New Astronomy 6 (2001) 487–491www.elsevier.com/ locate /newast

Component thermalization time-scale estimate for the advection*dominated accretion flow around Sgr A

David TsiklauriSpace and Astrophysics Group, Physics Department, University of Warwick, Coventry CV4 7AL, UK

Received 18 January 2001; accepted 10 July 2001Communicated by R. McCray

Abstract

We report here on a calculation of thermalization time-scale of the two temperature advection dominated accretion flow(ADAF) model. It is established that time required to equalize the electron and ion temperatures via electron-ion collisions inthe ADAF with plausible physical parameters greatly exceeds the accretion time, which corroborates validity one of thecrucial assumptions of the ADAF model, namely the existence of a hot two temperature plasma. This work is motivated bythe recent success of the ADAF model [Nature 394 (1998) 651; MNRAS 304 (1999) 501] in explaining the emittedspectrum of Sgr A*, and it is complementary to the similar analysis of Mahadevan and Quataert [ApJ 490 (1997) 605]. 2001 Elsevier Science B.V. All rights reserved.

PACS: 98.62.M; 97.60.L; 98.35.JKeywords: Accretion, accretion discs; Black hole physics; Galaxy: center; Plasmas

1. Introduction that the luminosity of the central object should be40 21more than 10 erg s , provided the radiative

Identification of the true nature of enigmatic radio efficiency is the usual 10%. However, observationssource Sgr A* at the Galactic center has been a indicate that the bolometric luminosity is actually

37 21source of debate since its discovery. Observations of less than 10 erg s . This discrepancy has been astellar motions at the Galactic center (Eckart and source of exhaustive debate in the recent past.Genzel, 1997; Genzel et al., 1996) and low proper The broad-band emission spectrum of Sgr A* can

21motion ( # 20 km s ; Backer, 1996) of Sgr A* be reproduced either in the quasi-spherical accretion24~indicate that, on the one hand, it is a massive model (Melia, 1992, 1994) with M . 2 3 10 M(

6 21(2.560.4)310 M object dominating the gravita- yr or by a combination of disk plus radio-jet(

tional potential in the inner # 0.5 pc region of the model (Falcke et al., 1993a,b). As pointed out bygalaxy. On the other hand, observations of stellar Falcke and Melia (1997), quasi-spherical accretionwinds and other gas flows in the vicinity of Sgr A* seems unavoidable at large radii, but the low actual

~suggest that the mass accretion rate M is about luminosity of Sgr A* points toward a much lower26 216310 M yr (Genzel et al., 1994). This implies accretion rate in a starving disk. Therefore, Sgr A*(

can be described by a model of a fossil disk fed byE-mail address: tsikd@astro.warwick.ac.uk (D. Tsiklauri). quasi-spherical accretion. Recently, Tsiklauri and

1384-1076/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PI I : S1384-1076( 01 )00074-4

488 D. Tsiklauri / New Astronomy 6 (2001) 487 –491

Viollier (1998) have proposed an alternative model the assumption (b) its validity would be hard tofor the mass distribution at the galactic center in verify as it is related to the yet unknown mechanismwhich the customary supermassive black hole is of viscosity in the accretion flow. The latter problemreplaced by a ball composed of self-gravitating, stands on its own in astrophysics. As concerns thedegenerate neutrinos. It has been shown that a assumption (a) it is from the field of plasma physics

6neutrino ball with a mass 2.5310 M , composed of were we seem to have more comprehensive in(

neutrinos and antineutrinos with masses m $ 12.0 comparison to astrophysics understanding of then2 2keV/c for g 5 2 or m $ 14.3 keV/c for g 5 1, underlying basic physical phenomena. Thus, moti-n

where g is the spin degeneracy factor, is consistent vated by the recent success (Mahadevan, 1998,with the current observational data. See also 1999) of ADAF model in explaining the emittedMunyaneza et al. (1998, 1999) for future tests of the spectrum of Sgr A*, we set out with the aim to checkmodel. Tsiklauri and Viollier (1999) have performed validity of the assumption (a). It should be noted thatcalculations of the spectrum emitted by Sgr A* in the similar previous analysis exists (Mahadevan andframework of standard accretion disk theory, assum- Quataert, 1997). These authors have investigated theing that Sgr A* is a neutrino ball with the above- form of the momentum distribution function formentioned physical properties, and established that at protons and electrons in an ADAF. They have shownleast part of the calculated spectrum, were the that for all accretion rates, Coulomb collisions areobservational data is most reliable, is consistent with too inefficient to thermalize the protons. The protonthe observations. distribution function is therefore determined by the

Probably the most successful model which is viscous heating mechanism, which is unknown. Theconsistent with the observed emission spectrum of authors also have established that the electrons canSgr A* has been developed by Narayan et al. (1995, exchange energy quite efficiently through Coulomb1998) (see also Manmoto et al., 1997). This model is collisions and the emission and absorption ofbased on the concept of advection dominated accre- synchrotron photons. Using the ADAF model, andtion flow (ADAF), in which most of the energy the fact that time required for the electrons andreleased by viscosity in the disk is carried along with protons to reach thermal equilibrium depends on thethe plasma and lost into the black hole, while only a accretion rate and also using standard estimate of thesmall fraction is actually radiated off. Recent papers accretion time, Mahadevan and Quataert (1997) haveby Mahadevan (1998, 1999) have significantly ad- set bounds on the critical accretion rate above whichvanced ADAF model. Inclusion of additional emis- electrons and protons become thermally well coupledsion component, namely synchrotron radiation from and so the assumption of the two-temperature plasma

6e created via decay of charged pions, which in turn is no longer valid, prohibiting existence of theare produced through proton–proton collisions in the ADAF.ADAF, has significantly improved fitting of the Sgr Our study is complementary to that of MahadevanA* spectrum in the low frequency band. After and Quataert (1997) as we take plausible physicalremoving the latter discrepancy ADAF model of the parameters which were obtained from the fit of theSgr A*, apart from the size versus frequency con- ADAF model to the observed spectrum and thenstraints (Lo et al., 1998 and references therein) which calculate thermalization time-scale for the electronsremain problematic for all current emission models and ions. Comparison of this time-scale to theof the radio source anyway, seems to be the most accretion time allows us to check the validity of theviable alternative. Thus, basic assumptions of the assumption (a).ADAF model should be carefully examined from thepoint of view physical consistency. As appropriatelypointed out by Mahadevan (1999), in order for the 2. The modelADAF solutions to exist two basic assumptions inplasma physics must be satisfied: (a) existence of a It would be reasonable to believe that if the timehot two temperature plasma, and (b) the viscous required to equalize the electron and ion tempera-energy generated primarily heats the protons. As to tures appears to be sufficiently large in respect to the

D. Tsiklauri / New Astronomy 6 (2001) 487 –491 489

accretion time, then one might be confident that the n ) and finally as the SI system of units is used,i212assumption of the existence of the two temperature ´ 5 8.8541878 3 10 F/m.0

plasma in the ADAF is physically justified. The Now, writing the thermal velocities V as V 5e,i e,i]]]relevant time scale for the temperature equalization k T /m in Eq. (3), these become closed set ofB e,i e,iœcan be calculated using well formulated methods ordinary differential equations for T and T . We sete i

known in plasma physics. The rate at which tempera- q 5 e and m as a mass of proton. We solvei i

ture equilibrium between the electrons and ions is numerically Eqs. (1)–(3) using Numericalapproached is determined by (see e.g. Melrose, Recipes Software, namely odeint driving1986): routine with fifth-order Cash–Karp Runge–Kutta

215method (tolerance error 10 ). Calculations weredTe (e,i) performed for the values of the number densities]5 n (T 2 T ), (1)eq i e 13 16 23dt ranging from 10 to 10 m . The upper end of the

number density’s range is taken according to thedT i ( i,e)]5 2 n (T 2 T ), (2) actual value of the n in the ADAF around Sgr A*eq i edt(Manmoto et al., 1997; Manmoto, 1999). In Fig. 1

23with the number density (in cm ) profile is given. Asestablished by Manmoto et al. (1997, 1999) such

2 2 (e,i)e q n ln Li(e,i) 2 2 23 / 2 ADAF number density profile corresponds to the]]]]]n 5 (V 1V ) . (3)eq 1 / 2 2 e i case when best fit of the ADAF model to the3(2p) pm m ´e i 0

observed emission spectrum is achieved. We gather(e,i)Here ln L is the Coulomb logarithm for electron- from this plot that the maximal value of the n

(e,i) 10 23ion collisions given by ln L 5 22.0 2 0.5 ln n 1 actually attained is somewhat less than 10 cme5 16 23ln T , (T . 1.4 3 10 K), e and q are charges of (10 m ) — the value we also use in our calcula-e e i

electrons and ions respectively, m , m and T , T are tions as a highest value of the number density.e i e i

their masses and temperatures (in Kelvin), V and V Naturally, as more dilute plasma is as much time wille i

are thermal velocities of the electrons and ions, n is be required to equilibrate electron and ion tempera-the number density of plasma (we have assumed the tures via electron-ion collisions.global charge neutrality of the ADAF, i.e. n 5 n 5 The results of our calculations are presented ine

23Fig. 1. The number density (n in cm ) versus radius profile which corresponds to the case when the best fit of the ADAF model (Manmotoet al., 1997; Manmoto, 1999) to the observed emission spectrum of the Sgr A* is achieved.

490 D. Tsiklauri / New Astronomy 6 (2001) 487 –491

Fig. 2. For the initial values of the temperatures we equalize the temperatures of ions and electrons9.5 12have used T 5 10 K and T 5 10 K respectively significantly exceeds the accretion time. For thee i

16 23(Mahadevan, 1998, 1999). At this point we need to number density 10 m which corresponds to theestimate the accretion time for the ADAF. The maximal value attained in the ADAF model of Sgrrelevant expression for the accretion time can be A*, the time of electron-ion temperature thermaliza-found in Mahadevan and Quataert (1997): tion is about 20 yr which is still about three orders of

magnitude larger than the estimated accretion time.25 21 3 / 2t . 1.8 3 10 a m r s. Therefore we conclude that the assumption of thea

existence of a hot two temperature plasma is valid,Here, a is the Shakura–Sunyaev’s viscosity parame- or at least initial temperature difference will not beter, which is usually 0.3 for the ADAF, m is the mass washed out by the electron-ion collisions within theof central object in solar masses and r is the radius in accretion time.Schwarzschild units (R ). Since we have done calcu- It should be acknowledged that the formulas useds

lations using parameters relevant for the Sgr A*, for in this paper, strictly speaking, are valid for non-6the m we take 2.5 3 10 . For the r we take mean relativistic plasma regime (Melrose, 1999) while the

4value of inner (3 R ) and outer (10 R ), i.e. (R 1 electron and ion temperatures concerned are re-s s in

R ) /2 5 5001.5 R . Using these values yields t 5 lativistic (for the electrons g . 200). However, theout s a5 227.5 3 10 s52.4 3 10 yr. temperature equalization time-scale obtained is so

We gather from the plot that the time required to large, that even inclusion of the relativistic effects in

Fig. 2. Solutions of Eqs. (1)–(3) for the various values of the number density (log–log plot). The thickest pair of lines corresponds to16 23 15 14 13 23n 5 n 5 n 5 10 m , then thinner ones with progressive order to n 5 n 5 n 5 10 , 10 , 10 m respectively. Solid lines corresponde i e i

to the T (t)’s whereas dashed lines correspond to the T (t)’s. Note that equalization of the temperatures occurs more and more later as thee i

plasma becomes more dilute, which is according to general physical expectations.

D. Tsiklauri / New Astronomy 6 (2001) 487 –491 491

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