Analytic fitting formulae for relativistic electronelectron thermal bremsstrahlung
S. Nozawa^{1}  K. Takahashi^{2}  Y. Kohyama^{2}  N. Itoh^{2}
1  Josai Junior College, 11 Keyakidai, Sakadoshi, Saitama, 3500295,
Japan
2 
Department of Physics, Sophia University, 71 Kioicho, Chiyodaku,
Tokyo, 1028554, Japan
Received 1 November 2008 / Accepted 18 February 2009
Abstract
We present accurate analytic fitting formulae for electronelectron thermal bremsstrahlung emissivity. The fitting formulae have an overall accuracy of higher than 0.5% for a wide electron temperature range
MeV, which will be useful when analyzing the precision observational data from the nonrelativistic regime to mildlyrelativistic regime. Assuming the thermal plasma, the present formulae will be also applicable to the relativistic regime.
Key words: cosmology: theory  galaxies: clusters: general  radiation mechanisms: thermal  relativity  Xrays: galaxies: clusters
1 Introduction
Following the launch of the Chandra XRay Observatory, the XMMNewton Observatory, and the Suzaku Xray Observatory, Xray astronomy has entered its own era of precision science. In particular, these satellites have revolutionized the accuracy of the observational data available for galaxy clusters (for example, Allen et al. 2001; Schmidt et al. 2001; Sato et al. 2007). The precision observational data taken by these satellites require precision basic physics data for their analysis.
The present authors have published a series of papers in which they have completed accurate calculations for the electronion thermal bremsstrahlung process. These are particularly suited to the analysis of radiation process in hightemperature galaxy clusters (Nozawa et al. 1998; Itoh et al. 2000, 2002b). In Itoh et al. (2000, 2002b), the present authors presented accurate analytic fitting formulae for the electronion thermal bremsstrahlung Gaunt factors, which have a general accuracy of about 0.1% for the ranges , , where Z_{j} is the charge of the ion and T is the electron temperature. These electronion thermal bremsstrahlung Gaunt factors are also suitable for the analysis of the precision Xray data of galaxy clusters taken by the Chandra, the XMMNewton and the Suzaku Observatories.
It has been known for some time that the electronelectron thermal bremsstrahlung makes a percentorder contribution to the thermal bremsstrahlung at low temperature K (Maxon & Corman 1967; Maxon 1972; Haug 1975a,b; Svensson 1982; Dermer 1986). In particular, Haug (1975a) and Haug (1975b) carried out a relativistic calculation of the electronelectron thermal bremsstrahlung emissivity and found that the nonrelativistic electronelectron thermal bremsstrahlung emissivity calculated Maxon & Corman (1967) and Maxon (1972) agree with the relativistic results to within about 1% accuracy at K. Itoh et al. (2002a) calculated the precision analytic fitting formula for the nonrelativistic electronelectron thermal bremsstrahlung Gaunt factor, which has an overall accuracy of higher than 1% for the temperature range .
On the other hand, it is known that the electronelectron bremsstrahlung dominates the electronion bremsstrahlung for high energy electrons. In active galactic nuclei, gammaray bursters and compact binary sources, the electron energies may be greater than 100 keV. Stepney & Guilbert (1983) studied several important processes contributing to the hightemperature thermal plasmas. Among other processes, they presented analytic fitting formulae for the electronelectron thermal bremsstrahlung emissivity in the temperature range MeV. Their method was based upon the relativistic numerical calculation by Haug (1975a,b). Their fitting formulae have an overall accuracy that is higher than 5% compared with the numerical calculation by Haug (1975b).
In this paper, we present accurate analytic fitting formulae for the electronelectron thermal bremsstrahlung emissivity. The fitting formulae have an overall accuracy of higher than 0.5% for a wider electron temperature range MeV. The fitting formulae is suitable for the analysis of the precision observational data for electron energies from nonrelativistic regime to mildlyrelativistic regime. Assuming the thermal plasma, the present formulae will be also applicable to the relativistic regime. However, it should be remarked that the usefulness of the present formula is limited to relativistic plasma since we do not include the positron contribution. The photon production rate is dominated by annihilations for the electronpositron pair equilibrium plasmas. Therefore, more careful study including the pair annihilation process will be necessary for the analysis of relativistic plasmas. The present paper is organized as follows. In Sect. 2, we will calculate the relativistic electronelectron thermal bremsstrahlung emissivity. In Sect. 3, we will present the precision analytic fitting formulae for the emissivity. Concluding remarks will be given in Sect. 4.
2 Electronelectron thermal bremsstrahlung
The expression for the thermally averaged relativistic electronelectron bremsstrahlung crosssection was derived by Haug (1975a, b). In this paper, we follow the notation used in Haug (1975a) unless otherwise noted. The number of photons emitted per unit time, per unit volume, and per unit energy interval by electron gas of uniform number density
at temperature T is given by Eq. (2.1) in Haug (1975b) and also by Eq. (A.1) in Stepney & Guilbert (1983) as follows:
=  
=  
(1)  
=  
(2) 
where is the photon energy in units of the electron rest mass, , is the modified Bessel function of the second kind, is the fine structure constant, and is the Thomson cross section, respectively. In Eq. (2) , and , where p_{1} ( ) and p_{2} ( ) are four momenta in units of the electron rest mass for the initial (final) electrons. We call the photon production rate for brevity. The explicit expression for 1/ is given by appendix in Haug (1975a). In Eq. (2), is the Elwert factor, which takes into account the Coulomb corrections between two electrons (Elwert 1939). The explicit form is
=  (3)  
=  (4)  
=  (5) 
It is known that the Coulomb corrections are important at low energies, whereas they are negligible for high energies.
Following Stepney & Guilbert (1983), where the zaxis is chosen to be parallel to the sum of the initial electron momenta, we rewrite Eq. (1) as follows:
(6) 
where and is defined by
=  
(7)  
=  (8) 
where is the zenith angle of photon. The boundary values of T_{1}, , and are given in Appendix of Stepney & Guilbert (1983). Here it should be also noted that our definition of in Eq. (6) differs from that of Eq. (4) in Stepney & Guilbert (1983) by a factor . Namely,
(9) 
where SG denotes Stepney & Guilbert.
Similarly, the total energy emitted per unit time, per unit volume (we refer to this as the emissivity for short) is given by
=  (10)  
=  (11)  
=  (12) 
Equations (6)(12) are our basic equations in the present paper. We performed the fivedimensional integration of Eq. (7) in the following range of 1 keV MeV for the electron temperature, and for the photon energy in units of the electron temperature.
It is known that the nonrelativistic approximation is reliable at low temperatures, whereas the extremerelativistic approximation is accurate at high temperatures. Here we study the extent of the approximations for the photon production rate and the emissivity. To ensure that the present paper is selfcontained, we recall the expressions in these approximations. The photon production rate and the emissivity in the nonrelativistic (NR) approximation (Maxon & Corman 1967; Maxon 1972; Itoh et al. 2002a) are given by
=  (13)  
=  (14)  
=  (15)  
A(x, s)  =  
(16)  
=  (17)  
=  (18) 
In Eq. (14) the Coulomb corrections are taken into account by the Elwert factor .
Similarly, we recall the expressions of the photon production rate and the emissivity in the extremerelativistic (ER) approximation (Alexanian 1968) as follows:
=  (19)  
=  
(20)  
=  (21)  
=  (22)  
=  (23) 
where is the exponential integral function, and is Euler's constant.
Figure 1: Plotting of as a function of x for keV, 10 keV, 1 MeV, and 7 MeV. The solid curve is the full calculation of the present work. The dotted curve is the calculation in the nonrelativistic approximation . The dashed curve is the calculation in the extremerelativistic approximation . 

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To ascertain the extent of the approximations for the photon production rate , we plot as a function of x at keV, 10 keV, 1 MeV, and 7 MeV in Figs. 1ad, respectively. The solid curve is the full calculation of the present work. The dotted curve represents the calculation for the nonrelativistic approximation . The dashed curve is the calculation in the extremerelativistic approximation . It can be seen from Fig. 1 that the maximum errors for the photon production rate at keV, 10 keV, 1 MeV, and 7 MeV are 0.9%, 9%, 6%, and 0.8%, respectively. We find that the nonrelativistic approximation is accurate to within errors of 1% for keV, whereas the extremerelativistic approximation is accurate to within errors of 1% for MeV.
Figure 2: Plotting of the emissivity and the relative errors of the approximations. The solid curve is the full calculation of the present work. The dotted curve is the calculation in the nonrelativistic approximation. The dashed curve is the calculation in the extremerelativistic approximation. The electronproton bremsstrahlung emissivity is also plotted in the dashdotted curve. In Fig. 2b, the relative errors of these approximations are plotted compared with the full calculation. 

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Similarly, we show the extent of the approximations for the emissivity . In Fig. 2a, we plot . The solid curve is the full calculation of the present work. The dotted curve is the calculation in the nonrelativistic approximation. The dashed curve is the calculation in the extremerelativistic approximation. The curves of the approximations are continuous with the curve of the full calculation at both around keV and a few MeV. To compare the behavior of the electronelectron bremsstrahlung with the electronproton bremsstrahlung, we also plot the electronproton bremsstrahlung emissivity calculated by Stickforth (1961) in the dashdotted curve in Fig. 2a. One finds that and cross at keV. This value should be compared with the reported value of keV by Haug (1975b). For keV, the electronelectron bremsstrahlung dominates the emissivity.
In Fig. 2b, we show the relative errors of the approximations compared with the full calculation. The dotted curve is the calculation for the nonrelativistic approximation. The dashed curve is the calculation in the extremerelativistic approximation. It can be seen from Fig. 2b that the errors in the approximations for the emissivity are less than 1% for both keV and MeV. As seen from Figs. 1 and 2, precision numerical data are now available for the photon production rate and the emissivity in the entire electrontemperature regime.
Figure 3: Plotting of as a function x. The solid curve, dashed curve, dashdotted curve, and dotted curve are for keV, 10 keV, 100 keV, and 300 keV, respectively. 

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We now investigate the effect of the Coulomb corrections to the photon production rate
.
It has been known that the Coulomb corrections are important for low electron temperatures and negligible for high electron temperatures. To determine the effect of the Coulomb corrections explicitly, we define the ratio of
with and without Coulomb corrections as follows:
(24) 
where CC denotes the Coulomb corrections. In Eq. (24), corresponds to the photon production rate without Coulomb corrections, which is obtained by inserting in Eq. (7). In Fig. 3, we show as a function of x at keV, 10 keV, 100 keV, and 300 keV. The maximum corrections at keV and 10 keV are 17% and 5%, respectively. The Coulomb corrections are important at low electron temperatures. On the other hand, the maximum corrections at keV and 300 keV are 0.7% and 0.2%, respectively. Therefore, the Coulomb corrections can be safely neglected at keV for the present purposes.
For the study in this section, we classify the electron temperature region for the photon production rate
and the emissivity
into four regions:
(25) 
In region I, the nonrelativistic approximation with Coulomb corrections is sufficiently accurate within 1% errors. The full relativistic calculation with Coulomb corrections is necessary in region II. In region III, the full relativistic calculation is necessary, however, the Coulomb corrections are negligible. In region IV, the extremerelativistic approximation is sufficiently accurate within 1% errors. In the next section, we show accurate analytic fitting formulae for the photon production rate and the emissivity in each electron temperature region defined here.
3 Analytic fitting formulae
Equations (6) and (10) provide useful tool for the data analysis of the radiation processes in active galactic nuclei, gammaray bursters, and compact binary sources, where electron energies may exceed 100 keV. However, the calculation of the fivedimensional integral in Eq. (7) is extremely timeconsuming and therefore not practical for the analysis of the observational data. In this section, we present analytic fitting formulae for the electronelectron thermal bremsstrahlung emissivity. Stepney & Guilbert (1983) presented analytic fitting formulae for the temperature range 50 keV MeV, which have an overall accuracy that is higher than 5% compared with the numerical integration of Eq. (7). Therefore, one task of this paper is to obtain analytic fitting formulae both of higher accuracy and a wider range of electron temperature to be useful in analyzing the precision observational data at present and also in the future. As described in the previous section, we determine the analytic fitting formulae for the photon production rate and the emissivity in region I region IV separately.
3.1 Region I ( keV)
The analytic fitting formula was obtained by Itoh et al. (2002a, which we refer to IKN hereafter) for the electronelectron bremsstrahlung Gaunt factor in the nonrelativistic approximation. The fitting range in the IKN paper is 50 eV
keV for the electron temperature and
for the photon energy in units of the electron temperature. The analytic fitting formula for the photon production rate is given by
=  (26)  
=  (27)  
=  (28)  
X  =  (29)  
=  (30)  
=  (31) 
where and . The numerical values of the coefficients and are given in the IKN paper, but we include the values in Tables 1 and 2 to make the present paper more selfcontained. The accuracy of the fitting formulae is superior than 0.1%.
Table 1: Coefficients (taken from Itoh et al. 2002a).
3.2 Region II (1 keV keV)
In this temperature region, the full relativistic calculation with Coulomb corrections is necessary for the calculation of
.
We define the fitting formulae for the photon production rate as follows:
(32) 
where is the fitting function for the planewave (PW) contribution . The fitting formulae are given as follows:
=  (33)  
=  (34)  
=  (35) 
where is the exponential integral function defined by Eq. (21). The numerical values of the coefficients and are shown in Table 3. In Eq. (32), is the fitting function for the Coulomb corrections defined by Eq. (24). It is given by
=  (36)  
=  (37) 
The numerical values of the coefficients are shown in Table 4. The fitting region for and is given by 1 keV keV and . In Figs. 4a and b, we show the relative errors of and compared with the numerical values, respectively. The solid curve, dashed curve, dashdotted curve, and dotted curve represent keV, 10 keV, 100 keV, and 300 keV, respectively. One can see from Fig. 4 that the maximum errors are 0.5% and 0.2% for and , respectively. The errors in the photon production rate are found to be less than 0.5%.
Table 2: Coefficients (taken from Itoh et al. 2002a).
Table 3: Coefficients and .
Table 4: Coefficients .
Figure 4: Plotting of errors for the analytic fitting functions and . The solid curve, dashed curve, dashdotted curve, and dotted curve are for = 1 keV, 10 keV, 100 keV, and 300 keV, respectively. 

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In a similar way, the fitting function for the emissivity
is given as follows:
=  (38)  
=  
(39) 
where is the function. The fitting region is 1 keV keV. We find that the errors are less than 0.2%.
Table 5: Coefficients and .
3.3 Region III (300 keV MeV)
In this temperature region, the full relativistic calculation is necessary for the calculation of
,
although the Coulomb corrections are negligible. We define the fitting formula for the photon production rate as follows:
=  (40)  
=  
(41)  
=  (42)  
=  (43) 
where is the exponential integral function defined by Eq. (21). The numerical values of the coefficients and are shown in Table 5. The fitting region for is 300 keV MeV and . In Fig. 5, we show the relative errors of compared with the numerical values as a function x. The solid curve, dashed curve, and dotted curve are for keV, 1 MeV, and 7 MeV, respectively. One can see from Fig. 5 that the maximum error in the analytic fitting function is 0.2%. In Fig. 5, we also plot the relative error for at MeV compared with the full caluclation in dashdotted curve. The maximum error in the extremerelativistic approximation is 0.8% as described in the previous section.
Similarly, the fitting formula for the the emissivity
is given as follows:
=  (44)  
=  
(45) 
The fitting region is 300 keV MeV. We find that the error is less than 0.1%.
3.4 Region IV ( MeV)
In this temperature region, the extremerelativistic approximation (Alexanian 1968) is sufficiently accurate to within the 1% errors. The explicit forms of the photon production rate and the emissivity have already been shown in the previous section. They are as follows:
=  (46)  
=  (47) 
where and are Eqs. (20) and (23), respectively.
Figure 5: Plotting of the errors for the analytic fitting function and the extremerelativistic approximation . The solid curve, dashed curve, and dotted curve are for at keV, 1 MeV, and 7 MeV, respectively. The dashdotted curve is for at MeV. 

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4 Concluding remarks
We have studied the relativistic electronelectron thermal bremsstrahlung based upon the approaches given by Haug (1975a,b), and Stepney & Guilbert (1983). The photon production rate
and the emissivity
were calculated for the electron temperature 1 keV
MeV and for 10
,
where x is the photon energy in units of the electron temperature. We have calculated the analytic fitting formulae for the numerical data of both the photon production rate and the emissivity. The obtained analytic fitting formulae have reproduced the numerical integration of Eq. (7) to within 0.5% errors for 1 keV
MeV. We have also found that the nonrelativistic approximation can be used within its 1% errors for
keV, whereas the extremerelativistic approximation can be used within its 1% errors for
MeV. Combining the existing analytic expressions for the nonrelativistic approximations and the extremerelativisticapproximations, we have obtained analytic fitting formulae for
and
in the entire electron temperature regime. We summarize the expressions of the analytic fitting formulae for the photon production rate and the emissivity as follows:
(48) 
(49) 
=  
(50) 
(51) 
In Eqs. (49) and (51) the regions IIV are defined by Eq. (25). The present analytic fitting formulae will be useful for the analysis of the precision observational data acquired by the Chandra XRay Observatory, the XMMNewton Xray Observatory, the Suzaku Xray Observatory, and the next generation Xray and gammaray observatories for the electron energies from nonrelativistic regime to mildlyrelativistic regime. Assuming the thermal plasma, the present formulae will be also applicable to relativistic regime. Finally, the subroutines of all the fitting formulae will de downloadable from our website^{}.
Acknowledgements
We wish to thank our referee for many useful suggestions. We wish to thank Prof. Y. Oyanagi for allowing us to use the least squares fitting program SALS.
References
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 Allen, S. W., Ettori, S., & Fabian, A. C. 2001, MNRAS, 324, 877 [NASA ADS] [CrossRef] (In the text)
 Dermer, C. D. 1986, ApJ, 307, 47 [NASA ADS] [CrossRef] (In the text)
 Elwert, G. 1939, Ann. Phys., 34, 178 [CrossRef] (In the text)
 Haug, E. 1975a, Z. Naturforsch., 30a, 1099 [NASA ADS] (In the text)
 Haug, E. 1975b, Z. Naturforsch., 30a, 1546 [NASA ADS]
 Itoh, N., Sakamoto, T., Kusano, S., Nozawa, S., & Kohyama, Y. 2000, ApJS, 128, 125 [NASA ADS] [CrossRef] (In the text)
 Itoh, N., Kawana, Y., & Nozawa, S. 2002a, Il Nuovo Cimento, 117B, 359 [NASA ADS] (In the text)
 Itoh, N., Sakamoto, T., Kusano, S., Kawana, Y., & Nozawa, S. 2002b, A&A, 382, 722 [NASA ADS] [CrossRef] [EDP Sciences]
 Maxon, M. S. 1972, Phys. Rev. A, 5, 1630 [NASA ADS] [CrossRef] (In the text)
 Maxon, M. S., & Corman, E. G. 1967, Phys. Rev., 163, 156 [NASA ADS] [CrossRef] (In the text)
 Nozawa, S., Itoh, N., & Kohyama, Y. 1998, ApJ, 507, 530 [NASA ADS] [CrossRef] (In the text)
 Sato, K., Yamasaki, N. Y., Ishida, M., et al. 2007, PASJ, 59, 299 [NASA ADS] (In the text)
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Footnotes
All Tables
Table 1: Coefficients (taken from Itoh et al. 2002a).
Table 2: Coefficients (taken from Itoh et al. 2002a).
Table 3: Coefficients and .
Table 4: Coefficients .
Table 5: Coefficients and .
All Figures
Figure 1: Plotting of as a function of x for keV, 10 keV, 1 MeV, and 7 MeV. The solid curve is the full calculation of the present work. The dotted curve is the calculation in the nonrelativistic approximation . The dashed curve is the calculation in the extremerelativistic approximation . 

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In the text 
Figure 2: Plotting of the emissivity and the relative errors of the approximations. The solid curve is the full calculation of the present work. The dotted curve is the calculation in the nonrelativistic approximation. The dashed curve is the calculation in the extremerelativistic approximation. The electronproton bremsstrahlung emissivity is also plotted in the dashdotted curve. In Fig. 2b, the relative errors of these approximations are plotted compared with the full calculation. 

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In the text 
Figure 3: Plotting of as a function x. The solid curve, dashed curve, dashdotted curve, and dotted curve are for keV, 10 keV, 100 keV, and 300 keV, respectively. 

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In the text 
Figure 4: Plotting of errors for the analytic fitting functions and . The solid curve, dashed curve, dashdotted curve, and dotted curve are for = 1 keV, 10 keV, 100 keV, and 300 keV, respectively. 

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In the text 
Figure 5: Plotting of the errors for the analytic fitting function and the extremerelativistic approximation . The solid curve, dashed curve, and dotted curve are for at keV, 1 MeV, and 7 MeV, respectively. The dashdotted curve is for at MeV. 

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In the text 
Copyright ESO 2009