arXiv:1110.2614v2 [astro-ph.HE] 13 Oct 2011 oh crto fsascoet asv lc holes black Dai Lixin massive to close stars of Accretion Roche o.Nt .Ato.Soc. Astron. R. Not. Mon. † ⋆ is near continuum from model The come (ISCO). line indeed Orbit lines iron Circular 2010; Fe Stable the the Innermost al. that the with et certain Shcherbakov not problem is 2009; The it that 2011). al. et al. Li et model Gou blackbody 2004; Nowak & multi-temperature al. Reynolds a et the (e.g., (Yuan using lines fitting by iron by to or the either is (2003)), of disc: method shapes accretion One a and the measure. shifts is from be it radiation to since pa- use harder measured relativistic much a easily is spin, more rameter, the is However, The hole parameter. approaches. black Newtonian various a using been of topic has There mass this galaxies. on of progress evolution informatigreat and reveal formation and the regime, general on of strong-field tests the to in lead relativity the can over measurements topics Such research decade. - astrophysics past spin important and an mass - been parameters has hole black of measurement The INTRODUCTION 1 3My2018 May 23 c 1 , -al d3safr.d (RB) [email protected] E-mail: -al oiosafr.d (LD) [email protected] E-mail: 2 09RAS 2009 al nttt o atceAtohsc n omlg,S Cosmology, and Astrophysics Particle for Institute Kavli 1 ⋆ n oe Blandford Roger and , 000 1– , ?? aais cie aais niiul EJ1034+396 RE individual: galaxies: active, galaxies: ABSTRACT eidi h -a msino EJ1034+396. words: RE inter Key of an of emission interpretation X-ray possible the an a in obse hole with period The the along frequency. orbiting discussed orbital s spot is stellar mass hot the modulation the a with that forms luminosity envisage it the We where ste modulate ev regime. disc, the “Roche accretion relativistic how the the the calculate hits through in We change stars lobe. will of Roche rate types its mass-transfer filling t and while thermal period hole and, adiabatically, the evolve the will than hole. from star faster the the recede Roc scale, scale to the time dynamical time by (L1) (I the a than enclosed point Orbit on volume Lagrange Circular happens the inner Stable mass-transfer for t the Innermost formulae If through interpolation the regime. flow relativistic outside relativistic will lobe presented the gas Roche are hole, in its tes formulae torque fills a radiation-reaction fitting star provide the New eq and and and radiation. hole circular time be gravitational black quasi-period to the resultant to presumed is the of due orbit from stellar spin hole. the regime and starts, black strong-field accretion mass massive the a the in orbiting detect relativity star to a is by goal formed system binary (EMRI) 20)Pitd2 a 08(NL (MN 2018 May 23 Printed (2009) nti ae ecnie oh crto na xrm Mass-Ratio Extreme an in accretion Roche consider we paper this In -as iais tr:wns uflw,acein aitv tr radiative accretion, outflows, winds, stars: binaries, X-rays: 2 † on afr nvriy el ak A905 USA 94025, CA Park, Menlo University, tanford aito.We h trfil tRceLb,ms sstripped is mass gravitationa Lobe, Roche of cir- it action equatorial fills star an the the in under When radiation. hole inward the spirals orbiting orbit In star cular model. a to case, al. hard other but et luminous the Zauderer Sw very recent are 2011; events the These is al. 2011). eventu- disruption et 2011). tidal observation(Bloom and Loeb a apart J16444+57 & of Stone torn instance 1988; is possible (Rees One hole star very vol- the the by a Roche swallowed and or the ally orbit periastron exceeds unbound at volume an stellar ume star in a The hole when orbit. black happen eccentric a can by This disruption. close tidal passes a is case first star- by induced of effects way passages. tidal close another using SMBH propose spins we mod- SMBH Here these detecting complicated. However more Znajek (2011)). are & Blandford els al. commonly (e.g., et are Tchekhovskoy spins Jets region absorbed. hole (1977); disc be black can inner with and the associated range UV from the Black in radiation mass is Super-Massive the stellar for of (SMBHs), However, spin Holes effectively. the obtain holes to black employed be can method hr r w xrm ae fsc ia ffcs The effects. tidal such of cases extreme two are There A T E tl l v2.2) file style X m cl u slower but scale ime aoil n shrinks and uatorial, lto”frvarious for olution” vblt fsc a such of rvability nms ae,will cases, most in csgas Before signals. ic a ultimately may d ra eventually tream o h inspiral the for elb.I this If lobe. he itn hour 1 mittent einspiralling he h ultimate The egv new give We C)o the of SCO) fgeneral of t lrorbital llar Inspiral ansfer, l 2 Lixin (Jane) Dai, & Roger Blandford stably from the star on a time scale of 10, 000 100, 000 within the star is frozen and the evolution is as described ∼ − years. This steady mass-transfer is not as luminous as tidal in our previous paper. In the Newtonian regime it is the disruption, but lasts for much longer. Also the second case is sign of (∂ρ¯⋆/∂m⋆)s, where m⋆ is the total remaining mass easier to model with high precision for the mass and spin of of the star, andρ ¯ is the mean stellar density, that determines the hole. In this paper we will provide techniques for doing whether or not an EMRI becomes an EMRO. so. The gas flowing through L1 will generally have enough Mass-transfer in binary systems has been well studied angular momentum to form a torus orbiting the hole. A hot for a variety of stellar components and mass ratios. This in- spot will be formed where the stream hits the disc. In ad- vestigation is mostly focused on binary systems composed dition, it is quite possible that one or two spiral arm shock of a stellar-mass compact object and a companion star, waves will be created in the disc. The hot spot and shocks e.g. Webbink (1985). In particular, Hjellming & Webbink will orbit with the orbital frequency of the star and an in- (1987) and Soberman et al. (1997) explored the stability of clined observer should see a modulation of the radiation adiabatic mass-transfers based on polytropic stellar mod- from the disc, most likely in the X-ray range. Clearly the els, and showed it could be dynamically unstable. However, modulation will be the greatest when the observer is in the these studies are Newtonian calculations, since the compan- equatorial plane, observing maximal Doppler boosting. ion star reaches the Roche limit far away from the ISCO One observation that may possibly be explained by this of the stellar-mass compact object. In Extreme Mass-Ratio model is the quasi-periodic XMM-Newton X-ray signal from Inspiral (EMRI) systems, stars can get close to the ISCO a nearby active Seyfert I galaxy RE J1034+396 observed by before being tidally disrupted. Therefore, the signals from Gierlinski et al. (2008). As shown in Fig. 1 in their paper, such binary systems can in principle be used to constrain the rest-frame X-ray emission has a periodicity of 1 hr. ∼ the mass and spin of SMBHs, and to test general relativity The X-ray flux has a modulation of 0.1 associated with ∼ in the strong-field regime. In this paper, we present an accu- the nucleus of the galaxy, and this modulation could be well rate general relativistic framework for studying such mass- fitted by a cosine function. We will show in this paper that a transfer. dwarf star or brown dwarf or red giant can produce emissions 6 7 We shall be especially concerned when the mass-transfer with such a periodicity filling a 10 10 M⊙ SMBH un- ∼ − is “adiabatic” - in other words it happens on a time scale der Roche accretion, though the mass accretion rate under short compared with the thermal (Kevin-Helmholtz) time adiabatic condition and pure gravitational radiation torque for the star yet long compared with the dynamical time. This is too low to produce the observed X-ray flux. problem has previously been considered by Hameury et al. In section 2, we will illustrate the mechanism by com- (1994) in connection with the (erroneous) identification of puting the evolution in the Newtonian limit. In particular, periodicity in NGC6814 (Madejski et al. 1993). This pa- we will show how the star spirals in, when it reaches the per builds upon the Hameury et al. (1994) discussion and Roche limit, and we will calculate how the stellar orbital presents a more comprehensive treatment of stars and plan- period and mass-transfer rate evolve taking a dwarf star as ets. In particular, we will show for various stellar types how representative. We will also discuss the final fate of stars the stellar Roche volume and evolution change with the spin after they have undergone Roche evolution. In section 3, parameter of the hole and also the rotational angular veloc- these calculations will be repeated in the relativistic regime, ity of the star in the relativistic regime. We will also con- and be compared with Hameury et al. (1994)’s. In section 4, clude that such accretion events are stable over a dynamical we will use the data from RE J1034+396 to fit our model, time scale, in contrast to the suggestions of Hameury et al. and to make predictions of the future signals. In particular, (1994). The differences in results are mainly caused by more we will discuss how spins can be constrained from signals. accurate calculation of the relativistic Roche potential (and Discussion and future research directions will be laid out in the resultant Roche volume), as well as more accurate grav- the last section. In our forthcoming paper in preparation, itational radiation power calculation. we will use these new results to construct new models of In an EMRI system, the central black hole is so massive EMRI/EMRO evolution. that its mass is effectively constant. The stellar orbit, on the other hand, shrinks due to the loss of energy and angular mo- mentum through gravitational radiation and other torques. This radiation is a prime target for space-borne gravitational radiation observatories intended to test the general theory of relativity in the strong field regime. An EMRI system can also emit electro-magnetic radiation (principally X-rays and 2 NEWTONIAN TREATMENT optical), and its observation can also provide the basis of a good test of general relativity, if measurements are carried A Newtonian treatment brings out the salient features of out accurately. the problem, e.g., Hameury et al. (1994). Consider a fully We restrict our discussion of EMRI systems to stars in convective lower main sequence star with a mass of m⋆ ≡ circular, equatorial orbits. If and when such a star overflows mM⊙ as a representative, and assume that it orbits a hole 6 its Roche lobe close to the ISCO, gas will flow through the with a mass M 10 M6M⊙, in a circular, equatorial orbit ≡ inner Lagrange point (L1), and the stellar orbit will usu- with radius r. (The stellar orbit circularizes quickly either ally expand - an Extreme Mass Ratio Outspiral or EMRO! through gas dynamical drag or gravitational radiation as We shall assume initially that the stellar evolution is adia- shown in Dai & Blandford (2011). We will take G = c = 1 batic, and discuss later the limitations of this assumption. henceforth. Under these conditions, the entropy variation with mass From Kepler’s third law, the orbital radius (measured

c 2009 RAS, MNRAS 000, 1–?? Roche Accretion of stars close to massive black holes 3 in units of the gravitational radius Rg (M6) = 1.47 near the star, and note that the linear term is canceled 11 × 10 M6 cm) is: by a linear term in the expansion of the centrifugal po- tential, the Newtonian tidal tensor contributes quadratic 1 2 2 2 2 2 r − 3 terms 2 (φ,xxx + φ,yyy + φ,zzz ), where φ,xx = 2φ,yy = r˜ = = 23.9 P 3 M , (1) − 3 − 6 2φ,zz = 2M/r . (Note that the contribution to the Lapla- Rg − cian vanishes as it should.) The expansion of the centrifugal where the orbital period P is measured in hours. 1 2 2 2 term also contributes a quadratic form 2 ω (x + z ), or The orbital angular momentum: 1 2 2 2 3 − 2 kω (x + z )M/r . 2 1 − 3 A co-rotating star will expand to fill an equipotential L = m √Mr = 4.9 mM P 3 (2) ⋆ 6 surface. However, it is rather unlikely that an inspiralling 54 2 measured in units of M⊙ c Rg(M6 = 1) = 8.8 10 g cm /s. star will be able to maintain corotation. If it rotates with × 2 Likewise, the orbital energy, in units of M⊙c = 1.79 angular frequency ω < Ω, the quadratic centrifugal term × 1054 erg, is is approximately 1 ω2(x2 + z2). which includes the effects − 2 2 2 of Coriolis force. (There may be some internal circulation −1 3 − 3 E = 0.5Mm⋆r = 0.021 mM P . (3) − − × 6 which will vitiate adiabaticity, but we do not consider this possibility further in this paper.) 2.1 Pre-Roche evolution The gradient of the Roche potential vanishes at the first and second Lagrange point. Using this we can calculate for We assume that the star follows a circular orbit which two extreme cases that the first and second Lagrange points shrinks slowly due to gravitational radiation. In the New- (L1 and L2) are located at tonian limit, the torque is 1 m⋆ 3 xR = 0.794 (kω) r (7) 4 7 4 7 32 3 2 2π 3 44 2 3 − 3 2 −2 ± X M τ = M m⋆ ( ) = 1.7 10 m M6 P g cm s .   5 P × from the star, where (kω) decreases from 1 to 0.87 as kω X (4) goes from 0 to 1. The time it takes the star starting from radius r to The transverse extents of the equipotential Roche sur- plunge into the hole is (assuming this gravitational radiation face are: 1 torque, and r rISCO, which is the radius of the ISCO, see m⋆ 3 ≫ yR = 0.497 (kω) r (8) below): ± Y M 4   − 2 8 and 5 r −1 3 3 tinspiral = 2 987 m M6 P yr. (5) 1 256 M m⋆ ∼ m⋆ 3 zR = 0.497 (kω) r (9) (e.g. Misner et al. (1973)). ± Z M   Typically if the orbit is at least mildly relativistic, the where (kω) goes from 1 to 0.89 and (kω) goes from 1 to Y Z inspiral time is less than a Kelvin-Helmholtz time. 0.93 as kω goes from 0 to 1. Furthermore, the volume contained by the Roche sur- 2.2 Roche limit face is numerically calculated to be: −1 kω m⋆ 3 In order to obtain the Roche volume of the star, it is con- vN 0.683 + 1 r (10) ≃ × 2.78 M venient to transform from an inertial frame to a frame of   reference rotating with the binary system. Combining this with Eq. [1], we obtain the relationship The Roche potential Φ(~r) for an arbitrary point with between the orbital period and the mean density of the star position vector ~r in the rotating frame is given by: when the star reaches the Roche limit:

1 m⋆ M 1 2 2 Φ(~r)N = ~ω ~r (6) kω −1/2 − ~r ~r⋆ − ~r ~rBH − 2 | × | P0 8.18 + 1 ρ¯ (11) | − | | − | ≃ 2.78   Frank et al. (2002), where ~r⋆ and ~rBH are the position vec- tors of the centers of the star and the hole, and ω is the independent of the mass of the hole,where P0 is the stellar rotational angular velocity of the star in the inertial frame. orbital period in hours, andρ ¯ the mean density of the star in cgs unit. For a dwarf star with mass 0.3M⊙ , the Roche ω satisfies 0 6 ω = kωΩ . Ω, where Ω is the stellar orbiting ∼ angular velocity. condition is satisfied when the orbital period decreases to a value of 1.96 2.27 hr depending on kω. If we suppose The first term in this equation in the Roche potential is − the gravitational potential of the star itself, making the ap- that the star plunges when its orbital radius reaches 6Rg , the maximum black hole mass for there to be mass-transfer proximation that the stellar mass is centrally concentrated 7 from a 0.3M⊙ dwarf star is 1.6 1.9 10 M⊙ (kω = 0 1). and only retaining the monopolar term. The second term is ∼ − × − the gravitational potential term of the hole. The third term is the potential in the rotating frame due to the centrifugal 2.3 Roche evolution force. We erect a locally orthonormal coordinate system To set the stage for the relativistic treatment below, let us x,y,z centered at the star and directed along the r,θ,φ consider the classical Roche model for a dwarf star that is co- { } { } coordinate axes. If we expand the gravitational potential rotating. Its mean density decreases monotonically as mass

c 2009 RAS, MNRAS 000, 1–?? 4 Lixin (Jane) Dai, & Roger Blandford is tidally stripped, leading to expanding orbits. Assuming 100 that the star has initial mass m⋆0 0.3M⊙, we found that ∼ 80 2.2 ρ¯⋆ m⋆ , (12) 60 ρ¯⋆0 ≃ m⋆0

  hr 

(Dai & Blandford (2011)), whereρ ¯ is the initial mean den- P ⋆0 40 sity of the star. Combining Eq. [11] and [12], the orbital period P of the 20 dwarf star, in terms of the initial period P0, varies with its remaining mass as: 0 −1.1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 P m⋆ . (13) møMŸ P0 ≃ m⋆0   We assume that the orbit of the star will change in a Figure 1. This figure shows the period-mass relationship of a way that the angular momentum of the system is conserved non-rotating lower main-sequence star with initial mass ∼ 0.3M⊙ 7 so that accreting onto a Schwarzschild SMBH with a mass of 10 M⊙. The dL vertical dashed line represents the evolution prior to the mass- τ + = 0, (14) transfer, when the mass does not change, but the orbit shrinks. dt The black curve describes the subsequent Roche evolution in the where we continue to assume that gravitational radiation Newtonian limit, when the stellar orbit expands. The blue curve provides the only torque as in Eq. [4]. shows how stellar orbital period changes with mass in the rela- Using Eq. [2], [4], [13], and [14], we see the time elapsed tivistic limit. since the onset of mass-transfer goes as: 3.6 2 20 3 − 3 2.7 P 8 ´ 10 tR 4.1 10 M6 P0 1 yr. (15) ∼ × P0 −   ! The associated rate of mass transfer of Roche accretion, 6 ´ 1020

also obtained assuming conservation of angular momentum, s  g is:  4 ´ 1020

2 8 m 22 3 2 − 3 m˙ ⋆ 1.23 10 M m P ∼ − × 6 23 −2.7 0.7 4.9 −1 2 ´ 1020 3.62 10 P0 M m g s . (16) ∼ − × 6 We show in Figs. 1 and 2 the Newtonian evolution of 7 0 this low-mass star in orbit around a hole with mass 10 M⊙. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 We see that the stellar orbital period can increase to around møMŸ five times the Roche period as the star is stripped. How- Figure 2. This figure shows how fast a non-rotating lower ever, the majority of the Roche evolution happens on a time ∼ ⊙ 5 main-sequence star with initial mass 0.3M accretes onto a scale 10 years, and the stellar mass-transfer rate deceases 7 ∼ Schwarzschild SMBH with a mass 10 M⊙ through the Roche evo- rapidly with time. lution. The black and blue curves are Newtonian and relativistic As discussed by Dai & Blandford (2011), the angu- calculations correspondingly. lar momentum of the binary system as in Eq. [2] follows 1 − 1 3 6 L m⋆P m⋆ρ¯ . If the mass accretion is stable on ∼ ∼ ⋆ 2.4 The final fate of stars under Roche evolution dynamic time scales, it mean L should decrease as mass is accreted onto the hole. If we define a parameter η = After mass-transfer begins, most stars will recede from the d(lnρ ¯⋆)/d(ln m⋆), then η < 6 is the criteria for Newtonian black hole so that they will continue to fill their Roche dynamical stability. lobes as their density falls. The star loses mass less and A more intuitive method of calculating the dynamical less rapidly. Eventually the star will cool and its structure stability, as Hameury et al. (1994) suggested, is to ensure will evolve. The motion of the star under the action of grav- that the star will not exceed the Roche surface under the itational radiation will then cause the orbit to shrink. Ad- assumption of no angular momentum loss through gravi- ditional excursions are possible depending upon the struc- tational radiation. In other words, dv⋆/dm⋆ > dvN/dm⋆, ture of the star. The star might cool down adiabatically or where v⋆ = m⋆/ρ¯⋆ is the stellar volume, and vN is the Roche on a thermal time scale. Therefore, it might or might not volume in Eq. [10]. This leads to the same stability criteria fill it Roche Lobe on its return. The equilibrium proper- η < 6. ties of these stars will be even more complicated than the Continuing to consider a 0.3M⊙ dwarf star, we have ones we calculated in this paper. Ultimately the star crosses ∼ η = 2.2 (Eq. [12]), ensuring that the star recedes from the the ISCO and plunges into the hole. When more than one hole stably under Roche accretion. star is in a “parking orbit” there is the possibility of reso- For many purposes this Newtonian treatment applied nant dynamical interaction analogous to that which occurs to a broader range of stars will be adequate. However, as in planetary systems. In addition interaction with the disc we want to exhibit relativistic effects, we will turn to a rel- can alter the evolution. ativistic treatment, again building upon the discussion in We illustrate the scenario in the following Fig. 3 Around Hameury et al. (1994). the hole, there will be stars migrating inward and mostly

c 2009 RAS, MNRAS 000, 1–?? Roche Accretion of stars close to massive black holes 5

0.066 0.330.627 1.0561.1881.419 1.6171.518 1.584 1.4851.65 1.551 1.452 1.32 1.023 30 1.0 1.287 1.155 1.386 1.254 1.353 1.221 1.122 1.089 0.792

0.231 0.495 20 Parking Belt 0.66 0.924 0.5 0.132 0.396 Roche Limit

0.594 10 0.264 0.825

0.462

Ž 0.099 S 0.0 0.957 0 0.726

0.297

0.528 0.891 ISCO -10 - 0.165 0.5 0.693 0.429

0.858 0.198 -20 0.561

-1.0 0.033 0.363 0.759 0.99 0.8 1.0 1.2 1.4 - 30 ArctanHrR L -30 -20 -10 0 10 20 30 ISCO Figure 4. The relativistic correction term T to the plunge time Figure 3. For a ∼ M⊙ star orbiting a Schwarzschild SMBH with 7 as a function of the distance to the hole and the spin parameter mass 10 M⊙ as discussed, the star reaches the Roche limit when of the hole. We see that as the radius goes to infinity, T becomes the orbital radius is ∼ 12R , and then spiral out. When the evo- g 1 no matter what the spin is. Also T = 0 when r = R . lution time scale is comparable with Kevin-Helmholtz time scale, ISCO the star stops receding in the “parking belt”, which is located at around ∼ 25Rg from the hole. Here we calculated the Kevin- This radius equals Rg, 6Rg, and 9Rg for a = 1, 0, and 1 Helmholtz time scale of the star using the initial stellar internal − respectively. The upper limit for a has been argued to be energy and luminosity. 0.998 (Thorne 1974) for which rISCO 1.237Rg . ∼ ∼ Let us now consider how to modify the treatment of outward under Roche evolution, stars that have gone at least gravitational radiation. Finn & Thorne (2000) computed a one Roche evolution “parked” in a belt or cloud (if the stars relativistic inspiral time scale for a star at an orbital radius came in on inclined orbits) waiting to move in again, new r: stars coming in under gravitational radiation still burning, and stars coming in from the “parking cloud” and not burn- 5 M 2 ing. tinspiral = 8 (19) 256 m⋆Ω˜ 3 T in the Boyer-Lindquist coordinate, where is the relativistic T 3 RELATIVISTIC TREATMENT correction term. They presented their results of in tabu- T lar form. We have used these results to give a convenient 3.1 Gravitational radiation emission interpolation formula for : T A star (with a mass m⋆) orbiting a massive hole (with a mass M and a spin parameter a) will lose angular momentum and (x 1)2 + b (x 1) (x,a) = − − , (20) energy due to gravitational radiation. The orbit is circular- T x2 + d x + f ized quickly, as we have shown in Dai & Blandford (2011). b = 1.06 (1 a)−0.11 0.66, However the rate of alignment of the spin and orbital angu- × − − d = 0.57 (1 a)0.23 2.22, lar momentum is slower than the inspiral rate (e.g., Hughes × − 1.08− −0.018 (2001)). For the moment we restrict attention to circular f = [ 1.08 (1 a) 3.18 (1 a) ] − × − − × − equatorial orbits. e−0.23×(1−a) + 4.91, We define a dimensionless orbital angular velocity Ω=˜ × where x = r/rISCO. Clearly T (x = 1,a)= 0, and T (x,a) MΩ, where Ω is the regular orbital angular velocity. The → 1 as x + (since at infinity the relativistic effects are relativistic Kepler’s law is: → ∞ negligible). This formula holds true with a maximum error 1 Ω=˜ 3 , (17) of 2% for x > 2. A contour plot of (x,a) is shown in r˜2 a ∼ T ± Fig. 4. where refers to prograde and retrograde orbits. The results for the evolution of the 0.3M⊙ dwarf star ± The star plunges into the hole when the orbit shrinks are shown in Fig. 5. It is clear that the standard ”Landau- to the ISCO. The ISCO of a black hole of mass M and spin Lifschitz” form equation is adequate for r > 100Rg . parameter a has a radius rISCO that satisfies the equation The gravitational radiation reaction torque is readily as given in Bardeen et al. (1972): computed from:

2 rISCO rISCO rISCO 2 ∂L ∂tplunge 6 + 8a 3a = 0. (18) τ(r, a)gr = / , (21) M − M M − − ∂r a ∂r a   r     c 2009 RAS, MNRAS 000, 1–?? 6 Lixin (Jane) Dai, & Roger Blandford

7 1.0 0.42 10 0.504

105 0.8

0.392

1000 0.6

0.56 Ω 0.476 k year  t 10 0.4

0.616 0.2 0.1 0.588

0.448

0.532 0.0 0.644 0.001 1.0 1.2 1.4 1.6 1.8 2.0 1 2 5 10 20 50 rRg k Figure 6. Figure 5. The total inspiral time in years as a function of r/Rg Contour plot of Roche volume as a function of k and 7 for a 0.3M⊙ dwarf star orbiting a 10 M⊙ hole. This plot is in kω for a = 0.5. k = 1 is the Newtonian limit. k = 2 when the star the log-log scale. The three colors indicate three different spin is at ISCO. kω = 0 when the star is non rotating, and kω = 1 parameters of the hole: -0.99 (green), 0 (black), and 0.999 (blue). when the star is co-rotating. The three solid curves are accurate inspiral times obtained using Eq. [19]. The red dotted curve is the Newtonian inspiral time using Eq. [5]. All other curves approach the Newtonian curve Here kω is the stellar rotational parameter in the inertial when the star is far away from the hole. frame. Similarly to the Newtonian treatment, we can locate the first and second Lagrange points and the transverse extent where the orbital angular momentum of the Roche surface, and calculate the volume contained by 3 1 (1 2˜r−1 + a r˜− 2 ) r 2 this surface. When k = 2, the Roche volume at the ISCO is L = m⋆M − ∗ 3 1 (22) calculated to be: −1 − 2 2 (1 3˜r + 2a r˜ ) −1 − ∗ kω m⋆ 3 (e.g., Bardeen et al. (1972)). VISCO = 0.456 + 1 r . (26) × 4.09 M   And more generally, an adequate approximation to the 3.2 Roche limit Roche volume is: We use the same approach as the Newtonian treatment. The VN(r, kω) VISCO(r, kω) local tidal tensor is evaluated in Boyer-Lindquist coordinates V (r,a,kω)= VN(r, kω) 0.5 − , − r r + F (a,kω)( 1) and then transformed into an orthonormal coordinate basis rISCO rISCO − freely falling with the star. This “local” coordinate system   (27) has spatial coordinates directed along the r,θ,φ axes. The { } only non-zero spatial elements of the tidal potential can be where expressed in the form: 13.9 F (a,kω) = 23.3+ 1+ k k 1 − kω + 2.8 φ = ,φ = ,φ = , (23) ,xx 3 ,yy 3 ,zz 3 14.8 − r r r + 23.8 (1 a)0.02 − 2.8+ kω × − where   1 r˜2 4ar˜ 2 + 3a2 0.4 −0.16 k = . (24) + 0.9 (1 a) . − 1 − 2.6+ kω × − r˜2 3˜r + 2ar˜ 2   − This satisfies Laplace’s equation as it must. Note the re- As an example, Fig. 6 is the contour plot of the rela- 3 markable prospects that k 2 for r rISCO independent tivistic Roche volume (in units of m⋆/Mr ) around a black → → of a. Also k 1 as r as must also be true in the hole with a = 0.5 as a function of k and kω. This volume → → ∞ Newtonian limit. decreases from 0.683 to 0.456 from the Newtonian limit to Combining the stellar gravity and rotation with the the most extreme relativistic limit (r = rISCO) for a non- tidal contribution, the Roche potential becomes: rotating star. For a co-rotating star, the variation is from 1 0.503 (k = 1) to 0.368 (k = 2). These volumes at the four 2 2 2 − 2 Φ(x,y,z) = m⋆(x + y + z ) corners of the plot are independent of the spin. − M ky2 + z2 (k + 1)x2 Comparing with Fishbone (1973)’s Roche volume used + − r3 2 by Hameury et al. (1994), our model is more accurate in the M k (x2 + y2) relativistic limit, including black hole’s spin and the stellar ω . (25) − r3 2 rotation. We plot the ratio of two Roche volumes in Fig.7

c 2009 RAS, MNRAS 000, 1–?? Roche Accretion of stars close to massive black holes 7

8 M = 10 MŸ , a = 0.9, PIsco = 2.04 hr 1.4 50

1.2 30 Fishbone V 

DB 1.0 20 V

15 0.8 hr

 10 Jupiter 0 20 40 60 80 100 ø P rRg

Figure 7. The ratio of our Roche volume to Fishbone (1973)’s Roche volume, as a function of r/Rg. The our four solid curves are for non-rotating stars orbiting a black hole with a = −1 (red), 0 (blue), 0.9 (black), and 0.998 (green). The lower four dashed ones are for co-rotating stars correspondingly. Earth

Using Eq. [17] and Eq. [27], we obtain a relationship 10-7 10-6 10-5 10-4 0.001 between the stellar mean density and the orbital radius when Mø  MŸ the star fills its Roche lobe: Figure 8. Roche evolutions of a perfect solid planet with the −1 −1 M 0.683 kω + 1 0.456 kω + 1 mass of Earth (green) and a perfect gas planet with the mass × 2.78 − × 4.09 3 = 0.5 of Jupiter (pink) near a maximally spinning SMBH with M = ρ¯⋆r − r r +
F (a,kω)( 1)
8 rISCO rISCO − 10 M⊙. We plotted four curves for each planets for comparison: − the relativistic calculation for a co-rotating star (colored solid  k  1 +0.683 ω + 1 . (28) curve), the relativistic calculation for a non-rotating star (colored × 2.78   dashed curve), the Newtonian calculation for a co-rotating star Then we know how the stellar properties govern the orbital (corresponding black solid curve), and the Newtonian calculation for a non-rotating star (corresponding black dashed curve). The evolution during the Roche accretion phase. vertical blue dashed lines represent the phase of inspiral due to gravitational radiation. 3.3 Roche evolution SMBH with a = 0.9 using Newtonian calculation so it can We studied how a star evolved adiabatically in go through Roche evolution. However, using relativistic cal- Dai & Blandford (2011), and calculated how the stel- culation this star will just plunge into the hole. And in plot lar mean density changes as mass is stripped for a (l), only with general relativity a co-rotating white dwarf representative set of stars. As the star approaches a SMBH with initial mass 0.8M⊙ can start the Roche accretion phase in a circular, equatorial orbit, it might plunge into the near a maximally spinning M6 black hole. SMBH directly, go through full Roche evolution, or start We also plot the Period-Mass relationship for planets as the Roche evolution then plunge into the hole. The detailed in Fig. 8. A solid planet like Earth has the same period as evolution depends on the mass and spin of the black hole, mass decreases due to unchanging density. A gas giant like the stellar type, and the rotational parameter k of the star. ω Jupiter evolves so that the period will increase. Using Eq. [28], we can check when the star will start Roche From the conservation of angular momentum, the stel- accretion and if so, how its orbit will change through the lar angular momentum L satisfies: process.We summarize these inspiral/outspiral scenarios in

Fig. 9. ∆L =∆LGR +∆LISCO +∆Lwind +∆Lrad. (29) Each plot shows how different stars behave as they spi- Here L is the angular momentum of the star as in Eq. ral to within 50Rg of a SMBH with given mass and spin. General relativity plays an important role in this region. [22]. ∆LGR is caused by the gravitational radiation torque For example, in plot (f), the non-spinning SMBH has mass τgr calculated from out new model (Eq .[21]). ∆LISCO repre- 7 sents the loss of angular momentum of materials falling into 10 M⊙. Brown dwarfs and white dwarfs will plunge into the hole. A Sun-like star, an upper main-sequence star with an the ISCO. ∆Lwind is the angular momentum of materials lost from an accretion disc. In this inner region of an AGN initial mass 7.9M⊙, and a lower main-sequence star with ∼ accretion discs, there are mainly two kinds of wind: radia- an initial mass 0.3M⊙ will go through full Roche accre- ∼ tion pressure and magnetic driven. The radiation pressure tion. A red supergiant with an initial mass 11.85M⊙ will ∼ start Roche accretion and will plunge into the hole. induced wind carries its own specific angular momentum as The plots also show how the stellar orbital periods on the disc. The magnetic induced wind carries away very lit- change in the Roche phase. The Period-Mass curves can tle mass but very large angular momentum which can be 10 be quite distinct with or without relativity or if the star times as large as its specific angular momentum on the disc. is co-rotating with the black hole or not. In plot (c), a Sun- The last term is the disc radiation torque due to beaming 8 effect, which is relatively small compared with the others. like co-rotating star barely crosses the ISCO of a 10 M⊙

c 2009 RAS, MNRAS 000, 1–?? 8 Lixin (Jane) Dai, & Roger Blandford

8 8 8 8 HaL M = 10 MŸ , a = -1, PIsco = 22.35 hr HbL M = 10 MŸ , a = 0, PIsco = 12.64 hr HcL M = 10 MŸ , a = 0.9, PIsco = 3.81 hr HdL M = 10 MŸ , a = 0.998, PIsco = 2.04 hr

WD 100.0 RG RG 100.0

100.0 50.0 HM HM 100.0 50.0 hr hr hr hr

 BD LM  BD LM  BD LM  BD ø ø ø ø P P 50.0 P P

50.0 Sun Sun 10.0 HM 10.0 WD WD WD HM 5.0 Sun RG LM RG 5.0 Sun 0.1 0.2 0.5 1.0 2.0 5.0 10.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0 0.1 0.2 0.5 1.0 2.0 5.0 10.0

Mø  MŸ Mø  MŸ Mø  MŸ Mø  MŸ

7 7 7 7 HeL M = 10 MŸ , a = -1, PIsco = 2.24 hr HfL M = 10 MŸ , a = 0, PIsco = 1.26 hr HgL M = 10 MŸ , a = 0.9, PIsco = 0.38 hr HhL M = 10 MŸ , a = 0.998, PIsco = 0.2 hr

10.0 10.0 HM HM 10.0 5.0 HM HM 10.0 5.0 Sun Sun hr hr hr hr    

ø ø ø ø LM P P 5.0 Sun P LM P RG RG BD 5.0 Sun BD 1.0 RG LM RG WD 1.0 BD BD WD 0.5 LM WD WD 0.5

0.01 0.05 0.10 0.50 1.00 5.00 10.00 0.01 0.05 0.10 0.50 1.00 5.00 10.00 0.01 0.05 0.10 0.50 1.00 5.00 10.00 0.01 0.05 0.10 0.50 1.00 5.00 10.00

Mø  MŸ Mø  MŸ Mø  MŸ Mø  MŸ

6 6 6 6 HiL M = 10 MŸ , a = -1, PIsco = 0.22 hr HjL M = 10 MŸ , a = 0, PIsco = 0.13 hr HkL M = 10 MŸ , a = 0.9, PIsco = 0.038 hr HlL M = 10 MŸ , a = 0.998, PIsco = 0.02 hr

LM LM LM LM BD 1.0 BD 1.0 BD BD 1.0 0.5 1.0 0.5 RG hr hr hr hr

 WD RG  WD RG  WD RG  ø ø ø ø P P 0.5 P P

0.5 0.1

0.1 WD

0.01 0.05 0.10 0.50 1.00 5.00 10.00 1.01.52.03.0 5.07.010.015.0 0.01 0.05 0.10 0.50 1.00 5.00 10.00 0.01 0.05 0.10 0.50 1.00 5.00 10.00

Mø  MŸ Mø  MŸ Mø  MŸ Mø  MŸ

5 5 5 5 HmL M = 10 MŸ , a = -1, PIsco = 0.022 hr HnL M = 10 MŸ , a = 0, PIsco = 0.013 hr HoL M = 10 MŸ , a = 0.9, PIsco = 0.0038 hr HpL M = 10 MŸ , a = 0.998, PIsco = 0.002 hr

0.20 0.20 0.20

RG RG RG 0.10 RG 0.10

0.10 0.05 0.1 0.05 hr hr hr hr    

ø WD ø WD ø WD ø WD

P P P P 0.02 0.05 0.02 0.01

0.01 0.02

0.01 0.05 0.10 0.50 1.00 5.00 10.00 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00

Mø  MŸ Mø  MŸ Mø  MŸ Mø  MŸ

8 7 6 5 Figure 9. Roche evolutions of different stars close to SMBHs with M = 10 M⊙, 10 M⊙, 10 M⊙, 10 M⊙, and a = −1, 0, 0.9 and 0.998. In each plot, the x-axis is the stellar mass ranging from 0.005M⊙ to 15M⊙, and the y-axis is the stellar orbital period from PISCO to the orbital period at 50Rg (close enough to the SMBH so these are the relativistic regions). We use different colors to denote different types of stars: a Sun-like star (red), a lower main sequence star with a mass ∼ 0.3M⊙ (purple), an upper main sequence star with a mass ∼ 7.9M⊙ (blue), a red super giant with a mass ∼ 11.85M⊙(orange), a white dwarf with a mass 0.8M⊙ (cyan), and a brown dwarf with a mass 0.05M⊙ (brown). We plotted four curves for each star for comparison: the relativistic calculation for a co-rotating star (colored solid curve), the relativistic calculation for a non-rotating star (colored dashed curve), the Newtonian calculation for a co-rotating star (corresponding black solid curve), and the Newtonian calculation for a non-rotating star (corresponding black dashed curve). The vertical blue dashed lines represent the phase of inspiral due to gravitational radiation.

c 2009 RAS, MNRAS 000, 1–?? Roche Accretion of stars close to massive black holes 9

In order to understand the details of ∆Lwind, a three- dimensional dynamical simulation will need to be employed. In this paper, for simplicity, we consider only the first two ∆L’s, the mass accretion rate onto the hole will be the same as the stellar mass deposition rate. We then have: dL + τ +m ˙ j = 0. (30) dt gr ⋆ ms jms is the specific angular momentum at the ISCO, which equals 0 in the Newtonian limit. For the same dwarf star with mass 0.3M⊙, we can ∼ Figure 10. Simulated X-ray period emission from a hot spot or- calculate how its P andm ˙ change in the relativistic limit biting a black hole is shown for various black hole spin parameter in comparison to the Newtonian treatment, as in Fig .1 and a and observer inclination angle i. In each figure the hot spot Fig. 2. With general relativity, the star does not recede from orbits the hole in equatorial circular orbits with the same radius. the hole as far as in the Newtonian limit, and the mass The units are arbitrary but consistent. accretion rate has dropped to about half the Newtonian rate. We can apply the same method as in Section 2.3 to check if the stellar volume will exceed the Roche volume 5 DISCUSSION AND FUTURE RESEARCH in the evolution. Contradictory to Hameury et al. (1994)’s DIRECTIONS result, the Roche evolution of a dwarf star is actually stable in the relativistic limit on dynamical time scales. Observation of quasi-periodic X-ray emissions from stars or- biting black holes in Active Galactic Nuclei (AGN) is a po- tential probe of general relativity. There is an intriguing pos- sibility that the X-ray modulation of RE J1034+396 does come from an orbiting star filling its Roche lobe and be- 4 CONSTRAINING SPINS FROM THE ing swallowed b the black hole, in a similar fashion to the SIGNALS ”black widow” pulsar systems (e.g. Thorsett & Chakrabarty (1999)) like PSR1957+20 (e.g. Kulkarni & Hester (1988); Returning to the observation of RE J1034+396, the 1hr Kulkarni et al. (1992); Arons & Tavani (1993)). It is reason- ∼ periodic X-ray signal can be associated with a red giant able to search the existing AGN X-ray data base for periodic or lower main-sequence star or brown dwarf orbiting and signals with relevant periods within range 1 20hr. 6 7 ∼ − accreting onto a 10 M⊙ black hole or a 10 M⊙ hole (the Although we are mostly interested in this paper in ob- ∼ ∼ latter with a high spin). 16 cycles of signals are insufficient servable tidally disrupted or accreting stars, it is quite rea- to be confident of a stable period. However, if more signals sonable to suppose that most dormant AGN black holes are can be observed, information on spin could be extracted. orbited by a stable system of stars with a common orbital In principle, the closer the star is to the hole, or the larger plane in much the same way as the sun is orbited by the the spin of the hole, the stronger the emission would be. planets. Sgr A⋆, which already has over 20 closely orbiting This could introduce an important selection effect as rapidly stars, might eventually evolve to such a configuration. Other spinning holes would be more likely to be observed. orbiting stars can have also dynamical interactions with the Fig. 10 shows simulated periodic signals from a hot spot accreting star and make its evolution more complicated. orbiting around the hole circularly using our numerical pack- The probability of observing such phenomena relates age. The modulation of intensity comes from beaming effect to the stellar distribution in accretion discs and the mecha- - as we discussed in Dai et al. (2010), the bolometric inten- nisms of bringing stars close to ISCO. When material accrete sity varies as the fourth power of the Doppler-gravitational onto the black holes, stars can be borne in the accretion disc, redshift factor g = E/E0 (Cunningham 1975). Here E is which could go away eventually leaving only the stars. Such the energy of the photon in the observer’s frame and E0 stars would orbit the hole in bound orbits in the equatorial in the hot spot’s inertial rest frame.The shape of the curves plane with low eccentricity. On the other hand, if an accre- and the relative intensity are different for different spins and tion disc still exists when the star gets close to ISCO, the observer inclination angles. star will have interaction with the disc as well, or might be Also a major difference between producing a hot spot traveling in the disc already producing a gap and perturba- on the disc using the Roche accretion model and other QPO tion patterns on the disc. models (e.g. Wagoner et al. (2001)) is that the periodicity A series of other interesting future projects can also be of the modulation will follow that of the stellar orbit al- done on this model. If the star is on a slightly eccentric or- though there is likely to be phase noise. Since the stellar bit or inclined orbit, the accretion rate could be larger than orbit evolves on an adiabatic time scale, the periodicity of the case of a circular orbit, but perturbations will be added the hot spot will also change very slowly but in a consistent to the sinusoidal behavior of the flux. A main-sequence star way. For example, an accreting 0.3M⊙ dwarf star’s orbital on its way to become a red-giant can also increase the mass ∼ periodicity can change by as much as 0.1s in one year, in accretion rate. A jet can also be formed due to the Roche ac- ∼ which time we could observe 8000 cycles of modulation. cretion - the radio component will also be very useful in pro- ∼ A P P˙ analysis similar to the double pulsar’s could in viding information on the black hole parameters. Between a − principle be done to extract information of the stellar type total tidal disruption and a steady mass-transfer, interme- and black hole mass and spin. diate phases can also exist and we think much can be done

c 2009 RAS, MNRAS 000, 1–?? 10 Lixin (Jane) Dai, & Roger Blandford on that. In order to understand the stellar evolution and Misner C. W., Thorne K. S., Wheeler J. A., 1973, Gravi- radiation mechanisms better, we also need to do a three- tation. San Francisco: W.H. Freeman and Co., 1973 dimensional general relativistic dynamical simulation. The Rees M. J., 1988, Nature, 333, 523 time it would take to run such evolutions on existing numer- Reynolds C. S., Nowak M. A., 2003, Phys. Rep., 377, 389 ical codes would be too long. New numerical methods need Shcherbakov R. V., Penna R. F., McKinney J. C., 2010, to be designed. ArXiv e-prints However, the most exciting and immediate prospect is Soberman G. E., Phinney E. S., van den Heuvel E. P. J., to open up a new line of investigation using existing X-ray 1997, Astronomy and Astrophysics, 327, 620 databases, to seek evidence for stars in bound orbits around Stone N., Loeb A., 2011, ArXiv e-prints massive black holes. Discovering such systems would elu- Tchekhovskoy A., Narayan R., McKinney J. C., 2011, cidate the evolution of AGN and also open the possibility ArXiv e-prints of a new measurement of black hole spin and ultimately of Thorne K. S., 1974, APJ, 191, 507 concluding that the Kerr metric correctly describes spinning Thorsett S. E., Chakrabarty D., 1999, APJ, 512, 288 black holes. Wagoner R. V., Silbergleit A. S., Ortega-Rodr´ıguez M., 2001, APJ, Part 2 - Letters, 559, L25 Webbink R. F., 1985, Stellar evolution and binaries. pp ACKNOWLEDGMENTS 39–+ Yuan F., Quataert E., Narayan R., 2004, APJ, 606, 894 This work was supported by the U.S. Department of Energy Zauderer B. A., Berger E., Soderberg A. M., Loeb A., contract to SLAC no. DE-AC02-76SF00515. We would like Narayan R., et. al. 2011, Nature, 476, 425 to give special thanks to our collaborator P. Eggleton for providing the stellar models, R. Wagoner for valuable com- ments, and D. Chakrabarty, J. Faulkner, L. Gou, J. Grind- lay, S. Hughes, A. Loeb, H. Marshall, J. McClintock, M. Nowak, S. Phinney, R. Remillard, P. Schechter, R. Simcoe, A. Tchekhovskoy, and N. Weinberg for useful discussions.

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c 2009 RAS, MNRAS 000, 1–??