D O I: 10.1051/0004-6361/201116702
© ESO 2011
Astrophysics
Spinning up black holes with super-critical accretion flows
A. Sadowski1, M. Bursa2, M. Abramowicz1,2,3,4, W. Kluzniak1, J.-P. Lasota5,6, R. Moderski1, and M. Safarzadeh3,7
1 Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland e-mail: [as;w lo d e k ;m o d e rsk i]@ ca m k .ed u .p l
2 Astronomical Institute, Academy of Sciences of the Czech Republic, Bocm II 1401/1a, 141-31 Praha 4, Czech Republic e-mail: b u r s a @ a s tro .c a s .c z
3 Department of Physics, Goteborg University, 412-96 Goteborg, Sweden e-mail: m arek.ab ram o w icz@ p h y sics.g u .se
4 Department of Physics, Silesian University at Opava, Bezrucovo namesfi 1150/13, 74601 Opava, Czech Republic 5 Institut d’Astrophysique de Paris, UMR 7095 CNRS, UPMC Univ. Paris 06, 98bis Bd Arago, 75014 Paris, France
e-mail: la s o ta @ ia p .f r
6 Jagiellonian University Observatory, ul. Orla 171, 30-244 Kraków, Poland
7 Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street Baltimore, MD 21218, USA e-mail: m ts@ pha.jhu.edu
Received 11 February 2011 / Accepted 24 May 2011
ABSTRACT
We study the process of spinning up black holes by accretion from slim disks for a wide range of accretion rates. We show that for super-Eddington accretion rates and low values of the viscosity parameter a (<0.01) the limiting value of the dimensionless spin parameter a, can reach values higher than a, = 0.9978 inferred by Thorne in his seminal study. For M = 10 MEdd and a = 0.01, spin equilibrium is reached at a, = 0.9994. We show that the equilibrium spin value depends strongly on the assumed value of a. We also prove that for high accretion rates the impact of captured radiation on spin evolution is negligible.
Key words. black hole physics - accretion, accretion disks
1. Introduction
A strophysical b la ck holes (BH s) are very sim ple objects - they can b e described by ju st tw o param eters: m ass M and angular m om entum J (usually described b y th e dim ensionless spin p a ram eter a , = a / M = J /M 2). In isolation, BH s conserve the birth values of these param eters but are often surrounded by accretion disks and experience both m ass and angular m o m en tum change, e.g., in close binaries or in active galactic nuclei.
A ccretion o f m atter always increases the B H ’s irreducible mass and m ay change its angular m om entum . T he sign of this change and its value depend on th e (relative) sign o f accreted angular m om entum and the b alan ce b etw een th e accretion o f m atter and various processes extracting the B H ’s rotational energy and an
gular m om entum .
T h e question about th e m axim al possible spin o f an object represented by th e K err solution o f the E instein equation is o f fundam ental and practical (observational) interest. First, a spin a , > 1 corresponds to a nake d singularity and not to a b la ck hole.
A ccording to the P enrose cosm ic censorship conjecture, naked singularities cannot form through actual p hysical processes, i.e.
singularities in the U niverse (except for th e initial one in the Big B ang) are alw ays surrounded b y event horizons (W ald 1984).
T his hypothesis has yet to b e proven.
In any case, the “third law ” o f B H therm odynam ics (B ardeen et al. 1973) asserts that a B H cannot b e spun-up in a finite tim e to the extrem e spin value a , = 1. D eterm ining th e m axim um value o f B H spin is also o f practical interest b ecau se the radiative e f
ficiency o f disk accretion depends on the B H ’s spin value. For exam ple, for the “can o n ical” value a , = 0.9978 (see below ) it is
about n ~ 32% , w hile for a , ^ 1 one has n ^ 42% . B anados et al. (2 0 0 9 ) show ed that the energy o f the center-of-m ass co lli
sion o f tw o particles colliding arbitrary close to th e B H horizon, grow s to infinity 1 w hen a , ^ 1.
A definitive study o f the BH spin evolution w ill only b e p o s
sible w hen reliable, non-stationary m odels o f accretion disks and je t em ission m echanism s are established. F or now, one has to use sim plified analytical or num erical m odels.
T horne ( 1974) u sed the m odel o f a radiatively efficient, g eo m etrically thin accretion disk (N ovikov & T h o rn e 1973) to eval
uate B H spin evolution taking into account the decelerating im p act o f d isk-em itted photons. T he m axim um value obtained to date a , = 0.9978, has been regarded as the canonical value for th e m axim al B H spin. In this w ork, w e g eneralize T h o rn e’s ap proach, using h y drodynam ical m odels o f advective, a-viscosity, optically thick accretion disks ( “slim disks” ) to calculate m a x im um B H spin values for a large ran g e o f accretion rates.
Follow ing T horne ( 1974), w e assum e that accretion o f m atter and radiation captured b y the B H are the only m echanism s af
fecting its rotation. Thus, w e neglect any im pact o f large-scale m agnetic fields (a discussion o f the applicability o f slim disks is presented in Sect. 6 ) . W e show th at for sufficiently high accretion rates the lim iting B H spin differs from the canonical value.
W e begin w ith a short discussion o f previous w o rk devoted to the evolution o f B H spin. In Sect. 2 , w e p rese n t form ulae
1 Of more fundamental interest is that the proper geodesic distances D between the marginally stable orbit (the innermost stable orbit, ISCO) and several other special Keplerian orbits relevant to accretion disk structure tend to infinity D when a, ^ 1 (Bardeen et al. 1972).
for a general tetrad (an orthonorm al set o f four vector fields, one tim elike and three spacelike) o f an observer com oving with th e accreting gas along th e arbitrary ph o to sp h ere surface. In S ect. 3 , w e give basic equations describing th e B H spin evo
lution. S ection 4 describes th e m odel o f slim accretion disks. In S ect. 5 , w e presen t and discuss the term inal spin values for all the m odels considered. Finally, in Sect. 6 w e sum m arize our results.
1.1. Previous studies
A num ber o f authors have studied the B H spin evolution re su lt
ing from disk accretion. B ardeen ( 1970) initiated this field o f re search b y solving equations describing the B H spin evolution for accretion from the m arginally stable orbit. N eglecting the effects o f radiation, h e proved that this accretion could spin-up the BH arbitrarily close to a, = 1. O nce th e classical m odels o f accre
tion disks w ere form ulated (S hakura & Sunyaev 1973; N ovikov
& T h o rn e 1973), it w as possible to account p roperly for th e d e celerating im pact o f radiation (fram e dragging m akes counter- rotating p hotons m ore likely to b e captured by th e BH ). A s m e n tioned above, T h o rn e ( 1974) perfo rm ed this study and obtained for an isotropically em itting thin disk the term inal B H spin a , = 0.9978, in dependently o f the accretion rate. T he original study b y T horne w as follow ed by m any papers, som e o f w hich are briefly m entioned below.
T he first to ch allenge the universality o f T h o rn e’s lim it w ere A bram ow icz & L aso ta ( 1980), w ho show ed that geom etrically th ick accretion disks m ay spin up B H s to term inal spin values m uch closer to u n ity than the canonical a , = 0.9978. T heir sim p le argum ent w as b ased on m odels by K ozlow ski et al. ( 1978), w ho show ed that for h igh accretion rates the inner edge o f a disk m ay b e located inside the m arginally stable orbit, and in fact, w ith increasing accretion rate, arbitrarily close to the m arginally bound orbit. H ow ever, this conclusion assum ed im plicitly a low viscosity param eter a , w hereas for high viscosities th e situation is m ore com plicated (see A bram ow icz et al. 2 0 1 0 , and references therein).
M oderski et al. ( 1998) assessed the im pact o f possible in ter
action betw een the d isk m agnetic field and th e BH through the B landford-Z najek process. T hey show ed that the term inal spin value m ay b e d ecreased to any, arbitrarily sm all value, if the disk m agnetic field is strong enough. G iven th e current lack o f self- consistent and reliab le m odels o f accretion disks w ith large-scale m agnetic fields, a m ore detailed study cannot b e perform ed. T he situation m ay b e further com plicated b y energy extraction from th e inner p arts o f accretion disks and th e m agnetic transport o f angular m om entum (see L ivio et al. 1999; G hosh & A bram ow icz 1997; and com pare w ith M cK inney & N arayan 2 0 0 7 ).
P opham & G am m ie ( 1998) studied the spinning-up o f BHs by optically thin advection dom inated accretion flows (ADA Fs).
They neglected th e contribution o f radiation to B H spin because such accretion disks are radiatively inefficient. T hey found that th e term inal value o f B H spin is very sensitive to the assum ed value o f the v iscosity param eter a and m ay vary betw een 0.8 and 1.0. G am m ie et al. (2 0 0 4 ), in addition to com prehensively su m m arizing the different w ays o f spinning up superm assive BHs, p resented results b ased on a set o f relativistic m agnetohydro- dynam ical (G R M H D ) sim ulations (w ith no radiation included) obtaining a term inal spin o f a , = 0.93.
T he cosm ological evolution o f th e spins o f superm assive B H s caused by hierarchical m ergers and thin-disk accretion episodes has been intensively studied. A lthough V olonteri et al.
(2 0 0 5 ) arrived at the conclusion that accretion tends to spin-up B H s close to a , = 1, as opposed to m ergers, w hich, on the
average, do not influence th e spin, subsequent studies by, e.g., V olonteri et al. (2 0 0 7 ), K ing et al. (2 0 0 8 ) and B erti & Volonteri (2 0 0 8 ) show ed that th e situation is m ore com plex, the final spin values depending on the details o f th e history o f the accretion events (see also Fanidakis et al. 2 0 1 1 ).
B elczynski et al. (2 0 0 8 ) applied p opulation synthesis m e th ods to estim ate B H spins in coalescing com pact star binaries.
B asing their calculations on results o f rad iation-hydrodynam ic sim ulations o f th ick accretion disks by O hsuga (2 0 0 7 ), they n e glected the im pact o f radiation on B H spin and assum ed that gas is transferred from th e innerm ost stable orbit conserving K eplerian angular m om entum . T hey show ed that the spin p a ra m eter a , resulting from th e co alescence is not expected to exceed 0.5 for those B H s that are not spun-up during the star collapse.
L i et al. (2 0 0 5 ) included the radiation returning to the disk in th e thin-disk m odel o f N ovikov & T horne and calculated the spin-up lim it for th e B H assum ing radiation crossing th e eq u a
torial p lan e inside the m arginally stable orbit is advected onto the b H . T heir resu lt (a, = 0.9983) differs slightly from T h o rn e’s result, thus show ing that returning radiation has only a slight im p act on th e process o f spinning-up BH s. In our study, w e u se ad- vective, optically th ick solutions o f accretion disks and account for p hotons captured b y the B H in detail. H ow ever, w e neglect the im pact on th e d isk structure o f the returning radiation.
2. The tetrad
W e base this w ork on slim accretion disks, w hich are not razor- thin and h ave an angular m om entum profile th at is not K eplerian (for details about the assum ptions m ade and th e d isk appear
ance see Sect. 4 ) . T herefore, p hotons are not em itted from m atter in K eplerian orbits in th e equatorial p lan e and th e classical ex pressions for photon m o m enta (e.g., M isner et al. 1973) cannot be applied. To properly d escribe th e m om entum com ponents o f em itted photons, w e need a tetrad for the com oving observer in stantaneously located at th e d isk photosphere. B elow w e give the explicit expression for th e com ponents o f such a tetrad assum ing tim e and axis sym m etries. A detailed derivation is given in A p pendix A .
W e choose the follow ing com oving tetrad
4 ) = u , S ' ] , (1)
w here u' is th e four-velocity o f m atter, N ' is a unit vector in the [r, 9] plane that is orthogonal to the photosphere, K is a unit vector in the [t, 0] plane th at is orthogonal to u' , and S ' is a unit vector orthogonal to u' , N ' , and K .
T he tetrad com ponents are given by
(2)
(3)
(4)
(5)
(6) d r g e g \ d r )
r lAa \21-1/2
N ° = i - g e e r 112 1
+ — ^ >
L g e e \ d r ) , n + q p + v s i
u = . ~ = ,
V gtt + ^ g # ( ^ - 2w) - v2
, { W + ? )
K = --- J72 ’ [- g # (1 - ^ l)(1 -
S l = {l + A2v2) -1/2 {AvU + S [ ) ,
w here 9 = 9,(r) defines the location o f th e p hotosphere, n and £ w here I0S = I0(r)S (a, b) is th e intensity o f the em itted radiation, are th e K illing vectors, l = u ^ / u t, Q = u4/ u \ u is the frequency a and b are the angles betw een the em ission vector and the N 1 o f fram e-dragging, and the expressions for v and S , are given in
E qs. (A .1 0 ) and (A .4 ), respectively.
3. Spin evolution
3.1. Basic equations
T he equations describing th e evolution o f dim ensionless B H spin a , w ith resp ect to th e B H energy M and th e accreted rest-m ass M 0 are (T h o rn e 1974)
(7)
(8)
(9)
(10)
and S 1 vectors, respectively, C is th e capture function defined in Sect. 3.4, the factor 2 occurs b ecause the d isk em ission com es from both sides o f th e disk, and n (a) = p (a) / p (0) are the no rm al
ized com ponents o f the p h oton four-m om entum in the com ov- ing fram e. T he last set o f param eters are given b y the relations (T horne 1974)
n (0) = 1, n (1) = cos a , n (2) = sin a cos b ,
(3) = sin a sin b . n
T he energy and angular m om entum o f BH in crease due to the capture o f photons according to th e form ulae
w here e(a) is our local fram e tetrad given b y E q. ( 14) . Taking into account the relations
(17)
(18)
(19)
w here nk and £k are the K illing vectors connected w ith tim e and axial sym m etries, respectively, T ik is the stress-energy tensor of photons, w hich is taken to b e n on-zero only for p hotons crossing th e B H horizon, and dS is the “volum e elem en t” in th e h y p er
surface orthogonal to N 1, w hich is given b y Eq. (B .8 ) . F ro m E qs. ( 9 ) and ( 10), it follow s that
(11)
(12)
n (a) = n j e(a ) ,
e (a)e i = P e j e (a) = ° j
w e h ave
n (a)e(a)N i = n (a)d(a)) = n (1) = cos a ,
n<J3)ek(p )n( = n Je f )ek(p) nk = n j<5k nk = n knk = n , n (fi)ek(p)£k = n ^ f e ^ £ = n J6k£k = n k£k =
w here,
(20)
(21) (22) (23)
(13)
(24) (25) 3.2. Stress energy tensor in the comoving frame
W e select the tetrad given in Eq. ( 1)
(14)
(15)
(16) T he d isk properties, e.g., th e em itted flux, are usu ally given in th e com oving fram e defined b y Eq. ( 14) . T he stress tensor co m ponents in the tw o fram es (B oyer-L indquist and com oving) are related in the follow ing w ay:
w here
(26)
(27)
(28) d a , _ d J / M 2 _ 1 M 0u<p +
~~ / \ jLdlk%
dinM d inM M +
d M I d M \
—
Ut + I — Mq.
d M 0 ' \ d t ) J
(dM )rad — f T ikm N i d S , Jdisk
(dJ)rad — f T % N i d S , disk
( s l
w here
i 2 \ 1/2 I dO* \ 2
dS = d 0 d r (gr^ - g„ g^ J -J grrr + gfee I — 1 •
e(0) = m1' e(i) = N
e(2) — K e(3) — S (
T ik — T (a)(P)e i ek T — T e (a) e(fi)-
T he stress tensor in th e com oving fram e is pn/2 r- 2n
T (a)(® — 2 h S C n (a)n (l3) sin a da db,
0 0
I d J \ r "out r 2n r n/2
( « L = 4d I 1 w c *
x cos a sin a d a db yf§ dr,
= (sfy " cJng<w) ^ g n- + 9 ee{ ~ j ~ j ' I d M \ r ro“‘ r 2n r /2
— = 4 n \ I o S C n ,
\ dt /rad Jra JO JO
x cos a sin a d a d b ^ J Ę dr,
T herefore, Eqs. ( 18) and ( 19) m ay b e finally expressed as nt — n (() e (i}gtt +
— n(() + n(()e%gw-
E quations ( 11) and ( 12) take the form
d t rad disk
( i t ) = [ T " w ' < » 4 f y w , d t rad disk
3.3. E m issio n
T he intensity o f local radiation m ay b e identified w ith th e flux em erging from th e d isk surface
I0 = F (r). (29)
T he angular em ission factor S w as taken by T h o rn e ( 1974) to be
(30)
for isotropic and lim b-darkened cases, respectively. In this work, w e assum e that the radiation is em itted isotropically.
3.4. Capture function
T he B H energy and angular m om entum are affected only b y p h o tons crossing the B H horizon. Follow ing T h o rn e ( 1974), w e d e
fine the capture function C
C =
1 if the photon hits the B H ,
0 in the opposite ca se. (31)
(32)
(33)
perform ing vertical integration. T he form alism w e u se here was adopted from Sadow ski et al. (2 0 1 1 ).
In the structure equations, w e assum e that G = c = 1, and use expressions involving the B H spin given by
A = r2 - 2 M r + a2, A = r4 + r2a 2 + 2 M r a 2, C = 1 a ' -1
D
H = 1 - 4 a . r ; 3/2 + 3 a 2 r ;2, w here a , = a / M and r, = r /M .
W e also define
4 r 3 C
(34)
(35)
H erein, w e calculate C in tw o w ays. F irst, w e u se the original T h o rn e ( 1974) algorithm m odified to account for em ission out of the equatorial plane. F or this purpose, w e calculate th e constants o f m otion, J and k, for a geodesic orbit o f a photon using
and a dim ensionless accretion ra te m = M /M Edd, w here M Edd = 16LEdd/ c 2 is th e critical accretion rate corresp o n d ing approxim ately to th e E ddington lum inosity (LEdd = 1.25 x
1038 M / Mq erg/s) for a disk around a non-rotating b lack hole.
T he equations d escribing slim disks w ritten in th e cylindrical coordinates are:
X
+ hh p dz is th e disk surface density, w hile v denotes the gas velocity as m easured by an observer co- rotating w ith the fluid and is related to the four-velocity ur b y R u r = A 1/2v / V l - v 2;(ii) for radial m om entum conservation
(36)
w hich replaces T h o rn e’s E q. (A 10). This approach does not take into account th e effects o f returning radiation, i.e., a photon h it
ting the disk surface is assum ed to continue its m otion. This treatm ent is inappropriate for optically thick disks - returning photons are m ost likely absorbed or advected tow ards the BH.
To assess th e im portance o f this inconsistency, w e adopt two additional algorithm s for calculating C. U sing p h oton equations o f m otion, w e determ ine w hether the photon hits the d isk surface (B ursa 2 0 0 6 ). W e then m ake one o f tw o assum ptions, either the angular m om entum and energy o f all “retu rn in g ” photons are advected onto the B H (C i), or all are re-em itted carrying away their original angular m om entum and energy, and never h it the B H (C2). In this way, w e establish tw o lim iting cases allow ing us to assess the im pact o f th e returning radiation.
W e note that for a fully consistent treatm ent o f th e returning radiation (as in L i et al. 2 0 0 5 , for geom etrically thin disks) it is not enough to m odify the capture function, but that finding a so
lution for th e w hole structure o f a self-irradiated accretion d isk is instead necessary. T he latter has not yet been done for lum inous and g eom etrically thick disks. W e are currently w orking on im plem enting such a schem e and w ill study its im pact on BH spin evolution in an u p com ing paper.
4. Slim accretion disks
4.1. E q u a tio n s
W e now in troduce slim disk equations. T hey w ere derived o rigi
nally b y L asota ( 1994) and im proved e.g., b y A bram ow icz et al.
( 1996) and G am m ie & P opham ( 1998). H ere, w e follow Kato et al. (2 0 0 8 ) and assum e th e p olytropic equation o f state w hen
(37)
(38)
and O = u ^ / u t is th e angular velocity w ith respect to a stationary observer, 00 = O - m is the angular velocity with respect to an inertial observer, 0 ± = ± M 1/2/(R 3/2 ± a M 1/2) are th e angular frequencies o f the co-rotating and counter- rotating K eplerian orbits, R = A /(R 2A 1/2) is the radius o f
X
+ hh p dz is th e vertically integrated total pressure;(iii) for angular m om entum conservation M „ „ A1/2A1/2r
— U - £ m) = --- aP,
2n R (39)
w here L = u<p, L in is a constant, and r is th e L orentz factor (G am m ie & P opham 1998)
(40)
(41)
, ( 1 /n isotropic
S ( a , b) = \
' ' ( ( 3 / 7 n ) ( 1 + 2 c o s a ) lim b darkening
j = a l + a ( n ^ / M n ^ ,
k = — — r \ttq - (n$ + cuM n, sin 6»,)“ / sin2 d„
( M n tf \ - ° '
A = r 2 - 2 M r + a2, A = r4 + r 2a 2 + 2 M r a 2, C = 1 - 3 r-1 + 2 a „ r-3/2, D = 1 - 2 r -1 + 2 a2 r-2 , H = 1 - 4 a „ r-3/2 + 3a2r-2 ,
(i) for m ass conservation M = - 2 n Y A 1/2 — --- .
V 1 — n2
u dc JA 1 d i3 1 - u2 ciR “ ¥ ~ S d t f ’ w here
_ MA
= A O +O - 1 - Cl~R~
^ 1 -C2r 2
T“ = ---7 + --- , 1 - y - A (iv) for vertical equilibrium
2 2 ^
H - i l l = ( 2 N + 3 ) - ,
Fig. 1. Flux profiles for M = 10 M0 and a, = 0.0. Fig. 2. Photospheric profiles for M = 10 M0 and a, = 0.0.
(v) for energy conservation
M
(42)
(43)
(44) T he equations given above form a tw o-dim ensional system of ordinary differential equations w ith a critical (i.e., sonic) point.
F or each set o f disk param eters, a regular solution exists for only one specific value o f L in, w hich is an eigenvalue o f the problem . T he appropriate value m ay b e found using either th e relaxation or the shooting m ethod. F or details o f th e num erical procedures, w e refer to S adow ski (2 0 0 9 ) and S adow ski et al. (2 0 1 1 ).
4.2. D isk a p p e a r a n c e
W e now briefly describe the properties o f slim d isk solutions. For a m o re detailed discussion, w e refer to e.g., S adow ski (2 0 0 9 ), A bram ow icz et al. (2 0 1 0 ), and B ursa et al. (in prep.).
T he radial profiles o f th e em itted flux for a non-rotating BH are p resented in F ig. 1. For low accretion rates (m « 1), they alm ost coincide w ith th e N ovikov & T horne solutions (the sm all dep artu re is due to the angular m om entum taken away by p h o tons, an effect that is neglected in our slim disk schem e). W hen th e accretion rate becom es high, advective cooling starts to play a significant ro le and the em ission departs from that o f th e ra diatively efficient solution. This departure is visible as early as
for m = 1 at w hich th e em ission extends significantly inside the m arginally stable orbit. F or super-critical accretion rates, the flux increases m onotonically tow ards th e B H horizon. D ifferent c o l
ors in Fig. 1 denote solutions for different values o f the viscosity p aram eter a . A lthough the solutions are very sim ilar, one can see that the h igher th e value o f a, the low er the accretion rate at w hich advection starts to m odify the em ission profile.
In F ig. 2 , w e plot d isk thickness profiles (cos 0 H = H / r ) for a ran g e o f accretion rates and tw o values o f a . F or m > 0.1, the inner region o f th e accretion d isk is p u ffed up by th e radiation pressure and the d isk surface corresponds to the location w here the radiation p ressu re force (w hich is p roportional to the local flux o f em itted radiation) is balan ced by th e vertical com ponent of the gravity force. F or the E ddington accretion ra te (m ~ 1), th e hig h est H / R ratio equals ~ 0 .3 (cos 0 H ~ 0.3), w hile for the hig h est accretion rate considered (m = 100) it reaches ~ 1 .5 (cos &h - 0.83).
In the thin d isk approxim ation, th e accreting fluid has a K eplerian angular m om entum . This condition is not satisfied for advective accretion disks w ith significant radial p ressu re g rad i
ents. In F ig. 3, w e present angular m om entum profiles for disks w ith different accretion rates, a = 0.01 (left) and a = 0.1 (right panel). It is clear that th e higher the accretion rate, the larger th e departure from th e K eplerian profile. H owever, the quan tita
tive b ehavior depends strongly on a . For a < 0.01, th e flow is super-K eplerian in the inner p art (e.g., betw een r = 4.5 M and r = 14 M for m = 100). F or larger viscosities ( a > 0.1) and h igh accretion rates, the flow is sub-K eplerian at all radii. A s a result, the value o f the angular m om entum at th e B H horizon (X in) also depends strongly on a , decreasing w ith increasing a . D ep en d en ce o f the flow topology on the v iscosity p aram eter was studied in detail b y A bram ow icz et al. (2 0 1 0 ).
5. R esults for bh spin evolution
U sing the slim d isk solutions d escribed in the previous section, w e solve E qs. (7 ) and (8 ) using a regular R unge-K utta m ethod o f the 4th order. To calculate the integrals (Eqs. ( 11) and ( 12)), w e u se the alternative extended S im p so n ’s rule (Press 2 0 0 2 ) with 100 grid points in a, b, and radius r. W e carried out tests to verify that this num ber is sufficient for convergence.
In F igs. 4 and 5, w e presen t th e BH spin evolution for a = 0.01 and 0.1, respectively. T he red lines show the results for d if
ferent accretion rates, w hile th e b la ck line indicates the classical
dv _ _ M _ l P d l n S
Q 2 ^ 2 V 3 2 d ln 7 ? m 2 din/?
d in R ± l a i n R j w here the am ount o f h eat advected Qadv is
adv A y2 dO 64<tTC
A ssum ing the poly tro p ic index N = 3, w e have 1 CH
Vi = —Ą \ T 4 d z = 1 2 8/315 H, T0 J 0
2 r H
m = p T d z = 4 0 /4 5 , LT0 J0
1 I 1 k 40 256 4 \
P \ 5 /3 - 1 p 45 315 C / 1 r H
r]4 = - p z 2 dz = 1/18 H 2.
l J 0
Fig.3. Profiles of the disk angular momentum for a = 0.01 (left) and a = 0.1 (rightpanel) at different accretion rates in the Schwarzschild metric.
The spin of the BH a, = 0.
Fig. 4. Spin evolution for a = 0.01.
Fig. 5. Spin evolution for a = 0.1.
T h o rn e ( 1974) solution b ased on the N ovikov & T h o rn e ( 1973) m odel o f thin accretion disk. O ur low accretion ra te lim it does not p erfectly agree w ith the black line as th e slim d isk m odel does not account for the angular m om entum carried aw ay by
Fig. 6. The rate of spin-up or spin-down by “pure” accretion (radiation neglected) for a = 0.01. Profiles for five accretion rates are presented.
Their intersections with the red line (marked with blue crosses) cor
respond to equilibrium states. For the two lowest accretion rates, the equilibrium state is never reached (a, ^ 1).
radiation. A s a result, the low -lum inosity slim disk solutions slightly overestim ate the em itted flux (by no m o re than a few percent) leading to stronger deceleration o f th e BH b y rad ia
tion. T he T h o rn e ( 1974) resu lt is the p ro p er lim it for the low est accretion rates. W hen the accretion rate is high enough (e.g., m > 0.1), the im pact o f th e om itted angular m om entum flux is overw helm ed by the m odification o f the d isk structure in tro duced b y advection.
To study th e im pact o f radiation on B H spin evolution in d e tail, w e calculated th e rate o f B H spin-up for th e “p u re” accretion o f m atter (w ithout accounting for th e im pact o f radiation). In that case, the B H spin evolution is given b y (com pare Eq. (7 ) )
(45)
In F igs. 6 and 7, w e plot w ith b la ck lines the first term on the right h an d side o f th e above equation for different accretion rates and values o f a . T he red lines in these plots show the absolute value o f th e second term . T he intersections o f the b lack and red lines d enote th e equilibrium states, i.e., th e lim iting values
d a 1 Us „
— --- --- 2t/*.
dln M M ut
Table 1. BH spin terminal values.
Capture function:
model:
C A
C T
C V
C1 A
C2
A NR
thin disk 0.9978 0.9978 0.9978 0.9981 0.9978 -> 1
m -= 0.01 0.9966 0.9966 0.9966 0.9975 0.9966 -> 1
m - 0.1 0.9967 0.9967 0.9967 ^ 1 0.9967 ^ 1
a = 0.01 m - 1 0.9988 0.9988 0.9988 ^ 1 0.9988 0.9998
m - 10 0.9994 0.9994 0.9994 ^ 1 0.9994 0.9996
iii == 100 0.9995 0.9995 0.9995 -> 1 0.9995 0.9995 m -= 0.01 0.9966 0.9966 0.9966 0.9975 0.9966 -> 1
m - 0.1 0.9975 0.9975 0.9975 ^ 1 0.9975 ^ 1
a = 0.1 m - 1 0.9924 0.9924 0.9923 ^ 1 0.9927 0.9948
m - 10 0.9846 0.9846 0.9845 0.9901 0.9847 0.9951
iii == 100 0.9800 0.9800 0.9800 0.9803 0.9800 0.9801
Notes. C - Thorne’s capture function, C1 - all returning photons advected onto the BH, C2 - all returning photons neglected; A - our fiducial model, T - emission from the equatorial plane, V - zero radial velocity, NR - pure accretion, radiation neglected.
Fig. 7. Same as Fig. 6 but for a = 0.1.
o f B H spin for p u re accretion. T hese values differ significantly from the previously d iscussed results only at low accretion rates.
In contrast, at h igh accretion rates radiation has little im pact on th e spin evolution and the value o f term inal spin is m ostly deter
m ined b y th e properties o f th e flow. In F ig. 8 , w e p lo t the ra d i
ation im pact p aram eter £, defined as the ratio o f th e disk-driven term s on the rig h t h an d sides o f E qs. (7 ) and (4 5 )
(46)
I f th e captured radiation significantly decelerates the B H spin- up, this ratio drops below unity. O n th e other hand, it is close to unity w hen the B H spin evolution is unaffected by the radiation.
A ccording to Fig. 8 , the latter is the case for th e hig h est accretion rates, independently o f a .
In Table 1 , w e list the resulting values o f the term inal B H spin for all the m odels considered. T he first colum n gives th e results for our fiducial m odel (A) including T h o rn e’s capture function and em ission from the ph o to sp h ere at th e appropriate radial velocity.
T he second colum n presents results obtained assum ing the sam e (T h o rn e’s) capture function and profiles o f em ission, angular m om entum , and radial velocity as in m odel A , but as
sum ing the em ission takes p la ce from the equatorial p lan e in stead o f th e photosphere. T he resulting term inal spin values are
Fig. 8. Radiation impact factor £ (Eq. (46)) for different accretion rates and values of a. The dotted line corresponds to the thin-disk induced spin evolution. For £ « 1, spin evolution is unaffected by radiation.
Stars denote the equilibrium states (compare Table 1).
equal, up to 4 decim al digits, to the values obtained w ith the fidu
cial m odel. This resu lt is as expected for the low est accretion rates, w here the p h otosphere is located very close to the eq u a
torial plane. F or the highest accretion rates, th e location o f the em ission has no im pact on the B H spin-up, as the spin evolution is driven by the flow itself and th e effects o f radiation are neg
ligible. H ow ever, for m o derate accretion rates one could have expected significant change in the term inal spin. W e find that the location o f the ph o to sp h ere has little im pact on the resulting B H spin regardless o f the accretion rate.
O ur third m odel (V ) neglects the flow radial velocity w hen the radiative term s are evaluated. S im ilar argum ents to those given in the previous paragraph apply. F or th e low est accretion rates, th e radial velocity is negligible and therefore should have no im pact on the resulting spin. F or the highest accretion rates, the spin-up process depends only on the properties o f the flow.
O nce again, how ever, th e im pact o f this assum ption on m oderate accretion rates is not obvious. T he radial velocity turns out to be o f little im portance for the calculation o f th e term inal spin (only for a = 0 .1 and m o derate accretion rates the difference betw een m odels A and V is h igher than 0.01% ).
In th e fourth and fifth colum ns o f Table 1 , results for m odels w ith the sam e assum ptions as th e fiducial m odel, but w ith differ
ent capture functions are presented. T he first alternative capture
function (C 1) assum es that th e angular m om entum and energy o f all photons returning to th e disk are added to that o f th e BH.
This assum ption has a strong im pact on th e spin evolution - the term inal spin values are higher, som etim es approaching a , = 1 . This m ay seem surprising b ecause in the classical approach the captured photons are resp o n sib le for decelerating the spin-up.
This deceleration occurs b ecause the cross-section (w ith respect to the BH ) o f photons m oving “ against” th e fram e dragging is larger than o f photons follow ing th e BH sense o f rotation. As fram e dragging is involved, this effect is significant only in the v icinity o f the B H horizon. F or our m odel C 1, how ever, the p ro b ability o f photons returning to the d isk does not differ ap precia
bly for co- and counter-rotating photons, as they both h it th e disk surface m ostly at large radii.
T he other capture function (C2) assum es, in contrast, that all returning photons are re-em itted from the d isk w ith their orig i
nal angular m om entum and energy (and never fall onto the BH ).
This assum ption cuts off the photons th at w ould h it the B H in th e fiducial m odel after crossing the disk surface, thus leading to a sm aller radiative deceleration and h igher values o f the ter
m inal spin param eter. H ow ever, these changes are insignificant, b ecau se m ost o f the original photons h it the B H directly, along slightly curved trajectories. O nly for a = 0.1 and m o derate ac
cretion rates do the term inal spin values differ in th e 4th decim al digit.
N either o f the m odels w ith a m odified capture functions is self-consistent. To account pro p erly for th e returning radiation, one has to m odify the d isk equations b y introducing ap propri
ate term s for the outgoing and incom ing fluxes o f angular m o m entum and additional radiative heating. N o such m odel for ad- vective, optically thick accretion disk h as been constructed. The em ission profile should b e significantly affected (especially in side the m arginally stable orbit) b y th e returning radiation, le ad ing to different rates o f deceleration by photons. In view o f our results for m odels C 1 and C2, as w ell as th e results o f L i et al.
(2 0 0 5 ), one m ay expect th e final spin values for super-critical accretion flows to b e slightly h igher than th e ones obtained in this w ork.
T he last colum n o f Table 1 gives term inal spin values for
“p u re” accretion (radiation n eglected). U n d er these assum ptions, th e BH spin could reach a , = 1 for sub-E ddington accretion rates as there are no p hotons that could decelerate and stop th e spin-up process. A s discussed above, for the hig h est accretion rates the resulting B H spin values agree w ith the values obtained for the fiducial m odel as radiation has little im pact on spin evolution in this regim e.
6. Discussion
W e have studied the spin evolution o f b la ck holes undergoing disk accretion assum ing that th e angular m om entum and energy carried by both th e flow and the em itted photons are th e only factors affecting th e B H rotation. W e have g en eralized the o rig inal study o f T h o rn e ( 1974) to high accretion rates by apply
ing a relativistic, advective, optically th ick slim accretion disk m odel. A ssum ing isotropic photon em ission from the disk (no lim b darkening), w e have show n that:
(i) the term inal value o f B H spin depends on the accretion rate for m > 1;
(ii) th e term inal spin value is very sensitive to the assum ed value o f the viscosity param eter a - for a < 0.01 th e B H is spun up to a , > 0.9978 for high accretion rates, w hile for a > 0.1 to a , < 0.9978;
(iii) w ith a low value o f a and h igh accretion rates, th e B H m ay b e spun up to spins significantly h igher than the canonical value a , = 0.9978 (e.g., to a , = 0.9994 for a = 0.01 and m = 10) but, u nder reaso n ab le assum ptions, B H cannot be spun up arbitrarily close to a , = 1;
(iv) B H spin evolution is h ard ly affected by the em itted rad ia
tion for high (m > 10) accretion rates (the term inal spin value is determ ined b y th e flow properties only);
(v) for all accretion rates, neither th e p h otosphere profile nor the profile o f radial velocity significantly affects the spin evolution.
We poin t out that th e inner edge o f an accretion disk cannot be uniquely defined for super-critical accretion (A bram ow icz et al.
2 0 1 0 ), as opposed to g eom etrically thin disks w here th e inner edge is u niquely located at the m arginally stable orbit (Rms). In the thin-disk case, the B H spin evolution is determ ined b y the flow properties at this p articular radius (as there is no torque b e tw een the m arginally stable orbit and B H horizon) and the p ro file o f em ission (term inating at R ms). F or super-critical accretion rates, how ever, one cannot distinguish a p articular inner edge that is relevant to studying B H spin evolution. O n the one hand, the values o f the specific energy (ut) and the angular m o m e n tum U ) rem ain co nstant w ithin the stress inner edge. O n the other, th e radiation is em itted outside the radiation inner edge.
T hese inner edges do not coincide as they are related to different p hysical processes (A bram ow icz et al. 2 0 1 0 ).
O ur study w as b ased on a sem i-analytical, hydrodynam ical m odel o f an accretion disk that m akes a num ber o f sim plify
ing assum ptions such as stationarity, no returning radiation, a - v iscosity prescription, no w ind outflows, and neglects in te rac
tions o f large-scale m agnetic fields interactions w ith BH s. O ne has to b e aw are that th e p recise values o f th e term inal spin p a ram eter are very sensitive to the flow and em ission properties, as w ell as to th e im pact o f m agnetic fields (e.g., by m eans o f the B landford-Z najek process). T he slim d isk m odel only approx
im ates th e real accretion flows driven by m agnetically-induced turbulence - in this respect it is no different from M H D sim u
lations. Its applicability is lim ited by the adopted assum ptions.
T he lack o f m agnetic fields m ay resu lt in im proper description of th e innerm ost p art o f the flow w here the d isk m ay b e b e m ag n eti
cally supported (N arayan et al. 2 0 0 3 ; Igum enshchev et al. 2 0 0 3 ; M eier 2 0 0 5 ; F rag ile & M eier 2 0 0 9 ). T he m odel also does not account for th e returning radiation that m ay affect the accretion flow. H owever, super-critical accretion is expected to be ra d ia tively inefficient and th erefo re th e im pact o f radiation should not b e large. D esp ite these lim itations, our study show s that T h o rn e’s canonical value for B H spin (a , = 0.9978) m ay b e exceeded u n der certain conditions.
Acknowledgements. This w ork was supported in part by Polish M inistry of Science grants N203 0093/1466, N203 304035, N203 380336, and N N203 381436. J.P.L. acknowledges support from the French Space Agency CNES, MB from GACR 205/07/0052.
A p p en d ix A: The tetrad for an observer instantaneously located at the photosphere O ur aim is to derive the tetrad o f an observer m oving along the p h otosphere that w ould depend only on the quantities that are typically calculated in accretion d isk m odels, i.e., on the radial and azim uthal velocities o f gas and the location o f the d isk p h o tosphere.
T he m etric considered h ere is the K err geom etry g k in the B oyer-L indquist coordinates [ t,0 , r,9 ]. T he signature adopted is + . A s in C a rter’s L es H ouches lectures (C arter 1972), w e consider tw o fundam ental planes; the sym m etry plan e So = [t,0 ] and th e m eridional p lan e M , = [r,9]. (F our)-vectors that belong to the plane S 0, are d enoted by the subscript 0, and v ec
tors that belong to th e p lan e M , , w ill b e d enoted by the sub
script , . F or exam ple, the tw o K illings vectors are n0, ^0. We note that for any pair X ‘0, Y, one has,
X0 Yk g ik = (X0 Y,) = 0.
A.1. Stationary and axially sym metric photosphere A .1.1. T he photosphere
N um erical solutions o f slim accretion disks provide the location o f th e ph o to sp h ere given b y H Ph(r) = r cos 9. This m ay b e sub
stituted into r cos 9 - H Ph(r) = F(r, 9) = 0. T he norm al vector to th e p h o to sp h ere surface has th e [r, 9] com ponents
(A.1)
(A.2)
is th e derivative o f th e angle defining th e location o f th e p h o tosphere at a given radial co ordinate [cos 9,(r) = H Ph( r ) /r ] . Its non-zero com ponents after norm alization [(N ,N , ) = - 1 ] are
(A.3)
(A.4)
It is u seful to construct a spacelike vector (k0) confined to the [t,0 ] plane, that is p erpendicular to both u and u0. F rom (k k) = - 1 and, e.g., (k u0) = 0, w e have
(A.8)
w here l = u $ /u t is th e specific angular m om entum . W e note that the set o f vectors [u0, N , , K0, S ,] already form s th e desired tetrad that is valid for the p u re rotation (ur = 0) case.
T he norm alization condition (uu) = 1 gives
~ r 91-1/2
A = | gtt + Q g ^ ( Q - 2m) - v J , (A.9) w here v is related to the radial com ponent o f the gas four- velocity ur by
(A .10)
T he vectors w e have ju st calculated (u , k0) are both orthogonal to N , since ( N , S ,) = 0. To com plete the tetrad, w e need one m ore spacelike vector ( S ) that is orthogonal to these three. We decom pose this into
S ' = a u ' + J3k0 + y N l + ÓS',. (A.11)
T he orthogonality conditions (k0S ) = 0 and ( N , S ) = 0 im m ed i
ately im plies that y = /3 = 0. T he only non-trivial condition is that ( u S ) = 0. Together w ith ( S S ) = - 1 , it im plies that
S ' = (1 + A2v2) 1/2 [Avu' + S , ) . (A .12)
T he vectors u' , N ,, K0, and S ' form an orthonorm al tetrad in the K err spacetim e
e (A) = [ u ' , N , ,K 0, S ' ] . (A.13)
T here are tw o uniq u e vectors S , confined to the [r, 9] p la n e that are orthogonal to N , (and therefore are tangential to the surface).
F ro m ( S , N , ) = 0 and ( S , S ,) = - 1 , one obtains the non-zero com ponents o f one o f them .
A .1.2. T he four-velocity of m a tte r and the tetrad
T he four-velocity u o f gas m oving along the p h otosphere m ay be decom posed into
u ' = A (u0 + v S , ) , (A.5)
w here
u0 = A0 ( i + o e ) (A.6)
is the four-velocity o f an observer w ith azim uthal m otion only.
T he n orm alization constant A 0 com es from (u0u0) = 1 and equals
This tetrad is kn o w n directly from th e slim d isk solutions, as it depends on the calculated quantities (ur, O , l and 9,(r)) only.
A ny spacetim e vector X could b e uniquely d ecom posed into this tetrad w ith X(A) = X , e(A).
A.2. The general case
W e h ere assum e nothing about the four-velocity o f m atter u and the location o f photosphere. B oth m ay b e non-stationary and non-axially sym m etric. Follow ing the sam e fram ew ork as in Sect. A .1 , w e d escribe how to obtain the tetrad o f an observer instantaneously located at th e ph o to sp h ere that depends only on th e quantities calculated b y accretion d isk m odels.
A .2.1. T h e fo u r velocity
A s in Sect. A .1 .2 , w e m ay always un'q uely d ecom pose u' , a g en eral tim elike u n it vector, into
u' = A(m0 + v S ' ^ , (A.14)
w here u0 is a tim elike u n it vector, and S , is a spacelike unit vector. E quation (A .1 4 ) u niquely defines the tw o vectors u'0, S , and the tw o scalars A, v. T he vectors and scalars
(A .7) A ,v ,u 0 , S ' (A.15)
: ~ \ d F d F 1 d e, 1 K = N * - 7 -, -TT = K - r , 1 ,
d r dd d r
w here
dfl.(r) _ _(FF I d F d r d r / 89
d r gee \ d r /
r lAa \2T-1/2
N t = ( - g o e r 112 1
+ — ^ '
L 9 e e \ d r j
1 1 j ^ a \21-1/2 5 * = (^ ) _ 1 [ g „ g e e \ d r /T T ’
. , , j d e , \ r 1 1 j d e , \ 2l -1/2
S* - - ( g e e ) "h- '
\ d r / [ g „ g e e \ d r j
~ [ 1- 1/2
Ao = [gtt + Q g ^ (Q - 2&0J .
2 u / S r )2 [gtt + Q g ^ ( Q - 2 ^ i r = ---i---
1 + (ur/ S r)2
.
(/// + ę )K° ~ [ ] l / 2 ’
[- g # (1 - ^ D (1 - ^ l)]
can b e calculated from kn o w n quantities given b y slim disk m odel solutions.
T he four-velocity (A .1 4 ) also defines the instantaneous 3- space o f th e com oving observer w ith th e m etric y ik and th e p ro jection tensor h'k
(A.16) (A.17) yik = g ik u i uk,
h i = ó i - U Uk.
(kqUq) = 0, ( S
.
N.
) = 0.A s b efo re, th e four vectors
ei(A)
=K
N. ,k0>
Sl]
.N ' =
dt ’ dtp' d r ’ dd
N = N
[ a (u0) + 1 (N .) + y (k0) + S ( S . ) ] . T he com ponentsN
, a , y , S are know n.u' = A [1 (u0) + 0 ( n .) + 0 (K0) + V (s .) ] , (A .24)
N ' = N [a (u 0 ) + 1 (N .) + y ( 4 ) + S ( S . ) ] , (A .25) K = K [0 (u0) + + 1 (k0) + 0 ( S . ) ] , (A.26)
S ' = S [A (uq) + B ( N ^ + C (k .) + 1 ( S Q ] . (A .27) T he four unknow n com ponents, b, A, B, C one calculates from th e follow ing four non-trivial orthogonality conditions ((u k) = 0 by construction, cf. (A .2 4 ) and (A .2 6 ))
( u S ) = 0, ( N S ) = 0, ( Sk) = 0, (Nk) = 0, (A.28)
and the tw o unknow n factors K and
S
from th e follow ing two n orm alization conditions(k k) = - 1 , ( S S ) = - 1 . (A.29)
hk = Sk - u' Uk'
W e define the tw o u n it vectors k0 and N ' b y th e u n iq u e condition
T he conditions (A .2 8 ) and (A .2 9 ) are given b y linear equations.
E quations (A .2 4 )- (A .2 9 ) define th e tetrad e(A) o f an observer com oving w ith m atter, and instantaneously located at the p h o to sphere:
(A.18)
(A.19)
e(A) = [ui, N i, K, S ' \ . (A.30)
form an orthonorm al tetrad o f an observer w ith the four- velocity u. can calculated from the solutions o f th e slim -disk equations.
A .2.2. T he photosphere
In th e m ost general case o f a n o n-stationary and non-axially sym m etric p h o to sp h ere, th e location o f the ph o to sp h ere m ay be described by the condition
F(t, ¢, r, ff) = 0. (A .20)
T he vector N norm al to the ph o to sp h ere has the com ponents ' d F d F d F d F
(A.21)
w hich m ay b e calculated from slim disk solutions.
W e p roject N into th e instantaneous 3-space o f the com oving o bserver (A .1 7 ) and norm alize to a unit vector after th e p ro je c tion to obtain
B oth the m atter and the p h otosphere m ove in a general m anner.
T he zenithal direction in the local o b serv er’s sky is given b y N ' .
A p p en d ix B: Integration over the w orld-tube of the photosphere
F or stationary and axially sym m etric m odels, w e define:
N ' = N . = unit vector orthogonal to th e photosphere, w hich is in th e [r, ff] plane;
S . = unit vector orthogonal to N ', w hich is in the [r, ff] plane;
u' = four-velocity o f m atter, w hich is in the [t, ¢, r, ff] space
tim e;
k' = k‘0 = unit vector orthogonal to u ', w hich is in th e [ t , ^ plane;
S ' = u n it vector orthogonal to U', N ' and k', w hich is in the [t, ¢, r, ff] space-tim e;
e(A) = [u', N ' , K , S '] = th e tetrad com oving w ith an observer located in the photosphere.
T he integration o f a vector ( ...) over the 3D h y p ersu rface H or
thogonal to N ' (i.e. the 3D w orld-tube o f the photosphere) m ay b e sym bolically w ritten as
(A .22) (B.1)
In term s o f the tetrad in E q. (A .1 9 ), a vector N ' constructed in this w ay has th e decom position
(A.23)
w here dS is th e “volum e elem ent” in
H
.O bviously, th e h y p ersu rface
H
is spanned by the three vectors [u',K, S '] N. E ach o f them is a linear com bination o f [n ',£ ', S .] N, and each o f the three vectors from [n ',£ ', S .] N is orthogonal to N'.T herefore, one m ay say that the hyp ersu rface
H
is spanned b y [n', ^ , S .] N. It is convenient to w riteA .2.3. T he tetrad
W e now d ecom pose the four vectors, th e first tw o o f w hich w e have derived, the next tw o guessed (but the guess should b e ob vious):
(B.2)
w here d ^ is the line elem ent along the vector S ., i.e. along the p h o to sp h ere in the [r, ff] plane, w ith ff = ff.(r) defining the lo c a
tion o f the p hotosphere, and d A being the surface elem ent on the [t, ¢1 plane.
To calculate d A w e im agine an infinitesim al p arallelogram w ith sides that are located along th e t = const. and ¢ = const.
lines. T he p roper lengths o f th e sides are du = |gtt|1/2d t and dv =
|g ^ | 1/2d ^ respectively, and therefore dA, w hich is ju st the area o f th e p arallelogram , is given by
dA = du dv sin a = dt d ¢ |gtt|1/2| g ^ | 1/2 sin a, (B.3) w here a is the angle betw een th e tw o sides. O bviously, th e c o sine o f this angle is given b y the scalar p ro d u ct o f th e tw o unit
i N ^ ~k ,
N ' = --- r---, N ‘ = N k hi.
I ( n n ) |1/2
f (■■■)iNid S , Jh
d S =
dAdR,d
R = dr ^ jg rr + cjmj >
vectors n i and x i pointing in th e [t, 4 ] p lan e into th e t and 4 d i
rections respectively. T hese vectors are given b y (note that n i = Z A M O )
(B.4)
(B.5)
(B.6)
and
(B.7)
Inserting this into the form ula for dA, w e get dA = d t d 4 (g24 —
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„ (ViO (V,-0)
~ / \ ? i — i /9 *
igjk ( v t ) ( v kt ) \ 1/2 | g jk (V (Vk 0 ) |1/2
B ecause (V,t) = 6, and (V ,]) = 6 ] , one m ay w rite
$ 6t
Hi \gtt\U2’ Xi m l/2' Therefore,
ik gt] gt]
cos a = m x kcjh = --- = ---— ---— ,
\g"\ I g ^ l \gtt\l l 2 M l
{
g% - gtt g # ) / sin a = --- T-rr—1 / 2 || || 1 / 2igtt \ \g]]\
g tt g ] ] ) 1/2. T he final form ula for dS is,
dS = df d ^ d r - g„ g ^ ^ g„- + g ee
j
• (B.8)References
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