aa r X i v : . [ a s t r o - ph ] M a y Draft version November 17, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
GAS SHEPHERDING BY AN INFALLING SATELLITE
Philip Chang
Draft version November 17, 2018
ABSTRACTI calculate the action of a satellite, infalling through dynamical friction, on a coplanar gaseousdisk of finite radial extent. The disk tides, raised by the infalling satellite, couple the satellite anddisk. Dynamical friction acting on the satellite then shrinks the radius of the coupled satellite-disksystem. Thus, the gas is “shepherded” to smaller radii. In addition, gas shepherding produces a largesurface density enhancement at the disk edge. If the disk edge then becomes gravitationally unstableand fragments, it may give rise to enhanced star formation. On the other hand, if the satellite issufficiently massive and dense, the gas may be transported from ∼
100 pc to inside of a 10 to 10s ofparsecs before completely fragmenting into stars. I argue that gas shepherding may drive the fuelingof active galaxies and central starbursts and I compare this scenario to competing scenarios. I arguethat sufficiently large and dense super star clusters (acting as the shepherding satellites) can shepherda gas disk down to ten to tens of parsecs. Inside of ten to tens of parsecs, another mechanism mayoperate, i.e., cloud-cloud collisions or a marginally (gravitationally) stable disk, that drives the gas . Subject headings: galaxies: nuclei – galaxies: starburst – galaxies: star clusters – accretion, accretiondisks INTRODUCTION
The tidal interaction between small satellites and thegas or particle disks in which they are embedded is a sub-ject of wide study in the planetary community. Thesesatellites are tidally coupled via disk tides, i.e., satellitesexcite spiral density waves at the Lindblad resonances inthe disk (Goldreich & Tremaine 1978, 1980; Artymow-icz 1993). As a result of these tidal interactions, gapscan be opened up in the embedded disks as in the caseof the shepherding moons of Saturn’s rings (Goldreich& Tremaine 1978) or in type-II migration (Ward 1997).Alternatively, if a gap does not open up, satellite maymigrate inward rapidly due to tidal torque imbalances,i.e., type-I migration (Ward 1997).The wide applicability of satellite-disk interactions inthe planetary community raises an interesting questionof whether the same physics may be applicable at largerscales, i.e., galactic scales. In this paper, I will addressthis question by considering a very simple model, whichillustrate the modification of the physics of satellite-diskinteraction when applied on a galactic scale. Most no-tably, the critical difference is that in addition to tidaltorques, the satellite will experience dynamical friction.The inclusion of dynamical friction produces a non-trivialeffect. Namely, dynamical friction on the satellite pro-vides a sink of angular momentum in the system. As aresult, the coupled satellite-disk system will continuallylose angular momentum and sink toward the center.To study this physics, I consider a very simple model.In my model, a satellite starts out in a circular orbit ata large radius and sinks toward the central mass concen-tration because of dynamical friction on the backgroundstars. Along its infall, it encounters a coplanar gaseous Astronomy Department and Theoretical Astrophysics Center,601 Campbell Hall, University of California, Berkeley, CA 94720;[email protected] Miller Institute for Basic Research disk or ring, which initially has a finite radial extent, r d , .Tides begin to couple the satellite with the disk. Becausethe satellite-disk system continues to suffer an ongoingloss of angular momentum from dynamical friction, itwill shrink in radius. As a result gas can be transportedon the dynamical friction timescale to smaller radii. Inaddition, I find that this satellite-disk interaction alsobuilds a substantial surface density enhancement at thedisk edge. This may lead to enhanced star formation atthe disk edge. This process which I call gas shepherding may be a generic feature of gaseous disks around galaxies,if a sufficiently massive and dense satellite is available.The physics of satellite-disk interactions in galaxiesor gas shepherding is not just an interesting exercise inmathematical physics, but may be important in the fu-eling of active galaxies and central starbursts. First, theaction of dynamical friction on the satellite-disk systemwill shrink radius of the disk, thereby transporting gasto smaller radii. This shepherding of gas will continueuntil the gas is forced into the center or the shepherdingsatellite is destroyed. I will discuss this scenario for nu-clear fueling and how it compares to competing scenariosin this paper.I have structured this paper as follows. In §
2, I dis-cuss the basic picture of gas shepherding and give a feworder of magnitude estimates. I estimate the timescalefor gas shepherding and the expected surface density en-hancement. I present the physics of gas shepherding andsolve numerical models in §
3. I also present an approx-imate analytic solution to this problem. I then discussthe application of gas shepherding to the feeding of ac-tive galactic nuclei and central starbursts in §
4. FinallyI present my conclusions in § BASIC PICTURE
The basic picture of gas shepherding is illustrated byFigure 1. A gas disk with a radius of r d and surfacedensity of Σ orbits about a spherical mass distribution Chang r s r d SatelliteEnclosed Mass M enc ( r d ) M s Gas Disk M d Fig. 1.—
Schematic of gas shepherding. A satellite with mass, M s , and orbital radius, r s , infalls via dynamical friction and inter-acts with a gaseous disk with mass, M d = π Σ r , that sits in a halowith total enclosed mass, M enc . with enclosed mass of stars or dark matter of M enc ( r d ).Due to dynamical friction on the stellar background, anexternal satellite with mass M s slowly spirals in on a cir-cular orbit with a radius of r s in the same orbital planeof the gas disk. The radial velocity at which the satel-lite initially spirals into the gas disk due to dynamicalfriction is v df ∼ ( M s /M enc ) v orb , where v orb is the orbitalvelocity of the satellite. As the satellite approaches thedisk edge, i.e., r s approaches r d , the satellite excites spi-ral density waves at the Lindblad resonances (Goldreich& Tremaine 1980), which transfers angular momentumfrom the disk to the satellite. This angular momentum isin turn transferred to the background stars or dark mat-ter via dynamical friction on the satellite. Since angu-lar momentum of the satellite-disk system is continuallybled, the radius of the satellite-disk system will shrink atthe shepherding velocity, v shep .I first give some simple order of magnitude estimates ofthis process. The dynamical friction timescale for a satel-lite is (Chandrasekhar 1943; Binney & Tremaine 1987) t df ∼ M enc M s t dyn , (1)where t dyn = Ω − is the dynamical time, Ω s = p GM enc /r is the orbital frequency of the satellite, and G is Newton’s constant. The torque due to dynamicalfriction on the satellite, T df , is T df ∼ M s r Ω s t − ∼ GM r s (2)The torque due to the excitation of spiral density wavesin a disk is (Goldreich & Tremaine 1980; Lin & Pa-paloizou 1986; Ward & Hourigan 1989; Artymowicz 1993;Ward 1997) T d ∼ GM r s M d M enc (cid:18) r s r s − r d (cid:19) , (3)where M d = π Σ r is the mass of the disk, I have pre-sumed that radial separation between satellite and diskis small compared to the radius, i.e., r s − r d ≪ r s .For now, I assume that there is no internal rearrange-ment of angular momentum in the disk, i.e., an inviscid disk. Dynamical friction torques down the satellite ( T df ),while the disk, whose edge sits at r d < r s , torques up thesatellite ( T d ). When T df ∼ T d , the satellite and the diskare well coupled for M s ≤ M d , i.e., the mass of the satel-lite is smaller or comparable to the mass of the disk.Setting T d ∼ T d gives r s − r d r s ∼ (cid:18) M d M enc (cid:19) / , (4)which I define as the “Hill” radius of the disk, r H , d = r s ( M d /M enc ) / . This gives the natural scale for theradial separation between disk and satellite, i.e., r s − r d ∼ r H , d .For spiral density waves that damp locally, the radialextent of the disk over which the satellite exerts torquesis also ∼ r H , d . Assuming no viscous spreading, the disksurface density will be enhanced at the disk edge due tothe piling up of material from the initial radius of r d , to the current radius of r d . This enhancement due to asatellite that sinks a distance ∼ r d isΣΣ ∼ r d /r H , d , (5)where Σ is the surface density and Σ is the initial surfacedensity. For M d /M enc ∼ − − − , the enhancementis a factor of a few to ten. GAS SHEPHERDING
Basic Equations
To study this problem in greater detail, I begin bywriting the equations for a viscous disk that is evolvingunder the influence of an external torque. The equationof continuity is ∂ Σ ∂t + 1 r ∂ ( r Σ v r ) ∂r = 0 , (6)where r is the radial coordinate, and v r is the radial com-ponent of the velocity (Frank, King, amd Raine 2002).The angular momentum equation isΣ ∂ ( r Ω) ∂t + v r Σ ∂ ( r Ω) ∂r = − πr (cid:18) ∂T visc ∂r − ∂T d ∂r (cid:19) , (7)where Ω = p GM enc ( r ) /r is the orbital frequency, M enc ( r ) is the mass enclosed inside r , and T visc = − πr ν Σ ∂ Ω /∂r is the viscous torque, ν is the viscos-ity (Frank, King, amd Raine 2002). Using equation (7),I find v r = − πr Σ (cid:18) ∂ ( r Ω) ∂r (cid:19) − (cid:20) ∂∂r ( T visc − T d ) (cid:21) . (8)Plugging equation (8) into (6), I find ∂ Σ ∂t = 12 πr ∂∂r (cid:18) ∂ ( r Ω) ∂r (cid:19) − (cid:20) ∂∂r ( T visc − T d ) (cid:21) . (9) I use quotation marks because disks do not have Hill radiiin the traditional sense. Rather I have adopted this terminologybecause of the familiar scaling, i.e., the 1/3 power law. The angular frequency, Ω is a function only of the coordinate, r , assuming that the mass enclosed, i.e., M enc is not an explicitfunction of time. Therefore, the first term completely disappearsfrom equation (7). as Shepherding 3For the torque density due to an orbiting satellite on adisk, ∂T d /∂r , I take a form suggested by Ward & Houri-gan (1989) (see also Goldreich & Tremaine 1980; Lin &Papaloizou 1979a, 1979b, 1986): ∂T d ∂r = sgn ( r − r s ) β G M Σ r (Ω − Ω s ) ( r − r s ) , (10)where β is a constant of order unity. For | r − r s | ≪ r , Iexpand Ω − Ω s ≈ ( ∂ Ω /∂r )( r − r s ) to get ∂T d ∂r ≈ sgn ( r − r s ) β G M Σ r Ω ( r − r s ) , (11)where I have taken a density profile for the enclosed massof the form ρ ∝ r − , so that M enc ( r ) = M enc , rr , (12)where M enc , is the enclosed mass at r . Note that forsuch a mass distribution, the orbital velocity, v orb = p GM enc , /r , is a constant. Using (12), I find the fol-lowing relations ∂T visc ∂r = 2 πv orb ∂∂r (Σ νr ) , (13) ∂T d ∂r ≈ sgn ( r − r s ) β GM Σ r r M enc , ( r − r s ) , (14)Plugging equations (13) and (14) into (9), I find ∂ Σ ∂t = 1 r ∂ ( rν Σ) ∂r + β π r v orb r GM M enc , ∂∂r " Σ r ( r − r s ) , (15)where I assume r < r s , which fixes the sign. Note thatthe term in equation (8) and (9), ∂ ( r Ω) /∂r = v orb , is aconstant using the prescribed mass distribution in equa-tion (12). I chose the viscosity law suggested by Lin &Papaloizou (1986): ν = ν (cid:18) ΣΣ (cid:19) . (16)The time-rate change of angular momentum of thesatellite is M s ∂ ( r Ω s ) ∂t = − T d + T df , (17)where − T d = − R dr∂T d /∂r . The velocity of infall, i.e.,the shepherding velocity, is v shep = ˙ r s . Thus, I find ∂r s ∂t = GM s v orb " − β Z sgn ( r − r s )( r − r s ) Σ r r M enc , dr − ln Λ 1 r s . (18)Equations (15) and (18) constitute a complete set ofequations which governs the behavior of the satellite-disksystem. These equations are exactly the same as thosethat govern the migration of protoplanets in protoplan-etary nebula (Lin & Papaloizou 1979ab, 1986; Hourigan& Ward 1984; Ward & Hourigan 1989; Ward 97; Rafikov2002), except with the addition of another term on theRHS of equation (18), which is due to dynamical friction. To simplify equations (15) and (18), I rescale the vari-ables σ = ΣΣ , (19) t ′ = tt df = Ω t ln Λ M s M enc , , (20)where Σ is the initial surface density of the disk, whichI assume to be constant and Ω is the orbital frequencyat r = r . Equations (15) and (18) become ∂σ∂t ′ = ν q Ω ln Λ r ∂ ( rσ ) ∂r + β ′ qr r ∂∂r (cid:18) σr ( r − r s ) (cid:19) , (21) ∂r s ∂t ′ = 2 β ′ π Σ r M enc , r Z σr ( r − r s ) dr − r r s (22)where q = M s /M enc , and β ′ = β/ (2 π ln Λ), and I havetaken the viscosity to be ν = ν σ (see eq.[16]), where ν = α ( h d , /r d , ) r d , v orb , where r d , is the initial ra-dius, h d , is the initial vertical scale height of the disk,and α dimensionless viscosity parameter (Frank, King,& Raine 2002). The initial scale height of the diskcan be written as h d , /r d , = Q q d / √ , where Q isthe initial Toomre Q (Toomre 1964) of the disk and q d = M d /M enc , is the ratio of the disk mass to theenclosed mass. Without loss of generality, I now take r = r d , ≈ r s , to find ∂σ∂t ′ = α ′ q d q ′ r , r ∂ ( rσ ) ∂r + β ′ q ′ q d r , r ∂∂r (cid:18) σr ( r − r s ) (cid:19) , (23) ∂r s ∂t ′ = 2 β ′ q d r d , Z σr ( r − r s ) dr − r , r s (24)where q ′ = M s /M d and α ′ = αQ / (2 ln Λ). Equations(23) and (24) are the master equations governing mymodel which I solve numerically in § § Numerical Results
Equations (23) and (24) consists of four dimension-less parameters: q d , q ′ , α ′ and β ′ . For the dimension-less parameters, β ′ , which represent the relative powerbetween disk tides and dynamical friction, I choose afiducial value for β ′ = 1, assuming fiducial values ofln Λ ∼ O (1) and β ∼ O (1). For α ′ , which representsthe relative power between viscosity and dynamical fric-tion, I take α ′ = 10 − − − to study the effects ofviscosity. Finally, I set q d = M d /M enc = 10 − .I solve equations (23) and (24) using standard ex-plicit finite-difference methods (Press et al. 1992). Forgood spatial resolution, I select 750 grid points between Using the standard Shakura & Sunyaev (1973) α prescriptionfor the viscosity, i.e. ν = αc s h , where c s is the sound speed and h is the disk scale height, I write c s = v orb h/r and I take h = h d , at r = r d , . To get this relation, take the Toomre Q parameter for a gaseousdisk, Q = c s κ /πG Σ (Binney & Tremaine 1987). The epicyclicfrequency, κ = (4 + 2 d ln Ω /d ln r ) / Ω = √ . Plugging c s = v orb h/r , I find h/r = ( M d /M enc )( Q/ √ Chang
Fig. 2.—
Evolution of the surface density as a function of radiusfor a satellite disk system where q ′ = 0 . α ′ = 0 . β ′ = 1.The snapshots are separated in ∆ t = 0 . t df , increments. Thesatellite position is marked by the black dot below the surface den-sity plot. Each point and each curve correspond to one snapshot.The labeled points, A-D, are explained in the text. r/r d , = 0 to 1 . r s = 2 r d , .The satellite begins to fall inward via dynamical friction.As the satellite approaches the disk, the disk exerts atorque on the satellite, which slows its infall. At thesame time, the surface density of the disk increases atthe disk edge.I show this evolution for α ′ = 0 . β ′ = 1, and q ′ = 0 . r s /r d , = 2. When the satellite sinks to r s /r d , = 1 .
25, Iset t = 0 and plot the evolution of the satellite-disk sys-tem in intervals of ∆ t = 0 . t df , for a total of 1 . t df , where t df , is the dynamical friction timescale at r d , . At t = 0 the disk is undisturbed except for a small amount ofspreading due to viscosity at the disk edge. The satelliteposition is at r/r d , ≈ .
25 as noted by the black dot la-beled A. I evolve in intervals of ∆ t and note that the disksatellite system has begun to respond (point labeled B).The disk begins to develop a surface density enhancementand the satellite slows down a bit. Evolving for another∆ t , i.e., the point labeled C, shows a substantial change.The surface density increases near the disk edge by amaximum factor of ≈
2. Note that the distance betweenpoint B and C is substantially smaller than the distancebetween A and B, which results from the backreactionof the disk tides. By point D, the density enhancementis ≈
5. Note also that the distance between point Dand its neighbor is somewhat larger than the other sep-arations. This is due to the decrease in the dynamicalfriction timescale compared to the dynamical timescalewhen the mass enclosed decreases relative to the satellitemass.In Figure 3 and 4, I show the case for q ′ = 0 .
01 and 1respectively. For q ′ = 0 .
01 (Fig. 3), the initial infall ofthe satellite is stopped by the disk because of the largemass ratio between the satellite and the disk. In addition,there is no surface density enhancement at the disk edge.For q ′ = 1 (Fig. 4), on the other hand, the initial infallof the satellite is only slightly slowed because the mass of Fig. 3.—
Same as Figure 2 but for q ′ = 0 .
01. The larger mass ofthe disk stops the body from infalling toward the center. In thiscase there is no surface density enhancement.
Fig. 4.—
Same as Figure 2 but for q ′ = 1. The surface densityis enhanced over the case where q ′ = 0 .
1. In addition the disk hasa much reduced effect on the infall velocity of the satellite becausethey have equal masses. the disk and satellite are the same. In this case, a hugesurface density enhancement is produced.One remarkable aspect of Figures 2 and 4 is that theyappear to approach quasi-steady state solutions. This isespecially clear in Figure 2. Motivated by the appearanceof these quasi-steady state solutions, I find an approxi-mate kinematic wave solution in § v df , = v df ( r d , ) (lower plot) and the localinfall velocity, v df (upper plot), as a function of r s for q ′ = 0 .
01, 0.1, and 1. I plot it for α ′ = 0 . v df ( r s ). For r s > r d , , v shep ≈ v df becausetidal coupling between the disk and satellite has not yetas Shepherding 5 Fig. 5.—
Plot of the normalized shepherding velocity as a func-tion of r s for q ′ = 0 .
01 (bottom set of curves), 0 . α ′ = 0 .
001 (short-dashed lines), 0.01 (dottedlines), and 0.1 (solid lines) normalized to the initial infall velocity, v df , . The local dynamical infall velocity, v df (dot-dashed line), isalso plotted for purposes of comparison. Finally, the estimate for v shep (eq.[41]) is plotted for r s < r d , − r H , d to allow a kinematicwave solution to be established. been achieved. Once the coupling has been achieved, i.e. r s < r d , , the shepherding velocity depends mainly onthe mass ratio between the satellite and disk, q ′ , andis independent of the viscosity, α ′ . This highlights thefact that dynamical friction dominates the dynamics. Ialso plot an estimate for migration velocity (eq.[41]) for r s < r d , − r H , d , i.e., I allow the shepherding satellite tofall inside of one Hill radius of the disk to allow a kine-matic wave solution to be set up (see § .
10% forthe fidicial cases of q ′ = 0 . q ′ = 0 . § Approximate Solution
Motivated by the numerical results of § § r H , d .Thus, I now change variables in equations (23) and (24) from r to x , where x = r − r s r H , d0 = r − r s q / r d , , (25)where r H , d0 = q / r d , is the initial Hill radius of the disk.Since r H , d0 ≪ r , I take the approximation r − r s ≪ r tofind ∂σ∂t ′ = α ′ q / q ′ ∂ σ ∂x + q ′ q / β ′ (cid:18) x s x d , (cid:19) ∂∂x (cid:16) σx (cid:17) , (26) ∂x s ∂t ′ = x d , " β ′ (cid:18) x s x d , (cid:19) Z σx dx − x d , x s , (27)where x s = r s /r H , d and x d , = r d , /r H , d . I now assume the kinematic wave ansatz : σ = σ w ( x − v shep t ′ ), where σ w denotes my kinematic wave solutionfor σ . Inserting this ansatz into equation (26), I find: − v shep ∂σ w ∂x = α ′ q / q ′ ∂ σ ∂x + q ′ q / β ′ (cid:18) x s x d , (cid:19) ∂∂x σ w x , (28)Integrating once and ignoring the constant of integration,I find − v shep = 3 α ′ q / q ′ ∂σ ∂x + q ′ q / β ′ (cid:18) x s x d , (cid:19) x , (29)where I have cancelled one factor of σ w from each term.I now integrate once more to find σ = C + q ′ α ′ q / − v shep x + 2 β ′ (cid:18) x s x d , (cid:19) q ′ q / (cid:18) x (cid:19)! , (30)where C is the constant of integration. The two terms inthe parenthesis are both negative because x ∝ r − r s < v shep < x becomesmore negative, i.e., away from the disk edge, while thesecond term cuts off the surface density as it approachesthe disk edge due to tidal interactions. The peak, i.e.,where σ is maximal, is found from equation (29) bysetting ∂σ /∂x = 0. This occurs when x = x peak ≡ − q ′ q / β ′ (cid:18) x s x d , (cid:19) − v shep ! / . (31)Hence, σ w ( x peak ) = σ max , so given an estimate for σ max ,I can determine the constant of integration, C , via C = σ − q ′ α ′ q / (cid:18) − v shep x peak To aid in the derivation of equation (26) and (27), it is helpfulto consider which terms will change rapidly for small x and whichterms will change slowly. For instance, the term ( r − r s ) − → x − changes rapidly with small x , whereas r → r = r s + r H , d x ≈ r s changes very slowly. Hence, I made the following series of substi-tutions r → r s and r − r s → r H , d x . To clarify the signs in equation (31), note that x <
0, i.e.,the satellite is at larger radii than the gas and v shep <
0, i.e., thesatellite-disk system moves inward.
Chang+ 2 β ′ (cid:18) x s x d , (cid:19) q ′ q / x !! , (32)In the appendix, I calculate σ max by balancing thetorque from the satelite-disk interaction with the torquefrom dynamical friction. I find the following cubic equa-tion (eq.[A13]):2 β ′ σ max x + 2 (cid:18) x d , x s (cid:19) α ′ q d q ′ σ − (cid:18) x d , x s (cid:19) = 0 , (33)which I solve numerically. This gives an estimate for σ max and hence, C , given v shep . Therefore, to completethe calculation, I now give an estimate for v shep . BeforeI do so, I note that the RHS of equation (30) can benegative. To join this kinematic wave solution, σ w , ontothe background solution, I set σ = σ if σ > x > x peak σ if σ > x < x peak σ < x > x peak σ < x < x peak . (34)I now proceed to calculate v shep . I estimate the torqueusing equation (2). The satellite raises tides on the disk,which do not operate over the entire disk, i.e. the tidesare most acute at the edge of the disk due to the ( r − r s ) − scaling of the torque density. Rather they act onthe accumulated mass at the disk edge, M edge , which Iestimate to be: M edge ≈ M d " − (cid:18) r d , edge r d , (cid:19) , (35)where r d , edge is the edge of the disk. To estimate r d , edge for equation (35), I presume the following fit r d , edge = r s − ψr H , d0 , (36)where ψ is a constant of order unity and is determinedbelow. The total angular momentum of the satellite-disksystem over which these torques are effective is L = v orb ( M s r s + M edge r d ) . (37)The difference between r d and r s is of order the Hill ra-dius of the disk, r H , d = r d ( M d /M enc ) / (see § → Σ ≈ M edge / πr d r H , d , where I as-sume that the mass of the edge is spread over a radialthickness of r H , d . Therefore, I define an effective Hillradius of the disk by r H , eff r d = (cid:18) π Σ( r d ) r M enc (cid:19) / , (38)= r H , d r (cid:18) M edge πM d r H , d (cid:19) / . (39)The disk edge is separated from the satellite by ∼ r H , eff .Thus, r d ≈ (1 − ηr H , eff /r d ) r s , where η is a constant of In the nomenclature of this paper, r d , edge should really be r d .However, I will introduce this variable as I will fit for this and r H , eff . Fig. 6.—
The surface density enhancement Σ / Σ as a function of( r − r s ) /r H , d for the approximate solution (dashed lines) and thenumerical solution (solid lines) for a satellite disk system where q ′ = 0 . α ′ = 0 . β ′ = 1. The snapshots are separatedin ∆ t = 0 . t df , increments and the corresponding position of theshepherding satellite (from top to bottom) is r s /r d , = 0 .
95, 0 . . order unity. Equating the time derivative of L (eq.[37])and (2), I find GM r s ∼ ˙ L = v orb (cid:20) M s + M edge (cid:18) − η r H , eff r d (cid:19)(cid:21) v shep , (40)where I ignore the time derivatives on M edge . This gives v shep ∼ v orb M s M enc ( r s ) M s M s + M edge (1 − ηr H , eff /r d ) . (41)Despite it crudeness, this velocity estimator gives reason-ably good agreement ( . η = 1 and ψ = 2 via trial and error. I plot the comparison in Figure5. Equations (30), (32), (33), (34), and (41) constitute acomplete analytic solution once the evolution of the diskcan be modeled as kinematic wave, i.e., after the satellitesinks down a few r H , d . This approximate solution agreeswell with the exact numerical solution as shown in Figure6 for q ′ = 0 .
1. The agreement for q ′ = 1 and q ′ = 0 . q ′ = 1, the assumption that T d ≈ T df breaks down and the agreement can be improved byestimating T d ≈ T df / (1 + q ′ ). For q ′ = 0 .
01, kinematicwave solutions are not established because the infall ofthe satellite is stopped.Finally, from the estimate for v shep given in equa-tion (41) and Figure 5, I note v shep is a factor of afew larger than ( M s /M d ) v df , where v df is taken at r d , .Hence I estimate the timescale for shepherding to be t shep ∼ f ( M d /M s ) t df or t shep ∼ f M d M s M enc M s t dyn ln Λ , (42)where f < t df ∼ ( M enc /M s ) t dyn / ln Λ. THE FUELING OF CENTRAL STARBURSTS ANDACTIVE GALACTIC NUCLEI as Shepherding 7As the satellite-disk system should continually shrinkdue to dynamical friction, it raises the interesting ques-tion of whether or not such a mechanism may be delivergas into the nuclear regions of galaxies. The fueling ofcentral starbursts and active galactic nuclei remains anopen question. It is generally believed that the fueling ofcentral starbursts and active galactic nuclei (AGNs) re-quires the delivery of gas from large radii to small radii(for a review see Shlosman, Begelman, & Frank 1990;hereafter SBF90). Gas transport is mediated at smallradii ( ∼ & . − . − ∼ ∼ . − Thin accretion disks, which aresufficient at small scales, would become gravitationallyunstable and would fragment to form stars outside of ∼ gaseous gravitational instabilities, i.e., the“bars within bars” model (Shlosman, Frank, & Begel-man 1989). Secondly, the gas may form a thick tur-bulent marginally stable accretion disk powered by starformation (Paczynski 1978; Kolykhalov & Sunyaev 1980;Shlosman & Begelman 1987; Goodman 2003) via radia-tion pressure (Goodman & Tan 2004; Thompson et al.2005) or supernova (Wada & Norman 2002). Third, ifthe gas exist in the form of molecular clouds, collisionsbetween them may lead to episodic feeding of the nu-clear regions, i.e., the chaotic accretion scenario (Krolik& Begelman 1988; SBF90; Hopkins & Hernquist 2006;King & Pringle 2007; Nayakshin and King 2007). Fi-nally, the fueling of these central regions by be one ofbrute force, where major or minor mergers drive gas fromthe disk of the galaxy to the nuclear region (Hernquist1989; Bekki & Noguchi 1994; Mihos & Hernquist 1994;Hernquist & Mihos 1995)In the “bars within bars” scenario, Shlosman, Frank,& Begelman (1989) argued that if a gravitationally un-stable gas disk is sufficiently massive such that it fulfillsthe Ostriker-Peebles criterion (Ostriker & Peebles 1973),it forms a secondary gaseous bar which funnels gas in-ward on a dynamical time. Numerical simulations con-firmed the viability of “bars within bars” (see for instanceFriedli & Martinet 1993; Levine et al 2007). It demandsa substantial mass in gas, M gas ∼ M enc , where M enc isthe enclosed mass. While such large gas concentrationsare observed (see for instance Jogee et al. 2005), de-livered into the nuclear region by large scale bars, it isunclear if this mechanism drives the bulk of AGN fuel-ing and quasar activity. Namely, the observed correlationbetween large-scale bars, which transports gas into thecentral region of galaxies, and Seyfert activity is statisti-cally marginal (see the review by Ho 2008 and references As pointed out by the referee, ILRs may or may not exist,depending on the nuclear mass concentration. However, these barsstill become inefficient at small scales, i.e., typically 10% of the sizeof the bar (Shlosman, Frank, & Begelman 1989). therein; see the review by Combes 2003 and referencestherein; but also see Laine et al 2002). In addition, thereis no evidence that Seyferts have a higher fraction of nu-clear bars (see Knapen 2004 and references therein). On the other hand, if the disk is gravitationally un-stable but lacks sufficient mass to fulfill the demands of“bars within bars”, the gas will fragment and form stars.The process of star formation may feed back on the diskto provide vertical support, maintaining marginal sta-bility, i.e., Q ∼
1, where Q is the Toomre Q (see forinstance Shlosman & Begelman 1989; Goodman 2003;Goodman & Tan 2004; Thompson et al. 2005). Thesethick marginally stable disks may have a sufficiently largescaleheight such that accretion via viscous processes ispossible before all the gas is consumed.Third, perhaps the gas exist primary in the form ofmolecular clouds. Collisions between these molecularclouds provides something akin to a viscosity (Krolik &Begelman 1988; SBF90). These clouds would then feedthe nuclear region in a chaotic fashion (Hopkins & Hern-quist 2006; King & Pringle 2007; Nayakshin and King2007). Typically, the viscosity due to cloud-cloud col-lision appears to be insufficient for fueling the centralregions from hundreds of parsecs. However, it may benecessary around the sphere of influence of the centralblack hole where large scale instabilities may not oper-ate (SBF90; see § Fueling via Gas Shepherding As pointed out by the first referee, however, the nuclear barsmay be very short lived and therefore, may be very difficult toobserved.
ChangGas shepherding is another scenario by which gas istranported from hundreds of parsecs. The gas shepherd-ing scenario, which I present is as follows: large scalebars drive gas toward the central regions of a galaxywhere they collect in a nuclear ring, i.e. the gas tran-sitions from X to roughly circular X orbits (Binney &Tremaine 1987) at a radii of a few hundred parsecs to 1-2kiloparsecs (Buta & Combes 1996; Knapen 2004). Thisgaseous ring becomes gravitationally unstable and formsgiant molecular clouds which in turn collapse to formstar clusters in coplanar orbits (see for instance, Maoz etal. 2001; Jogee et al. 2002). If the star cluster formed issufficiently massive, it may shepherd in the remnant gasin the gaseous ring from a few hundred parsecs into thecenter.I first compare the timescale of shepherding and vis-cous processes. For an α viscosity, the viscous timescaleis t visc ∼ α − ( r/h ) t dyn (Frank, King, & Raine 2002).For a marginally stable disk, i.e., Q ∼ h/r ∼ M d /M enc .Therefore the ratio between the shepherding timescale, t shep , and the viscous timescale, t visc , is t shep t visc ∼ f α ln Λ M d M enc (cid:18) M d M s (cid:19) , ∼ − (cid:16) α . (cid:17) (cid:18) f / ln Λ0 . (cid:19) (cid:18) M d /M s (cid:19) (cid:18) M d /M enc . (cid:19) . (43)These timescales become comparable to each other forthicker disks, i.e. h/r ∼ M d /M enc ∼
1, or lower masssatellites, i.e., M d /M s ∼
100 or a combination of thetwo. Thus when shepherding operates, it dominates overviscous processes for low mass disk and large satellites.Therefore, for this scenario to work, the star cluster thatis formed must be sufficiently massive compared to themass of the gas disk. I would expect it to be at least 10%or greater in order for shepherding to be efficient basedon equation (43) and on my calculations in § Disk Fragmentation and Star Formation
Marginally stable disks with sufficiently rapid cooling( t cool < t dyn ) will tend to fragment (Gammie 2001). Inthe galactic context this fragmentation will lead to starformation, which may feed back on the disk, maintain-ing marginal stability. This feedback may be due to mo-mentum driving (Murray, Quataert, & Thompson 2005;Thompson et al. 2005), cosmic ray feedback (Socrates,Davis, & Ramirez-Ruiz 2008), or supernovae driven tur-bulence (Wada & Norman 2002). A surface density en-hancement of a few to ten would enhance local star for-mation. I take the standard Kennicutt law (Kennicutt 1998) for the star formation rate to be˙Σ ∗ = ΣΩ η, (44)where ˙Σ ∗ is the star formation rate per unit area, Σis the gas surface density, η ∼ .
01 is the empiricallydetermined efficiency. Equation (44) suggests that thestar formation rate would also be enhanced by the samefactor as that of the surface density.Equation (44) implies that the timescale for the gas tofragment into stars is t ∗ ∼ Σ˙Σ ∗ = η − t dyn . (45)The ratio between the timescale for shepherding and thetimescale for the gas to fragment into stars is t shep t ∗ ∼ η f ln Λ (cid:18) M d M s (cid:19) M enc M d (46) ∼ (cid:16) η . (cid:17) (cid:18) f / ln Λ0 . (cid:19) (cid:18) M d /M s (cid:19) (cid:18) M enc /M d (cid:19) . (47)Thus, sufficiently large satellites can shepherd a consider-able fraction of the gas into the central regions before thegas completely fragments into stars. For typical numbers( M enc ∼ M ⊙ ), this implies a satellite mass ∼ M ⊙ .For less massive satellites, the disk would completelyfragment into stars. Thus, the evolution would likelybe different. As the satellite continues to fall inward, itcaptures these newly-formed stars into mean-motion andsecular resonances, which increases the star’s eccentric-ity and inclination. Indeed, Yu, Lu, & Lin (2007) havesuggested that this exact process may be responsible forthe distribution of the young stars in the Galactic Cen-ter (GC). In this case, an intermediate-mass black hole(IMBH) or dark cluster falls into a disk of stars in thesame orbital plane. The stellar disk is heated up as thestars are captured into the previously-mentioned reso-nance. Some of these stars are forced to migrate withthe infalling IMBH, while others are forced into closeencounters. Some of these stars that are subjected tothese close encounters may be ejected out of the GCat large velocities, producing hypervelocity stars (HVSs)or hypervelocity binaries (HVBs) (Lu, Yu, & Lin 2007).Hence stars, whose semi-major axis is larger than thecurrent radius of the infalling IMBH would be excitedinto a torus-like structure, while stars interior to the ra-dius of the infalling IMBH would remain in a cold disk.Such a dynamical distribution may result from the effectsof shepherding, but at much larger scales, i.e., ∼
100 pcrather than ∼ . Destruction of the Shepherd
If this gas does not fragment into stars (which is thecase for sufficiently massive satellites), gas shepherdingmay transport a substantial amount of gas from ∼ . ∼ M ⊙ ) limits the number of possible sys-tems that can serve as a shepherd.as Shepherding 9A super star cluster (SSC) is one such candidate. SSCsrange up to mass of ∼ − M ⊙ with a radius of par-secs (see for instance McCrady & Graham 2007; Hageleet al. 2007) so they are sufficiently massive to trans-port gas before it completely fragments into stars. How-ever, as they sink deeper into the bulge, tidal forces maydisrupt them, stopping the shepherding process. TheSSC is tidally disrupted when the enclosed backgrounddensity ρ enc ∝ r − is equal to their mean stellar den-sity, ρ SSC ∼ M SSC /a , where a SSC is the radial sizeof the SSC. Plugging in a few example numbers, I findthat ρ SSC ∼ ( M SSC / M ⊙ )( a SSC / − M ⊙ pc − .For typical numbers, i.e., M enc ∼ M ⊙ at 100 pc, ρ enc ∼ ( r/ − M ⊙ pc − so the radius where thesatellite is disrupted is r tid ∼ (cid:18) M SSC × M ⊙ (cid:19) − / (cid:18) a SSC (cid:19) / pc , (48)which would also be where the gas disk is shepherdeddown to. Note that I have ignored the effect of theSMBH, which begins to become important at such radii.The densities and density structures of these SSCs arenot very well constrained. McCrady and Graham (2007)presents a very detailed study of the size and mass distru-bution of super star clusters in the nearby starburstinggalaxy M82. The mean half power radius of these SSCsis 1.8 pc (McCrady and Graham 2007, their Figure 7),and there are quite a few clusters around 5 × M ⊙ .By equation (48), this gives a tidal disruption radius ofabout 10 pc. Lets consider two examples in more de-tail. One example is SSC L in M82 which has a virialmass of ≈ × M ⊙ , a projected half-light radius of ≈ . r tid ≈ ≈ × M ⊙ , a projected half light radius of ≈ . ≈ . − . ⊙ and radii of 1 . − . < ∼ years, and thus theirlifetime may be substantially longer. The vulnerability ofSSCs to these internal processes strongly depend on theirstellar initial mass functions, i.e., they must have a suffi-cient number of low mass stars (Meurer et al 1995). Top-heavy IMFs are vulnerable to the disruptive processes ofstellar mass loss and other processes (Chernoff & Wein-berg 1990). Young, dense stellar clusters, the closestlocal analogues to SSCs, show a significant population of low mass stars. For instance, the 2 × M ⊙ (Walbornet al 2002) stellar cluster, R136 in the Large MagellanicCloud has a large population of low mass stars down to0 . ⊙ (Brandl et al 1996). In addition, McCrady et al.(2003) found SSC 9 was consistent with a normal IMF,though SSC 11 appeared to have a top heavy IMF.Aside from SSCs, the nucleated remains of an accretedsatellite galaxy may also be the shepherds that herds thatgas into the central regions of the galaxy for sufficientlysmall mass ratios. For mass ratio between the mass ofthe satellite galaxy’s remnant and the dynamical mass is of order 10% or greater, long range tidal interactionsmay disrupt a central disk before the shepherding sce-nario outlined aboved can be set up and would be akinto the minor merger scenario studied previously (Hern-quist 1989; Bekki & Noguchi 1994; Mihos & Hernquist1994; Hernquist & Mihos 1995; Tanguchi & Wada 1996;Taniguchi 1997, 1999). Using the nucleated remains ofsatellite galaxies as shepherds is interesting, but a de-tailed study of this scenario is left to future work. The Final Ten (or Tens of ) Parsecs
As I argue in § α cl − cl = (cid:26) C for C ≪ C − for C ≫ , (49)where C is the filling fraction of the clouds (see SBF90).It is unclear if this is more effective than magnetic (seethe review by Balbus 2003) or gravitationally driven vis-cosities (Gammie 2001).The effects of cloud-cloud collisions on the velocity dis-persion of the clouds is not known. The internal struc-ture, i.e., magnetic fields, will play a huge role in de-termining the efficiency of cloud-cloud collisional drivenviscosity (see for instance Krolik & Begelman 1988). An-other way by which this final “10 pc problem” can besolved is via a disk which is supported either by ra-diation pressure (Thompson et al. 2005), supernovadriven turbulence (Wada & Norman 2002; Kawakatu &Wada 2008), or cosmic ray pressure (Socrates, Davis, &Ramirez-Ruiz 2008). The gas disk or ring which resultsfrom the deposition of gas at ten to tens of parsecs mayform a marginally stable disk which is supported by starformation/supernova which can then viscously accrete.One-zone calculations by Thompson et al. (2005) suggestthat a star formation rate of ∼ ⊙ yr − is sufficient to0 Changsupport a marginally stable disk at ten to tens of par-secs (see Thompson et al. 2005, their Figure 5), whilealso allows for a nontrivial accretion rate onto the centralblack hole. I note that the strengths of the “bars withinbars” or gas shepherding scenarios and that of the star-burst supported disk scenario are complementary to oneanother. Inside of tens of parsecs, modest star formationrates can support a radiation pressure supported diskrather than requiring 200 −
600 M ⊙ yr − star formationrates from 200 pcs inward (again see Thompson et al.2005, their Figure 5). Recent observation of NGC 3227by Davies et al. (2006) support this notion of star for-mation in a gas disk at small radii. However, the currentstar formation rate of ∼ . ⊙ yr − appears to be in-sufficient to support the vertical scale of the disk. Otherforms of pressure support, i.e., cosmic ray pressure (seeSocrates, Davis, & Ramirez-Ruiz 2007 their appendix E),may be important in this context. DISCUSSION AND CONCLUSIONS
I have calculated the action of an infalling satellite on acoplanar gaseous disk. The satellite falls inward becauseof dynamical friction and couples to the gaseous disk viadisk tides. Due to the unrelenting loss of angular momen-tum, the satellite-disk system shrinks in radius. Thus,gas is transported from larger radii to smaller radii. Icalculate the structure of the evolving disk as a functionof radius, both numerically and analytically and show asignificant density enhancement at the disk edge. Thisdensity enhancement is due to the transport of gas ini-tially at larger radii to a ring that is r H , d thick radially.The fate of this shepherded gas is not known. I pre-sented two possible outcomes: 1. the gas will fragmentand form stars or 2. the gas is shepherded into thenuclear region. In the latter case, I suggest that thegas shepherding scenario, which I have presented maybe one means by which the “100 parsec” problem maybe resolved. For this to occur, gas must be shepherdedsufficiently rapidly such that the gas is not completelyconsumed in star formation, which requires a sufficientlymassive satellite ( ∼ M ⊙ ). I argue that such massivesatellites exist on the form of SSCs.In the model presented, the physics of star forma-tion is a disk environment has not been fully addressed.Observationally, galactic star-forming disks appear tobe marginally unstable to fragmentation, i.e., Q ∼ I note, however, that they presume some sort of global torquevia spiral arms or the like in their calculations. into stars on a few dynamical times. It is also unclearif the Kennicutt law would continue to hold at the diskedge. Additional insights from future research on self-gravitating star-forming galactic disks will be needed tohelp formulate a more self-consistent calculation of thegas shepherding scenario, which will be needed to fullyaddress the fate of the shepherded gas.Assuming that the gas is not fully consumed in star for-mation, it would be shepherd down to 10 to 10s of parsecs(see § r H , d /r d , i.e., the satel-lite stays within one “Hill radius” of the disk of its outeredge, then the shepherding calculation presented shouldbe a reasonable estimate of the shepherding timescalesand velocities.Compared to a coplanar satellite, an inclined satel-lite would couple less well to the gaseous disk. For asufficiently large inclination, the gas disk and satellitewould be effectively decoupled. However, its inclinationcould be sufficiently rapidly damped via induced bend-ing waves in the gas disk in addition to spiral densitywaves (see for instance Ostriker 1994; Terquem 1998).Once its inclination is damped below the critical value,where the disk and satellite become well coupled, theshepherding scenario presented in this paper would likelyfollow. I would expect this critical inclination to be oforder sin i = r H , d /r d , i.e., the satellite stays within oforder a “Hill radius” of the disk above and below it.Extending the present calculation to satellites of mod-est inclination would be fruitful and would allow a studyof using the nucleated cores from minor mergers as shep-herding satellites (see § §
4, minormergers may drive nuclear activity. The initial inclina-tion of minor mergers is arbitary, but their inclination isdamped as they interact with the large scale stellar andgaseous disk. As a result, they may approach the shep-herding scenario presented in this paper. Such a studyis a subject for future work.In addition, dynamical friction would also act on thesatellite-induced spiral density waves. Dynamical fric-tion of the extended mass perturbation, i.e., the satellite-induced spiral density waves, may enhance the transferof angular momentum between the gas disk and back-ground stars (Tremaine & Weinberg 1984). A study ofthese effects, while fruitful, is beyond the scope of thiswork.The physics of gas shepherding is also important inother areas as well. For instance, it may shape radialdistribution of star clusters in nuclear gas rings. In someof these nuclear ring systems, the location of the starclusters is exterior to the gas ring from which they arepresumably formed. For instance, NGC 4314 clearlyshow the star clusters exterior to the gas (Benedict etal. 2002). Martini et al. (2003) found that the eightgalaxies with strong bars all have star formation exterioras Shepherding 11to the dust ring in their sample of 123 galaxies. How gasshepherding drives this morphology is the subject of aforthcoming work. (Van der Ven & Chang, in prepara-tion).I thank N. Murray for his early encouragement anduseful discussions. I thank A. Socrates for his adviceand continuing encouragement. I thank R. Levine andA. Kratsov for making a early copy of their manuscriptavailable. I thank E. Quataert for useful discussions anda careful reading of this manuscript. I also thank G. Van der Ven and E. Chiang for useful discussions. Ithank M. Jones for a careful reading of this manuscript. Ithank the anonymous referees for very useful commentswhich improved the presentation of this paper. I alsothank the Kavli Institute for Theoretical Physics andthe Max-Planck Institute for Astrophysics for their hos-pitality during the initial and final stages of this project,respectively. I gratefully acknowledge the support of theMiller Institute for Basic Research. This research wassupported in part by the National Science Foundationunder Grant No. PHY05-51164
APPENDIX
CALCULATION OF σ MAX
I now discuss the calculation of σ max and hence derive the constant of integration C . I first plug equation (30) intoequation (27) to find x − , ∂x s ∂t ′ = 2 β ′ (cid:18) x s x d , (cid:19) Z vuut C + q ′ α ′ q / − v shep x + 2 β ′ (cid:18) x s x d , (cid:19) q ′ q / (cid:18) x (cid:19)! dxx − x d , x s . (A1)I make the approximation that the torque due to the satellite-disk interaction and the torque from dynamical frictionnearly cancel each other out, i.e., x − , ∂x s /∂t ′ ≪
1. Hence I find,2 β ′ Z vuut C + q ′ α ′ q / − v shep x + 2 β ′ (cid:18) x s x d , (cid:19) q ′ q / (cid:18) x (cid:19)! dxx = (cid:18) x d , x s (cid:19) . (A2)It is helpful to break the integral in equation (A2) into two parts. I rewrite the integral, I , as I ≡ Z → Z x peak −∞ + Z x max x peak , (A3)where x max < x s is defined as where σ w ( x > x max ) = 0, i.e., the disk is cutoff by disk tides above this radius, andconsider each integral in turn. The first integral, I , is I ≡ Z x peak −∞ vuut C + q ′ α ′ q / − v shep x + 2 β ′ (cid:18) x s x d , (cid:19) q ′ q / (cid:18) x (cid:19)! dxx . (A4)At x peak , σ w = σ max and it declines as x becomes more negative, i.e., away from the satellite, because of the v shep x term in the square root. However, the torque density declines even faster, so I may approximate σ w = σ max for thepurposes of this integral, i.e., I ≈ Z x peak −∞ σ max x dx = 13 σ max x . (A5)The second integral, I , is I ≡ Z x max x peak vuut C + q ′ α ′ q / − v shep x + 2 β ′ (cid:18) x s x d , (cid:19) q ′ q / (cid:18) x (cid:19)! dxx . (A6)Here, the disk torque increases as x approaches x max , but σ w declines due to the x − term in the square root. As thelinear term v shep x is overwhelmed by x − as x approaches smaller absolute values i.e. x → x max , I ignore the linearterm and approximate this integral by I ≈ Z x max x peak s C + q ′ α ′ q d β ′ (cid:18) x s x d , (cid:19) (cid:18) x (cid:19) dxx . (A7)Changing variables to y = C + q ′ α ′ q d β ′ (cid:18) x s x d , (cid:19) (cid:18) x (cid:19) (A8)2 Chang dy = − β ′ q ′ α ′ q d (cid:18) x s x d , (cid:19) x dx, (A9)and performing the integral, I find I ≈ β ′− (cid:18) x d , x s (cid:19) α ′ q d q ′ C + q ′ α ′ q / − v shep x + 2 β ′ (cid:18) x s x d , (cid:19) q ′ q / (cid:18) x (cid:19)!! / x peak x max . (A10)At x peak the term in brackets, which I identify as my approximation for σ w , is σ , while at x max , the term in bracketsis 0. Thus I find I = β ′− (cid:18) x d , x s (cid:19) α ′ q d q ′ σ (A11)Hence equation (A3) becomes I = I + I ≈ σ max x + β ′− (cid:18) x d , x s (cid:19) α ′ q d q ′ σ . (A12)Plugging this result into equation (A2) gives a cubic equation for σ max β ′ σ max x + 2 (cid:18) x d , x s (cid:19) α ′ q d q ′ σ − (cid:18) x d , x s (cid:19) = 0 , (A13)which I may solve numerically. REFERENCES
Artymowicz, P. 1993, ApJ, 419, 166Bath, G. T. & Pringle, J. E. 1981, MNRAS, 194, 967Bekki, K., & Noguchi, M. 1994, A&A, 290, 7Bekki, K. 1995, MNRAS, 276, 9Benedict, G. F., Howell, D. A., Jørgensen, I., Kenney, J. D. P., &Smith, B. J. 2002, AJ, 123, 1411Binney, J. & Tremaine, S. 1987, “Galactic Dynamics”, (PrincetonUniversity Press: Princeton)Brandl, B., Sams, B. J., Bertoldi, F., Eckart, A., Genzel, R.,Drapatz, S., Hofmann, R., Loewe, M., & Quirrenbach, A. 1996,ApJ, 466, 254Buta, R., & Combes, F. 1996, Fundamentals of Cosmic Physics,17, 95Chandrasekhar, S. 1943, ApJ, 97, 251Chang, P., Murray-Clay, R., Chiang, E., & Quataert, E. 2007, ApJ,668, 236Chernoff, D. F. & Weinberg, M. D. 1990, ApJ, 351, 121Combes, F. 2003, SF2A-2003: Semaine de l’AstrophysiqueFrancaise, meeting held in Bordeaux, France, June 16-20, 2003.Eds.: F. Combes, D. Barret, T. Contini, and L. Pagani., 243Davies, R. I., et al. 2006, ApJ, 646, 754Frank, J., King, A., & Raine, D. 2002, “Accretion Power inAstrophysics”, (Cambridge University Press: Cambridge)Friedli, D., & Martinet, L. 1993, A&A, 277, 27Gammie, C. 2001, ApJ, 553, 174Goldreich, P. & Tremaine, S. 1978,
Icarus , 34, 240Goldreich, P. & Tremaine, S. 1980, ApJ, 241, 425Goldreich, P. & Tremaine, S. 1982, ARA&A, 20, 249Goodman, J. 2003, MNRAS, 339, 937Goodman, J. & Tan, J. C. 2004, ApJ, 608, 108H¨agele, G. F., D´ıaz, ´A. I., Cardaci, M. V., Terlevich, E., &Terlevich, R. 2007, MNRAS, 378, 163Hernquist, L. 1989, Nature, 340, 687Hernquist, L. & Mihos, J. C. 1995, ApJ, 448, 41Hills, J. G. 1975, Nature, 254, 295Ho, L. C. 2008, to appear in ARA&A, arXiv:0803.2268Hopkins, P. F. & Hernquist, L. 2006, ApJS, 166, 1Hopkins, P. F., Hernquist, L., Cox, T. J., Di Matteo, T., Robertson,B., & Springel, V. 2006, ApJS, 163, 1Hourigan, K. & Ward, W. R. 1984,
Icarus , 60, 29Jogee, S., Scoville, N., & Kenney, J. D. P. 2005, ApJ, 630, 837Kawakatu, N., & Wada, K. 2008, to appear in ApJ, arXiv:0803.2271Kennicutt, R. C., Jr. 1998, ApJ, 498, 541King, A. R., & Pringle, J. E. 2007, MNRAS, 377, L2 Knapen, J. H. 2004, in Penetrating bars through masks of cosmicdust : the Hubble tuning fork strikes a new note, Ed. D. L.Block, I. Puerari, K. C. Freeman, R. Groess, and E. K. Block(Dordrecht: Kluwer)Kolykhalov, P. I. & Sunyaev, R. A. 1980,
Sov. Ast , 6, 35Krolik, J. H., & Begelman, M. C. 1988, ApJ, 329, 702Lada, C. J. & Lada, E. A. 2003, ARA&A, 41, 57Laine, S., Shlosman, I., Knapen, J. H., & Peletier, R. F. 2002, ApJ,567, 97Levin, Y. 2007, MNRAS, 374, 515Levine, R., Gnedin, N. Y., Hamilton, A. J. S., & Kratsov, A. V.2007, submitted to ApJLi, C., Kauffmann, G., Wang, L., White, S. D. M., Heckman, T.M.; Jing, Y. P. 2006, MNRAS, 373, 457Lin, D. N. C. & Papaloizou, J. C. B. 1979a, MNRAS, 186, 799Lin, D. N. C. & Papaloizou, J. C. B. 1979b, MNRAS, 188, 191Lin, D. N. C. & Papaloizou, J. C. B. 1986, ApJ, 307, 395Lu, Y., Yu, Q., & Lin, D. N. C. 2007, ApJ, 666, L89Martini, P., Regan, M. W., Mulchaey, J. S., & Pogge, R. W. 2003,ApJS, 146, 353Maoz, D., Barth, A. J., Sternberg, A., Filippenko, A. V., Ho, L. C.,Macchetto, F. D., Rix, H.-W., & Schneider, D. P. 1996, AJ, 111,2248Maoz, D., Barth, A. J., Ho, L. C., Sternberg, A., & Filippenko,A. V. 2001, AJ, 121, 3048McCrady, N., Gilbert, A. M., & Graham, J. R. 2003, ApJ, 596, 240McCrady, N. & Graham, J. R. 2007, ApJ, 663, 844Meurer, G. R., Heckman, T. M., Leitherer, C., Kinney, A., Robert,C., & Garnett, D. R. 1995, AJ, 110, 2665Mihos, J. C. & Hernquist, L. 1994, ApJ, 425, L13Miralda-Escud´e, J., & Kollmeier, J. A. 2005, ApJ, 619, 30Murray, N., Quataert, E., & Thompson, T. A. 2005, ApJ, 618, 569Nayakshin, S. & King, A. 2007, submitted to MNRAS,arXiv:0705.1686Nayakshin, S., Cuadra, J., & Springel, V. 2007, MNRAS, 379, 21Ostriker, E. C. 1994, ApJ, 424, 292Ostriker, J. P. & Peebles, P. J. E. 1973, ApJ, 186, 467Paczynski, B. 1978,
Acta Astron. , 28, 91Papaloizou, J. C. B. & Lin, D. N. C. 1995, ARA&A, 33 505Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P.1992, “Numerical recipes in FORTRAN. The art of scientificcomputing” (Cambridge University Press: Cambridge)Rafikov, R. R. 2002, ApJ, 569, 997Rees, M. J. 1988, Nature, 333, 523Roos, N. 1981, A&A, 104, 218 as Shepherding 13
Sanders, D. B., Soifer, B. T., Elias, J. H., Madore, B. F., Matthews,K., Neugebauer, G., & Scoville, N. Z. 1988, ApJ, 325, 74Shakura, N. I., & Syunyaev, R. A. 1973, A&A, 24, 337Shlosman, I. & Begelman, M. C. 1987, Nature, 329, 810Shlosman, I., & Begelman, M. C. 1989, ApJ, 341, 685Shlosman, I., Begelman, M. C., & Frank, J. 1990, Nature, 345, 679(SBF90)Shlosman, I., Frank, J., & Begelman, M. C. 1989, Nature, 338, 45Socrates, A., Davis, S. W., Ramirez-Ruiz, E. 2006, submitted to
ApJ , astro-ph/0609796Springel, V., Di Matteo, T., & Hernquist, L. 2005, MNRAS, 361,776Taniguchi, Y. 1997, ApJ, 487, L17 Taniguchi, Y. 1999, ApJ, 524, 65Taniguchi, Y. & Wada, K. 1996, ApJ, 469, 581Terquem, C. E. J. M. L. J. 1998, ApJ, 509, 819Thompson, T. A., Quataert, E., & Murray, N. 2005, ApJ, 630, 167Toomre, A. 1964, ApJ, 139, 1217Tremaine, S. & Weinberg, M. D. 1984, MNRAS, 209, 729Wada, K., & Norman, C. A. 2002, ApJ, 566, L21Walborn, N. R., Maz-Apellniz, J., & Barb, R. H. 2002, AJ, 124,1601Ward, W. R. & Hourigan, K. 1989, ApJ, 347, 490Ward, W. R. 1997,