Sgr A* as probe of the theory of supermassive compact objects without event horizon

0.24 MB·English

by  L. V. Verozub

Astron.Nachr./AN32X(200X)X,XXX–XXX
Sgr A* as probe of the theory of supermassive compact objects
without event horizon
L. VEROZUB
6
KharkovNationalUniversity,Kharkov,Ukraine
0
0
2
Receiveddatewillbeinsertedbytheeditor;accepteddatewillbeinsertedbytheeditor

n
a
J

9
2
Abstract. Inthe present paper some consequences of the hypothesis that the supermassive compact object in the Galaxy

centrerelatestoaclassofobjectswithouteventhorizonareexamined.Thepossibilityoftheexistenceofsuchobjectswas
1
substantiatedbytheauthorearlier.Itisshownthataccretionofasurroundinggascancausenuclearcombustioninthesurface
v
layerwhich,asaresultofcomptonizationofthesuperincumbent hotterlayer,maygiveacontributiontotheobservedSgr
2
6 A*radiationintherange1015 1020Hz.Itisfoundacontributionofthepossiblepropermagneticmomentoftheobjectto
÷
6 theobservedsynchrotronradiationonthebasisofBoltzmann’sequationforphotonswhichtakesintoaccounttheinfluence
1 ofgravitytotheirmotionandfrequency.WearriveattheconclusionthatthehypothesisoftheexistenceintheGalaxycentre
0 oftheobjectwithsuchextraordinarygravitationalpropertiesatleastdoesnotcontradictobservations.
6
0 Keywords:densematter,blackholephysics,gravitation,accretion,radiationmechanism:non-thermal
/
h
p
- 1. Introduction fieldequationsofgravitation(Verozub1991;Verozub2001).
o
Accordingtothephysicalprinciplesfoundingtheequations,
r
t An analysis of stars motion in the Galaxy centre gives gravity of a compact object manifests itself as a field in
s
a strong evidence for the existence here of a compact object Minkowski space-time for a remote observer in an inertial
v: with a mass of (3 4)106M⊙ associated with the radio frameofreference,butbecomeapparentasaspace-timecur-
÷
i source Sgr A* (Genzeletal.2003a; Scho¨deletal.2003; vature for the observer in comoving reference frames asso-
X
Ghezetal.2003; Ghezetal.2005). There are three kind ciated with particles freely moving in this field . If the dis-
r of an explanation of observed peculiarities of the object tance from a point attractive mass is many larger than r ,
a g
emission: the physical consequences, resulting from these equations,
–gasaccretionontothecentralobject-asupermassiveblack are very close to General Relativity results. However, they
hole(SMBH)(Coker&Melia2000;Narayan&Yi1995), areprincipallyotheratshortdistancesfromthecentralmass
–ejectionofmagnetisedplasmafromanenvironmentofthe . The spherically- symmetric solution of these equations in
SchwarzschildradiusoftheSMBH(Falcke&Markoff2000; Minkowski space-time have no event horizon and physical
Melia&Falcke2001), singularityinthecentre.
– explanations, based on hypotheses of other nature of the
Since these gravitational equations were successfully
central object ( clusters of dark objects (Maos1998) , a
tested by post-Newtonianeffectsin the solar system and by
fermion ball (Viollier2003), boson stars (Torresetal.2000;
thebinarypulsarPSR1913+16(Verozub&Kochetov2000),
Yuan,Narayan&Rees2004).
and the stability of the supermassive configurations pre-
dicted by them was sufficiently strictly substantiated
In the present paper some consequencesof the assump- (Verozub&Kochetov2001a), it is of interest to investigate
tion that emission of Sgr A* caused by the existence in the possibility of the existence of such a kind of an object
theGalaxycentreofasupermassivecompactobjectwithout in the Galaxy centre as an alternativeto the hypothesisof a
eventhorizonareexamined.Theexistenceofsuchstablecon- supermassiveblackhole.
figurationsofthedegeneratedFermi-gasofmasses102 1010
÷ ThegravitationalforceofapointmassMaffectingafree-
M⊙,radiusesofwhicharesmallerthantheSchwarzschildra- fallingparticleofmassmisgivenby(Verozub1991).
diusr ,isoneofconsequences(Verozub1996)ofthemetric-
g
Correspondenceto:[e-mail]; F = m c2C′/2A+(A′/2A 2C′/2C)r·2 , (1)
[e-mail] − −
(cid:20) (cid:21)
L.V.Verozub:SgrA*asprobeofthetheoryofsupermassivecompactobjectswithouteventhorizon 1
where We assume hereforcalculationsthatthe massM of the
A=f′2/C, C =1 r /f, f =(r3+r3)1/3. (2) centralobjectisabout3106M⊙. TheradiusR oftheobject
− g g canbefoundfromthedistributionofthedensityρmofmatter
insidetheobject.Itcanbeobtainedbysolvingtheequations
Inthisequationristheradialdistancefromthecentre,r =
g
2GM/c2, G is the gravitational constant, c is the speed of ofthehydrodynamicalequilibrium:
lightatinfinity,theprimedenotesthederivativewithrespect dp /dr =ρ g , dM /dr=4πr2ρ , (4)
m m m r m
tor. p =p (ρ ).
· m m m
Forparticlesatrest(r =0)
IntheseequationsM isthemassofmatterinsidethesphere
r
of the radius r, g = F/m is the force (3) per unit of
F = GmM 1 rg (3) mOarss, r = 2GM /mc2 is the Schwarzschild radius of the
− r2 " − (r3+rg3)1/3# sphere g and p =r p(ρ ) is the equation of state of mat-
r m
O
ter inside the object. The equation of state, which is valid
Fig. 1 shows the force F affecting particles at rest and
for the density range of (8 1016)gcm−3, is given by
particles,freefallingfrominfinitywithzeroinitialspeed,as
÷
(Harrisonetal.1965)
thefunctionofthedistancer =r/r fromthecentre.
g
n
p = m ρ c2, (5)
m m
∂n /∂ρ −
(cid:18) m m (cid:19)
whereinCGSunits
−4/9
6 1 n =Q ρ 1+Q ρ9/16 , (6)
2 m 1 m 2 m
4 (cid:16) (cid:17)
Q = 6.02281023 andQ = 7.748310−10 . Forthe gravi-
1 2
2
tationalforceunderconsiderationthere aretwo typesof the
0 solutionsofEqs.(4).Besidesthesolutionsdescribingwhite
dwarfsandnewtronstars,therearesolutionswithverylarge
F −2
masses.Forthemassof3106M⊙ theobjectradiusresulting
−4 fromEqs.(4)isabout0.4R⊙ =0.034rg =31010cmwhere
r istheSchwarzschildradiusoftheobject.
−6 g
Itseemsatfirstglancethataccretionontotheobjectwith
−8
a solid surface must lead to too large energy release which
−10 contradicts a comparatively low bolometric luminosity (
0 2 4 6 8 10 1036 erg s−1 ) of Sgr A*. However, it must be taken in∼to
r / r
g
account that the radial velocity of test particles free falling
frominfinityisgivenbytheequation(Verozub1991)
Fig.1.Thegravitationalforce(arbitraryunits)affectingfree-
fallingparticles(curve1)andparticlesatrest(curve2)near 1/2
C
apointattractivemassM. vff =c (1 C) . (7)
A −
(cid:20) (cid:21)
Thevelocityfastdecreasesatr <r .Asaresult,thevelocity
g
ItfollowsfromFig.1thatthegravitationalforceaffecting atthesurfaceisonlyabout3.3108cms−1.Consequently,at
free fallingparticleschangesits sign atr 2r . Although ·
we still never observed particles motion a≈t distgances of the the accretion rate M = 10−7 M⊙ yr−1 the amount of the
·
orderofr ,wecanverifythisresultforveryremoteobjects releasedenergyMv2/2isequalto3.41035ergs−1.
g
in the Universe – at large cosmological redshifts . Indeed,
considerasimplemodel-homogeneousselfgravitatingdust-
2. Atmosphere
ballsofdifferent-sizeradiusesr .Theforceactingonparti-
b
cles at the surface of such a ball is given by Eq. (1) where,
There are reasons to believe that the rate of gas accretion
inthiscase,r = (8/3)πc−2G̺ r3 istheSchwarzschildra-
g b b onto the supermassive object in the Galaxy centre due to
diusof the balland ̺ is its density.Ifthe balldensityis of
b star winds from surrounding young stars is of the order of
t(he 1o0rd−e2r9gofcmth−e3o)b,saenrdvetdhedeitnssritaydioufsmratitsernionttlhesesUthnaivnetrhsee 10−7 M⊙ yr−1 (Coker2001). Therefore,if the objecthasa
∼ b solidsurface,itcanhaveanatmosphere,basicallyhydrogen.
radius of the observed matter ( 1028cm ), then r r .
∼ b ≤ g During10Myr (anestimatedlifetimeofsurroundingstars)
It follows from Fig. 1 that under the sircumstances the ra-
dial accelerations F/m of speckle particles on the ball sur- the mass Matm of the gaseous envelope can reach 10M⊙.
If the density of the atmosphere is ρ , and the pressure is
facearepositive.Therepulsiveforcegiveriseadeceleration a
p = 2ρ kT/m , where k is the Boltzmann constant,T is
of the expansion of such a selfgravitating dust-like mass ( a a p
theabsolutetemperatureandm istheprotonmass,thenthe
Verozub2002 ). More accurate calculations at the redshift p
heightofthehomogeneousatmosphereis
0 < z 1 (Verozub&Kochetov2001b) give a good ac-
≤
cordancewithobservationdata(Riessetal.1998). h =2p /ρ g (R)=kT/m g (R), (8)
a a a m p m
2 Astron.Nachr./ANXXX(200X)X
whereg (R)isg atr =R.AtthetemperatureT =107K
m m 600
theatmosphereheighth 108cm.Theatmosphericdensity
a
ρ =M /4πR2h 1∼03gcm−3.Undersuchacondition,
a atm a
∼
ahydrogencombustionmustbeginalreadyinourtime.
Ofcourse,theaccretionrateinthepastcouldbeofmany 3D 400
- l
m 1
ordersmoreifsurroundingstarshavebeenbornina molec- c
ular cloud which was located in this region. (See a discus- @rg l2
sion in (Genzeletal.2003a; Ghezetal.2005). In this case 200 l
3
thecombustioncouldbeginmanytimeagoandmustbemore l
4
intensive. l5
A relationship between the temperature and density can 0
befoundfromthethermodynamicalequilibriumequations 1107 3107 5107 7107
T@KD
1 dL M˙ c2 dS
4πr2ρ drr =ǫpp+λM −T dt, (9) Fig.2.Relationshipbetweentemperatureandpressureofthe
a atm
homogeneousatmosphereforthevaluesoftheparameterλ:
where λ = 1,λ = 1/6,λ = 10−2,λ = 10−3,λ = 10−4 at
1 2 3 4 5
ac d M =3106M⊙andM˙ =10−7M⊙yr−1
L =4πr2 T4. (10)
r
3κρ dr
a
In these equations M˙ is the rate of the gas accretion , ǫpp ρa(R) = ρ0, T(R) = T0, Lr(R) = 0. At T0 = 107K the
istherateofthenuclearenergygenerationperunitofmass, luminosity L = 2.41031erg s−1 at ρ0 = 102g cm−3 and
λ is the portion of the energythermalized at accretion, S is L=5.61033ergs−1atρ0 =103gcm−3.
entropy,andaistheradiationconstant.Fortheproton-proton Thetemperatureatthesurfacecannotbemuchmorethan
cyclethevalueofǫppcanbetakentoafirstapproximationas the above magnitude. At the temperature T0 = 3107K and
(Schwarzschild1977;Bisnovatyi-Kogan2001) ρ =103gcm−3theluminosityL=1.01036ergs−1,i.e.is
0
avalueoftheorderoftheobservedbolometricluminosityof
−2/3
T SgrA*.
ǫ =2.5104X2
pp H 109 Due to gravitational redshift a local frequency (as mea-
(cid:18) (cid:19) (11)
−1/3 suredby a localobserver)differsfromthe onefor a remote
T
exp 3.38 ergg−1s−1, observerbythefactor1
× "− (cid:18)109(cid:19) #
(1+z )−1 =√C, (15)
whereX isthehydrogenmassfractionwhichisassumedto g
H
beequalto0.7,and
wherethe functionC = C(r) is definedby Eqs.(2). Fig. 3
κ=0.2(1+X ) showsthedependencyoftheredshiftfactoronthedistanced
H
(12) fromthesurfaceoftheobject.
+41024(1+X )ρ T−3.5 cm2g−1
H a The difference between the local and observed fre-
isthe absorptioncoefficientcausedbytheThomsonscatter- quency can be significance for the emission emerging from
ingandfree-freetransitions. small distances from the surface. For example, at the
In the stationary case for the homogeneous atmosphere above luminosity L = 5.61033ergs−1 the local frequency
wehave of the maximum of the blackbody emission is equal to
5.71014Hz. The observed frequency of the maximum is
·
acT4 Mc2 equal to 2.71012Hz. The correspondingspecific luminosity
=ǫ +λ . (13)
3κρ2h2 pp M is 21021ergs−1Hz−1.AtL=1036ergs−1thefrequency
a a atm ∼
Fig. 2 showsthe relationshipbetweenT andρ forsev- 1 Thesimpestway toshow itisfollows. Fromtheviewpoint of
a
eralvaluesoftheparameterλ.Itfollowsfromthefigurethat aremoteobserver,theenergyintegralofthemotionofaparticlein
the hydrogenburningcan occuratthe densityof 102 103 spherically-symmetricfieldis(Verozub1991)
gcm−3. ÷
∂ ∂
Aluminosityfromtheburninglayeranddensitydistribu- r˙∂Lr˙ +ϕ˙∂Lϕ˙ −L=E
tioncanbefoundbysolvingthedifferentialequationssystem
where istheLagrangefunction
dM dp L
drra =4πr2ρa, dra =ρag(R), = mc c2C Ar˙2 Bϕ˙2 1/2
(14) L − − −
dL dT 3κρ L
drr =4πr2ρaǫpp, dr =−16πacar2rT3, andEisth(cid:0)eenergyofthepartic(cid:1)le.Intheproperframeofreference
oftheparticletheaboveequationstaketheform: =E,and =
where Mra is the mass of the spherical layer from the sur- mc2√C.Photonscanbeconsideredasapartic−leLsoftheeffecLtive
facetothedistancerfromthecentretheobject.Thebound- p−ropermassmeff = hν/c2.Ityieldstherelationshipν√C = ν∞,
aryconditionsonthesurfaceareoftheform:Mra(R) = 0, whereν∞isthefrequencyatinfinity.
L.V.Verozub:SgrA*asprobeofthetheoryofsupermassivecompactobjectswithouteventhorizon 3
where
250
3K (x)+K (x)
3 1
φ=3+x 1
4K (x) −
200 (cid:18) 2 (cid:19) (18)
3K (y)+K (y)
3 1
+y 1 ,
4K (y) −
150 (cid:18) 2 (cid:19)
g
1+z x = mec2/kT,y = mpc2/kT andKj (j = 2,3)isthe ith
100 ordermodifiedBesselfunction,me istheelectronmass
The sound velocity v as a function of r can be found
s
(Service1986)as
50
1/2
ΓP
v =c , (19)
0 s ρc2+P
0 0.5 1 1.5 2 2.5 3 (cid:18) (cid:19)
d / R
where ρ = m n is the gas (fully ionised hydrogenplasma)
p
Fig.3. The gravitational redshift close to the object surface
density.Theadiabaticindex
of the mass M = 3106M⊙ and radius R = 0.04rg as the
functionofthedistancedfromthesurface(inunitsofR). Γ=c /c , (20)
p v
where
of the maximum is equal to 21015Hz , the observed fre-
quency is equal to 9.71012Hz, and the specific luminosity c =5ςΘ−1 (ς2 1)Θ2),
p
is 11023ergs−1Hz−1.Thesemagnitudesareclosetoob- − −
∼ cv =cp 1, (21)
servationdata.(Melia&Falcke2001;Falckeetal.2000) −
ς =K (Θ−1)/K (Θ−1),
3 2
3. Peculiarityofaccretion andΘ=kT/m c2.Then
e
Themainpeculiarityofasphericalsupersonicaccretiononto
supermassive objects without event horizon is the existence P Θ
= , (22)
of thesecondsonic point– ina vicinityoftheobjectwhich ρc2+P η
is not connected with hydrodynamicaleffects. The physical
where η = K (Θ−1)/K (Θ−1). Having used results by
reason is that as the sound velocity v in the incident flow 3 2
s
Pressetal.1986 Service (Service1986) have obtained the
growstogetherwiththetemperature,thegasvelocityv,like
followingpolynomialfittingforΓandP/(ρc2+P):
the velocity of free falling particles (7 ) , begin to decrease
closetor =r .Asaresult,theequalityv =v takeplaceat
g s P/(ρc2+P)=0.03600y+0.0584y2 0.1684y3,
somedistancers >R,notfarfromthesurfaceoftheobject. −
According to the used gravitation equations Γ=(5 1.8082y (23)
−
(Verozub1991), the maximal radial velocity of a free +0.8694y2 0.3049y3+0.2437y4),
−
fallingparticledoesnotexceed0.4c.Therefore,theLorentz-
factor is nearly 1, and to find rs we can proceed from the where y = Θ/(0.36 + Θ). We assume that M˙ =
simplehydrodynamicsequations: 10−7M⊙ yr−1 andthatatthedistancefromthecentrer0 =
1017cm the velocity v(r ) = 108cm s−1 and the tempera-
0
ture T(r ) = 104K . Under these conditions r = 8R. At
· 0 s
4πr2v =M, vv+n′/n=g this point the solutions of the equations have a peculiarity.
ε ′ n′ (16) The temperature T(rs) 1010K. The value of rs weakly
P =0. dependens on M˙ , howev∼er, it is more sensitive to physical
n − n
(cid:16) (cid:17) conditionsat large r. For example, it varies from the above
Intheseequations.nistheparticlesnumberdensity,εis magnitude8Rupto22RatthetemperatureT(r )=106K.
0
the energydensityof the gas, g = F/m is the gravitational
The postshock region stretches from r to such a depth
s
acceleration. The gas pressure P is taken as 2nkT, and the
where protons stopping occurs due to Coulomb collisions
primedenotesthederivativewithrespecttor. Therefore,as
withatmosphericelectrons.(Wedonottakeintoaccountcol-
inthecaseoftheBondiaccretionmodel,weneglectradiation
lective effectsin plasma ). The changein the velocityv of
p
pressure,anequipartitionmagneticfieldwhichmayexistin
infallingprotonsinplasmaisgivenbythesolutionofthedif-
theaccretionflow,viscosityandlossofenergyduetoradia-
ferentialequation
tion.However,theenergydensityoftheinfallinggasin(16)
iscalculatedmoreaccurate(Coker&Melia2000):
dv
ε=mpc2n+φnkT, (17) vp drp =g+w, (24)
4 Astron.Nachr./ANXXX(200X)X
wherewisthedecelerationofprotonsduetoCoulombcolli- The first-order moment equation of the standard transfer
sions. Thismagnitudeis of the form(Alme&Wilson1973; equation for a spherically- symmetric gray atmosphere can
Li&Petrasso1993;Deufel,Dullemond&Spruit2001) bewrittenas(Schwarzschild1977)
4πNe4 ′ 2
w = f(x ) lnΛ. (25) H + H +cκρU jρ=0, (33)
e
− m m v r −
e p p
In this equation e is the electron charge, N is the particles whereU isthedensityoftheradiationenergy,jandκarethe
number density. At r < 1.3R the last magnitude is taken meanemissionandadsorbtioncoefficients,respectively.
hereasthequantityn = ρ /m resultingfromthesolution Eqs.(32)and(33)yield
a a a
ofEqs. (14),andastheparticlesnumberdensityin afree-
fallinggas L′4πr2ρ=(j cκU). (34)
r −
M˙ Therefore,theappropriateforthepurposeofthispaperequa-
n= , (26)
4πr2v m tionoftheenergybalanceisoftheform
ff p
atr >1.3R,wheren>na.Themagnitude ǫ =j cκU. (35)
cul
−
3 (kT)3/2
lnΛ=ln (27) To a first approximation, j and κ can be taken as (
2√πe3 N1/2
Bisnovatyi-Kogan2001)
is the Coulomblogarithm.The functionf(x ) is taken here
e j =51020ρT1/2+6.5UT
as
(36)
+2.3TB2 ergg−1s−1
f(xe)=ψ(xe) xe(1+me/mp)ψ′(xe), (28) and
−
where κ=0.4+6.41022ρT−7/2
(37)
x = mevp2 1/2, (29) +6.5c−1Tγ +2102B2T−3 cm2g−1,
e
2kT
! where T = (U/a)1/4 is the radiation temperature and B
γ
ψistheerrorfunction is a possible magnetic field close to the object surface (See
Section 5). Terms in (36) describe free-free, Compton and
2 xe
ψ(x )= exp( z2dz (30) synchrotronemission, respectively.Terms in ( 37 ) describe
e
√π Z0 − theinverseprocesses,andthefirsttermtakesintoaccountthe
andψ′(x )isthederivativewithrespecttor. Thomsonscattering.
e
Thevelocityvff iscloseto thegreatestpossiblevelocity The rate ǫcul of the energy release due to Coulomb col-
of the infalling gas. It leads to the minimally possible dis- lisions per unit of mass is given (Alme&Wilson1973) by
tance of a point of the full proton stopping from the object
surface. A numerical solution of Eq. ( 24 ) shows that this
4πe4
magnitude is equal to (0.3 0.4)R. The particles number ǫcul =f(xe) lnΛ. (38)
densityinthisregionis 10÷14cm−3 (Weneglectsomedif- mpvp
∼
ferencebetweentheheightofprotonsandelectronsstopping IftoneglectadsorbtiontermsinEq.(35)andthegrav-
(Bildstenetal.1992)).Abovethisregionthemostpartofthe itationinfluenceinsidethestoppingregion, itispossibleto
kineticenergyprotonsisreleased. obtain some simple analytic relationships between the tem-
The distribution of the temperature T(r) in this region perature T and the energy E of a decelerating photon. In-
p
can be obtained from an equation of the energy balance. deed,the value of N at the distance r = 6R is of the order
(Suchaquestionfornutronstarshasbeenconsideredearlier of108cm−3.Atthetemperatures 1010Kthefree-freeand
by Zel’dovich & Shakura (1969) , Alme & Wilson (1973), Compton radiations are predomin∼ated. Taking into account
Turollaetal. (1994),Deufeletal.(2001),and,formagnetic thatx 1,weobtain
e
whitedwarfs,byWoelk&Beuermann(1992)).Supposethat ≪
allprotonsstoppingpowerisconvertedintoradiation.LetH 4 m 3/2 E 3/2
f(x ) e p , (39)
be the radiativefluxand Lr = 4πr2H – the fluxtroughthe e ≈ 3√π (cid:18)mp(cid:19) (cid:18)kT(cid:19)
sphereoftheradiusr.Thentheenergybalancemeansthat
whichyieldstherateoftheenergyreleaseperunitofmass:
L′ =4πr2ǫ ρ (31)
r cul
NE
where ǫcul is the rate of the energyrelease due to Coulomb ǫcul =ξT3/2p, (40)
collisions,ρ=m N isthegasdensityandtheprimedenotes
p
where
thederivativewithrespecttor.Theleft-handsidein(31)is
L′ =4πr2 H′ + 2H . (32) ξ = 16√πmee4 . (41)
r r 3√2m2k3/2
(cid:18) (cid:19) p
L.V.Verozub:SgrA*asprobeofthetheoryofsupermassivecompactobjectswithouteventhorizon 5
Then, if the largestcontributionprovidesfree-freeradiation
1010
( the field B is weak ) , it follows from the equation of the
B = 0
energybalancethat B = 103 (R/r)3 G
T =(δ E )1/2 (42) B = 104 (R/r)3 G
1 p 109
whereδ1 =2.361024K2erg−1.Forexample,atr =6Rthe K ] )
protonenergyEp = 4.810−6ergwhichyieldsthetempere- T [
tculoreseTto=r3=.3110.49KR., SainshcearEppdriospaifnastthedetcermeapseirnagtufruentcatkioens Log ( 108
placeatthedistance 0.4Rfromthecentre.
∼
Ifthemagneticfieldissufficientlystrong,themaincontri-
butionintheenergybalanceprovidessynchrotronradiation.
107
In this case the relationship between the temperature T and
theenergyE oftheprotontakestheform 0.5 1 1.5 2
p
r [cm] x 1011
2/5
T = δ NEp (43) Fig.4. ThetemperatureT asthe functionof r inside there-
2 B2 gion of protons stopping for the object of the radius R =
(cid:18) (cid:19)
0.04r . The boundary conditions are: at the distance from
whereδ2 =1024K5/2erg−1cm3andB 6=0. the sugrface 5R the luminosity L0 = 1036erg s−1, the tem-
Ingeneralcase,assumingthatatthedistancer0 =6Rthe perature T = 5.4109K, the velocity of protons v =
luminosity L(r0) = 1036ergs−1 , the energyflux U(r0) = 2.32109cm0s−1. The plots for three value of the magp0netic
L(r )/4πr2candthe protonsvelocityv (r ) = v (r ), we
0 0 p 0 ff 0 fieldBareshown.
findfromEq.(35)thatthetemperatureT(r )=5.4109K.
0
On the other hand, if the absorption is not ignored, Eq.
(35)yieldsthedifferentialequation
To find the temperature distribution from the surface to
∂Φ(r,T)/∂r distances d R more correctly, equations of the structure
T′ = , (44) ∼
ofloweratmosphere,accretionandenergybalanceshouldbe
−∂Φ(r,T)/∂T
solved simultaneously. It can affect the value of the atmo-
whereΦ(r,T) = ǫcul j+cκU.Asimultaneouslysolving spheric density at the surface. However, as it was noted in
−
ofthisdifferentialequationtogetherwiththeequations Section2,ourknowledgeofthismagnitudeinitiallyisrather
uncertain.Thedensityofρ 103gcm−3(whichcorresponds
1U′ =κ Lr tothetemperatureT 107∼K)isthemostprobablesincean
3 4πr2c ∼
essentiallygreatervalueofthedensityortemperatureofthe
and(31)allowstoobtainthedistributionT(r)atthedistances bottomatmosphereleadstotheluminositywhichcontradicts
rwhereǫcul =0. theobservablebolometricluminosityofSgrA*.Atthesame
6
Fig.4showsatypicaltemperatureprofileintheregionof time the existence of a high temperature at distances where
protonsstopping. protonsstoppingoccurshardlyis a subjectto doubt.There-
Itisinterestingtoestimatethetemperatureatthesurface fore,thejumpofthetemperatureoftheorderof(102 103)K
÷
proceedingfromthesimplifyingassumptionthatallluminos- isaninevitableconsequenceoftheSgrA*modelundercon-
ityofSgrA*(L 1036ergs−1)iscausedbynuclearburning sideration.
∼
at the surface and stopping of incidentprotonsat some dis-
tance d R from the surface. Under such a condition, an
≤
effectivetemperatureT onthetopofthedenseatmosphere(
0
d 0.4R)isabout104K.Theradialdistributionofthetem-
∼ 4. Comptonization
peratureinthediffusionapproximationcanbedeterminedby
thedifferentialequation
(T4)′ = 3χH, (45) AnemergentintensityI ofthelow-atmosphericemissionaf-
4b terpassagethroughahotsphericalhomogeneouslayerofgas
s
isa convolutionofthe PlankianintensityI oftheemission
whereχ=κρ,b =5.7510−5ergcm−2K−4istheStephan- 0
s ofaburninglayer,andthefrequencyredistributionfunction
Boltzmannconstant,andH istheradiativefluxwhichissup-
Φ(s)ofs =lg(ν/ν ),whereν andν arethefrequenciesof
0 0
posedtoconstant.Itgivesapproximately
theincomingandemergingradiation,respectively:
χ
T4 T4 = H∆r, (46)
− 0 b
where ∆r = 0.4R . Setting χ = 1 we find that at the sur-
facethetemperatureisabout107Kwhichcoincideswiththe ∞
I(x)= Φ(s)I (s)ds. (47)
magnitudeusedinSection2. 0
−∞
Z
6 Astron.Nachr./ANXXX(200X)X
At the temperatureof the orderof 1010K the dimensionless
19
parameterΘ=kT/m c2 1.Undertheconditionthefunc-
e
∼
tionΦ(s)canbecalculated(Birkinshaw1999)asfollows: -1LDz 18 Τ=0.5
H
m e−ττk -1 17 +
Φ(s)= P (s), (48) s
k g
Xk=0 k! HEr 16 Τ=0.1
where @LΝ +Τ=0.03
1 og 15
P (s)= ϕ(β )P(s,β )dβ . (49) L Τ=0.01
1 e e e
Z(βe)min
14 15 16 17 18 19 20
In the last equations β is the electron-light velocity speed
e Log@ΝHHzLD
ratio,
Fig.5. Emergent spectra of nuclear burning after comp-
γ5β2exp( γ/Θ)
ϕ(β )= − , (50) tonizationbyahomogeneoushotlayerfortheopticalthick-
e ΘK (Θ−1)
2 nessτ =0.01, 0.03,0.1,0,5.Thecrossesdenotetheobser-
γ =(1 β2)−1/2., vation data (Baganoffetal.2003; Porquetetal.2003) (The
− e upperandlowerlimitsarespecified).
epsp 1
(β ) = − (51)
e min epsp+1
5. Transfer equation andsynchrotron radiation
and
3 µ2 ofSgrA*
P(s,β )= (1 β µ)−3
e 16γ4β − e
e Zµ1 Some authors (Robertson&Leiter2002) find in spectra of
1
1+β µ′)(1+µ2µ′2+ (1 µ2)(1 µ′2) dµ, candidatesintoBlackHolessomeevidencesfortheexistence
e
× 2 − −
(cid:18) (cid:19) ofa propermagneticfield of these objectsthatis incompat-
iblewiththeexistenceofaneventhorizon.Inordertoshow
es(1 β µ) 1
µ′ = − e − , (52) howthemagneticfieldnearthesurfaceoftheobjectinques-
βe tion can manifests itself, we calculate the contribution of a
magnetic field to the spectrum of the synchrotronradiation.
1 s 0 Weassumethatthemagneticfieldisoftheform
µ1 = 1−−e−s(1+βe) s≤ 0 (53)
( βe ≥ R 3
B =B , (55)
int 0
r
1−e−s(1−βe) s 0 (cid:18) (cid:19)
µ2 =(1 βe s ≤0. (54) whereB0 =1.5104G.
≥
To find the emission spectrum from the surface vicin-
Takingintoaccountscatteringupto3rdorder( m = 3)
ity, the influenceof gravitationon the frequencyof photons
an approximate emergent spectrum of the nuclear radiation
and their motion must be taken into account. An appropri-
aftercomptonizationhasbeenobtainedforthecasewhenthe
ate transfer equation is the Boltzmann equation for photons
temperatureatthesurfaceis107Kandρ=3103gcm−3.The
which uses equationsof their motionin stronggravitational
results are plotted in Fig. 5 for several values of the optical
fields resulting from the used gravitation equations. (Such
thicknessτ .
an equation in General Relativity has been considered by
It follows from this figure that the comptonization
Lindquist(1966),Schmidt-Burgk(1978),Zaneet al. (1996)
may give a contribution to the observed Sgr A* radia-
andPapathanassiou&Psaltis(2000))TheBoltzmannequa-
tion (Baganoffetal.2003; Porquetetal.2003) in the range
tionisoftheform
1015 1020Hz.
÷
Itisnecessarytonotethatthetimescale∆tofavariable d
F =St( ,t,x), (56)
processisconnectedwith thesize ∆ofits regionbythere- dt F
lationship ∆ c ∆t/(1+z ). For example,the variations
g
in the radiatio≤n intensity of 600 s on clock of a remote where = (t,x,p) is the phase-space distribution func-
observer occur at the distanc∼e d 6 1.7R from the surface. tion, xF= (xF1,x2,x3) and p = (p1,p2,p3) are the photon
Forthisreasonhigh-energyflashescanbeinterpretedaspro- coordinatesandmomentums,d/dtisthederivativealongthe
cessesnearthesurfaceofthecentralobjects. photonpath,St( ,t,x)isacollisionsintegral.Thesolution
It is possible that the diffuse, hard X-ray emission from of this equationFis related to the intensity I as = β−1I
the giant molecular clouds of central 100pc of our Galaxy whereβ =2h4ν3c−2andν isthelocalfrequencyF.
(Parketal.2004; Cramphorn&Sunyaev2002) is a conse- Inthestationaryspherically-symmetricfieldthedistribu-
quenceofthe moreintensiveX-rayemission ofSgr A*due tionfunction = (r,µ,ν)whereµ = cosθ isthecosine
F F
tonuclearburstsinthepast. betweenthe directionof the photonandthe surfacenormal.
L.V.Verozub:SgrA*asprobeofthetheoryofsupermassivecompactobjectswithouteventhorizon 7
Thepathderivativetakestheform where,K (1/Θ)is the modifiedBessel functionofthe sec-
2
ondorder,
d ∂ ∂ ∂ν ∂ ∂µ
F =v F + F + F , (57)
dt ph(cid:18)∂r ∂ν ∂r ∂µ ∂r(cid:19) (ζ)= 4q1 1+ 0.4q2 + 0.53q3 e−1.8896ζ1/3, (65)
M ζ1/6 ζ1/4 ζ1/2
wheretheradialcomponentofthephotonvelocitydr/dt = (cid:18) (cid:19)
vph isgivenby(Verozub1996) ζ =2ν/3νbΘ2andνb =eB/2πmecisthenonrelativisticcy-
clotronfrequency.Theparametersq ,q andq arefunctions
1 2 3
C Cb2 ofthetemperatureT.TheyhavebeenobtainedfromTable1
vph =c A 1− f2 , (58) of the above paper. The true adsorbtion opacity χth is sup-
(cid:20) (cid:18) (cid:19)(cid:21) posedtoberelatedtoη viaKirchofflawasχ =η /B ,
th th th ν
andbisanimpactparameter.Thefrequencyasafunctionofr whereB isthePlanckfunction.
ν
isgivenbytheequationν =√Cν∞.Thefunctionµ=µ(r) For the population of non-themal electrons we use
satisfythedifferentialequation a power-law distribution ( O¨zel,Psaltis&Narayan2000 )
which is proportionalto γ−α where γ is the Lorentz factor
dµ dθ
=(1 µ2)2 , (59) andtheparameterαissupposedtobeequalto2.5.Forsuch
dr − dr
a distribution the emissivity of the non-thermal electrons is
where(Verozub1996) approximatelygivenby
dθ = cCb. (60) e2N ν −0.25
dr f2 η =0.53δ ϕ (Θ)ν , (66)
nth 1 b
c ν
(cid:18) b(cid:19)
In the above equations the geometrical relationship b2 =
where
r2(1 µ2)mustbetakenintoaccount.
−
Withregardstoallstatedabove,theBoltzmannequation 6+15Θ
ϕ (Θ)=Θ (67)
alongthephotonpathscanbetakenintheform 1 4+5Θ
d 1 andthefractionδ oftheelectronsenergyinthethermaldis-
v F = (η+σJ) (χ+σ) . (61)
ph dr β − F tributionissupposedtobetheconstantinradiusandequalto
0.05.
Theright-handsideofthisequationistheordinarysimplified
The adsorbtion coefficient for non-thermal electrons is
collisionsintegralforanisotropicscattering(Mihalas1980)
takenas
intermsofthefunction .Inthisequation
F e2N ν 2.75
1 χ =1.371027δ ϕ(θ)ν . (68)
J =(1/2) β dµ, (62) nth cν b ν
F (cid:18) b(cid:19)
Z−1
ForacorrectsolutionofEq.(61)isnecessarytotakeinto
isthemeanintensity,ηisthesynchrotronemissivity,χisthe
accountthatfromthepointofviewoftheusedgravitational
trueadsorbtionopacity,σistheThomsonscatteringopacity.
equationstherearethree typesofphotonstrajectoriesin the
If (r,µ) is a solutionof the integro-differentialequa-
Fν spherically-symmetricgravitationfield.ItcanbeseeninFig.
tion(61),thespecificradiativefluxis
6.Thefigureshowsthelocusinwhichradialphotonveloci-
1 tiesareequaltozero.Itisgivenbytheequation
H =(1/2) β (r,µ)µdµ (63)
ν ν
Z−1 F b= f . (69)
Therefore,bysolvingEq.(61)forallparameters0<b< √C
∞
anemergentspectrumofSgrA*canbeobtain.Foranumeri-
The minimal value of b is b = 3√3r /2. The corre-
calsolvinganiterativemethodcanbeusedwithnoscattering cr g
spondsing distance from the centre is r = √319r /2. The
asastartingpoint. cr g
photonswith an impact parameter b > b , the trajectories
Weassumethatatthesurfaceoftheobject(duetononsta- cr
ofwhichstartatr < r (type1inFig.6),cannotmoveto
tionaryMHD-processes)andalsohigher,intheshockregion, cr
infinity. It can do photons with b > b , if their trajectories
asmallfractionsoftheelectronsmayexistinanon-thermal cr
startatr >r (type3),andallphotonswithb<b (type
distribution. Therefore,following Yuan, Quatert & Narayan cr cr
2).
(2004),wesettingη =η +η andχ=χ +χ ,where
th nth th nth
In order to find the solutions of equation (61), de-
indexesthandnthrefertothemalandnonthemalpopulation, F+
scribingthe outgoingemission providedby photonsof type
respectively.
2,theboundarycondition (r = R,ν,µ) = 0canbeused.
There is a fitting function for the synchrotron emis- F
Theboundarycondition (r ,ν,µ) = 0allowstofind
sivity of the thermal distribution of electrons which F → ∞
the solutions , describingthe emission ingoingfromin-
is valid from T 108K to relativistic regimes ( F−
∼ finity.Thevaluesofthissolutioncanbeusedasaboundary
Madehavan,Narayan&Yi1996):
conditionstofindtheoutgoingemissionprovidedbythepho-
Ne2ν tonsoftype3(Zaneetal.1996).Ofcourse,thevalueofthe
η = (ζ), (64)
th √3cK2(1/Θ)M functionF = β−1Bν,whereBν isthePlanckfunction,can
8 Astron.Nachr./ANXXX(200X)X
It follows from the figure that the proper magnetic field
10
canmanifestitselffortheremoteobserveratthefrequencies
largerthat(2 4)1012Hz–inIRandX-rayradiation.
÷
8 ItfollowsfromFigs.5 and7 thatthe perculiarityofSgr
A*spectrumatν >1014Hzcanbeexplainedbybothphys-
ical processes at the object surface and by the existence of
6
a non-thermal fraction of electrons in the gas environment.
g
b / r We do not know at the moment which of these sources is
4 3 predominant.However,theopportunityoftheexplanationof
1 flareactivityofSgrA*byphysicalprocessesatthesurfaceis
ratherattractive.
2
2 ThemeasurementsofthepolarisationofSgrA*emission
offer new possibilities for testing of models of this object.
0
0 1 2 3 4 5 6 7 AccordingtoobservationsbyBoweretal.(2003)atthefre-
r / r quency 2.31011Hz, the Faraday rotation measure (RM) is
g
2106radm−2. In our case at this frequency the optical
Fig.6.Thefunctionb(r).Thereare3typesofphotonstrajec- ≤
depth τ of the atmosphere become equal to 1 at a distance
tories:1)Theimpactparameterb>b ,thetrajectorybegins
cr of r 7r from the centre. The influence of the inter-
atr <rcr,2)b<bcr,3)b>bcr,thetrajectorybeginsatr > nal mmiangn≈etic figeld is insignificant at here. The value of the
r .
cr RM is 107radm−2. This magnitudeis notdifferedfrom
∼
that obtained in the model by Yan, Eliot & Narayan (2004)
based on the SMBH - theory. Taking into account that we
beusedasaboundaryconditiontofindemissionfromlarge
knowthe distributionofthe densityandmagneticfield very
opticaldepths.
approximately,we cannotsay that the obtained value of the
Fig.7showsthespectrumofthesynchrotronradiationin
RMcontradictobservationdata.
thebandof1011 21018Hzforthreecases:
–forthemagneti÷cfieldB (55)oftheobjectatthegibrid At increasing of the frequency the value of rmin de-
int creases, and influence of the proper magnetic field must in-
distribution of electrons with the parameter δ = 0.05 (the
creasebecausetheradiationcomefrommoredepthoftheat-
dashedline),
moshere.Atfrequenciesmorethan21012Hztheatmosphere
–forthesumofB andanexternalequipartitionmagnetic
int becomethetransparentcompletelytothesynchrotronradia-
fieldB = (M˙ v r−2)1/2 whichmayexistintheaccretion
ext ff tion. Therefore, the full depolalization of the radiation may
flow(Coker&Melia2000)(thesolidline),
givesomeevidencefortheexistenceofamagneticfieldclose
– for the magnetic field B without non-themal electrons
int thesurface.
(thedottedlinewhichatthefrequencies<1014Hzcoincides
with the dash line). in the presence of the proper magnetic
field 6. Conclusion
The clarification of nature of compact objects in the galac-
1025 ticcentresisoneofactualproblemsoffundamentalphysics
δ=0.05, B=Bint+Bext and astrophysics.The results obtained above,of course, yet
δ=0, B=B +B
−1Hz ] ) δ=0.05, Bi n=t Binetxt dthoeynosthaolwlowthatot tdhreawpotshseibwilietlyl,-dinevfiensetdigcaotendclhuesrieo,nas.tHlaoswt,deoveesr
−1Erg s 1020 sntoutdcyo.ntradict the existing observations, and require further
L [ ν
g (
Lo References
1015
Alme,M.,Wilson,J.:1973,ApJ186,1015
12 14 16 18
Log ( ν [ Hz ] ) Baganoff,F.,Maeda,Y.,Morris,M.etal.:2003,ApJ591,891
Birkinshaw,M.:1999,PhysicsReports310,97
Fig.7.Thespectrumofthesynchrotronradiation.Thedashed
Bildsten,L.,Salpeter,E.,Wasserman,I.:1992ApJ384,143
line shows the luminosity Lν for a remote observer for the Bisnovatyi-Kogan, G.: 2001 Stellar Physics, v. I and II, Springer
possible proper magnetic field Bint. The parameterδ of the Verlag
hybrid distribution of electrons is equal to 0.05. The dotted Bower,G.,Write,M.,Falcke,H.&Backer,D.:2003,ApJ588,331
lineshowstheluminositywithoutnon-themalelectrons.The Coker,R.,Melia,F.:2000,ApJ534,723
Deufel,B.,Dulemond,C.,&Spruit,H.:2001,A&A377,955
solidlineistheluminosityatthepresencebothB andthe
int
Coker,R.:2001,A&A375,18
external equipartition magnetic field B . The short lines
ext Cramphorn,C.&Sunyaev,R.:2002,A&A389,252
with the double arrows show observation data according to
Falcke,H.,Markoff,S.:2000,A&A362,113
O¨zeletal(2000) Falcke,H.,Melia,F.&Agol,E.:2000,ApJ528,13
L.V.Verozub:SgrA*asprobeofthetheoryofsupermassivecompactobjectswithouteventhorizon 9
Genzel,R.,Scho¨del,R.,Ott,T.,etal.:2003a,ApJ594,812
Genzel,R.,Scho¨del,R.,Ott,T.etal.:2003b,Nature425,934
Ghez,A.,Salim,S.,Horstein,S.,etal.:2005,ApJ620,744
Ghez,A.,Ducheˆne,G.,Matthews,K.etal.:2003,ApJ586,127
Harrison,B.,Thorn,K.,Wakano,M.,&Wheeler,J.:1965,Gravita-
tionTheoryandGravitationCollapse,Univ.ChicagoPress
Lindquist,R.W.:1966,Ann.Phys.37,487
Li,C.&Petrasso,R.:1993,Phys.Rev.Lett.70,3063
Madehavan,R.,Narayan,R.&Yi,I.:1996,ApJ465,327
Maoz,E.:1998,ApJ494,181
Melia,F.,&Falcke,H.:2001,ARA&A39,309
Mihalas, D.: 1980, Stellar Atmospheres, Freeman and Company,
SanFrancisco
Narayan,R.&Yi,I.:1995ApJ444,231
O¨zel,F.,Psaltis,D.,&Narayan,R.:1996,ApJ541,234
Papathanassiou,H.&Psaltis,D.:2000,e-printastro-ph/0011447
Park,S.,Muno,M.,Baganoff,F.,etal.:2004,ApJ603,548
Porquet,D.,Predehl,P.,Aschenbach,B.etal.:2003,A&A404,17
Press,W.H.etal.:1986,NumericalRecipes:TheArtofScientific
Computing,New-York,CambridgeUniv.Press
Riess,A.:1998,AJ116,1009
Robertson,S.&Leiter,H.:2001,ApJ565,474
Service,A.:1986ApJ,307,60
Schwarzschild , M.: 1977, Structure and Evolution of the stars,
Dover
Scho¨del,R.,Ott,T.,Genzel,R.etal.:2003,ApJ596,1015
Schmidt-Burgk,J.:1978,APSS56,191
Torres,D.,Capozziello,S.&Lambiase,S.:2000,Phys.RevD62,
104012
Turolla,R.,Zampieri,I.,Colpi,M.,etal.:1996,ApJ496,35
Verozub,L.:1991,Phys.Lett.A156,404
Verozub,L.:1996,AN317,107
Verozub,L.&Kochetov,A.:2000,Grav.andCosm.6,246
Verozub,L.&Kochetov,A.:2001a,AN322,143
Verozub,L.:2001In:”Gravitation,CosmologyandRelativisticAs-
trophysics”,KharkovUniv.,44
Verozub, L. & Kochetov, A.: 2001b In: ”Gravitation, Cosmology
and Relativistic Astrophysics”, Kharkov Univ., 90 (e-print:
astro-ph/0010596)
Verozub,L.:2002AIPConferenceProc.624,371.
Viollier,R.:2003,AN324,50
Woelk,U.&BeuermannK.:1992,A&A256,498
Yuan,F.,Quataert,E.,&Narayan,R.:2004,ApJ612,724
Yuan,Y-F.,Narayan,R.,&Rees,M:2004,ApJ606,1112
Zane,S.,Turolla,R.,Nobili,L.,&Erna,M.:1996,ApJ466,871
Zel’dovich, Ya. B. & Shakura, N. I.: 1969 Soviet. Astron. AJ 13,
175