMB frame. Another possible application concerns the models of outflowing accretion disk-coronae or jets (Beloborodov, 1999; Malzac et al., 2001). If the outflow velocities are non-relativistic, the radiation transport can be considered directly in the accretion disk frame following recipe by Poutanen & Svensson (1996), with the Lorentz transformed electron distribution.

Weak anisotropies in the electron distribution can arise in high-energy sources with ordered magnetic field, because of the pitch angle-dependence of the synchrotron cooling rate and/or anisotropic injection of high-energy electrons (e.g. Bjornsson, 1985; Roland et al., 1985; Crusius-Waetzel & Lesch, 1998; Schopper et al., 1998). An anisotropic electron distribution is also a very natural outcome of the photon breeding operation in relativistic jets with the toroidal magnetic field (Stern & Poutanen, 2006, 2008), because the electron-positron pairs born inside the jet by the external high-energy photons move perpendicularly to the field.

Our method is also extendable to the polarized radiation using the techniques developed in Nagirner & Poutanen (1993). It is also in principle possible to calculate the scattering redistribution function in the case when the electron distribution is expressible as an arbitrary order expansion over the polar angle cosine. Unfortunately, in the latter case the analytical expressions become extremely cumbersome and the advantage over direct numerical integration becomes small.

Although here we consider only photon scattering it is also possible to apply the same method for electrons interacting with the photons in the case when photon angular distribution is expressible as an expansion of powers of the polar angle cosine. This can be of interest only in the deep Klein-Nishina regime where the electron can lose or gain a large fraction of its initial energy in one scattering and continuous energy loss approach is not applicable.

Our paper is organized as follows. In Section 2 we introduce the relativistic kinetic equation for Compton scattering and define the redistribution function and total cross-section. The expressions for the total cross-section, the mean energy and dispersion of scattered photons, and the radiation pressure force are given in Section 3. The exact analytical formulae for the redistribution function for mono-energetic anisotropic electrons as well as approximate formulae valid in Thomson regime are derived in Section 4. We present the numerical examples of redistribution functions in Section 5, where we also develop the relativistic theory of Sunyaev-Zeldovich effect. We summarize our findings in Section 6.

2. Relativistic kinetic equation

Let us define the dimensionless photon four-momentum as , where is the unit vector in the photon propagation direction and . The photon distribution will be described by the occupation number . The dimensionless electron four-momentum is , where is the unit vector along the electron momentum, and are the electron Lorentz factor and its momentum in units of , and is the electron velocity in units of speed of light. The momentum distribution of electrons is described by the relativistically invariant distribution function (see Belyaev & Budker 1956; NP94).

The interaction between photons and electrons via Compton scattering (in linear approximation, i.e. ignoring induced scattering and electron degeneracy) can be described by the explicitly covariant relativistic kinetic equation for photons (Pomraning 1973; Nagirner & Poutanen 1993; NP94):

(1)

where is the four-gradient, is the classical electron radius, is the Klein-Nishina reaction rate (Berestetskii et al., 1982)

(2)

and

(3)

are the four-products of corresponding momenta. The second equalities in both equations (3) arise from the four-momentum conservation law. The invariant scalar product of the photon four-momenta can be written in the laboratory frame as well as in the frame related to a specific electron

(4)

where is the cosine of the photon scattering angle in some frame and is the corresponding cosine in the electron rest frame.

In any frame, the kinetic equation can be also put in the usual form of the radiative transfer equation (Nagirner & Poutanen, 1993):

(5)

where is the electron density in that frame. Here we have defined the photon redistribution function

(6)

and the total scattering cross-section (in units of Thomson cross-section )

(7)

2.1. Electron distribution and scattering geometry

Let us now consider a specific frame . Our basic assumption is that the anisotropy of the electron distribution in this frame can be expressed as a second order polynomial expansion in the cosine of the polar angle in some coordinate system :

(8)

where is the electron density in frame , is the cosine of the polar angle of the electron momentum, are the Legendre polynomials, and we now switched to the dimensionless distribution function normalized to unity:

(9)

The moments , and determine the energy spectrum of electrons and the relative magnitudes of the isotropic and anisotropic components. The distribution function for mono-energetic electrons of energy can be described by

(10)

where the ratios and are constants.

Figure 1.— The scattering particles’ momentum vectors and the vector . Note that the shown angular variables are the cosines of the respective angles.

The directions of photons in this coordinate system (see Fig. 1) are given by

(11)
(12)

so that the cosine of the scattering angle is

(13)

3. Total cross section and mean powers of energies

3.1. Total cross-section

Let us simplify the expression for the total cross-section. We follow here the approach described in NP94. We rewrite the cross-section as

(14)

where

(15)

Using the identity

(16)

and taking the integral over in equation (15), we get

(17)

where we used invariant and the fact that does not depend on azimuthal angle . Substituting from Equation (2) we get (Berestetskii, Lifshitz, & Pitaevskii 1982; NP94)

(18)

Putting , we, of course, get the total Klein-Nishina cross-section for a photon of energy on electrons at rest.

To obtain the total scattering cross-section on an anisotropic electron distribution, we have to calculate the angular integrals over incoming electron directions in Equation (14). We introduce the cosines between electron momentum and photons:

(19)

so that

(20)

We choose the spherical coordinate system and measure the polar angle from the direction of the initial photon , we get

(21)

Azimuth is now defined as the difference between the azimuths of the electron momentum direction and the vector in a frame with -axis along . The angular variable in the expansion (8) is expressed in this frame as

(22)

The physical meaning of Equation (21) can be also understood if we consider a monoenergetic beam of electrons along axis: . Then

(23)

where is the photon energy in the electron rest frame (we omitted subscript 0 in and ). The factor in Equation (23) accounts for the reduced number of collision per unit length.

Figure 2.— (a) Total cross-section for isotropic mono-energetic electrons of various momenta (from top to bottom, dot-dashed, solid, dotted, dashed, and triple-dot-dashed curves) as a function of photon energy . (b) Relative correction to the cross-section arising from the dipole term in the electron distribution (8) for the same as in panel (a). At small the curves approach the asymptotic value in Thomson limit . (c) Relative correction to the cross-section arising from the quadrupole term .

When calculating the azimuthal integral in equation (21) we just have to integrate with given by Equation (22). The properties of the Legendre polynomials give us the average

(24)

so that the averaged distribution function becomes

(25)

The cross-section now can be written as

(26)

where

(27)

Changing the integration variable and expressing through using Equation (20), we get

(28)

where

(29)

and

(30)

The zeroth function coincides with the function from NP94. When electron distribution is isotropic (i.e. ), expression (26) for the total cross-section is reduced to equation (3.4.1) from NP94 and the dependence on obviously disappears. Functions of two arguments can be presented through the functions of one argument:

(31)

where

(32)

The explicit expressions for the functions () can be found in Appendix B (see also NP94). Thus the total cross-section is given by a single integral over the electron energy (26) with all functions under the integral given by analytical expressions. Numerical calculations of functions can be separated into three regimes: (1) in Thomson regime, , the series expansion (see Appendix D) can be used; (2) for , but not sufficiently small for regime (1), we numerically take the integral in Equation (27) using 5-point Gaussian quadrature to reach accuracy better than 1%; (3) in other cases, we use the sum in Equation (28) and analytical expressions (31) for .

For mono-energetic electron distribution of Lorentz factor given by Equation (10), we can introduce the cross-section analogously to Equation (26):

(33)

For isotropic mono-energetic electrons, the total cross-section is shown in Figure 2a. The relative corrections arising due to the dipole and quadrupole term in the electron distribution are shown in Figures 2b and 2c, respectively. These have to be multiplied by the angle- and, possibly, the energy-dependent factor to obtain the final correction. In the Thomson limit, at small , the cross-section takes the form (see Appendix D)

(34)

where the correction to unity term can be easily obtained by averaging the transport cross-section over electron directions (i.e. integrating over the angles). This corresponds to the flattening in Figure 2b at . The correction from the quadrupole term in this regime as well as for non-relativistic electrons becomes negligible:

(35)

3.2. Mean powers of scattered photon energy

In some situations, the full relativistic kinetic equations can be substituted by the approximate one obtained in Fokker-Planck approximation. This requires knowledge of various moments of the redistribution function, such as total cross-section, the mean energy and dispersion of the scattered photons (see NP94, Vurm & Poutanen 2009). It is often time-consuming to compute numerically the integrals of the redistribution function and instead direct calculations of the moments are preferable. Below we obtain analytical expressions for the mean energy and dispersion of the energy of scattered photons in frame as a function of the initial photon energy and the direction of its propagation relative to a symmetry axis of the electron distribution .

Following NP94, we define the mean of powers of energy of scattered photons:

(36)

where now

(37)

Averaging over photon directions

Quantities (37) are not invariants (except for ), and we have to compute the scattered photon energy is a certain frame, which we choose to be frame . Because of the additional term under the integral, a simple change of variables to the electron rest frame as in Equation (17) is not possible. Instead, we use the -function to take the integral over :

(38)

Now we change the variables to those in the electron rest frame (with subscript 0). We choose the coordinate system with the polar axis along the direction of the incoming photon . In this frame, the cosine of the angle between the electron momentum and the incoming photon is . The cosine of the angle between the outgoing photon momentum and the electron is then .

We use invariants and the energy conservation law in the electron rest frame , to get (see NP94)

(39)

Finally, we have

(40)

Because is the energy of scattered photon in the electron rest frame, the Doppler effect gives us

(41)

where we now can substitute

(42)

which are consequences of the conservation law and of the Lorentz transformation , respectively. The terms containing a linear combination of square roots and will disappear after averaging over .

We now introduce moments of the invariant cross-section

(43)

For , we get of course the total cross-section given by Equation (18). NP94 derived the corresponding expressions for :

(44)
(45)

where , and .

For the mean energy of the scattered photon we have then

(46)

and for the mean square of energy

(47)

where

(48)

All functions are elementary. In addition, they are defined in such a way so that not to become zero at . The series expansion of functions and for small arguments are presented in Appendix A.

Figure 3.— (a) Mean energy of scattered photons in units of the incident photon energy as a function of for isotropic mono-energetic electrons of various momenta (from bottom to top, dot-dashed, solid, dotted, dashed, and triple-dot-dashed curves). The asymptotic value at small in Thomson approximation is . (b) A correction to the mean energy (in units of ) arising from the linear term in the electron distribution (8) with . Solid, dotted and dashed curves correspond for , respectively. The curves from top to bottom correspond to . At small , the curves approach the limiting value given by Equation (58). (c) Same as (b), but for the quadrupole term in the electron distribution (8) with . These are even functions of , the curves from the bottom to the top correspond to . At small , the curves approach the limiting value given by Equation (59).

Averaging over electron directions

We need to integrate in Equation (36) over anisotropic electron distribution. We follow the derivation of the total cross-section that lead from Equation (21) to Equation (26). Representing integral over electron momentum we get:

(49)

where

(50)

and

(51)

Functions coincide with functions introduced by NP94, while functions are given by Equation (31). The explicit expressions for the function for (which are analogous to functions and from NP94) can be obtained using expression for mean powers of energies (46) or (47):

(52)
(53)

where

(54)

These are related to functions

(55)

because functions are expressed through . The explicit expressions for both type of these functions as well as their series expansions for small arguments are given in Appendices B.

As in the case of functions , for calculating , we consider three regimes: (1) , when we use the series expansion (see Appendix D); (2) for we numerically take the integral in Equation (50) using Gaussian quadrature; (3) in other cases, we use the sum in Equation (50) and analytical expressions for .

For mono-energetic electron distribution (10) of Lorentz factor , we can introduce the mean powers of photon energy analogously to Equation (49):

(56)

The mean energy of scattered photons for such electrons for a scattering act is given by the ratio of Eqs. (56) and (33). It is shown in Figure 3a. In the low-energy (Thomson) limit the energy gain factor is given by a well known expression , which translated to at large . The relative corrections arising due to the dipole and quadrupole term in the electron distribution are shown in Figures 3b and 3c, respectively. Using asymptotic expansions of in the Thomson limit (see Appendix D), we get the asymptotic value

(57)

Thus in non-relativistic limit , the correction is negligible. In the relativistic limit , the relative corrections arising from the two terms are

(58)
(59)

Figure 4.— (a) Dispersion of the energy of scattered photons (in units of ) for isotropic mono-energetic electrons of (from bottom to top). The asymptotic value of at small in Thomson approximation is . (b) A correction to the dispersion (in terms of the isotropic quantity) arising from the linear term in the electron distribution (8) with . Solid, dotted and dashed curves correspond for , respectively. The curves from top to bottom correspond to . (c) A correction to the dispersion arising from the quadrupole term in the electron distribution (8) with for the same and as in panel (b). These are even functions of , the curves from bottom to top correspond to .

3.3. Energy exchange and dispersion

The difference of the photon energies before and after scattering is of course just the energy transfer to the electron gas. For the fixed angle between electrons and incident photons (and fixed electron energy ), the energy loss averaged over the directions of scattered photons is . The product is then the energy loss on a unit length. From Equation (46) we can easily get (see also NP94):

(60)

The corresponding energy loss (per unit length and in units ) averaged over the electron directions (and integrated over electron energies) becomes [see Eqs. (26) and (49)]:

(61)

The heating rate per unit volume is then

(62)

where is the specific intensity of radiation in a given direction. This expression can be positive (so called Compton heating) when the photons typically have larger energies than the electron gas, or negative (Compton cooling) when one considers cooling of the relativistic electron gas by soft radiation.

The dispersion of the scattered photon energy is given by the usual expression , which of course depends on the electron momentum distribution. For mono-energetic electrons we can define the dispersion as

(63)

where are given by Equation (56). The dispersion for isotropic electrons is shown in Figure 3a. The low-energy (Thomson) limit for is (see NP94)

(64)

The relative corrections arising due to the dipole and quadrupole term in the electron distribution reach about 50 per cent and are shown in Figures 4b and 4c, respectively.

3.4. Radiation force

Now we would like to derive analytic expression for the radiation force acting on the electron gas. NP94 have developed a formalism appropriate for isotropic electron distribution, when the averaged transferred momentum is along the momentum of the incoming photons, because of the symmetry. For the electron distribution described by Equation (8), the momentum is transferred in the plane containing the initial photon momentum and the symmetry axis . If the incident photons are axially symmetric around , then obviously, the total momentum transferred to the electrons has be parallel to by symmetry. We derive here more general formulae for the total momentum transferred by a beam of photons propagating along direction (such as ), as well as its projections to and perpendicular direction.

Let us introduce the vector basis: