The Astrophysical Journal, 581:1061–1070, 2002 December 20 # 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.
THE ORIGIN OF DIFFUSE X-RAY EMISSION FROM THE GALACTIC RIDGE. I. ENERGY OUTPUT OF PARTICLE SOURCES Vladimir A. Dogiel,1,2,3 Hajime Inoue,1 Kuniaki Masai,4 Volker Scho¨nfelder,3 and Andrew W. Strong3 Received 2002 May 23; accepted 2002 August 19
ABSTRACT We analyze processes for the hard X-ray emission from the Galactic disk, whose origin has remained enigmatic for many years. Up until now, no model has been able to explain the physical origin of this emission. Even the most plausible mechanism of bremsstrahlung radiation requires an energy output in emitting particles higher than the luminosity provided by known Galactic sources. We show that this energy enigma can be resolved if the emission comes directly from regions of particle acceleration. In this case, a broad quasi-thermal transition region of particle excess is formed between the thermal and nonthermal energy regions. The necessary energy output for production of electrons emitting 10 keV X-rays is of the order of 1041 ergs s1, which can definitely be supplied by supernovae or other known Galactic sources of energy. The temperature of the accelerating region is restricted to a value of a few 100 eV, and plasmas with these temperatures are hydrostatically stable in the Galaxy. Since only background electrons are supposed to be accelerated, the acceleration process does not violate the state of hydrostatic equilibrium in the Galactic disk. Subject headings: acceleration of particles — cosmic rays — X-rays: general — X-rays: ISM
this model, the iron 6.4 keV line must be present in the emission spectrum. However, this line is not observed in the direction of the Galactic ridge, except in the direction of the source Sgr A. Recent analysis of the data obtained with ROSAT (Tanaka, Miyaji, & Hasinger 1999), ASCA (Sugizaki et al. 2001), and Chandra (Ebisawa et al. 2001) showed convincingly that the ridge emission cannot be due to unresolved discrete sources and that most of the disk flux is really diffuse. The mechanism of X-ray photon production by IC scattering of relativistic electrons on background photons is very attractive, since the necessary energy output in relativistic electrons does not exceed 1038–1039 ergs s1. As discussed in Skibo, Ramaty, & Purcell (1996) and Strong, Moskalenko, & Reimer (2000), however, the IC scattering of relativistic electrons is unlikely to produce the bulk of hard X-ray emission below 100 keV, since this would imply a sharp upturn in the electron spectrum below 100 MeV. Hence, we consider here only bremsstrahlung models for the origin of the X-ray flux and present a solution in the form of a quasi-thermal particle spectrum. The line and continuum emission from this quasi-thermal model is studied in more detail in the accompanying Paper II.
1. INTRODUCTION
Decades ago, diffuse X-ray flux from the Galactic disk of the order of 1038 ergs s1 was detected in the energy range above several keV (Bleach et al. 1972), but its origin remains unknown. This is surprising, especially since there are few classes of sources of this emission in the Galaxy that could provide this flux: unresolved Galactic X-ray sources, thermal emission of hot plasma, nonthermal bremsstrahlung of subrelativistic particles, and inverse Compton (IC) of relativistic electrons. Breitschwerdt & Schmutzler (1994) claimed that hard X-ray emission could arise from a relatively cold plasma by delayed recombination of ions. However, Itoh & Masai (1989) showed that the emission decreases rapidly in such a recombining plasma. Therefore, it is not clear whether this effect can explain the ridge X-ray flux, especially since an extended power-law component of the ridge spectrum can hardly be described by recombination processes of thermal (in the past) heavy ions. For more discussion, see Masai et al. (2002, hereafter Paper II). The diffuse hard X-ray emission might also be due to hypothetical X-ray sources (e.g., a supermassive black hole) active in the past (see Cramphorn & Sunyaev 2002). The emission from flares of such sources, scattered by Galactic neutral hydrogen, might be seen today, in spite of the fact that the sources have now disappeared and their direct emission is unseen today. According to
2. BREMSSTRAHLUNG MODELS OF THE RIDGE X-RAY EMISSION
2.1. Nonthermal Bremsstrahlung The main process of energy losses by subrelativistic particles in plasmas is Coulomb losses. The characteristic time of Coulomb losses can be written as (see, e.g., Hayakawa 1969) rffiffiffiffi 3 0:5 m mE 1 : ð1Þ i ’ 4 ^ e n ln i 2 m
1 Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan;
[email protected]. 2 I. E. Tamm Department of Theoretical Physics, P. N. Lebedev Physical Institute, 119991 Moscow, Russia;
[email protected]. 3 Max-Planck-Institut fu ¨ r extraterrestrische Physik, Postfach 1312, D-85741 Garching, Germany;
[email protected],
[email protected]. 4 Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan;
[email protected].
^ is the rest mass of the emitting particles, e.g., m ^ ¼m Here m 1061
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for electrons and M for protons, and i ¼
T3
pffiffiffiffiffiffiffiffi : e3 8n
ð2Þ
All definitions used in this paper are presented in Table 1. In the subrelativistic energy range, any charged particle loses almost all its energy after time i by transferring it to the background plasma. In a stationary state, sources of particles should compensate these losses. Therefore, the flux of particles dN=dt supplied by sources in order to support the particle density at the value N is dN N : ð3Þ dt s i In the acceleration region, interactions of nonthermal particles with electromagnetic fluctuations dominate over Coulomb collisions (collisionless regime). Particle energy gain can be achieved if the time of acceleration is less than any other characteristic time. The flux of bremsstrahlung photons LX from any spatial region can be estimated as ; ð4Þ LX L br where L is the rate of particle production, is the particle lifetime, and br is the characteristic time for the production TABLE 1 Table of Definitions Symbol
Definition
n............................. kT .......................... E ............................ p............................. v............................. u = (2E/kT )1/2 ....... EX .......................... f .............................
Density of background gas Temperature of background gas Kinetic energy of particles Momentum of particles Particle velocity Dimensionless particle velocity (momentum) Energy of bremsstrahlung photons Function of particle distribution over particle momenta Equilibrium (Maxwellian) particle distribution Nonequilibrium (Non-Maxwellian) particle distribution Function of particle distribution over particle energies Density of thermal particles Density of quasi-thermal particles Density of nonthermal particles Energy output of particles emitting X-rays Luminosity of bremsstrahlung X-ray photons Energy output of nonthermal particles emitting X-rays in the framework of the thick target model (TTM) Energy output of particles emitting X-rays in the framework of the in situ acceleration model Characteristic time of ionization loss Characteristic time of bremsstrahlung loss Characteristic particle lifetime for either losses or acceleration processes Frequency of Coulomb collisions of thermal particles Frequency of particles in situ acceleration Dimensionless frequency of particles in situ acceleration Injection energy of particle acceleration
fM ........................... fneq ......................... N............................ Nth ......................... Nqth ........................ Nnth ........................ W........................... WX ......................... WTTM .....................
WISA ...................... i ............................ br .......................... ............................. ............................. 0 ........................... = 0/ ................ Einj .........................
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of bremsstrahlung photons. In acceleration regions, the lifetime of nonthermal particles is determined by acceleration processes: ’ acc , since the acceleration time acc is less than the time of Coulomb collisions i. Outside acceleration regions, the particle lifetime is completely determined by Coulomb losses ¼ i . Since acc < i , it is clear that the flux of bremsstrahlung photons from acceleration regions produced by nonthermal particles is relatively small in comparison to that from surrounding regions. It is clear that nonthermal particles generate bremsstrahlung emission in regions where Coulomb collisions are significant, i.e., when these particles have left the acceleration region and lose their energy by ionization/heating of the surrounding plasma. The maximum efficiency of bremsstrahlung photon production can be reached if charged subrelativistic particles lose all their energy in the emitting region (i.e., they do not escape from the Galactic disk: the so-called thick target model [TTM]). The differential flux of bremsstrahlung X-ray photons generated by nonthermal particles is Z ðm=m^ ÞEm dLX dN dbr ¼ nv dE : ð5Þ dEX ^ ÞEX dE dEX ðm=m Here br is the bremsstrahlung cross section, EX and E are energies of bremsstrahlung photons and parent subrelativistic particles, respectively, Em is the maximum energy of the particles, and v is the particle velocity. We can roughly present this equation for the total flux of bremsstrahlung photons in the form (see eq. [3]) N dN i ; ð6Þ LX br dt s br where br ðbr nvÞ1 is the characteristic time for photon production by bremsstrahlung (see, e.g., Ginzburg 1979): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi 3 3 m Eðmc2 Þ hc br pffiffiffi : ð7Þ ^ e2 e4 nc 16 2 m The energy flux of bremsstrahlung photons, WX ’ LX EX , is proportional to the energy output of subrelativistic particles, W ’ ðdN=dtÞs E: WX W
EX i : E br
ð8Þ
As follows from equations (1) and (7), the ratio in equation (8) in the framework of the TTM is a constant: WX EX e2 16 ’ 2 105 ð9Þ W TTM mc hc 3 ln i (here we took EX ¼ 10 keV). This ratio is almost independent of the background gas density and temperature and the energy and rest mass of the emitting particles. Therefore, over a wide range of these parameters, we obtain the same energy output in subrelativistic particles, 1043 ergs s1, to produce the 10 keV X-ray flux of 1038 ergs s1 from the Galactic disk. We note that more accurate numerical calculations give rather less than 1043 ergs s1: Valinia et al. (2000b) and Dogiel, Scho¨nfelder, & Strong (2002) obtained a value of about 3 1042 ergs s1, which, however, does not solve the energetics problem.
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This means that the energetics of the subrelativistic particle production is the most serious problem for the origin of the X-ray emission. The energy output of the order of 1043 ergs s1 cannot be supplied by known powerful Galactic sources. For comparison, we present the energy output from some classes of sources. These numbers were taken from Schlickeiser (2002, p. 181). 1. 2. 3. 4.
Supernova (SN) explosions: 1042 ergs s1. Neutron stars: 1041 ergs s1. Stellar winds from young hot O/B stars: 1041 ergs s1. Flare stars of spectral class K–M: 3 1040 ergs s1.
These all fall far short of the required power. If other models of the Galactic ridge X-ray emission (unresolved sources, IC) can be excluded by observations, the bremsstrahlung model survives, but with huge energetic problems. This leads us to the surprising conclusion that there is an unknown class of sources of subrelativistic particles with luminosity of the order of or higher than that of SNe. If this is the case, the question arises as to why these sources become apparent only in the hard X-ray energy range. 2.2. Thermal Bremsstrahlung Model In the case of thermal particles, Coulomb collisions redistribute particle energy, but the number of particles with any given energy E is constant. A slow dissipation of the total energy of thermal particles occurs because of emission of bremsstrahlung photons. Therefore, if the background plasma is in a state of hydrostatic equilibrium, the flux supplied by sources should be about dN N ; ð10Þ dt s br i.e., much less than the flux in equation (3), since i 5 br . However, the thermal interpretation is doubtful (for more details, see Paper II) because it cannot explain the heating and the confinement of this plasma in the Galactic disk (see, e.g., Tanaka et al. 1999, 2000; Tanaka 2002). Gaseous components can be confined to the Galactic plane by the Galactic gravity if their temperatures are below the average gravitational escape temperature of the Galaxy, Tesc 0:5 keV (Kaneda et al. 1997). We note that the escape velocity (temperature) is a function of the galactocentric radius, and its value increases toward the Galactic center (Breitschwerdt, McKenzie, & Vo¨lk 1991), but even in this case, the values in the central parts of the Galactic disk do not exceed 1 keV. Nevertheless, recent observations of the X-ray diffuse emission from the Galactic disk found an excess at EX 1 keV that is often described as thermal emission of a very hot gas whose temperature is higher than 1 keV. This model requires a too large SN rate in the Galaxy in order to compensate for a hot plasma outflow from the disk (see, e.g., Yamasaki et al. 1997). In order to reproduce the ridge continuum and line spectrum, the thermal model requires at least three components of the interstellar plasma with different temperatures, the highest of which is about 10 keV (see Tanaka 2002); each component is then responsible for a rather narrow range of the total spectrum. The ASCA observations show, however, that the ridge spectrum hardly varies along the Galactic plane (Kaneda et al. 1997; Tanaka 2002). Such a close similarity cannot be accidental and must imply that the X-ray
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emission from different parts of the disk is due to the same emission mechanism. However, it is rather difficult to accept that the relative thermal fluxes from independent plasma components are the same in different parts of the Galactic disk. It is more natural to assume that the whole spectrum is formed by a single radiation mechanism or by physical processes that are bound somehow with each other. 2.3. Summary of Bremsstrahlung Models One of the advantages of the thermal bremsstrahlung model is small particle losses. Particle losses occur only if in the course of collisions, bremsstrahlung photons are produced. Therefore, if the 10 keV plasma were hydrostatically stable in the Galaxy, we would need the luminosity of particle sources to be of the order of the X-ray luminosity, i.e., 1038 ergs s1. On the other hand, the thermal model is unlikely because of hydrostatic instability. If a significant fraction of the hard X-rays come from hot plasma with a temperature of several keV, the energy density of the hot plasma needs to be 1 1010 f 1=2 ergs cm3, and then the total energy in the Galaxy, EG, should be about 1 1056 f 1=2 ergs, where f is the filling factor (see Inoue 2001). Even if the filling factor is 1%, the energy density of hot plasma is 10 times as large as the currently known energy density of the interstellar medium. Therefore, the hot plasma should escape from the Galaxy within a time tcool
R R vs 105:5 yr ; vs 100 pc 107:5 cm s1
ð11Þ
where R is the thickness of the Galactic plane and vs is the sound velocity of hot plasmas. Then, the necessary energy supply rate Wth should be W th
EG ’ 1043 f 1=2 ergs s1 : tcool
ð12Þ
Even if the filling factor of hot plasmas were only 1% for the Galaxy, an energy supply rate as high as 1042 ergs s1 would be necessary. The nonthermal bremsstrahlung mechanism has the same energetics problem, but for another reason. Nonthermal particles transfer their energy to background plasma by Coulomb collisions. Therefore, they are lost from the nonthermal regime in each collision with thermal particles. The probability of generating a 10 keV bremsstrahlung photon in a collision is only of the order of 105 relative to a Coulomb collision. Thus, we need a particle luminosity of 1043 ergs s1 in order to generate the 1038 ergs s1 X-ray flux. Thus, the conclusion is that none of the models can explain the origin of the X-ray emission from the Galactic disk. On the other hand, we do not see any other mechanism that could generate this flux in the Galaxy. Below we suggest a new interpretation of the ridge emission that is free from the shortcomings of both the thermal and nonthermal bremsstrahlung models. We present an interpretation that is neither thermal nor nonthermal. We consider bremsstrahlung emission directly from regions of particle acceleration from the background plasma: in this case, a broad transition region of particle excess (defined below as the energy range of quasi-thermal particles) is formed between the thermal and the nonthermal parts of the particle spectrum, under the influence of both acceleration and Coulomb collisions. We show that this third
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(quasi-thermal) component of particles may solve the energetics problem of the X-ray flux. The presence of a quasi-thermal component in the spectra of particles accelerated from the background plasma can be seen from the kinetic equations (see Gurevich 1960 for details of the spatially uniform case and Bulanov & Dogiel 1979 for acceleration by shock waves: a spatially nonuniform case). We start from a brief review of parameters of background gas in the interstellar gas. 3. COMPONENTS OF THE GALACTIC INTERSTELLAR GAS
The interstellar medium is nonuniform, with a variety of gaseous temperatures and densities. One of the first models of the interstellar medium was presented by Field, Goldsmith, & Habing (1969), who assumed the heating of the interstellar medium by a hypothetical flux of 2 MeV protons. Their model contained two thermally stable phases, one (with a density of 0.2 cm3) at T ¼ 104 K and one (with a density of 10 cm3) at T < 300 K. The warm gas occupied most of the interstellar medium, but 75% of the total mass of the gas was compressed into cool dense clouds. This model was developed later by McKee & Ostriker (1977), who suggested a three-component model of the interstellar medium based on X-ray observations. They assumed that SN explosions evaporated the cold interstellar gas and thus generated a large amount of hot low-density interstellar gas (T 106 K, n 3 103 cm3). The outer edges of the cold clouds (which still contained the main part of the interstellar gas) ionized by the diffuse UV and soft X-ray background provided warm (T 104 K) ionized and neutral components. They estimated the densities of cold clouds and the warm component as 40 and 0.25 cm3, respectively, and their filling factors f 0:024 and 0.23. Thus, in this model, 75% of the volume was occupied by the hot gas. Numerous observations support the existence of the hot interstellar component. UV observations show the presence of interstellar gas at a few 105 K (see, e.g., Hurwitz & Bowyer 1996 and references therein). Soft X-ray observations provide information on slightly hotter (T 106 K) gas. Since this emission is largely absorbed by the cool interstellar gas, only information about regions at distances 100 pc can be obtained from these observations. Recent analyses have shown that McKee & Ostriker (1977) overestimated the fraction of the disk volume occupied by hot gas. It was concluded that the hot gas filling factor is no more than 15%–20% (Ferrie`re 1998b, 2001). The gas density of the hot component is estimated at ð3:4 6:5Þ 103 cm3 (Ferrie`re 1998a; Snowden et al. 1998), and the temperature of the gas is a few 106 K (Aschenbach 1988; McCammon & Sanders 1990). The contributions of other components are estimated as follows: dense (n 103 cm3) molecular clouds contain about half the mass of the interstellar gas, 30% is in cool neutral clouds (n 30 cm3), and 20% is in the warm neutral medium (n 0:4 cm3). A summary of the interstellar gas parameters is presented in Table 2, taken from Ferrie`re (1998a). The measurements provided by ASCA (Kaneda et al. 1997) and the Rossi X-Ray Timing Explorer (RXTE; Valinia & Marshall 1998; Valinia, Kinzer, & Marshall 2000a) found emission of plasma whose temperature (if the emission is thermal) exceeds 1 keV. In terms of thermal emission, the
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TABLE 2 Temperature, Density, and Pressure of Components of the Interstellar Gas
Phase
Temperature (K)
Density (cm3)
Pressure (eV cm3)
CM .................. WNM .............. WIM................ HM.................. SNM................ XHM ...............
80 8000 8000 (1–8) 106 2.6 107 (7–10) 107
40 0.4 0.21 (3–6) 103 8 102 3 103
0.3 0.3 0.2 1–4 260 20–30
Note.—Phases are CM: cold medium; WNM: warm neutral medium; WIM: warm ionized medium; HM: hot medium; SNM: hot medium heated by SNe; XHM: hot interstellar plasma assumed to reproduce the spectrum of the ridge X-ray emission.
ASCA flux from the Galactic ridge can be represented as emission of a two-temperature plasma with temperatures of 0.8 and 7 keV, respectively (Kaneda et al. 1997). The 7 keV plasma component is denoted below as XHM. Analysis of the X-ray spectrum carried out by Tanaka (2002) showed that at least three components of thermal plasma with temperatures of 0.75, 1.8, and 10 keV are required to obtain a reasonable reproduction of the observed ridge spectrum. The parameters of the 7–10 keV XHM plasma as well as the parameters of regions heated by SNe (SNM component) taken from Yamasaki et al. (1997) are presented in Table 2. 4. EVIDENCE FOR PARTICLE ACCELERATION IN THE GALACTIC RIDGE
Analysis of the observed X-ray flux in the range 3–16 keV with the Ginga satellite (Yamasaki et al. 1997) showed that there is a hard nonthermal component in the ridge spectrum in addition to the hot plasma component. The total estimated luminosity is 1038 ergs s1 in this energy range. The RXTE data also show a flux of nonthermal hard X-rays at energies higher than several keV (Valinia & Marshall 1998). The combination of the RXTE and Ginga spectra with the OSSE measurements at higher energies show that the emission spectrum can be represented as a power law over a very broad energy band. The spectrum of the diffuse emission is flat up to 35 keV. Valinia et al. (2000a) interpreted the ASCA data with a two-temperature distribution (temperatures of 0.6 and 2.8 keV) plus a hard X-ray bremsstrahlung emission generated by a flux of nonthermal electrons. The important conclusion of this analysis is that a significant part of the total flux from the Galactic ridge cannot be due to thermal emission only. A contribution from nonthermal processes is necessary. The continuum spectrum of X-ray emission is very hard. If it is interpreted as a power law, then the observed X-ray emission from the Galactic ridge at energies above 1 keV is (see, e.g., Valinia et al. 2000a) IX / EX2:1 :
ð13Þ
The large-scale association of the hard X-ray emission with thermal X-rays implies that these two components are linked. This leads to the idea that thermal particles in the hot plasma are in situ accelerated to produce the nonthermal particles responsible for the hard X-ray emission (Yamasaki et al. 1997).
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A model for in situ particle acceleration in regions heated by SNe, applied to the interpretation of the ridge emission, was given by Dogiel et al. (2002), who estimated the parameters of the acceleration processes from the observational data. This model includes hypothetical regions of very hot (T ’ 2:6 keV) and dense (n ’ 102 cm3) plasma heated by SNe (Yamasaki et al. 1997), where particles are accelerated. These particles are injected into the surrounding interstellar medium of cold neutral gas (atomic and molecular hydrogen), where they lose their energy and produce bremsstrahlung photons. However, the nonthermal interpretation requires a source luminosity of about 1042–1043 ergs s1. Below we analyze processes of bremsstrahlung radiation directly from regions of particle acceleration.
5. BREMSSTRAHLUNG EMISSION FROM REGIONS OF PARTICLE IN SITU ACCELERATION
The particle spectrum formed in regions of in situ acceleration differs from a simple sum of thermal+nonthermal components. There is a significant particle excess above the equilibrium Maxwellian distribution in the transition region between thermal and nonthermal particles. We present a solution of the kinetic equation for the electron distribution function f over particle momenta p with the normalization condition Z 1 f ðp0 Þp02 dp0 ; ð14Þ Nð> pÞ ¼ p
describing particle Coulomb collisions and stochastic in situ acceleration for electrons. The kinetic equation for protons is more bulky, since collisions with both background electrons and protons should be taken into account (for the general equation, see, e.g., Dogiel et al. 2002). The necessary energy output is estimated for both electrons and protons. The equation for stochastic (Fermi II) acceleration of electrons and their Coulomb collisions is @f @f 1 @ 2 AðuÞ þ e u4 þ BðuÞf ¼ 0 ; ð15Þ @ u @u @u where ðuÞ ¼ e u2 is the diffusion coefficient in momentum space, e ¼ e0 =e is the dimensionless characteristic frequency of stochastic acceleration, e0 is the dimensional frequency of acceleration (for details about this momentum diffusion, see Dogiel et al. 2002), e is the collisional frequency of background electrons at E ¼ kT, e ¼
4ne4 ðkTÞ3=2 m1=2
ln i ;
ð16Þ
n and T are the background gas density and temperature, E is the kinetic energy of particles, and u ¼ ð2E=kTÞ0:5 is the dimensionless particle velocity. The function AðuÞ describes the electron momentum diffusion, and the function BðuÞ the electron Coulomb losses [for AðuÞ and BðuÞ, see, e.g., Butler & Buckingham 1962; Dogiel et al. 2002]. Therefore (if no acceleration occurs), the solution of the stationary equation 1 @ @f AðuÞ þ BðuÞf ¼ 0 ð17Þ u2 @u @u is the Maxwellian function. The functions AðuÞ and BðuÞ
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can be represented for electrons as 1 AðuÞ ¼ G ðuÞ ; u
ð18Þ
BðuÞ ¼ G ðuÞ ;
ð19Þ
where 2 GðxÞ ¼ pffiffiffi
"Z
x
exp z
2
dz x exp x2
# ð20Þ
0
(for the kinetic equations describing charged particle Coulomb collisions, see, e.g., Sivukhin 1966). In the energy range E > kT, GðuÞ ’ 1; therefore, AðuÞ ¼ 1=u and BðuÞ ¼ 1, and in this range, equation (15) can be written as @f 1 @ 1 4 @f þ e u þf ¼0: ð21Þ @ u2 @u u @u From this equation, it follows that the total flow of electrons from thermal to nonthermal momenta is (Gurevich 1960) rffiffiffi Z 1 dN 2 u du ne exp ’ ; ð22Þ 5 dt e 0 1 þ e u and the density of electrons from equation (21) is described by Z 1 pffiffiffiffi dN 2 u du ¼ nðÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E exp 5 dE 0 1 þ e u ðkTÞ3 " ! # Z 1 u du exp pffiffiffiffiffiffiffiffiffiffiffi 1 : ð23Þ 5 2E=kT 1 þ e u Some more details of particle acceleration processes from a bulk plasma can be found in Berezinskii et al. (1990). The X-ray emissivity of the accelerated particles is Z dbr dN dE ; ð24Þ nv X ¼ dEX dE EX where the bremsstrahlung cross section is 2 pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 2 2 ^ EX ^þ E 2 2 E dbr 8 e e mc 6 7 ¼ ln 4 5: 2 ^ dEX 3 hc mc EX E EX
ð25Þ
^ ¼ Ee for electrons and E ^ ¼ m=MEp for protons. Here E With equations (23) and (24), we can calculate the intensity of bremsstrahlung radiation: Z IX ¼ X dl ; ð26Þ l
where l is the thickness of the emitting region. The intensity is a function of the temperature T of the gas, its column density along the path of view L ¼ ln, and the acceleration frequency : IX ¼ F ðT; L; e;p Þ ;
ð27Þ
where e is the dimensionless ratio of the characteristic frequency of electron acceleration to the thermal electronelectron frequency of collisions, e ¼ e0 =e , and p is the
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the background gas temperature T, the rest mass of the emitting particles, and the dimensionless acceleration frequency e,p: WX ¼ F1 ðT; e;p Þ : ð28Þ We;p ISA Since the observed value of WX 1038 ergs s1, the energy necessary for acceleration can be supplied by known Galactic sources if the ratio is less than 104. We can derive the boundary function e;p ðTÞ from the equality F1 ðe;p ; TÞ ¼ 104 :
ð29Þ
The energy output necessary for particle acceleration can be supplied by known Galactic sources if the range of parameters (, T) is below this boundary curve. Mechanisms of in situ acceleration can be different in different components of the interstellar gas (for a review, see Berezinskii et al. 1990). However, stochastic acceleration is possible, even inside dense and cold clouds of molecular hydrogen, where chaotic electromagnetic fluctuations are excited by turbulent motions of the gas (Dogiel et al. 1987). In Figure 2 we present the boundary functions e ðTÞ (left) and p ðTÞ (right) obtained from equation (29) for the case of electron and proton bremsstrahlung origin of the ridge X-ray emission. These functions are shown by the solid lines in the figures. The values e and p derived from equation (27) for different components of the interstellar medium are shown by circles. The ranges of (T, ) for cold fractions of the gas are far above the line. This means that a power much higher than 1042 ergs s1 is necessary in order to produce the ridge X-ray flux by in situ acceleration from the cold gaseous fractions. This is not surprising, since the emitting particles are nonthermal in this case, and as we mentioned already, they weakly interact with the background plasma of acceleration regions. On the other hand, if electron acceleration takes place in hot fractions of the interstellar medium with temperatures T 0:3 keV (see Fig. 2 [left]), the energy output needed for production of the observed X-ray flux does not exceed 1042 ergs s1, although these fractions occupy a small part of the Galactic disk. Our analysis for the proton bremsstrahlung model shows that the proton energy output needed to produce the ridge emission is much higher than that of electrons (see Fig. 2 [right]), and it cannot be less than 1043 ergs s1, since the emitting protons are nonthermal. Another problem of the proton bremsstrahlung model is the very high energy density of protons, 100 eV cm3 (Dogiel et al. 2002). This
Fig. 1.—Calculated bremsstrahlung emission of nonthermal and quasithermal particles from the hot gas component (T ¼ 0:3 keV) of the Galactic disk, with the gas column density of the HM component taken as L ¼ 1020 (e ¼ 1:5 103 : solid line), 3 1020 (e ¼ 1 103 : dashed line), and 1021 cm2 (e ¼ 0:7 103 : dot-dashed line). The crosses show the RXTE data taken from Valinia et al. (2000a).
dimensionless ratio of the characteristic frequency of proton acceleration to the thermal proton-proton frequency of collisions, p ¼ p0 =p , where p ¼ e ðm=M Þ1=2 . From equation (27), we derived the characteristic frequency of stochastic acceleration that can produce the X-ray spectrum observed with RXTE (Valinia et al. 2000a) in each component of the interstellar medium presented in Table 2. As an example, we present in Figure 1 the calculated bremsstrahlung X-ray flux produced by electrons for different gas column densities L of the hot medium plasma and the RXTE data taken from Valinia et al. (2000a). The values of e and p derived from equation (27) for different interstellar gaseous components are presented in Table 3. The ratio of the energy flux of X-ray photons WX to the energy output of emitting particles We,p for the case of in situ acceleration (ISA) can be obtained from equations (23) and (24) in the same way as was done for the TTM model. In contrast to the TTM case, in equation (9) the ratio ðWX =We;p ÞISA inside acceleration regions is a function of
TABLE 3 Dimensionless Acceleration Frequency of Electrons e ¼ e0 =e and Protons p ¼ p0 =p and the Necessary Electron We and Proton Wp Energy Outputs Derived for Different Components of the Interstellar Gas
ISM Phase
L (cm2)
e
We (ergs s1)
p
Wp (ergs s1)
CM ........................ WNM .................... HM........................ SNM...................... XHM .....................
1022 1022 (1–10) 1020 1020 1020
1.12 104 1.7 104 (1–2) 103 5.8 104 1.5 103
8 1047 1 1046 (2–5) 1041 2 1040 1 1040
1.2 103 1.75 103 6 103 3.9 103 9 103
3 1050 4.7 1047 2 1044 5.7 1042 3 1042
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Fig. 2.—Boundary functions for dimensionless e ðTÞ and p ðTÞ derived from the condition F1 ¼ 104 for the cases of electron and proton bremsstrahlung origin of the ridge X-ray emission (see eq. [29]). The circles show the values of e and p derived from the observed X-ray intensity for in situ acceleration of electrons in different fractions of the interstellar gas (see eq. [27]).
implies that the X-ray emission is probably generated by subrelativistic electrons. Although we do not know of a specific process that is responsible for particle acceleration in the interstellar medium, we present arguments in favor of preferential electron acceleration. In the case of Fermi acceleration, the characteristic acceleration frequency is proportional to 0 v=L (see Toptygin 1985), where v is the particle velocity and L is the particle mean free path. Since velocities of thermal electrons are ðM=mÞ1=2 times larger than velocities of thermal protons, we expect that e0 4p0 in the subrelativistic energy range. Although we do not discuss here processes of particle acceleration by shock waves, we note that from predictions of nonlinear shock acceleration models, it also follows that in the subrelativistic energy range, the density of electrons accelerated in SN remnants is higher than that of protons (see Baring et al. 1999; Ellison, Berezhko, & Baring 2000). This result is in favor of our speculations about preferential electron acceleration at these energies. Thus, we conclude that the possible solution of the energetics problem could be that the ridge emission is due to bremsstrahlung of electrons from regions where the acceleration occurs. The necessary temperature range of acceleration regions is strongly constrained. As we see from Figure 2, if the temperature is below a few hundred eV, then the emitting electrons are nonthermal. Therefore, a high energy output is needed to provide the emission. On the other hand, if the plasma temperature is higher than 1 keV, the ridge emission is almost thermal. As we mentioned already, this plasma is hydrostatically unstable, and a large energy output is necessary to compensate for the convective plasma outflow from the Galactic disk. Therefore, we conclude that the temperature of the acceleration region should be somewhere between 300 eV and 1 keV. We show in x 6 that in this case, the energies of particles emitting hard X-rays are in a rather specific energy range of quasi-thermal particles, which can exist in acceleration regions only. 6. SPECTRUM OF IN SITU ACCELERATED PARTICLES AND PRODUCTION OF BREMSSTRAHLUNG PHOTONS BY QUASI-THERMAL ELECTRONS
It is easy to see from equation (23) that there are three energy ranges in the particle spectrum formed in accelera2=5 tion regions. Indeed, at energies E < kT e , the spectrum
is described by the equilibrium Maxwellian thermal distribution: pffiffiffiffi dN 2 E E ’ nðÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp : ð30Þ dE th kT ðkTÞ3 The nonthermal power-law spectrum at energies 2=3 has the form E > kTe ! rffiffiffi dN 2 nðÞ 1 a exp 2=5 ; ’ ð31Þ dE nth 6 e E e where the constant a is Z a¼
1 0
x dx ’ 0:66 : 1 þ x5
ð32Þ
The condition Einj ¼ kT2=3 e
ð33Þ
determines the injection energy at which the rate of acceleration equals the rate of ionization losses. The energy range E > Einj corresponds to the collisionless part of the spectrum of electrons that weakly interacts with the background plasma of the acceleration regions. At these energies, the characteristic acceleration time, 1=e0 , is smaller than the time of Coulomb collisions, i. There is a broad transition region in between these energy ranges in which the electron spectrum can be described neither as thermal nor as nonthermal. In the 2=5 2=3 < E < kTe , the spectrum of energy range kTe thermal particles is strongly distorted by the flux ‘‘ running away ’’ into the range of high momenta. It has an exponential form that differs strongly, however, from the equilibrium Maxwellian distribution ðdN=dEÞth . This spectrum of quasi-thermal particles can be approximated from equation (23): 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 pffiffiffiffi 3 dN 2C E 1 kT 5 4 ; ð34Þ ’ nðÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp dE qth 3 E e ðkTÞ3 where C ¼ exp
a 2=5
e
! :
ð35Þ
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DOGIEL ET AL.
Fig. 3.—Total electron spectrum (thermal+quasi-thermal+nonthermal components) in a hot plasma with temperature T ¼ 0:6 keV and density n ¼ 0:0034 cm3, whose electrons are accelerated with frequency e ¼ 1:8 103 . The thermal+nonthermal spectrum is shown by the dotdashed line.
This spectrum of quasi-thermal electrons is flatter than the thermal distribution in equation (30) and mimics a thermal distribution with a very high temperature (see Paper II). As an example, we show in Figure 3 the total electron spectrum in a hot plasma with T ¼ 0:6 keV and n ¼ 0:0034 cm3, whose electrons are accelerated with frequency e ¼ 1:8 103 . The component Nqth can be seen as an excess above the thermal+nonthermal distribution (dotdashed line) in the energy range in which Coulomb collisions are still essential. It describes the nonequilibrium part of the thermal distribution formed by particle collisions, which is distorted due to the runaway electron flux toward the higher momenta. The runaway flux in equation (22) is constant in the energy range of quasi-thermal and nonthermal particles. Therefore, at these energies we can define the characteristic lifetime of electrons e at energy E as e ðEÞ
NðEÞ : dN=dt
ð36Þ
Let us estimate the energy output of emitting electrons that should be produced by acceleration processes in order to generate the observed Galactic X-ray flux. From equation (6), we have LX
dN e i dN e ¼ : dt br br dt i
ð37Þ
The power of the ridge X-ray emission has a maximum at photon energies EX 10 keV. This power drops at higher energies. It is important to note that the efficiency of photon production increases at higher energies (see eq. [8]). Both these factors make the energy problem most serious at EX 10 keV. From equation (9), the energy output of quasi-thermal electrons emitting 10 keV photons is (for electrons, we can put E EX ) br i i i WX ¼ WeTTM ’ 1043 ergs s1 ; ð38Þ We i e e e since for an energy of 10 keV, br =i 105 and WX 1038 ergs s1. In Figure 4 we present the ratio i =e in the energy range of quasi-thermal (5–50 keV) and nonthermal (>50 keV)
Vol. 581
Fig. 4.—Ratio i =e of quasi-thermal (energy range 5–50 keV) and nonthermal (>50 keV) particles calculated for in situ acceleration in the HM component, whose gas column density is taken as 1021 cm2. The solid line shows the ratio inside the acceleration regions. The straight horizontal dotted line shows the ratio e =i ¼ 1. The vertical straight lines mark the region of quasi-thermal electrons.
particles derived for the hot medium (HM) component. The power-law nonthermal spectrum sets in at energies Einj 50 keV. The straight dotted line defines i =e ¼ 1. The point of intersection of the solid line and the straight dotted line marks the boundary between quasi-thermal and nonthermal electrons . We can interpret this figure in the following way. If we slightly violate the equilibrium spectrum, then the electron distribution function tends to return to its equilibrium state by Coulomb collisions; in other words, collisions destroy the deviation from the equilibrium state fneq fM in the characteristic time of collisions i. Here fM is the equilibrium Maxwellian distribution, and fneq is the nonequilibrium electron distribution. If we support somehow this nonequilibrium state (in our case, by in situ acceleration), then we should supply a flux of electrons of the order of fneq fM ; i
ð39Þ
i.e., we can roughly define the lifetime of particles e in the nonequilibrium energy range as e
fneq i : fneq fM
ð40Þ
We see that when fneq ¼ fM , then e ¼ 1 (we neglect here radiative losses), and when fneq 4fM , then e ’ i . From Figure 4 we see that the bremsstrahlung flux produced by nonthermal electrons of E > Einj 50 keV in acceleration regions is negligible, since their lifetime is smaller than the characteristic time of Coulomb collisions (collisionless regime), i.e., i =e > 1. These particles generate a bremsstrahlung flux more effectively outside regions of their acceleration, where i =e ¼ 1. On the other hand, quasi-thermal electrons generate a bremsstrahlung flux more effectively just inside acceleration regions (solid line at E < 50 keV), since in those regions i =e < 1. Therefore, we expect that the necessary energy output of particles emitting 10 keV ridge flux can be less than 1042 ergs s1 if it is produced by quasi-thermal particles. We note here that the efficiency of 10 keV photon production (105) used in this paper and the energy output of the emitting particles (1043 ergs s1) are extreme limits. More accurate numerical calcu-
No. 2, 2002
DIFFUSE GALACTIC RIDGE X-RAY EMISSION. I.
lations show that the efficiency of bremsstrahlung photon production by 100 keV electrons is about 2:5 104 , which drops to 3:8 105 for 10 keV electrons (see Valinia et al. 2000a), and the energy output of 10 keV nonthermal electrons is of the order of 3 1042 ergs s1 (Valinia et al. 2000a; Dogiel et al. 2002). With these numbers and with the ratio e =i 3 102 at 10 keV (see Fig. 4), we expect that the necessary energy output of quasi-thermal electrons cannot be higher than 1041 ergs s1. It is interesting to note that just this value of the luminosity was derived by Moskalenko et al. (2002) for the cosmic-ray flux in the MeV to TeV energy range. Thus, we conclude that the hard X-ray emission is probably generated in acceleration regions of hot plasma with temperatures of about several hundred eV. This emission is produced by electrons that are neither thermal nor nonthermal. For these temperatures of plasma, the energy range of quasi-thermal electrons is between several keV and several tens of keV, so that only these particles produce the ridge X-ray flux. The energetics problem that is so critical for the thermal or nonthermal interpretations (see x 2) is avoided for the quasi-thermal component. The acceleration does not distort the proton equilibrium distribution. Therefore, the plasma pressure is completely determined by thermal protons whose temperature does not exceed a few hundred eV. Of course, these estimates do not have pretensions of high accuracy. More accurate estimates could be obtained if we knew a mechanism for particle acceleration in the Galactic disk. However, they definitely show a qualitative trend to a smaller energy output of quasi-thermal electrons in comparison to thermal or nonthermal electrons. 7. CONCLUSIONS
If nonthermal relativistic particles are assumed to be the sources of the Galactic diffuse X-ray ridge (around 10 keV), then there is no alternative to the conclusion that the particle energy output required is of the order of 1043 ergs s1. If the ridge X-ray flux is thermal, then the necessary energy output is again of this order to compensate for the hot plasma outflow from the Galactic disk. This forces us to assume that there is ‘‘ a class of unseen sources ’’ with an energy output at least 1 order of magnitude higher than that of known sources. In the case of an in situ acceleration model, the origin of the hard X-rays is neither thermal nor nonthermal. The
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important point is that a quasi-thermal component of particles is formed in addition to thermal and nonthermal components. The lifetime of the quasi-thermal particles is larger than the characteristic time of Coulomb collisions, and the density is far above the thermal distribution. Both these effects make this component efficient for the production of bremsstrahlung photons. The energetics problem can be resolved if the emitting particles are electrons and they generate bremsstrahlung photons directly in regions of their acceleration. In this case, the energy output depends strongly on the temperature of the background plasma. If this temperature is higher than a few hundred eV, then the necessary energy output is about 1041 ergs s1. The pressure of plasma is determined by thermal protons whose spectrum is supposed to not be distorted by the acceleration. This gives a solution to the problem of plasma hydrostatic instability that affects the thermal interpretation of the ridge X-ray flux. We also mention that a similar problem appears for the interpretation of the hard X-ray flux from clusters of galaxies. The nonthermal bremsstrahlung interpretation (see, e.g., Ensslin, Lieu, & Biermann 1999; Sarazin & Kempner 2000; Petrosian 2001) also leads to a huge energy output from particle sources. The quasi-thermal interpretation of the cluster X-ray emission may solve this problem too (see Liang, Dogiel, & Birkinshaw 2002). The authors are grateful to Professor Jun Nishimura, Masanobu Ozaki, and Tai Oshima, whose advice helped to perform the numerical calculations. They are grateful to Dieter Breitschwerdt, Stu Bowyer, Katia Ferrie`re, Atsushi Ichimura, and Hidehiro Kaneda and also members of the MPE Institute, particularly to Professors G. Hasinger and Y. Tanaka, for discussions and comments. The authors would like to thank the unknown referee, whose advice helped very much to improve the paper. V. A. D. would like to thank the administration of the Institute of Space and Astronautical Science (Sagamihara, Japan) and his colleagues from the High Energy Astrophysics Division, who helped his stay to be productive, and for their traditional hospitality, kindness, and useful discussions. V. A. D. is also grateful to the Max-Planck-Institut fu¨r extraterrestrische Physik and to the Alexander von Humboldt-Stiftung, whose permanent financial support during the last years allowed him to perform a series of studies focused on problems of X-ray emission from Galactic and extragalactic sources.
REFERENCES Ellison, D. C., Berezhko, E. G., & Baring, M. G. 2000, ApJ, 540, 292 Aschenbach, B. 1988, in IAU Symp. 101, Supernova Remnants and the Ensslin, T. A., Lieu, R., & Biermann, P. 1999, A&A, 344, 409 Interstellar Medium, ed. R. S. Roger & T. L. Landecker (New York: Ferrie`re, K. 1998a, ApJ, 497, 759 Cambridge Univ. Press), 99 ———. 1998b, ApJ, 503, 700 Baring, M. G., Ellison, D. C., Reynolds, S. P., Grenier, I. A., & Goret, P. ———. 2001, Rev. Mod. Phys., 73, 1031 1999, ApJ, 513, 311 Field, G. B., Goldsmith, D. W., & Habing, H. J. 1969, ApJ, 155, L149 Berezinskii, V. S., Bulanov, S. V., Dogiel, V. A., Ginzburg, V. L., & Ginzburg, V. L. 1979, Theoretical Physics and Astrophysics (New York: Ptuskin, V. S. 1990, Astrophysics of Cosmic Rays (Amsterdam: NorthPergamon) Holland) Gurevich, A. V. 1960, Soviet Phys.–JETP, 38, 1150 Bleach, R. D., Boldt, E. A., Holt, S. S., Schwartz, D. A., & Serlemitsos, Hayakawa, S. 1969, Cosmic Ray Physics: Nuclear and Astrophysical P. J. 1972, ApJ, 174, L101 Aspects (New York: Wiley) Breitschwerdt, D., McKenzie, J. F., & Vo¨lk, H. J. 1991, A&A, 245, 79 Hurwitz, M., & Bowyer, S. 1996, ApJ, 465, 296 Breitschwerdt, D., & Schmutzler, T. 1994, Nature, 371, 774 Inoue, H. 2001, Prog. Theor. Phys. Suppl., 143, 99 Bulanov, S. V., & Dogiel, V. A. 1979, Soviet Astron. Lett., 5, 278 Itoh, H., & Masai, K. 1989, MNRAS, 236, 885 Butler, S. T., & Buckingham, M. J. 1962, Phys. Rev., 126, 1 Kaneda, H., Makishima, K., Yamauchi, S., Koyama, K., Matsuzaki, K., & Cramphorn, C. K., & Sunyaev, R. A. 2002, A&A, 389, 252 Yamasaki, N. Y. 1997, ApJ, 491, 638 Dogiel, V. A., Gurevich, A. V., Istomin, Ya. N., & Zybin, K. P. 1987, Liang, H., Dogiel, V. A., & Birkinshaw, M. 2002, MNRAS, 337, 567 MNRAS, 228, 843 Masai, K., Dogiel, V. A., Inoue, H., Scho¨nfelder, V., & Strong, A. W. 2002, Dogiel, V. A., Scho¨nfelder, V., & Strong, A. W. 2002, A&A, 382, 730 ApJ, 581, 1071 (Paper II) Ebisawa, K., Maeda, Y., Kaneda, H., & Yamauchi, S. 2001, Science, 293, McCammon, D., & Sanders, W. T. 1990, ARA&A, 28, 657 1633
1070
DOGIEL ET AL.
McKee, C. F., & Ostriker, J. P. 1977, ApJ, 218, 148 Moskalenko, I. V., Strong, A. W., Ormes, J. F., & Potgieter, M. 2002, ApJ, 565, 280 Petrosian, V. 2001, ApJ, 557, 560 Sarazin, C. L., & Kempner, J. C. 2000, ApJ, 533, 73 Schlickeiser, R. 2002, Cosmic Ray Astrophysics (London: Springer) Sivukhin, D. V. 1966, Rev. Plasma Phys., 4, 181 Skibo, J. G., Ramaty, R., & Purcell, W. R. 1996, A&AS, 120, 403 Snowden, S. L., Egger, R., Finkbeiner, D. P., Freyberg, M. J., & Plucinsky, P. P. 1998, ApJ, 493, 715 Strong, A. W., Moskalenko I. V., & Reimer, O. 2000, ApJ, 537, 763 Sugizaki, M., Matsuda, K., Kaneda, H., Matsuzaki, K., Yamauchi, S., & Koyama, K. 2001, ApJS, 134, 77
Tanaka, Y. 2002, A&A, 382, 1052 Tanaka, Y., Koyama, K., Maeda, Y., & Sonobe, T. 2000, PASJ, 52, L25 Tanaka, Y., Miyaji, T., & Hasinger, G. 1999, Astron. Nachr., 320, 181 Toptygin, I. N. 1985, Cosmic Rays in Interplanetary Magnetic Fields (Amsterdam: Reidel) Valinia, A., Kinzer, R. L., & Marshall, F. E. 2000a, ApJ, 534, 277 Valinia, A., & Marshall, F. E. 1998, ApJ, 505, 134 Valinia, A., Tatischeff, V., Arnaud, K., Ebisawa, K., & Ramaty, R. 2000b, ApJ, 543, 733 Yamasaki, N. Y., et al. 1997, ApJ, 481, 821
