Astron. Nachr. 307 (1986) 3 , 183-189 183

Accretion disks in bimetric theory of gravitation. A strong field test of general relativity?

H. G. PAUL, Potsdam-Babelsberg

Zentralinstitut fur Astrophysik der Akademic der Wissrnschaftrn dcr DDR

With 3 figurcs (Received 1985 October 21)

The sensitivity of accretion disk models with respect to its inner boundary conditions, being located in the strong gravity region of a highly compact central body, is used to discuss a possibility of strong field tests of gravitation via compact source observations. Within the bimetric theory of gravitation the optically thin bremsstrahlung model is calculated as an example for this possibility. tising this model to describe the inncrmost region of the “bimetric disk”, significant differences with rcspect t o thc corresponding results in liin- stein’s theory were found. Thc more massiv and compact “bimetric disk” radiates softer with higher luminosity.

Die Empfincllichkcit von Akkrctionsscheibenmodellen in bezug auf ihre inneren Iiandbedingungen konnte fur rincn Test dcr A R T in1 starkrn Gravitationsfeld genutzt werden, da bei der Beachtung der inneren Energic dcr Scheibe diesc I<andbedingungen \wit nnterhalb der letzten stabilen Krcisbahn (innerer Rand der Standardmodclle) nahe Clem Zentralkorper liegen. Als ein Reispiel diescr lCIoglichkeit wurde das optisch dunne Rremsstrahlungsmodell in der bimetrischen Gravitationstheorie berechnet. Renutzt man diescs Model1 fur den innersten Bcreich tlcr ,,bimetrischen Sclieibc“, so findet man signifikantc Unterschiede zu den entsprechenden ,,Einsteinschen Scheiben“. Die massiverc rind kompaktere ,,bimetrische Scheibe“ strahlt ein wcichercs Spektrum aus dieser Region mit grdkrc r Hellig- keit ab.

1. Introduction

For testing general relativity there are now many experiments reaching from laboratory measured atomic processes, via relativistic effects in the solar system up to the astronomically observed phenomena in which relativity is important. But most significant and forcing experimental consequences ought to be expected in the last class of tests. Here phenomena take place in a strong gravitational field (appearing in consideration o f cosmology and compact galactic and extragalactic sources and objects, respectively). Unfortunately, these phenomena are tightly interwoven with the local hydro- and thermodynamics, There is little hope of separating the several effects suffi- ciently to get clean tests for general relativity.

But one of the most exciting possibilities seems to be connected with black holes or highly compact galactic sources. On the one hand, from the detection of black holes (by dynamic effects (e.g. GIBBONS, HAWKING 1971) and via the “no hair” behaviour of the compact component in single line binaries) and, on the other hand, from the observation of sufficiently strong phenomena in the immediate neighbourhood of high compact objects (e. g. PAUL 1985) tests could be inferred to the nature of gravitational theories. From observations of highly compact sources (especially galactic X-ray or y-sources) the last one possibility could arise in case that inner boundary processes of accretion disks, generally accepted to drive compact sources (see e.g. PRINGLE 1981, PAUL and RUDIGER 1985) with gravitational potential energy release dominance, would be situated in sufficiently strong gravity regions ( 2 = AAIA 2 0.4). These inner boundary processes seem to be found in form of a ring shaped inner cooling region ( I~ANG 1980, ZHANG and JIANG 1982, PAUL 1985, 1986) located on radii less than the radius of marginally stable test particle orbit (Y < Y~,), being the boundary of the standard disk models (eg. BARDEEN et al. 1972, NOVIKOV and THORNE 1973, PACE and THORNE 1974). In order to compare the relative values of the various effects, the inner cooling region is considered by PAUL (1985) without specifying the gravitational theory. There it was shown that gravity induced cooling located in the region of strong gravitation of the compact central object domi- nates the innermost disk region giving contributions which could be used to test theories of gravitation.

In order to visualize this possibility in the optically thin bremsstrahlung model (PRINGLE, REES, PACHOLCZYK 1973, PAYNE, EARDLEY 1977) the inner boundary behaviour (PAUL 1986) is considered in an alternative gravitation theory. The used alternative theory, formulated by KASPER and LIEBSCHER (1973) and containing the bimetric theory proposed by KOSEN (1973), flows out from a general variational principle for a large class of tetrad theories (LIEBSCHER 1977) as a Treder-type tetrade theory of gravitation (TREDER 1967). For this bimetric theory of gravi- tation the external static solution was found (KASPER and LIEBSCHER 1973, ROSEN 1973) necessary for the purpose of this note. In this external gravitational field of a compact central object a “bimetric disk” is constructed.

2.

Without substantial self-gravitation a thin accretion disk (thickness 12 <Y) orbits a compact object of mass M near and in the equatorial plane (10 - (n/z)l <I) . The central object is nonrotating and its external static, spherical

The optically thin bremsstrahlung disk in bimetric theory

184 Astron. Nadir. 307 (1986) 3

symmetric gravitational field is described by the line element

being the external static solution in the bimetric theory of gravitation (KASPER, LIERSCHER 1973, ROSEN 1973). (Sometimes the isotropic radius r = Ar is used as convenient quantity.) Of course, the central object has to be sufficiently compact for disks extending down into its strong gravity region. Such objects were found by ROSEN (15)77), SAHAKIAN, SARKISSIAN, KHACHATRIAN, CHOUBARIAN (1978), and PAUL (1982) in the bimetric theory. The corresponding crushing masses are shifted to higher values with respect to GRT in dependence on the used equation of state. Assuming ideal hadron gas for the core, the massive objects are much more compact than the corresponding one in Einstein’s theory (PAUL 1982). That means, for the purpose of this note, sufficient compact, massive objects could be assumed which lay within the radius of marginally bound test particle orbits.

Using the theoretical framework for calculating optically thin bremsstrahlung disks in general spherical symmetric space time (PAUL 1985), the equations of structure follow by substituting the metric functions (I) there, The disk matter can be described by the energy-momentum tensor TILv = eo(I + n) uupu,. + tllY + qpu, + qvzt/,. eo is the rest mass density and n = (3k/m,) T c 2.5. loR T <c2is the specific internal energy. t,,,. is the stress tensor and q/, is the energy flux vector (11 . q =- t . 21 = 0). The accretion disk matter moving with the velocity of orbit ‘u in the cffectivc potential

I -

possesses the specific angular momentum I:: the energy E ( L = Llm, angular velocity :

= E / m , m = unit rest mass) and the orbital

2”’ e ; M’i‘

Q y =T------ E 7 ( r - M)’i‘

\I -~ - r . y ,

- 7 2 0 1, -- - y s , E = c

The radial disk structure follows from the conservation law of rest mass

&lo -= 4n 1/-i / q 0 i 1 7 (2)

( r f - averaged radial accretion velocity measured in tlic coordinate frame of (I), h(r) - disk thickness), energy 7 E = o ( E = - 2 . 3,) and angular momentum 7 - J = o ( J = T ap). Together with the radial equations of structure, the vertical pressure balance equation, the equation of state for completely ionized hydrogen plasma, the vertical energy transport equation, and the a viscosity law (SHAKURA, S~JNYAEV 1972) one obtains for the tem- perature profil the differential equation

and for the density po(7), the thickness h(r), the pressure p ( r ) , and the vertical erg cm3

- - c0 =- 1.4432 . 10-27 5 Kli’

energy flux q(r) tlie expressions

po 2 (with the surface density 2’ 7 2 1 2 ~ ~ ) . (7) -- 22;

CO

The integral i = i r 2 Q d In can be calculated in the bimetric theory (I): 1 l ) l

I/:!

In contrast to the weak field approximation of (I) (Newtonian limit), describing the external gravitational field of the outer part of the disk, significant differences in the structure of the disk (3 , 4, 5, 6, 7) with respect to the corresponding one in the GRT can only be expected from the inner disk edge, being located in the strong field region. Therefore, the innermost part of the accretion disk is considered in the following investigation.

I’AUL, H. G : Liccrction disks 111 bimetric thcory 185

Tlie inner boundary Y,,, is determined by the sonic point Y , of tlic radial accretion flow, Y,,, = Y ~ , wliicli is defined bv the condition

(9)

(a5 is the velocity of sound.) The inner edge of the disk, rill, is located between the radii of tlie niarginally bound, Y,,,,, and the marginally stable, Y,,,, test particle orbit (PAUL 1985) in general, without to be confronted wit,li.tlic singular boundary conditions of the standard models (e.g. NOVIKOV, THORNE 1973, PAGE, THOI~NE 1974, STOEGEK 1976). Expressing the radii in units of the mass, Y* = ( Y / M ) , one gets

1 1 -

4.28 , r*il,s 2=1 f*,,,, er*mr - 6.34 y*,,,l, = y *inb erenib - < - - -

from drV2lrmsrmb = o, E2(Ylllb) = I and a~V2(Yll,J = o. In order to discuss tlie inner boundary region of thc disk the temperature T(r) and metric functions (I), determining the other functions of structure, are expanded in the neighbourhood of the inner edge i l l , . We express M,, M , and Y in units typical for compact galactic X-ray sources:

Substituting a power series 1‘ = A7{(Dl + b2 AY, -+ ...) , A?, = Y , - F*ll,l, < 1 (10)

in tlic equations (2 , 4, 5) tlic inner boundary nicasurccl in the radial polar coordinate ( I ) , is found by iiieans of tlic boundary condition (9). ( l l i c metric functions are given tlierc.) I t is

I’or a wide class of compact sources (with gravitational potential eiicrgy release dominance), deteriiiincd by the condition (AY, <I with (10))

(12)

the inner boundary Y, , , , is closely located near the margiiially bound orbit, Y * ~ , , ~ , and is do~iiinately determined by tlie viscosity parameter ix of the disk mattter.

By substituting tlie expansion (10) in the equation (3 ) wc get

Till N 5.48 1o’J I< OLIl;fi7 ’’’ ill, I

on the iiiner edge (11). By means of this temperature o ~ i c obtaiiis tlic density

on tlic inner boundary Y*;,,,, of the disk, because the functions 7, A42, and S2 are given there. Substituting tlie equations (11, 13, 14) in expression ( 5 ) the inner gap of the dissk is defined by thc thickness lz on the inner boundary:

/z,,, = 3.13 . 104 cm [A~17~~;oc11/3 . (15)

Tlic pressure $, the radial velocity a;, and tlie vertical energy q in the inner gap of the disk are obtained by substi- tuting the corresponding terms in the structure equations (6, 2, 7) :

With increasing mass Ad * of the central object the temperature l’,,,, the radial velocity vfn, tlie pressure pIl1, and the density ~,,,,, is decreasing because the inner gap of the disk grows due to increasing h,,,. On the other hand, a growing accretion rate M17 leads to a growth of all functions of structure TI,, PI,,, qIn, and the gap hin. An increasing viscosity (represented by a) sweels out the gap (15) and push up the temperature (13) and the radial velocity (17) a t decreasing density (14) there. Because tlie internal 7c was taken into consideration no singular boundary value problem appears in tlie inner gap of the disk at yin. The same was obtained for accretion disks cal- culated in the GRT. The singular boundary behaviour of the thin standard oc-disks (e.g. NOVIKOV, THORNE 1973, SHAPIRO, TEUKOLSKY 1984) can be removed there, too (PAUL 1986).

186 Astron. Kachr. 307 (1986) I

Of course, tlie deduced properties of tlie disk at the inner boundary are necessary conditions for the functions of structure only. These conditions are calculated on the assumption (12). That means accretion disks, fulfilling the inequality (12), possess necessarily the derived inner boundary behaviour. In the next higher order of Ai* (11) the values of the structure functions on the inner edge of the disk have to be corrected; and with respect to the parameters of structure a , &Il7, and M * the class of accretion disks are enlarged of course.

In order to compare tlie beliaviour of the innermost part of accretion disks with the corresponding one found in Einstein’s theory (PAUL 1986), the integrated quantities are calculated by numerical investigations of the equa- tions of structure ( 2 , 3, 4, 5 , 6, 7) throughout tlie inner cooling region and middle, hot region.These investigations are carried out without tlie restriction (12). But, of course, restrictions on the parameter of structureor, MI,, and M,, being valid throughout the disk, follow from the thin disk assumption h < Y and the subsonic flow condition g19 = (v;/a,) < I. By using the explicit expression of the thickness (5) we get

2 -

from tlic first one and I ~- , --- I -

a21’ < 5.45 . 10,’ K - I - z arc tg V Y * - I + z arc tan h f Y y , - I - 1.211 (Y* - 1)-2 ( z o ) from tlJc second one with \JJt, (20) and the corresponding explicitly expressed structure functions. Thc condition (IS)) is iulfilled into the gap for a wide class of disks,

d l N ,,_ - .17:.. 4 3 9 7 . 109 M,: (21 )

(witli (13)), and tlie incquality (9) exists as a definition there. With increasing tcmperature 1’ and thickness /z, turning up with increasing radius, stronger restrictions follow Iroin tlic inequalities (19, 10) on tlic parameter of structurc a, MI,, and N‘l’. Especially tlie inequality (19) ~

7~ -5.49 . 101, K ( 2 2 )

has to bc fulfilled in general, because at ?*,,,, = 11.197 (following from z = o in equatioii ( 3 ) ) T possesses a inaxi- nium T,,,;,, there. I n the neiglibourhood of I’,,,t,, the optically tliin brenisstraliluiig disk tends to bc self-inconsistence in the same way as in Einstein’s tlieory (PAUL 1986).

Taking values for tlie parameters of structure a , M17, and M,, in agreement with the bounds (IS), zo) , a disk model is presented as an example in the Einstein’s and bimetric tlieory, respectively. Both models are calculated for an accretion rate M,, = I and a viscosity parameter oc = 0.01. The models possess a central mass value of M , = I. For visuality the temperature profil 1’ of the disk is shown for tlie inner cooling and tlie middle hot region in 1;ig. I . Of course, for a more realistic model of disk parts laying outside of tlie innermost cooling region, the matter specifying assumptions have to be modified in tlie middle hot region. But tlie discussion of inner boundary pro- cesses, taking place in tlie innermost strong gravity region can be carried out witli the matter assumptions used liere and wliicli are accepted in the innermost boundary region (e.g. SHAIWIIA, SUNYAEV 1972. NOVIKOV, THORNE, PRICE 1975, PAYNE, EARDLBY 1977). Especklly the inner boundary region assumptions have to be corrected a t rtllUx, which leads to Q,, = p = o there (with i = o (8)).

3. Discussion

S o w the inner boundary beliaviour of the optically tliiii brenisstraliluiig disk, calculated in Einstein’s and the bimetric theory, respectively, will be compared for a discussion of the tlieory dependence of innermost disk phenomena, taken place in the region of strong gravitation of the central object.

10

6

06 0.8 I 0 Igr.

Fig. I . The temperature profil T and tlic radial accretion flow \doci ty ur (iiieasurctl in tlic orbiting coordiriatc lranic) is shown for the central body niass izf, : 1, an accretion rate i%I,, = 1, arid thc viscosity parameter a -= 0.01. Thc oyti- cally thin bremsstrahlung modcl calculated with thesc structure parameter is constructed in Einstein’s and the bimctric thcory of gravitation. The Einstein disk arid the binietric disk and its functions of structurc, rcspectivcly, are earmarked by “E” arid “b”, respcctively.

PAUL, H G. : Accretion disks it] biiiietric theory 187

0.6 0.7 0.8 0.9

liig. 2. 'l'lie luminosity for the niodel shown in l i g . I is pi-exntcd.

0.6 0.7 0.8 0.9

h'* - Fig. 3 . The surface density 2' and tlic tliickiicss h lor tlic Einstein and the bimctric disk is shown for tlie model prcsentcd in Fig. I .

On the inner boundary, 111 Einstein's theory tlic functions of structure of tlic disk are given by tlie expressions

188

and

(PAUL 1986). Writing down tlie inncir hounclary beliaviour (13, 14, 15, 16, 17, IS) in units of tlie corresponding quantities found in Einstein’s tlicory wc obtain

F(~)r a wi(lc range of structure ptra11iete . 1 0 1 (12). ’rliis iriccluality iollows fro111 iiiiicr clisk boundary value coiisidcrations in tlie biiiietric tlieory wliicli leads to stronger rcstrictiuiis tliaii i i i 13iu- stein’s tlicory. Tlie index 1) stands for “calculated in bimctric tlieory” and II; represents tlic corresponcliiig oiic in Eiiistciii’s tlieory.

I t is cviclciit from tlicsc expressions (23) tliat tlic “bimctric disk” is iiiucli liottcr tliaii tlic “l~iiistein disk’’ oii the iiincr boundary. This leads to a sinaller density poi,, and pressure pll, but a grcatcr gap lzlll there. This beliaviour is caused b y the greater local radial accretion stream uf,, in tlic inner gap of the “bimetric disk”. Tlicrefore, (from tlic inner edge) the vertical cncrgy flux, qill, giving the total energy flux, is greater tlian the corresponding ciicrgy flux in tlic “I<instein disk” because there tlie “bimctric disks” are more iiiassiv (I’Z > lczil,) and liotter but a little bit [ewer com1)act (1’/zi,, >lcIzi,,, I1poi,, <l;poil,), only. (See (7) connected with ( 2 3 ) . ) Tliat means, in suniinary, tlie inner boun- tlary of the “binietric disk” is Iiotter, niore massive and luminous but less compact with respect to the corresponding “Eiiistcin disks”. The deduced relations, between tlic “bimetric” and “Einstein disk” are indcpendciit of tlic para- iiicter of structure, and therefore they are valid for a wide range of disk models (rcstrictetl by the above mentioned incqu;ility (n4h~17/AJ;,) < 104 only). Tliis is caused by the fact tliat tlie qualitativc structure ol tlic disk (determined by the structure paraineter a, M17, M,: and followiiig from tlic local tliernio- and liydrodynamics) is not iiiflueiicccl by tlic glol)al gravitational field (rcpresentcd by (1.9)) in tlic case considered Iicre. But tlic quantitative features of the disk arc dctermined by tlie global actions of the gravitational field too, especially i n tlic rcgioii ol strong gmvi- tation near the central object.

I n order to compare the properties of tlic innermost boundary rcgion with the corresponding onc i n GICT tlie optically thin bremsstrahlung inodcls calculated numerically in this paper arc conlronted with tlic corresponding inodels in GR’r (PAUL 1986). For oiie model the comparison is presented in Fig. I , 2 and 3 as an example.

l‘irst of all, from the figures of the local physical quantities T , v;, h , and 2 it is evident that the distinction bc!tween tlie local properties of tlie binietric and Einstein’s disk can be neglectcd for increasing radius;. This beliaviour is forced by tlie approaching of tlie binietrjc and Einstein’s theory in great distances to the central object. We get the strongest difference iii the cooling region for Y 5 Y,,,. In this region the teinperature o f tlic biinetric disk is lower tlian in tlie Einstein’s disk caused by the lower local radial accretion flow (Fig. I). On tlie other hand, tlie bimetric disk is more compact (”h < “h) and massive(b2 >‘$Z) than the corresponding one of models in GKT which leads to a higher luminosity (lJL > ’{L, Fig. 2 and (7)). This seems to be in contradistinction with boundary behaviour cliaracterizecl by liotter radiation and less compactness ( 2 3 ) . (It holds ”Ti,, > ISTi,, and”/zill > li/ziI1.) But this contrast does not exist because the bimetric disk edge is located at greater radii possessing a larger gap there. Additionally, tlie cooling proces are stopped at greater radii near ”rInb > k 3 ~ l , l h with a higher temperature Till > Till Fig. I).

, and ‘11, ; it is ,xdAJl/iII7*:

>

4. Conclusions

C;alculatiiig ol)tically thiu 1)reuisstralrlung iiiotlcls for accretion tli in Einstein’s tlieory and bimctric gravitation theory, significant distinctions are found in tlie innermost cooling rcgioii only. Tlic more niassiv and compact “bimetric disk” radiates softer there with higher luminosity (’)L ’V 1.5 ”L). 1;rom tliis result tlit: argument could be inferred tliat the investigation o f inner boundary phenomena of accretion disks, taking place i n tlie strong gravity region of the very compact ccntral object, leads to a possibility for tests of the foundation of tlie GlCT in the strong field case. (Of course, for this reason, the tlirougliout optically thin bremsstralilung model has to be refined to astrophysical realistic approximations to observed compact sources.) Especially, this possibility is forced by the sensitivity of the properties of an accretion disk to the inner boundary conditions. And this conditions are situated in the strong gravity region where the nonlinear character of the field equations for the gravitation becames impor- tant.

I’AUL, IT. G : .kcrction tlihks in binictric tlieory 189

l‘lic sensitivity of the functions of structure of the accretion disk ought to bc very enhanced in tlic case of rotating central objects. The inner boundary conditions, fulfilled in the immediate neighbourhood of the marginally bound test particle orbit, is located much nearer to the horizon for high enough angular momentum of the central body in the case of the Kerr solution in Einstein’s theory. Unfortunately, the corresponding solution does not exist for a wide class of intcresting alternative gravitation theories.

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.~

.\ddrcs\ of tlic author:

H. G. P A U L Zcntrslinstitut iiir Astroiilivsik dcr Ad\\’ dcr 1)Uli _ - Sternwarte Babelsbcrg, UL)K-1502 l’otsda~n-Babelsbcrg Rosa-Luxemburg-Str. I 7 a German Democratic Republic