V iria l t e s t s for p o s t-N e w to n ia n sta tio n a r y b la ck -h o le—d isk s y ste m s
P io t r J a r a n o w sk i
W y d z ia l F izy k i, U n iw e rsy te t w B ia ły m sto k u , C iołkow skiego 1L, 1 5-245 B ia ły sto k , P o la n d E -m ail: p .ja r a n o w s k i@ u w b .e d u .p l
P a t r y k M a c h , E d w a r d M a le c , M ic h a ł P ir o g
I n s ty tu t F izy k i im . M a r ia n a Sm oluchow skiego, U n iw e rsy te t Jag iello ń sk i, L o jasiew icza 11, 30-348 K rak ó w , P o la n d
E -m ail: p a tr y k .m a c h @ u j .e d u .p l , m a l e c @ t h . i f . u j . e d u . p l , m i c h a l .p i r o g @ u j .e d u .p l
A b s t r a c t . W e in v e s tig a te d h y d ro d y n a m ic a l p o s t-N e w to n ia n m o d els o f s e lfg ra v ita tin g s ta tio n a r y b la c k -h o le -d isk sy stem s. T h e p o s t-N e w to n ia n schem e p re s e n te d h ere a n d also in o u r re c e n t p a p e r is a c o n tin u a tio n o f p rev io u s, p u re ly N e w to n ia n stu d ie s o f s e lfg ra v ita tin g h y d ro d y n a m ic a l d isk s r o ta tin g a c co rd in g to th e K e p le ria n r o ta tio n law . T h e p o s t-N e w to n ia n re la tiv is tic c o rre c tio n s a re sig n ifican t ev en a t th e 1P N level. T h e 1P N c o rre c tio n to th e a n g u la r v elo city c a n b e o f th e o rd e r o f 10% o f its N e w to n ia n value. I t c a n b e e x p ressed as a c o m b in a tio n o f g eo m etric a n d h y d ro d y n a m ic a l te rm s. M oreover, in c o n tr a s t to th e N e w to n ia n Poincaróe- W av re th e o re m , it d e p e n d s b o th o n th e d is ta n c e fro m th e r o ta tio n axis a n d th e d is ta n c e fro m th e e q u a to r ia l plan e.
In th e te c h n ic a l p a r t o f th is n o te we d eriv e v iria l re la tio n s valid u p to 1P N o rd er. W e show t h a t th e y are in d e e d sa tisfie d by o u r n u m e ric a l so lu tio n s.
1. I n tr o d u c tio n
In a recent p a p e r [1] we investigated p ost-N ew tonian m odels of selfgravitating gaseous disks th a t ro ta te according to th e K eplerian ro ta tio n law. T he analysis presented th ere is a continuation of our previous studies, w here such disk system s were investigated in N ew tonian th eo ry [2, 3].
In th e N ew tonian fram ew ork we posed th e following problem : Suppose one observes a selfgravitating sta tio n a ry gaseous disk around a central ob ject (m odeled by a point-m ass) th a t ro ta tes according to th e K eplerian ro ta tio n law, th a t is w ith th e ang ular velocity = w0/ r 3/2, w here r is th e distance from th e ro ta tio n axis, and w0 is a co n stan t. For th e disk consisting of te s t p articles we have w0 = y /G M c, w here M c is th e m ass of th e central object, and G is th e g rav itatio n al co n stan t. W h a t does th e observed value of w0 correspond to in th e case where th e m ass of th e disk is com parable w ith M c? Is it still th e central m ass M c, th e sum of th e two masses, or some n ontrivial com bination of them ? It tu rn s o u t th a t th e selfgravity speeds up th e ro ta tio n of th e disk— it ro ta tes faster th a n this would follow from th e K eplerian form ula involving th e central m ass M c only. M oreover, th e way in which w0 depends on th e central m ass and th e m ass of th e disk is prescribed by th e geom etry of th e disk. T hus, in principle, it
procedure was applied to th e accretion disk in th e AGN of N G C 4258, w here th e K eplerian ro ta tio n curve was m easured in th e m aser em ission [3].
In [1] we extended th e N ew tonian analysis to th e first post-N ew tonian appro xim atio n (1PN ).
Selfgravitating sta tio n a ry gaseous disks were investigated before in full relativ ity (cf. [4, 5]). We decided on th e post-N ew tonian scheme, because of its conceptual simplicity. In p articu lar, th e notion of th e K eplerian ro ta tio n has a clear m eaning in th e post-N ew tonian scheme.
T he m ain result o b tain ed in th e 1PN approx im atio n is th a t th e angu lar velocity profile is affected in two different ways— some p a rts of a disk can be speeded up and th e oth ers slowed down. Furtherm ore, th e sum of th e N ew tonian and post-N ew tonian com ponents of th e angular velocity is not anym ore a function of th e cylindrical radius only, b u t in general it depends on radial and vertical co ordinates [1].
In th is pap er we sketch briefly th e m ain equations th a t c o n stitu te th e 1PN m odel and th en derive virial-type relation th a t can be used to te s t th e obtain ed num erical solutions. Suitable virial te sts valid in th e N ew tonian case were presented in [6] and [2]. We discuss th em here for clarity. T h e post-N ew tonian virial identities given here are new and have not been discussed in [1]. In th e last section of this p ap er we also show th a t th ey are satisfied by our num erical m odels w ith th e accuracy sim ilar to th a t of N ew tonian solutions.
2. D e s c r i p t i o n o f t h e m o d e l
O ur 1PN black-hole-disk m odels are con stru cted assum ing th e m etric of th e form
d s2 = g,„ d X W - = ( - 1 - 2 U ( x f z) - 2<U < W » ! 1 (dx« ) 2 - 2 4 4 w )
+ ^ 1 - 2U ' ' O ' z ) ^ (dx2 + d y 2 + d z 2) , (1)
w here we use C artesian coordinates x0 = ct, x1 = x , x 2 = y , x 3 = z, and c is th e speed of light.
We w rite th e energy-m om entum ten so r as
T «y = ^IhU bH £(x ) + p (c2 + h )u a u ? + , (2) V 9 u BH
w here th e first com ponent describes th e point particle (it is p ro p ortion al to th e D irac d elta d istrib u tio n and m odels th e cen tral black hole) a t rest, located a t th e origin of th e coordinate system ; th e second one is th e energy-m om entum ten so r of th e disk m a tte r. Here M c denotes th e m ass of th e point particle; g is th e d e te rm in a n t of th e m etric g = - d e t(g MV). T he four-vectors uBH and u a d enote th e four-velocities of th e central point-m ass and th e fluid, respectively. T he sym bol p denotes th e baryonic rest-m ass density, h is th e specific enthalpy, and p is th e pressure.
In th e following sections we will also work w ith th e three-velocity, defined as v % = cu%/ u ° , i = 1, 2, 3.
In th e rem aining p a rt of th e article we use sta n d a rd cylindrical coordinates (r, z, 0). We consider a statio nary , selfgravitating, axially and equatorially sym m etric polytropic disk, ro ta tin g aro und a central point m ass M c according to th e K eplerian ro ta tio n a l law Vq = w0r -3 /2 . We assum e th a t th e disk is geom etrically bounded by th e inner and ou ter radius r-m and r out, respectively. We introduce th e n o tatio n according to which any q u a n tity £ (if it is necessary) is sep arated into its N ew tonian £ 0 and post-N ew tonian £ 1 p a rt according to th e general p a tte rn
£ = £o + £1/ c 2. Following [7] we derive basic equations which, separated into th e ir N ew tonian
and post-N ew tonian p a rts, read:
AUo = 4nG (po + MCS ( x ) ) , (3)
ho = -U o + ^ d r(v0^)2r + Co, (4)
A — = 2— — 0 — 1 6 n G r2p0v°, (5)
A U i = 4nG ^M cUoD(0)6(x) + pi + 2po + po (ho — 2Uo + 2r2(v °)2^ , (6) h i = —Ui — v0 —o + 2 h o (v °)2r2 — J d r (v (^ ) 4r ‘3 — 2 — 4hoUo — 2Uo2 — C i, (7) w here A denotes th e flat L aplacian w ith respect to coordinates (x i , x2, x 3), Co and C i are in teg ratio n con stan ts, and U D is th e g rav itatio n al p o ten tial due to th e disk only, i.e., Uo =
—G M c/ |x | + U D. T h ey follow from th e conservation law, V aT al3 = 0, th e continuity equation V a (pua ) = 0, and E in stein equations
R G
R^,v — — T^ v , (8)
w here R MV is th e Ricci ten so r and R denotes th e Ricci scalar.
T he above system of equations is closed by assum ing an equation of sta te . O ur num erical solutions are obtain ed for a polytropic eq u ation of sta te of th e form p = K p Y, or equivalently h o = K y / ( y — 1)p^- i (for th e 0th order solution) and h i = ( 7 — 1)hop i / p o (for th e 1PN p a rt), w here K and 7 > 1 are co nstants.
E q u atio n s (3-7) should be solved w ith respect to th e N ew tonian grav itatio n al p o ten tial Uo(r, z), th e p o st-N ew tonian g rav itatio n al p o ten tial Ui (r, z), th e ro ta tio n a l p o ten tial A o ( r , z ) and th e N ew tonian and post-N ew tonian enth alpy h o(r, z) and h i (r, z). Any o th er q u a n tity (the d ensity p(r, z), th e pressure p (r, z), etc.) can be o b tained from these five functions.
N um erical solutions are o b tain ed as follows. We use th e classic Self-Consistent Field (SCF) scheme (cf. [8]) to solve th e set of Eqs. (3) and Eq. (4). Given th e N ew tonian p o ten tial Yo and th e en th alpy h o, we can solve Eq. (5) a t once. Finally, we use again th e SC F scheme to solve th e set of Eqs. (6) and (7). For th e K eplerian ro ta tio n law th e above m eth od converges for all values of th e param eters th a t we have tested.
3. R e s u lts
T he m ain result of our p ost-N ew tonian scheme is th e correction to th e N ew tonian angular velocity profile. T he well-known theo rem by Poincare and W avre sta te s th a t N ew tonian sta tio n a ry b aro trop ic disks (or stars) ro ta te w ith th e angular velocity th a t can depend on th e distan ce r from th e ro ta tio n axis only [9]. It tu rn s o ut th a t already th e 1PN correction can be significant and in general it depends also on z. It can be expressed as
v ° (r, z) = — —00 dr v° + 2rhodr v ° . (9)
2rv o0
N ote th a t th e above form ula involves b o th geom etric and hydrodynam ical factors.
A cceptable 1PN solutions should satisfy th e following (quite strin gen t) conditions: i) 1 » |Uo|/ c2 » |Ui |/ c 4, ii) 2G M c/ c2 ^ r;n, iii) c » cs, w here cs is a speed of sound. Sam ple num erical m odels th a t do satisfy th e above conditions are presented in [1]. It tu rn s o ut th a t th e 1PN correction to th e velocity can be of th e order of 10% of th e N ew tonian value. T he reader
4. V i r ia l i d e n t i t i e s
In th is section we use C artesian and cylindrical coordinates. It is im plicitly assum ed th a t L atin indices refer to C artesian coordinates.
T h e virial id en tity th a t can b e used to test th e N ew tonian solution (including th e central point-m ass) was obtain ed in [6]. It reads
E pot + 2 E kin + 2 E therm = 0, (1 0)
w here Epot = \ f R3 d3x p(Uo — G M c/ |x |) is th e to ta l p o ten tial energy, Ekin = 2 Jr3 d3x pvoivO is th e bulk kinetic energy and E therm = | Jr3 d3x p is th e internal th erm al energy. We assum e, as a virial te st p aram eter, th e value
E pot + 2 E kin + 2E therm E pot + 2 E kin + 2E therm
E pot
In o rder to o b tain th e post-N ew tonian relations we rew rite Eqs. (5) and (6) in a slightly different form. E q u a tio n Eqs. (5) can be w ritte n in C artesian co ordinates as
A A % = —16nGpov0.
E q u a tio n (6) is split in two parts:
A U 1 = ^pi + 2po + po (ho — 2Uo + 2 r 2 ( v 0>)^ ) , (11)
A U f = 4nG M cU oD(0)^(x), (12)
w here Ui = U[ + U f. T he solution for U f is
TT„ = GMcUoD( 0)
Ul = i x ^ .
C onsider a vector
d = ( x i dt Ak + 2 A ^ j d iA k — 2 x idiAkd lA k . Its divergence reads
did1 = ( x l0iAk + 2 Ak^ A A k = — 16nG ^ x id A k + 2 A ^ povk. (13)
For a finite disk (po of com pact su p p o rt), A k ten ds to zero sufficiently fast, and
|x |2a i ^ 0, as |x| = \ / x2 + y2 + z2 ^ <x>.
Thus, by in teg ratin g Eq. (13) over R3, and m aking use of th e G auss theorem , we see th a t
0 = / ^ d3x ^ x idiAk + 2A ^ j povk.
In te g ratin g by p a rts, one can get rid of th e te rm w ith dtA k . T his yields
0 = ^ 3 d3^ —5 poAkvk — x ldi [povo j A ^ j ,
di b
i
= ( x l di U( + 2 U ?j A U [ , and analogously0 = j 3 d3x ( ^ x l d
i
U[ + 2 U ^(
p 1 + 2po + po {ho - 2Uo + 2 r 2(vQ)2) ) .M any different forms of th e above relatio n can be obtain ed by ‘playing’ w ith Eq. (7). A helpful relation th a t can be used here is
- x l d
l
U i = U f.In th e analogy to th e N ew tonian case we choose as virial te st param eters d = |(e^ + eb) / eal and d ' = |(e^ + d O/ d i l, where
e
'a
= f d3x 7 poA0v0 , (14)J R 3 2
eb = - / d3x x l di fpov0) Aq, (15)
7r3 v '
e'
a
= -J
^ d3x 2 U(
pi + 2po + po(
ho - 2Uo + 2 r 2(v0)2))
, (16) eb =J
g d3x x l di U((
p i + 2 p o + Po(
ho - 2Uo + 2 r 2(v0) 2))
. (17)In Table 1 we rep o rt results of th e convergence te sts for a sam ple system . In our im p lem entation, num erical precision is controlled by th e resolution of th e grid, th e m axim um num ber L of Legendre polynom ials used in th e angular expansion of th e solutions of th e scalar and vector Poisson equations, and a value of th e m axim al difference betw een d ensity d istrib u tio n s o b tained in th e last two consecutive iteratio n s ptoi. (In each ite ratio n we com pute th e q u a n tity perr = m axi ,j |p(k+ 1 ^ - P(k) |. Here index k num bers subsequent iterations; indices i and j refer to different grid nodes. T he ite ratio n procedure is stopped, w hen perr < ptol.)
where only 0th o rder term s are differentiated.
T he above relation can be also w ritte n in term s of th e vector com ponents in cylindrical coordinates. Because of sym m etry assum ptions, we have A kv k = Aqvq , and
x ldi (pov0^) A k = x lV i (poV l) A k = poA^v0 + x ldi [povQ) Aq. T his yields
0 = y g d3x ^ - 2PoAqv0 - x ldi (pov0 ) A ^ j .
T he virial relation following from Eq. (11) can be o b tain ed in a sim ilar way. It is enough to consider th e divergence of th e vector
bi = ( x ld iU + 1 U ^ j d i U[ - 2x idiU[d lU .
It yields
dibi = ^ x ldiU( + 2U ?j A U [ , and analogously
T a b le 1. Typical dependence of th e results on th e resolution of th e num erical grid, th e m axim um num ber of th e Legendre polynom ials L, and th e tolerance coefficient ptoi. These results are o b tain ed for a polytropic fluid w ith polytropic index y = 5 /3 and th e K eplerian ro ta tio n law v ° = wo/ r 3/2. T he m ass of th e disk is M d = 0.327M c, th e inner and o u ter radii are r;n = 50RS and r out = 500RS respectively.
R esolution L ptol e e' e"
1 0 0 x 1 0 0 1 0 0 1 0-6 2.53 x 10-5 2.29 x 10-4 3.51 x 10- 4
2 0 0 x 2 0 0 1 0 0 1 0-6 6.36 x 10-6 5.87 x 10-5 8.87 x 10- 5 400 x 400 1 0 0 1 0-6 1.56 x 10-6 1.93 x 10-5 2.29 x 10- 5 800 x 800 1 0 0 1 0-6 3.66 x 10-7 3.65 x 10-6 4.62 x 10-6
1 2 0 0 x 1 2 0 0 1 0 0 1 0-6 1.44 x 10-7 1.56 x 10-6 2.49 x 10-6 1600 x 1600 1 0 0 1 0-6 6 . 6 6 x 1 0-8 5.43 x 10-7 1.39 x 10-6 400 x 400 50 1 0-6 1.56 x 10-6 1.93 x 10-5 2.27 x 10- 5 400 x 400 75 1 0-6 1.56 x 10-6 1.93 x 10-5 2.29 x 10- 5 400 x 400 1 0 0 1 0-6 1.56 x 10-6 1.93 x 10-5 2.29 x 10- 5 400 x 400 125 10-6 1.56 x 10-6 1.93 x 10-5 2.30 x 10- 5 400 x 400 150 10-6 1.56 x 10-6 1.93 x 10-5 2.30 x 10- 5 400 x 400 1 0 0 1 0-5 1.52 x 10-6 1.94 x 10-5 2.29 x 10- 5 400 x 400 1 0 0 10-6 1.56 x 10-6 1.93 x 10-5 2.29 x 10- 5 400 x 400 1 0 0 1 0-7 1.59 x 10-6 1.93 x 10-5 2.29 x 10- 5 400 x 400 1 0 0 1 0-8 1.59 x 10-6 1.93 x 10-5 2.29 x 10- 5
A c k n o w le d g m e n ts
T his research was carried ou t w ith th e su p ercom puter ‘D eszno’ purchased th an k s to th e financial su p p o rt of th e E u ro p ean Regional D evelopm ent Fund in th e fram ew ork of th e Polish Innovation Econom y O peration al P ro g ra m (co n tract no. P O IG . 02.01.00-12-023/08). T he work of P J was p artially sup p o rted by th e Polish N C N g ran t N etw orking and R& D fo r the E in ste in Telescope.
P M and M P acknowledge th e su p p o rt of th e Polish M inistry of Science and H igher E d u catio n g ran t IP2012 000172 (Iuventus P lus).
R e f e r e n c e s
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