A&A 417, 381-389 (2004)
DOI: 10.1051/0004-6361:20031765
D.-B. Liu1 - L. Chen1 - J. J. Ling1 - J. H. You1 - X. M. Hua2
1 - Institute for Space and Astrophysics, Department of Physics, Shanghai Jiao Tong University,
Shanghai 200030, PR China
2 -
Laboratory for High Energy Astrophysics, NASA/GSFC Code 661, Greenbelt, Maryland 20771, USA
Received 13 May 2003 / Accepted 18 November 2003
Abstract
The original Kompaneets equation fails to describe down-Comptonization, which is the most
important radiative transfer process in hard X-ray and -ray astronomy, compared
with up-Comptonization. In this paper, we improve our previous derivation of the extended
Kompaneets equation and present it more clearly. The new equation can be used to describe
a more general Comptonization process, including up- and down-Comptonization, suitable for
any case,
,
and
.
The condition of the original Kompaneets equation
is no longer necessary. Using the extended equation,
we give some typical solutions in X-ray astronomy, and compare them with those of the
prevailing Monte Carlo simulations and the Ross-McCray equation. The excellent consistency
between the extended Kompaneets equation and Monte Carlo simulation or Ross-McCray equation
confirms the correctness of our extended Kompaneets equation. The numerical solution of
the extended Kompaneets equation is less expensive in terms of computational time than
the Monte Carlo simulation. Another advantage of the equation method is the simplicity
and the clarity in physics. The potential applications in X-ray and
-ray astronomy are also emphasized.
Key words: scattering - radiative transfer - methods: analytical - X-rays, -rays: ISM
There are two kinds of Comptonization processes in astrophysics. If the average energy of photons
in radiation field
is markedly larger than the average thermal energy of electrons
,
,
which often occurs in hard X-ray and
-ray astronomy, then the average energy of scattered photons decreases. This is known as down-Comptonization or
Comptonization-softening. When
,
which often occurs in radio and
infra-red astronomy, the average energy of scattered photons increases, which we call
up-Comptonization, or Comptonization-hardening. For the hardening process, if the condition
is satisfied, the equation of radiative transfer, which is
in the form of a diffusion equation, has been given by Kompaneets as follows,
However, we emphasize that the Kompaneets Eq. (2) is restricted to a limited application
area, only useful for the calculation of the up-Comptonization process owing to the severe condition
.
For down-Comptonization, which could be more important
in the hard X-ray or
-ray astronomy where the condition
is often
satisfied, the adoption of Eq. (2) will inevitably bring a large error in the resultant
calculation. Furthermore, even when the condition
is satisfied in the initial stage of scattering process, the error in the calculation of the up-Comptonization process may also be significant if the scattering depth is large,
.
In this case, the average energy
of the scattered photons could be markedly increased to the level
,
which destroys the necessary condition of Eq. (2), i.e.
.
Therefore, the original Kompaneets Eq. (2) should be extended
and improved in order to describe a more general non-elastic scattering of
photon-electron, particularly for the down-Comptonization process. Ross &
McCray obtained another diffusion equation for the case
to describe the down-Comptonization process.
Following Kompaneets (1956, 1957), we consider a "mixed gas'' consisting of
an electron-gas (the fully ionized plasma) and a photon-gas (the radiation
field) under a condition
and
(Chen et al. 1994). We make two assumptions: (i) the common thermal equilibrium
between the two gases has not yet been reached, but the electron-gas itself
is already in thermal equilibrium as the interaction between electrons is
the Coulomb long-range force. Therefore the Maxwellian distribution
can be used to describe
the electron gas; (ii) owing to the frequent Compton scattering, the frequency
distribution function of the photon-gas
is assumed to be isotropic,
independent of the direction of the photon-wavevector
,
and
will change with time t until the photon-gas reaches a thermal equilibrium with
the electron-gas. In the following, the change of distribution function
is treated as a diffusion of photon-gas in the "frequency space'' along the frequency
axis
.
The "diffusion rate'' of the "photon number''
in the "frequency space'' is determined by the diffusion
Eq. (5) given as follows.
Consider an individual collision between an electron with momentum and a photon with frequency
.
The energy and momentum conservations
in the non-relativistic limit are
In real astrophysical circumstances, the plasma is always tenuous enough. Therefore,
as a good approximation, the electron-gas can be regarded as a classical system
rather than a Fermi-gas, and the number density of electrons
in the interval
-
can be discribed by the classical formula, i.e.
is the Boltzman
distribution function. Denoting the transition probability of the elementary
collision
by
,
the total transition number in unit
volume is
,
where
,
are
the photon numbers before and after collision, respectively. The occurrence of
the factor
is due to the fact that the photon-gas is a boson
system for which the transition number depends on both n and
.
Similarly, the inverse-collision process
leads to an increase
of
.
The corresponding transition number is
,
where the
transition probability
is the same as that in the collision
process
,
because the Klein-Nishina cross-section
has the same value both for the scattering angle
and
(see Eq. (11) below). Therefore
can be written as:
Using the diffusion Eq. (21), we studied the effect of Comptonization
on the X-ray spectrum emergent from the scattering medium. The effect of
down-Comptonization, for which Eq. (2) is difficult to use, is specially emphasized.
Given the initial condition, i.e., the initial spectrum, and the diffusion time scale,
we can solve Eq. (21) numerically. For simplicity, we assume that the X-ray
source is stable, with a time-independent spectrum f(x). So the initial condition
of Eq. (21) at t=0 can be expressed as
In this section, we give some numerical solutions of the diffusion Eq. (21)
under given conditions (22) and (23) to show the time-evolution behavior
of a spectrum in the Comptonization, particularly in down-Comptonization process. We focus
our attention on the high-frequency end of the primary incident hard X-ray or -ray
spectrum, where
is satisfied, for which the original
Kompaneets Eq. (2) fails to give a correct results. We will compare our
solutions with those of a Monte Carlo simulation to confirm the correctness and the
effectiveness of our Eq. (21). Under the same conditions, we also calculated
the evolution of the X-ray spectrum by use of the RMC Eq. (3) to
compare with the results of our Eq. (21). For simplicity and to be specific,
we assume that the scattering medium is homogeneous in space with constant electron density
and temperature
.
We fix
and
respectively, which are the typical
values for the plasma in the accretion disk around a compact star, e.g., a neutron star,
a singular star or a black hole with stellar mass (Treves et al. 1989; Sunyaev et al. 1973).
Different from Chen et al. (1994) and Deng et al. (1998), in this paper Eq. (21)
is solved numerically by use of the three-point finite difference method. For generality,
we abandon the assumption of a "tenuous radiation field'',
,
adopted by Chen (1994)
and Deng (1998), and retain the n2 term in Eq. (21). This means that our results
can be used to discuss very luminous high-energy objects with a strong radiation field.
For comparison, we present the Monte Carlo results using Hua's program of Monte Carlo
simulation of Comptonization (Hua 1997).
The expression of the spectral intensity for an emission line with normal Gaussian profile
is given as:
![]() |
Figure 1:
The down-Comptonization evolution behavior of a Gaussian line; a), b) and c)
represent the calculated results by the extended Kompaneets equation, by the
Monte Carlo simulation and by the RMC equation respectively. The solid
line is the original spectrum, Dashed line, dash-dot-line and dotted line
represent the profile of the emergent spectrum with diffusion time scale
![]() ![]() ![]() ![]() |
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The Planckian formula of a black body spectrum gives
![]() |
Figure 2: The down-Comptonization evolution of a blackbody spectrum; a), b) and c) represent the calculated results using the extended Kompaneets equation, the Monte Carlo simulation, and the RMC equation respectively. |
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The power-law spectrum which often occurs in X-ray astronomy has the form:
![]() |
Figure 3: The down-Comptonization evolution of a power-law spectrum; a), b) and c) represent the calculated results using the extended Kompaneets equation, the Monte Carlo simulation, and the RMC equation respectively. |
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In this section, we show the evolution of the thermal bremsstrahlung radiation
originating from an optically thin plasma with temperature
.
When the radiation passes through a "cold'' plasma with a
lower temperature
,
down-Comptonization
occurs because of
,
particularly for the high
frequency portion of the incident bremsstrahlung spectrum where the condition
is satisfied.
The bremsstrahlung spectrum is given as (Rybicki & Lightman 1979; You 1998)
where
is the average
Gaunt factor at temperature
.
In the following calculation,
we adopt the same parameter group as above,
,
and
,
the frequency range in our calculation
is
.
The initial condition is
now expressed as:
The evolution spectra calculated by Eq. (21), by the Monte Carlo method, and by Eq. (3) are shown in Figs. 4a, b, c, respectively. From Fig. 4 we see that the evolution characteristics of the bremsstrahlung spectrum are quite similar to those of the power-law spectrum, shown in Fig. 3.
![]() |
Figure 4: The down-Comptonization evolution of a thermal bremsstrahlung spectrum; a), b) and c) represent the calculated results using the extended Kompaneets equation, the Monte Carlo simulation, and the RMC equation respectively. |
Open with DEXTER |
The original Kompaneets equation fails to describe down-Comptonization,
which is the most important radiative transfer process in hard X-ray and -ray
astronomy. Therefore in this paper, we present an extended Kompaneets Eq. (21), which can be used to describe a more general Comptonization process,
including down- and up-Comptonization, suitable for any case,
,
and
.
In other words, the condition
for the original Kompaneets Eq. (2) is no longer necessary.
Using Eq. (21), we give some typical solutions in X-ray astronomy,
and compare them with Monte Carlo simulation results. The excellent consistency of
the Compton-evolution spectra obtained by the use of the two different methods
confirmed the correctness of our extended Kompaneets equation. Though the prevailing
approach to a quantitative analysis of the Comptonization process is still the Monte
Carlo simulation, it has a remarkable defect in practice, namely the great quantity
of computations. In addition to the simplicity and the clarity in physics, a more
important advantage of solving Eq. (21) is that it is less expensive
in terms of computational time. The solutions of the extended Kompaneets equation
are also consistent with those of the RMC equation particularly for
down-Comptonization. Therefore Eq. (21) can be regarded as an important
improvement of the original Kompaneets Eq. (2), particularly for
hard X-ray and
-ray astronomy.
Our calculations show that the change of the emergent spectrum in Comptonization,
particularly in down-Comptonization, depends on both the difference between
and
,
and the scattering depth
(or equivalently, the diffusion time scale T). The larger the difference
,
and/or the larger the depth
(equivalently, the time scale T), the larger the change of the emergent
spectrum will be.
The structure of Eq. (21) has two marked characteristics which can
be regarded as a criterion of the correctness of our Eq. (21).
(i) It has the form of
,
which ensures the
conservation of number of photons (see the conservation Eq. (14)).
The invariance of the total number of photons is a basic requirement for the
electron-photon scattering process. (ii) Equation (21) contains a factor
which ensures
when the photon gas reaches the thermal equilibrium
distribution
.
This is also a necessary requirement for the
correctness of this equation because the thermal equilibrium will inevitably
be reached through the scattering and
,
which implies that the diffusion finally stops. Ross et al. (1978) also
noticed the restriction of the original Kompaneets Eq. (2) and
suggested an alternative equation to replace Eq. (2) to suit
the case
.
Their equation has a form
,
which is quite similar to
Eq. (21) but deviates from the necessary form
.
According to their
equation, the diffution will never stop, i.e.,
even when the thermal equilibrium,
,
has been reached.
Acknowledgements
We would like to thank the anonymous referee for his helpful suggestions. We are also grateful to Prof. McCray for his helpful discussions and suggestions when Dr. You visited JILA, Boulder in 2001. The work of D.B.L. is supported by the National Natural Science Foundation of China (Grant No. 10373010), and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20030248012).