astro-ph/ May 1995

Transcription

1 Mon. Not. R. Aston. Soc., { () Pinted 4 May 995 Satellites in Discs: Regulating the Accetion Luminosity D. Sye Canadian Institute fo Theoetical Astophysics, MacLennan Labs, 6 St. Geoge Steet, Toonto M5S A7, Ontaio. C.J. Clake Institute of Astonomy, Madingley Road, Cambidge CB HA; pesent addess: Queen May and Westeld College, Mile End Road, London. asto-ph/955 4 May 995 INTRODUCTION ABSTRACT The theoy of tidal inteactions between an accetion disc and embedded companion has been elaboated in a seies of papes by Lin & Papaloizou (979a & b, 986a & b) (see also woks by Houigan & Wad 984, Goldeich & Temaine 98). These papes wee pimaily concened with the ealy sola nebula, and the question of obital migation of the poto planets, in paticula of the giant planets. An annula gap may be fomed in the accetion disc due to the action of the tidal foces fom the planet, and the fomation of such a gap leads to a coupling between the obit of the planet and the viscous evolution of the disc. In the existing wok the satellites wee light compaed to the local disc, so they migated on a local viscous timescale. Thee was little modication of the disc popeties outside the gap. In this case, theefoe, the pesence of the satellite modies the disc spectal enegy distibution only though the emoval of a naow annulus of emitting mateial aound the satellite. In this pape we exploe the eects of a satellite that is massive compaed with the local disc. Table gives details of two astophysical situations in which this could be the case. The inetia of the satellite pevents it fom migating on the local viscous timescale. As a esult, disc mateial is banked up upsteam. Downsteam of the satellite, mateial is able to accete onto the cental object, so that the mass of the disc inteio to the satellite deceases with time. The satellite theefoe causes global changes in the disc stuctue, and enhances the elative contibution of mateial upsteam We demonstate, using a simple analytic model, that the pesence of a massive satellite can globally modify the stuctue and emission popeties of an accetion disc to which it is tidally coupled. We show, using two levels of numeical appoximation, that the analytic model gives easonable esults. The esults ae applicable to two astophysical situations. In the case of an active galactic nucleus, we conside the case of a M compact companion to the cental black-hole and show that it could modulate the emitted spectum on a timescale of 5 yeas. In the case of a T Taui accetion disc, a satellite such as a sub-dwaf o giant planet could modify the disc spectal enegy distibution ove a substantial faction of the T Taui sta lifetime. Key wods: accetion discs galaxies: active stas: fomation to the spectal enegy distibution emitted by the disc. We conside the coupled evolution of a satellite-disc system in the limit that the disc extends only upsteam of the satellite. Depletion of the inne disc means that the inuence of the inne disc on the satellite dynamics is elatively mino. The neglect of the inne disc implies eithe that the disc fomed fom mateial of highe specic angula momentum than the satellite (as in the case of a collapse scenaio whee the satellite fomed fom low angula momentum mateial, with the disc being fomed subsequently fom highe angula momentum), o else that the satellite is suciently massive fo the inne disc to have lagely dained onto the cental object. In any case, as the satellite migates adially, it becomes elatively moe massive than the local disc, so we anticipate that ou calculation of the evolution of the satellite and disc stuctue is insensitive to the omission of the inne disc. The esultant spectal enegy distibution at high fequencies may be moe sensitive to this assumption. The stuctue of the pape is as follows. In Section we eview the pocess of gap fomation and in Section we descibe an analytic model fo the evolution of the disc and the obit of the satellite. In Section 4 we descibe two numeical appoximations to the same poblem, and compae thei esults with the analytic pedictions. In the nal section we discuss some astophysical applications of ou models, in active galactic nuclei, and in potostella discs. In the AGN case we show that pesence of massive compact objects in the disc would modulate the emission on a timescale of up to 5 yeas. In the case of a potostella disc, we conside a

2 D. Sye & C.J. Clake satellite which is a massive giant planet o sub-dwaf. The long timescales fo the inwad tansit ( 5 yeas) implies that such satellites would be expected to cause a substantial faction of T Taui stas to show specta makedly dieent fom those geneated by standad steady-state accetion discs. THE EVOLUTION OF COUPLED DISC-SATELLITE SYSTEMS Suppose the satellite has a mass M s and its obit is at a adius s. Conside a `collision' between the satellite and a pacel of gas with mass. The eect of the tidal toque of the planet on the disc may be thought of by analogy with dynamical fiction: the uid at > s moves slowe than the satellite, and hence exets a dag on it, educing its angula momentum and tending to dive its obit to smalle adius. The uid at < s does the opposite. The eect on the disc of the back eaction to the fiction is to push the uid away fom the planet; to decease the specic angula momentum of the uid at small adii, and to incease it at lage adii. The esponse of the disc is detemined by a competition between these tidal petubations fom the satellite, tending to open up an annula gap aound the satellite, and pessue and viscous foces in the disc, tending to close the gap. We will deive an expession fo the toque using the analogy with dynamical fiction. This is essentially the calculation done by Lin & Papaloizou (979a), but in the pesent case, since ou pupose is illustative, we omit constants of ode unity. Lin & Papaloizou (986a) examine the validity of the impulse appoximation numeically. (See also Lin & Papaloizou 986b fo a discussion of the same point.) Conside a single collision with a uid paticle at an impact paamete p in the impulse appoximation (e.g. Binney & Temaine 987). If the elative velocity of the satellite and uid is v, then the angula momentum tansfeed to the satellite is L G M s s p v () whee G is the gavitational constant. Aveaging ove such impacts, we nd that the ate of change of L is Z dl dt L vdp Z which we wite as Z dl dt G Ms sdp; () p v T sdp () with the toque pe unit mass of the disc given by T = G M s p v : (4) Witing q fo the mass atio of the satellite to cental object, M s=m, and expanding v in a Taylo seies about p = we have T = q s s 4 s (5) p whee s is the angula velocity of the satellite, assuming a Kepleian otation law and cicula obits fo the satellite and disc. The specic viscous toque nea the edge of the gap, assuming that the adial scale length, is whee is the viscosity in the disc and is the local angula velocity. The adial extension of the gap,, can be evaluated appoximately by equating and T with p : (6) q : (7) Note that in the limit! equation (7) coectly pedicts that!, since in this case thee is no equilibium gap: the satellite continually eats into the disc, at a slowe ate as the gap gows. Following Lin & Papaloizou, we dene a dimensionless measue of the gap width A = q (8) which is the atio of the viscous to obital timescales, multiplied by the squae of the mass atio. Two citeia need to be fullled in ode fo the satellite to succeed in evacuating an annula gap in the disc. Fistly thee is the equiement that the gap be not e-lled by adial pessue gadients, which means that the disc vetical scale height, H, must be less than the width of the gap > H: (9) Secondly, thee is the equiement that the eect of the satellite is to tansfe angula momentum, athe than to accete fom the disc. This means that the gap width should be geate than the Roche adius of the satellite: > R L = q = s: () This educes to q > ; () i.e. q must exceed the local invese Reynolds numbe. In the elatively cool and inviscid discs which satisfy these citeia, the satellite sits in an annula gap in the disc of width, exchanging angula momentum with the disc mateial though the action of tidal toques concentated nea the gap's edge. In pinciple it is possible to aange inteio and exteio disc mateial such that the net toque acting on the satellite is zeo, but such a situation will not pesist once the disc has undegone signicant viscous evolution. The depletion of mateial inteio to the satellite by accetion will esult in an imbalance between the toques expeienced by the satellite. The net angula momentum tansfe is outwad fom the satellite, which theefoe spials in. Let us dene the tem `efeence disc' to mean the steady state disc in the absence of the satellite. We will efe to quantities in the efeence disc with the subscipt. Now we dene a dimensionless paamete measuing the mass of the satellite compaed to the efeence disc B = 4 s M s ; ()

3 Satellites in Discs whee is the suface density in the efeence disc. A elatively light satellite, in the sense that B >, does not substantially modify the suface density of the disc outside the gap. It behaves like a epesentative uid element in the disc, and theefoe its obit evolves adially on the viscous timescale (Lin and Papaloizou 986b). In this pape we ae inteested in the opposite case, whee B <. The disc is unable to push the satellite adially inwads on its own viscous timescale and mateial theefoe foms a esevoi upsteam of the gap. Let us dene f = ; () whee is the viscosity in the disc. Suppose that the themal and hydostatic equilibium elation between and has the powe law fom given implicitly by H =5 q q = / f a b ; (4) c B o whee a and b ae constants. The lagest contast in, ove the efeence value, allowed by condition (), is then q a=( a) (5) whee is the coesponding efeence value of. If the suface density behind the satellite ises to s then the gap will be ovewhelmed by viscous diusion. In a Shakua-Sunyaev (97) disc, whee the viscosity is given by H ; (6) with constant, the citeia (9) and () educe to, espectively, H q = < (7) and H < q =5 : (8) If (7) is violated, the gap will be closed by pessue; if (8) is violated, the gap will be closed by accetion onto the satellite. In most cases of astophysical inteest H= is an inceasing function of in the efeence disc, so once the satellite opens up a gap it pesists, as long as the disc stuctue is not modied by the satellite (i.e. if B > ). If the disc stuctue changes as a esult of the pesence of the satellite, and H= becomes a deceasing function of, the values of q and detemine whethe it is accetion o pessue which eventually closes the gap. If q = q =5 < ; (9) then if the gap closes, accetion will be esponsible. This is illustated in Figue in which we have dened thee impotant adii: o is the adius at which the citeion () is satised; B is the adius at which B = ; and c is the adius at which () is no longe satised in the petubed disc. Figue is entiely schematic, and no paticula viscosity law has been used to calculate H=. When s < o a gap is opened in the efeence disc, and when s < B the disc is banked up behind the satellite. Figue. The aspect atio H= of the disc as a function of. The dot-dashed line is the efeence disc. The long-dashed line shows the eect of a satellite in banking up the disc. The lowe and uppe shot-dashed lines ae the citical values fo closue of the gap by accetion and pessue espectively. The solid line is a possible tajectoy of H= immediately behind the satellite as it migates adially. When s < c o s > o the satellite cannot maintain an annula gap in the disc, so that the analysis involved in the pesent wok (which assumes tidal coupling between satellite and disc acoss such a gap) is invalid. Knowledge of the evolution of a satellite embedded in a disc ow equies twodimensional hydodynamic calculations beyond the scope of the pesent wok. Thus the following emaks ae necessaily speculative. Once the satellite eaches the adius c, we would expect that the inwad ow of mateial as the gap is ovewhelmed would esult in a bust of accetion luminosity. The faction of this mateial that is acceted by the satellite, athe than by the cental object, is howeve not clea. It is also not clea whethe the satellite would emain embedded in the disc theeafte o whethe the oveow of a small faction of the dammed up mass would allow the gap to e-open. In the latte case the satellite might continue to spial in, maintaining a state of maginal stability against gap closue, with intemittent oveow of the mateial upsteam. The maximum enegy which could be eleased when the gap is closed is E b GMc c : () This coesponds to accetion of the mass M cc by the cental object of mass M and adius. The expectations that a satellite would move as a epesentative uid element in the case B > was conmed by Lin and Papaloizou (986b) using one-dimensional numeical calculations. In thei calculation, the usual viscous diusion equation fo the disc was supplemented by an additional tidal toque tem (equation 5). The main motivation fo these calculations was to evaluate migation timescales fo the poto-jupite. The question of substantial damming up of the disc by a satellite consideably moe massive than the local disc was not extensively investigated. In the following section we conside this question in some detail.

4 4 D. Sye & C.J. Clake Table. The appoximate values of elevant physical paametes in a selection of astophysical situations. M Ms B t B AGN 6 M M 4 GM=c 5 y T Taui sta M M R 5 y M is the cental mass; M s is the satellite mass; B is the adius within which the satellite is moe massive than the local disc; t B is the adial migation timescale of the satellite at B. AN ANALYTIC MODEL FOR SATELLITE-DISC EVOLUTION In this section we st descibe the chaacteistics of the efeence disc. We then develop a model fo the suface density of the disc when inuenced and enhanced by a satellite with B < (equation ). The slow adial dift of the satellite enables viscous toques in the disc to communicate the pesence of the satellite to lage adii. We chaacteise the upsteam stuctue of the disc by dividing it into two sections: an oute egion whee the inuence of the satellite has not yet been felt, and one in the vicinity of the satellite. The bounday between these egions is denoted t. The oute egion, t <, is assumed to be in a quasi-steady state with a mass ux detemined by the fuelling ate at innity. The inne egion, s < < t, is assumed to be in a quasi-steady state the popeties of which ae detemined by the angula momentum tansfe between the satellite and the inne edge of the disc. This esults in a lowe mass ux in the inne egion, on account of the damming eect of the satellite. The limit of t! is analytic and explicit.. The efeence disc In a thin, Kepleian accetion disc with no extenal toques, f is (e.g. Pingle 98), and so in a steady @ f = : In a steady-state disc we assume an oute bounday = _ M = f;! ; () whee _M is the mass ux though the disc. At the inne bounday,, we assume the toque on the inne edge of the disc vanishes, whence f = ; = : (4) The geneal solution to () is f = X + Y (5) whee X and Y ae constants. With the bounday conditions () and (4) we nd that f = f = f f : (6) In the limit!, which is adequate fo ou puposes, f = f. : : : : Figue. The suface density s behind the satellite as a function of s (shot dashed line), in the analytic model with t!. The constants of the model ae chosen so that B = :4 at =. The solid lines ae snapshots of the suface density. Also shown is the efeence disc (long dashed line).. The satellite We conside the situation in which the disc inteio to s is empty ( = ). Such a situation would aise if the disc was fomed in this way, but would also be appoximately ealised in the limit of a suciently massive satellite: the satellite acts as a dam, allowing the inne disc to dain away in a time much shote than the migation timescale of the planet. We assume that the tidal coupling between the satellite and the disc is conned to a vey naow egion at the inne edge of the gap. In this case equation () adequately descibes the evolution of f. The oute bounday condition emains the same, on the assumption that fa enough upsteam of the satellite the disc must appoach the undistubed solution. The inne bounday condition is detemined by supposing that a) the satellite and disc edge move with a common velocity, and b) thee is oveall consevation of angula momentum. The adial velocity in the disc is given by consevation of mass as v f Evaluating this at = s, and equating it to the velocity of the satellite, v s, povides the inne bounday condition. (Povided emains consideably smalle than s, then v and v s cannot substantially die.)

5 Satellites in Discs 5 Angula momentum consevation is guaanteed by equation () eveywhee except at the inne edge of the gap. The viscous toque on the edge of the disc is G = f s s s (8) assuming a Kepleian otation law. The ate of change of the angula momentum of the satellite is d dt Ms s s = Ms s s v s s (9) again assuming a Kepleian otation law. Consevation of angula momentum is thus expessed as f s s s + Ms s s v s s = ; () so the value of v in the bounday condition (7) is v s = 6sfs M s : () We wite equation () as v s = B fs f whee s t ; () t = s () is the viscous timescale in the undistubed disc. In eality the disc will neve each a global steady state, since v s is always non-zeo. But, fo small B, the satellite moves slowly compaed to the local viscous timescale, so we make one moe appoximation: that the disc is able to each a steady state fo all values of s. In the oute egion of the disc, t <, we suppose that it has not had time to adjust to the pesence of the satellite, and thus that f = f = f ; t < : (4) In the inne egion, s < < t, we solve () with the bounday conditions (4) at = t, and a combination of (7) and () at = s. Using (4) and (5), we nd that the equied solution to () is s f = f a + (f s f a) (5) whee f a < M _ is the mass ux though the inne disc. Equating (4) and (5) at = t we nd that f a + (f s f a) s t = f ; (6) with =(+a) f f s = f B : (7) f a Once t is specied, equations (6) and (7) ae sucient to detemine f s and f a, given s. The model assumes that the disc is in a steady state eveywhee apat fom at t. At t the mass uxes in the inne and oute disc ae unbalanced, so mass is not conseved unless t moves at just the ight ate. The evolution of t would be specied by equations (6) and (7), togethe with an intego-dieential equation that ensued consevation of mass. We do not develop the latte in this pape, but etain the notion of an inne and oute disc fo the pupose of qualitatively undestanding ou numeical simulations.. The limit t! In the limit t!, the model becomes explicit and analytic. We expect that this limit should be appoached as B!, in which the satellite moves vey slowly, and is able to inuence the suface density vey fa upsteam. In this limit we nd that f s = f B =(+a) : (8) This is an appoximate desciption of the `banking up' of the suface density behind the satellite. The suface density just behind the satellite is given by s = B a=(+a) : (9) Povided that a > we ae eassued that the smalle the value of B, the geate is the suface density s. The suface density can thus be enhanced so that s. This is illustated in Figue, in which we have used a viscosity law (equation 4) with a = :65 and b = :65, which is appopiate to quiescent potostella discs (Clake, Lin & Pingle 99). In all the quantitative wok in this pape, and in all the gues (except Figue ), we use the same viscosity law. Combining (8) and () we can nd v s explicitly in tems of s: v s = B a=(+a) s t : (4) Integating this equation we nd the evolution of s with time h s = i c t i =c B a=(+a) ; (4) t i i whee t i and B i ae the values of t and B evaluated at = i, and c = + b + a : (4) Compaing equation (4) to the adial velocity at sin the undistubed disc v = s t ; (4) we nd that v s = v B a=(+a) : (44) Fo B <, a >, b >, we see that v s < v. Ou assumption of a quasi-steady state must beak down when v s > v but we expect v s to be at most v, when B. 4 NUMERICAL MODELS 4. Innitessimal bounday laye To test the steady-state assumption in the analytic model, we now dop that assumption, but etain the inne bounday condition (equations 7 and ), and then solve equation () numeically. The ighthand side of equation () is diffeenced in a standad implicit scheme, and a Lax aveage is employed fo the lefthand side (e.g. Pess et al. 99). An iteative pocedue is employed fo solving the implicit equations, as descibed fo example by Potte (987). A unifom gid in the vaiable = is employed, with 64 points in the ange = in = :5 to = out = 4. The initial

6 6 D. Sye & C.J. Clake 5 4 f f :5 y :5 Figue. The esult of dopping the steady-state assumption: snapshots of f(y) with B = :4 at = (solid cuves). The shot-dashed cuve shows the analytic pediction fo the ightmost snapshot with t = ; the long-dashed cuve shows the analytic pediction with t!. 5 f f 5 :5 y :5 Figue 4. As Figue but with B = : at =. The shot-dashed cuve in this case is with t = fo the ight-most snapshot; the long-dashed cuve shows the analytic pediction with t!. adius of the satellite obit is = 6, and the inne bounday condition is applied at the neaest gidpoint upsteam. At evey timestep, the new location of the satellite is computed fom equation (), and if it cosses a gidpoint, the inne bounday condition is applied at that gidpoint. In Figue we show two snapshots of the disc stuctue fo the case B = :4 at = ; we plot f=f as a function of y = =. The gue illustates that the disc is indeed well epesented by a sequence of quasi-equilibium states f(y) can be appoximated by two staight line sections. The adius of the intesection of these staight lines, t, does not change much as the satellite migates adially. (We have checked the independence of t on the position of the oute bounday by caying out additional calculations with out = 56, in which the esults wee the same.) Figue 4 (B = : at = ) shows the case of a heavie satellite. A moe massive satellite causes a lage suface density enhancement and is able to inuence the disc out to a lage adius. The analytic model is in each case able to t the numeical esults adequately, once t is specied. 4. Explicit toque As a second numeical compaison with the analytic model we dop the assumption of an innitessimal bounday laye, and include the tidal toque fom the satellite explicitly. Thus equation (5) appeas as a souce tem in the diusion @ (45) (Lin & Papaloizou 986b). Fo pactical puposes, T has to be softened on a scale which is smalle than. We use the pesciption of Lin & Papaloizou (986b), but since vanishes in the gap anyway, the details of the softening ae not impotant once the gap has fomed. Angula momentum consevation is ensued by setting the ate of change of the angula momentum of the planet equal to the integated contibutions of the tidal toque exeted by the disc: Z d Ms s s = dt t d: (46) The numeical esolution has to be high fo small A (small gap width). Hee we used 8 gid elements in the adial ange [:5; 56]. The esults of the explicit toque calculation ae shown in Figue 5. In common with Figue, we see that f is appoximately descibed by two staight line segments as in the analytic model, and that the location of t does not change much with time. This illustates that, as expected, the fom of the toque distibution has negligible eect on the disc stuctue at lage distances upsteam. The disc stuctue close to the satellite is somewhat dieent when the distibuted toque is included. The disc is eectively obeying the bounday condition (equation ) at a adius whee the tidal toque is maximum. But the tidal toque does not immediately dop to zeo outside this adius, and the suface density can continue to ise. As a esult, the peak suface density is geate than that implied by equation (). The distibuted toque model pedicts highe peak densities than the innitessimal bounday laye model. In Figue 6 we compae the esults of using two dieent values of A (equation 8). A smalle value (Figue 6a) leads to a naowe gap, and a shape tun ove of the suface density pole at the gap edge, but does not signicantly aect the value of the suface density at the tun ove. We conclude that the gap width ( A = s) has a mino effect on the suface density enhancement, which is contolled lagely by the atio of the disc mass to the mass of the satellite (B). 4. Summay Figue 7 shows a compaison of the evolution of s fo the numeical models with the analytic model. In the case of

7 Satellites in Discs f f s 6 4 :5 y :5 t 4 Figue 5. The esult of including an explicit tidal toque: the explicit toque model with B = :4 at = and A = :5. The dashed cuve shows the analytic pediction fo the ight-most snapshot with t!. the innitessimal bounday laye model, the suface density enhancement is smalle than pedicted by t!. The suface density enhancement detemines the adial evolution of the satellite though equation (), so the evolution of s is slowe than the analytic pediction. In the case of the explicit toque model it is slowe because the adius at which equation () is satised is close to the satellite than the peak suface density. The numeical esults show that the disc has a quasiequilibium stuctue with t vaying slowly. The limit t!, in which all the equations ae analytic and explicit should be appoached in the limit of a massive satellite. Since the numeical calculations ae consideably moe expensive in eot and computational esouces, the advantages of the analytic appoach ae obvious. 5 DISCUSSION We have deived analytic expessions fo the suface density enhancement in a disc which is modied by the pesence of Figue 7. The evolution of s as a function of time (in units of the viscous time t at = ), fo B = :4 at =. The long-dashed line is the explicit toque model; the shot-dashed line is the innitessimal bounday laye model; and the solid line is the analytic model with t!. a massive satellite at its inne edge (equations and 9). We have veied the accuacy of these expessions, to ode unity, using two levels of numeical appoximation. A satellite initially at lage adius, whee B >, moves likes a epesentative uid element and scacely modies the stuctue of the disc. Povided B deceases with deceasing adius (b >, equation 4), such a satellite may eventually each the adius, B, at which B =. The disc suface density is subsequently enhanced behind the satellite because it is elatively massive compaed with the local disc (B < ). In Figue 8 we compae the spectal enegy distibution, F, in a disc dammed up behind a satellite with that geneated by a efeence disc with a zeo toque bounday condition at its inne edge. In all cases we assume that the disc adiates eveywhee as a black body with local eective tempeatue xed by the local viscous dissipation ate. When the satellite is at adii consideably lage than the inne edge of the efeence disc, the assumed absence of ma- a) : : b) : : : : : : Figue 6. Snapshots of the suface density in the explicit toque model with B = :4 at = and a) A = :5, b) A = :5. The dashed cuves show the analytic pediction fo s with t!.

8 8 D. Sye & C.J. Clake F : : : : : Figue 8. The evolution of the spectal enegy density F fom the accetion disc, as a satellite difts adially inwads. The dashed line shows F fo the efeence disc, with inne adius = :5, out = and a = b = =5. The solid lines show F fom the banked-up disc, which is assumed to be empty inside the adius of the obit of the satellite. Fom left to ight, the solid cuves ae fo s =, and 5. In Each case t! and B = : at =. teial inteio to the satellite implies that the spectal enegy distibution is decient in highe fequency adiation compaed with the efeence disc. (This eect would be moe maked if the contibution fom bounday laye emission wee included in the case that the efeence disc acceted onto a stella suface at its inne edge). As the satellite pogesses to smalle adii, the emission is enhanced ove a wide ange of fequencies: the enegy lost by the inwad spialling satellite is adiated though the disc, and the enhanced emission fom just upsteam of the satellite compensates fo the absence of mateial inteio to it. The spectum of a classic steady state accetion disc may be chaacteised as a powe law F / = ove a wide dynamic ange. When a satellite with B < is pesent this changes to F / 5=7 (see Appendix fo deivation). We conside two applications of astophysical inteest: a disc aound a pe-main sequence (T Taui) sta with a sub-dwaf satellite; and a disc in an Active Galactic Nucleus whee the satellite is eithe of stella mass o is a compact object of mass M (see Sye, Clake and Rees 99 and Lin 994 fo aguments elating to the possible pesence of such objects in AGN discs). The T Taui model uses a cental mass of M, viscous alpha paamete and an accetion ate of 7 My (details of the disc stuctue in Clake, Lin and Pingle 99). The AGN model uses a cental mass of 6M, viscous alpha paamete and an accetion ate of one tenth the Eddington ate (details of the disc stuctue in Clake 987, 988). Fo a stella mass object in an AGN disc, the disc ovewhelms the gap ceated by the satellite at eectively all adii. When the mass of the satellite is M, and in the T Taui model the gap opening citeia ae satised and it is meaningful to calculate B, as listed in Table. A massive compact satellite in an AGN disc would bank up the disc behind it fo s < B 4 times the adius of the cental object (at this adius the disc is cool and unionised). In the pe-main sequence case, a sub-dwaf of mass M would bank up the suface density in the disc fo s < B R. The time taken fo the companion to spial in fom B is of ode the disc's viscous timescale at B; t s, the time to spial in fom a smalle adius is geate than the local efeence viscous timescale, t, because the inetia of the satellite is lage compaed with the local disc. At smalle adii the atio t s=t inceases. Howeve, t B povides a measue of the inteval of time ove which the satellite is able to signicantly modify the stuctue and spectal enegy distibution of the disc. Thus a massive black hole in an AGN disc would give ise to deviations fom classic steady-state disc emission duing its tansit of the inne disc. It may also aect the continuum emission, on the assumption that the continuum aises fom physical pocesses nea the cental mass, and theefoe elies on the disc to tanspot fuel to small adii. A ealistic calculation of the spectal enegy distibution would also have to include the emnant disc inteio to the satellite, and would equie knowledge of the physical pocesses that give ise to the continuum emission. We nd that fo a lage ange of AGN disc paametes, the disc is unable to close the annula gap ceated by the companion and that the banking up of the disc behind the satellite (and associated incease in luminosity) continues until the companion eaches the last stable obit and is swallowed by the cental black hole. At this point the spectal enegy distibution of the disc evets to that of the usual steady state disc on a timescale compaable with t B. In the pe-main sequence case the coesponding value of t B is long ( 5 yeas) so that the disc stuctue and spectum would deviate fom its steady state values ove a consideable peiod. Although, as above, the calculation of the spectal enegy distibution should also contain a component fom the disc inside the satellite (a contibution that becomes inceasingly unimpotant once the satellite is well inside B), Figue 8 gives some indication of the types of spectal enegy distibutions that might be geneated. Thus, contay to pevious attempts to model the eect of satellites on disc emission meely in tems of \missing emission" fom annula gaps (e.g. Mash and Mahoney 99), we nd that satellites massive compaed with the local disc poduce global modications in the spectal enegy distibution: the disc emission is enhanced upsteam of the satellite and this poduces a vaiety of signatues in the esultant spectal enegy distibution. While the satellite is at elatively lage adius, the inceased stength of the oute disc poduces a hump in emission at low fequencies. This is in line with the obseved tendency of T Taui stas to exhibit too much powe at low fequencies compaed with standad steadystate disc models. Once the satellite is vey close to the sta, its eect is to steepen the spectal enegy distibution fom the classic = to a 5=7 powe law. This eect would be masked obsevationally by the contibution fom the bounday laye between the depleted disc and the cental object. In the case of a satellite in a T Taui disc we nd that the disc is neve able to ovewhelm the gap ceated by the satellite, povided it emains cool and lagely unionised; if this is the case, the satellite continues to spial inwads until it is eithe enveloped o tidally disupted by the cental sta. Anothe possibility is that the tempeatue in the disc upsteam of the satellite eaches the theshold fo the hydogen ionisation; if this is the case then the associated apid ise in

9 Satellites in Discs 9 the viscosity could enable the disc to ovewhelm the gap and ow apidly inwads past the satellite. Such a scenaio may oe a mechanism fo the geneation of FU Oionis outbusts in pe-main sequence stas (Clake & Sye in pepaation). ACKNOWLEDGMENTS CJC is gateful fo the hospitality of the MPA, Gaching whee this wok was initiated. We ae indebted to Ds F. and E. Meye fo input duing the ealy stages of this poject, and, in paticula, fo suggesting the appoach adopted in Section 4.. DS acknowledges nancial suppot fom PPARC and NSERC of Canada. REFERENCES Binney, J., and Temaine, S., 987, Galactic Dynamics, Pinceton: Pinceton Univesity Pess. Clake, C.J. 987, Thesis, Oxfod. Clake, C.J. 988, MNRAS, 5, 88 Clake, C.J., Lin, D.N.C., and Pingle, J.E. 99, MNRAS, 4, 49 Goldeich, P., and Temaine, S. 98, ApJ, 4, 45 Hebig, G.H. 989, ESO Wokshop,, (ed B. Reiputh) Houigan, K., and Wad, R.W. 984, Icaus, 6, 9 Lin, D.N.C., 994, in Theoy of Accetion Disks, eds W.J.Duschl, Dodecht:Kluwe. Lin, D.N.C., and Papaloizou, J. 979a, MNRAS, 86, 799 Lin, D.N.C., and Papaloizou, J. 979b, MNRAS, 88, 9 Lin, D.N.C., and Papaloizou, J. 986a, ApJ, 7, 95 Lin, D.N.C., and Papaloizou, J. 986b, ApJ, 9, 846 Mash, K.A., and Mahoney, M.J. 99, ApJ, 95, L5 Potte, M.C. 987, Mathemetica Methods in the Physical Sciences, Englewood Clis, N.J.: Pentice-Hall. Pess, W.H., Teukolsky, S.A., Vetteling, W.T., and Flanney, B.P. 99, Numeical Recipes, Cambidge: Cambidge Univesity Pess. Pingle, J.E. 98, ARAA, 9, 7 Shakua, N.I., and Sunyaev, R.A. 97, AA, 4, 7 whee k is Boltzmann's constant. Thus the spectum of the disc as a whole is F / Z out in B (T ) d (49) whee in and out ae the inne and oute adii of the disc, espectively. Suppose that in T = T (5) in which case fo h kt we can ewite equation (49) as Z xout F / T = = x += dx e x (5); whee x = h=kt and x out = h=kt out. Fo fequencies such that kt out h kt we may take x out and hence F / =. In the efeence disc = =4 and we ecove the classic esult that F / =. When the disc is tuncated by a satellite we take in = s, and assume that the disc inteio to the satellite is empty. Fo B, equations (5) and (8) imply that = f B =(+a) s : (5) Thus we nd that = 7=8 and F / B 4=7(+a) 5=7 : (5) Thus, the spectum of the disc is steepe than the efeence disc, and the nomalisation inceases as B deceases (i.e. as s deceases) fo a >. This pape has been poduced using the Blackwell Scientic Publications TEX macos. APPENDIX : THE EMITTED SPECTRUM OF THE DISC The esponse of the disc to the tidal foces fom the satellite causes addtional viscous dissipation in the disc. If the inne disc is evacuated, then thee is also a decit of high fequency adiation compaed with the efeence disc. Hee we calculate the spectal enegy density of the adiation fom the disc in ou analytic model. We show that the fome effect dominates at late times, and the latte dominates when the satellite st begins to stem the adial ow in the disc. On the assumption that the disc is optically thick, at each adius thee is an eective tempeatue of the emitted adiatio T given by T 4 / f (47) whee is the Stefan-Boltzmann constant (e.g Pingle 98). The spectum fom each annula element of the disc is the blackbody spectum B (T ) / exp(h=kt ) (48)