arXiv:1005.0040v2 [astro-ph.GA] 7 Oct 2010 cl oqe ahrta ytobd eaain(e.g. relaxation two-body 2004). large- by Poon by & than dominated central Merritt be rather the to torques likely to is ar- stars scale galaxies Simple of many feeding in galaxies. the BHs long, that very luminous or- suggest be most of guments can scale the which time in time, a on particularly relaxation only central drives cen- but the also BH, the der central into scattering the way two-body into their stars stel- Gravitational find of BH. to eccentricities stars tral the allowing in orbits, changes lar present, gradual once induce universe. can triaxiality, the of that age the long for so persist relax- could are central galaxies, times star elliptical ongoing ation with luminous of associated in nuclei recurring, formation; the or features In recent non-axisymmetric be 2008). may al. the that et 1993; galaxies, bars, Seth distribution al. low-luminosity 2002; et including Sparke stellar (Shaw & axisymmetry, spirals Erwin the nuclear with revealed and in consistent also bars-within-bars, features smaller galaxies not On of of are centers wealth 2007). the al. a 1991; of et al. Cappellari imaging et 2004; (Franx scales, al. galaxies (kilo- et early-type large Statler in on models scales non-axisymmetry parsec) galaxy for triaxial accumulated grad- ually for evidence observational constructed 1982), 1979, (Schwarzschild be could libria [email protected] [email protected] hspprdsusstecaatro risna su- a near orbits of character the discusses paper This potential stellar the from torques nucleus, triaxial a In equi- self-consistent that demonstration the Following rpittpstuigL using 2018 typeset 1, Preprint November version Draft eateto hsc n etrfrCmuainlRelativ Computational for Center and Physics of Department iea h yai risaedpee;hwvrti iecnb of be can time this f however the depleted; dominate are can orbits orbits pyramid pyramid the on as stars time low- of a in capture relaxa inspirals that resonant extreme-mass-ratio show by We of we driven rate momentum. evolution the angular for to the consequences also change t which to apply at torques value should for the eccentricity time roughly eccentric the eccentricity: sufficiently to the BHs for equal to limit to precession upper matter the an c supplying dominates imposes fig the they in periapse triaxial space, at relaxation the the phase unity resonant of of of reaching with regions axis and and momentum short angles, repopulation angular the orientation lowest two about the orbit the occupy the of motion consis of function of motion axis t a the equations major property orbit-averaged integrable: the are interesting the orbits of derive the pyramid We have the triaxiality, BH. orbits, T weak the pyramid nonaxisymmetric. to is the cluster closely these, star surrounding of the one that case the in edsustepoete fobt ihnteiflec peeo s a of sphere influence the within orbits of properties the discuss We 1. A INTRODUCTION T E tl mltajv 11/10/09 v. emulateapj style X RISAON LC OE NTIXA NUCLEI TRIAXIAL IN HOLES BLACK AROUND ORBITS eee hsclIsiue eisypopk 3 Moscow, 53, prospekt Leninsky Institute, Physical Lebedev rf eso oebr1 2018 1, November version Draft ueeVasiliev Eugene ai Merritt David ABSTRACT 14623 t n rvtto,RcetrIsiueo ehooy Ro Technology, of Institute Rochester Gravitation, and ity uemsieB eg ert on20)adare and paper. 2004) current Poon to the & remnants of Merritt stellar focus (e.g. and the BH stars expected of supermassive are supply a the orbits the Exam- Centrophilic and dominate to 1999; 2001). 1999) BH. Merritt Valluri & 1997, & or- the Poon (Merritt Touma i.e., “pyramids” two-dimensional & to (Sridhar the three-dimensional orbits, close include orbits impor- centrophilic orbits arbitrarily “lens” centrophiliic the one of pass ples over orbit: that of passed bits class (2000) tant Sridhar & Sambhus constant- a in motion an nucleus. for triaxial (2000) in density Sridhar motion and & BH, Sambhus for massive de- by (1999) a was containing equations Touma approach nucleus el- & of axially-symmetric This Sridhar orbital forces. set by remaining perturbing a followed the the to of is Ke- due evolution result ements unperturbed slow The the the the such scribing with over with motion motion. associated Verhulst of deal plerian & scale equations Sanders to time the (e.g. way short averaging averaging standard i.e., orbit A 1985), via is or- can time. motion the grad- mass with change causes to which distributed ually eccentricity) perturbation (inclination, the small elements from a bital as force treated the be centrophilic. and be can plerian, nuclei BH fraction triaxial large in a orbits that mod- reveals the Self-consistent 2004) of Merritt BH. is & the emphasis to (Poon elling closest The come that the nucleus. orbits, orbits “centrophilic,” triaxial or a low-angular-momentum, on in BH permassive h ae sognzda olw.In follows. as organized is paper The nuclei, triaxial in motion of discussion their In Ke- nearly are orbits sphere, influence BH the Within uioiyglxe.I in galaxies, giant In galaxies. luminosity ris n eso htrelativity that show we and orbits, a hycnapoc arbitrarily approach can they hat eeaefu ao ri families; orbit major four are here opt ihcliinlls cone loss collisional with compete eigo H,a es ni such until least at BHs, of eeding ru htti pe ii othe to limit upper this that argue r,wt cetiiyvrigas varying eccentricity with ure, so w-iesoa libration two-dimensional a of ts in ihptnilyimportant potentially with tion, res eas yai orbits pyramid Because orners. erltvsi rcsintm is time precession relativistic he n hwta ntelmtof limit the in that show and eea eaiitcadvance relativistic General . prasv lc oe(BH), hole black upermassive re ubetime. Hubble a order Russia § epeeta present we 2 hse,NY chester, 2 Merritt and Vasiliev model for the gravitational potential of a triaxial nuclear treated as a small perturbation which causes the elements star cluster, which is more general than that studied in of the orbit (inclination, eccentricity etc.) to change Sambhus & Sridhar (2000), but which has many of the gradually with time. A standard way to deal with such same dynamical features. Then in 3 we write down the motion (e.g. Sanders & Verhulst 1985) is to average the orbit-averaged equations of motion,§ and in 4 present a equations over the coordinate executing the most rapid detailed analytical study of their solutions, with§ empha- variation, e.g., the radius. The result is a set of equations sis on the case where the triaxiality is weak and the eccen- describing the slow evolution of the remaining variables tricity large. In this limiting case, the averaged equations due to the perturbing forces. of motion turn out to be fully integrable. In 5 we derive We begin by transforming from Cartesian co- the equations that describe the rate of capture§ of stars on ordinates to action-angle variables in the Kepler pyramid orbits by the BH. Comparison of orbit-averaged problem. Following Sridhar & Touma (1999) and treatment with real-space motion is made in 6, to test Sambhus & Sridhar (2000), we adopt the Delaunay vari- the applicability of the former. In 7 we consider§ the ables (e.g. Goldstein et al. 2002) to describe the unper- effect of general relativity on the motion,§ which imposes turbed motion. an effective upper limit on the eccentricty. 8 discusses Let a be the semi-major axis of the Keplerian or- the connection with resonant relaxation: we argue§ that a bit. The Delaunay action variables are the radial action 1/2 similar upper limit to the eccentricity should character- I = (GM•a) , the angular momentum L, and the pro- ize orbital evolution in the case of resonant relaxation. jection of L onto the z axis Lz. The conjugate angle Finally, in 9 we make some quantitative estimates of variables are the mean anomaly w, the argument of the the importance§ of pyramid orbits for capture of stars in periapse ̟, and the longitude of the ascending node Ω. galactic nuclei. 10 sums up. In the Keplerian case, five of these are constants; the § exception is w which increases linearly with time at a 2. MODEL FOR THE NUCLEAR STAR CLUSTER rate Consider a nucleus consisting of a BH, a spherical star 2 3 ν = (GM•) /I (3) cluster, and an additional triaxial component. An ex- r pression for the gravitational potential that includes the In terms of the new variables, the Hamiltonian is three components is 2 2−γ 1 GM• GM• r Φ(r)= +Φ (1) = +Φp(I,L,Lz, w, ̟, Ω); (4) − r s r H −2 I 0 2 2 2 +2π Gρt Txx + Tyy + Tzz . the first term is the Keplerian contribution and Φp, the “perturbing potential”, contains the contributions from The second term on the right hand side is the potential of the spherical and triaxial components of the distributed a spherical star cluster with density ρ(r) = ρ (r/r )−γ ; s 0 mass. This transformation is completely general if we the coefficient Φ is given by s interpret the new variables as instantaneous (osculating) 4πG orbital elements. However if we assume that the per- Φ = ρ r2. s (3 γ)(2 γ) s 0 turbing potential is small compared with the point-mass − − potential, the rates of change of these variables (again The scale radius r0 may be chosen arbitrarily but it is with the exception of w) will be small compared with convenient to set r0 = rinfl, with rinfl the radius at which the radial frequency νr, and the new variables can be re- the enclosed stellar mass is twice M•: garded as approximate orbital elements that change little 1/3 over a radial period P 2π/νr. Accordingly, we average 3 γ M• the Hamiltonian over the≡ fast angle w: rinfl = − . (2) 2π ρs 2 1 GM• The third term is the potential of a homogeneous triaxial = + Φ , (5a) H −2 I p ellipsoid of density ρt; this term can also be interpreted as a first approximation to the potential of a more general, 2π dw 1 r inhomogeneous triaxial component. In the former case, Φp Φ= dE (1 e cos E)Φp( ). (5b) ≡ 2π 2π 0 − the dimensionless coefficients (Tx,Ty,Tz) are expressible I Z in terms of the axis ratios (p, q) of the ellipsoid via el- The final term replaces the mean anomaly w by the ec- liptic integrals (Chandrasekhar 1969). The x(z) axes are centric anomaly E, where r = a(1 e cos E) and the assumed to be the long(short) axes of the triaxial figure; − eccentricity is e = 1 L2/I2. After the averaging, this implies T T T . In what follows we will gener- x y z is independent of w, and− I is conserved, as is the semi-H ally assume ρ ≤ ρ (≤r ), i.e. that the triaxial bulge has t s 0 major axis a. We arep left with four variables and with a low density compared≪ with that of the spherical cusp Φp as the effective Hamiltonian of the system. at r = rinfl. The spherically symmetric part of Φp is 3. ORBIT-AVERAGED EQUATIONS 2−γ Within the BH influence sphere, orbits are nearly Ke- a Φ = F (e)Φ , (6) plerian1 and the force from the distributed mass can be s γ s r 0 1 3 γ 2 γ We consider general relativistic corrections in §7. F (e) F − , − , [1],e2 . γ ≡ 2 1 − 2 − 2 Orbital structure of triaxial nuclei 3
A good approximation to Fγ (e) is tions of motion describing the perturbed motion: Φ 3 23−γΓ( 7 γ) H p = ℓ2 + ǫ H + ǫ H , (13a) F (e) 1+ αe2 , α = 2 − 1 (7) ≡ ν I −2 b b c c γ ≈ √π Γ(4 γ) − p − dℓ ∂H d̟ ∂H dℓ ∂H dΩ ∂H = , = , z = , = .(13b) which is exact for γ = 0 and γ = 1; for 0 γ < 2, dτ − ∂̟ dτ ∂ℓ dτ − ∂Ω dτ ∂ℓz 0 < α 3/2. When γ > 1 and e is close to 1,≤ a better ≤ The renormalized triaxiality coefficients are ǫb,c approximation is (t) ≡ ǫb,c/(1 + A). 1−γ 5 If there were no spherical component (A = 0), this ′ 2 ′ 2 (2 γ) Γ( 2 γ) Fγ (e) 1+α+α (e 1) , α = − − . would reduce to the purely harmonic triaxial case studied ≈ − √π Γ(3 γ) 2 − by Sambhus & Sridhar (2000). Adding the spherically (8) symmetric cusp increases the rate of periapse precession, We adopt the latter expression in what follows. Good while at the same time reducing the relative amplitude of approximations are the triaxial terms; otherwise the form of the Hamiltonian is essentially unchanged. α 3 79 γ + 7 γ2 1 γ3, (9a) ≈ 2 − 60 20 − 30 A more transparent expression for the precession fre- ′ 3 29 11 2 1 3 quency νp is α 2 20 γ + 20 γ 10 γ . (9b) ≈ − − ′ Mt(a) 3Tx Ms(a) 2α Similar expressions can be found in νp = νr + (14a) Ivanov, Polnarev & Saha (2005) and M• 2 M• 3(2 γ) − − Polyachenko, Polyachenko & Shukhman (2007). 4π 4π a γ M (a) a3ρ ,M (a) a3ρ (14b) Expressions for the orbit-averaged triaxial harmonic t ≡ 3 t s ≡ 3 γ s r potential (excluding the spherical component) are de- − 0 rived in Sambhus & Sridhar (2000). Adopting their no- where M(a) denotes the mass enclosed within radius tation, the orbit-averaged potential in our case becomes r = a. From equation (13b), the precession rate of an orbit in the spherical cluster is ν d̟/dt = 3ℓν = ̟ ≡ | | p 3√1 e2 ν . Near the BH influence radius, it is clear 2−γ − p a ′ 2 2 that νp νr; hence the orbit-averaged treatment, which Φp =Φs (1 + α α ℓ )+2πGρtTx a (10a) ≈ r0 − × assumes only one “fast” variable, is likely to break down at this radius. 5 3 2 (t) (t) ℓ + ǫ H (ℓ,ℓ , ̟, Ω) + ǫ H (ℓ,ℓ ,̟) We note that ν̟ 0 as e 1. For the very eccentric × 2 − 2 b b z c c z → → orbits that are the focus of this paper, the rate of preces- 1 2 2 Hb = (5 4ℓ )(c̟sΩ + cicΩs̟) (10b) sion is much lower than for a typical, non-eccentric orbit 2 − of the same energy. This will turn out to be important, + ℓ2(s s c c c )2 , ̟ Ω − i Ω ̟ since the slow precession allows torques from the triaxial H = 1 (1 c2)[5 3ℓ2 5(1 ℓ2)c ], (10c) part of the potential to build up. c 4 − i − − − 2̟ (t) (t) 4. ORBITAL STRUCTURE OF THE MODEL POTENTIAL ǫb Ty/Tx 1 , ǫc Tz/Tx 1. (10d) ≡ − ≡ − 4.1. General remarks The shorthand sx,cx has been used for sin x, cos x. We The orbit-averaged Hamiltonian (13a) describes a dy- have defined the dimensionless variables ℓ = L/I and namical system of two degrees of freedom. The trajec- ℓz = Lz/I, both of which vary from 0 to 1; the orbital tories must be obtained by numerical integration of the inclination i is given by cos i ℓz/ℓ and the eccentricity equations of motion (13b). We begin by making some by e2 =1 ℓ2. ≡ − qualitative points about the nature of the solutions. The first term in equation (10b), which arises from the In the absence of the triaxial terms in equation (13a), spherically-symmetric cusp, does not depend on the an- 2 the effect of the distributed mass is to rotate the pe- gular variables, and has the same dependence on ℓ as the riapse angle ̟ in a fixed plane; this steadily rotating corresponding term in the harmonic triaxial potential. 2 elliptic orbit fills an annulus. The addition of a weak tri- So we can sum up the coefficients at ℓ and renormalize axial perturbation changes the rate of in-plane precession the triaxial coefficients ǫb,c to obtain the same functional slightly, and also causes the orbital plane itself to change, form of the Hamiltonian as in the purely harmonic case. as described by the last two terms in equation (13b). Dropping an unnecessary constant term (depending only In general, two angular variables ̟ and Ω can either on a) and defining a dimensionless time τ = νpt, where librate around fixed points or circulate, giving rise to νp is characteristic rate of precession, four basic families of orbits (Figure 1). Solutions to the equations of motion that are characterized by circula- 2 νp 2πGρtTxa (1 + A)/I, (11) tion in both ̟ and Ω correspond to tube orbits about ≡ −γ the short axis (SAT). Motion that circulates in ̟ but ′ ρ a A 4α s , (12) librates in Ω corresponds to tube orbits about the long ≡ 3(3−γ)(2−γ)Tx ρ r t 0 2 Equation (12) of Sambhus & Sridhar (2000) for ℓ˙ lacks a minus we obtain the dimensionless Hamiltonian and the equa- sign in front of the first term. 4 Merritt and Vasiliev
3 l 0 w W 0.2
-10 2 0.1
0 -20 1 lz -0.1
Long-axis tubes -30 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0.4 0 0
0.3 -10 -2
-20 -4 0.2 -30 -6 0.1 -40 -8
Short-axis tubes
0 -50 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 3 0 0.2 -10 2
-20 0.1 1 -30
Short-axis saucers 0 0 -40 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 10 3 10 10 6 0.2 4 2 2 0.1 0 0 1 -2
Pyramids -0.1 -4 0 -6 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 Fig. 1.— Four classes of orbits around a BH in a triaxial nucleus. Left column: time dependence of the dimensionless angular momentum ℓ (top/blue) and its component ℓz along the short axis of the figure (bottom/red). Middle column: argument of the periapse ̟. Right column: angle of nodes Ω. nw , n W to the tube orbits that are generic to the triaxial geom- 1 etry (Schwarzschild 1979). A subclass of the SAT orbits max= corresponds to motion that circulates in Ω and librates in ̟ (Sambhus & Sridhar 2000; Poon & Merritt 2001). These orbits resemble cones, or saucers; similar orbits ex- 0.1 ist also at r rinfl in oblate or nearly oblate potentials (Richstone 1982;≫ Lees & Schwarzschild 1992). If the degree of triaxiality is small (ǫb,c 1), then 0.01 as noted above, the dominant effect of the≪ distributed mass is simply to induce a periapse shift, at a rate ν̟ d̟/dτ = 3ℓ. If the additional mass is much less≡ than the mass− of the BH, then on short time scales −1 0.01 0.1 1 (comparable to the radial period νr ) the orbit resem- Fig. 2.— Dependence of the characteristic frequencies ν̟ (in- bles a nearly closed ellipse. On intermediate time scales −1 plane precession) and νΩ (nodal precession) on the value of the (of order the precession time ν̟ ), a steadily-rotating dimensionless angular momentum ℓ. Top (open) symbols: ν̟; bot- elliptic orbit fills an annulus in a fixed plane. On still tom (filled) symbols: νΩ. Magenta boxes: LATs; red triangles: −1 SATs; yellow diamonds: saucers; green circles: pyramids. For longer time scales νΩ , the orbital plane itself changes large ℓ ν̟ ∝ ℓ and νΩ ≪ ν̟, but for sufficiently low ℓ these two due to the torques from the triaxial potential. Similar are comparable, which gives birth to the pyramid and saucer orbits. considerations give rise to the concept of vector resonant Vertical line denotes the threshold in ℓ (equation 36), horizontal relaxation (Rauch & Tremaine 1996). line the characteristic frequency νx0 (equation 22). Triaxiality co- −3 The foregoing description is valid as long as the angu- efficients were set to ǫc = 10 , ǫb = 0.4ǫc. lar momentum is not too low. Since the precession rate axis (LAT). Both types of orbit are qualitatively similar is proportional to ℓ, for sufficiently low ℓ the intermedi- Orbital structure of triaxial nuclei 5 ate and long time scales become comparable (Figure 2). of equations (16) and using equations (13b), we find As a result, the triaxial torques can produce substantial changes in ℓ (i.e. the eccentricity) on a precession time e˙x =3ℓ(sin ̟ cosΩ+ cos ̟ sin Ω cos i), (18a) scale via the first term in (13b), and the circulation in ̟ e˙y =3ℓ(sin ̟ sin Ω cos ̟ cosΩ cos i) (18b) can change to libration. This is the origin of the pyra- − wheree ˙ de /dτ etc. Taking second time derivatives, mid orbits, which are unique to the triaxial geometry x ≡ x (Merritt & Valluri 1999). the variables describing the orientation and eccentricty of the orbit drop out, as desired, and the equations of motion for ex and ey can be expressed purely in terms of 4.2. Pyramid orbits ex and ey: Of the four orbit families discussed above, the first 2 three were treated, in the orbit-averaged approximation, e¨x = ex 6(H +3ℓ ) (19a) − 2 2 by Sambhus & Sridhar (2000). The fourth class of orbits, = ex [30ǫc 6H 30ǫcex 30(ǫc ǫb)ey], the pyramids, are three-dimensional analogs of the two- − − − − − e¨ = e 6(H +3ℓ2 5 ǫ ) (19b) dimensional “lens” orbits discussed by Sridhar & Touma y − y − 2 b (1997), also in the context of the orbit-averaged equa- 2 2 = ey [30ǫc 6H 15ǫb 30ǫcex 30(ǫc ǫb)ey]. tions. An important property of the pyramid orbits is − − − − − − that ℓ can come arbitrarily close to zero (Poon & Merritt From equations (18), (e ˙x, e˙y) = 0 implies ℓ = 0, i.e. the 2001) and Merritt & Poon (2004). This makes the pyra- eccentricity reaches one at the “corners” of the orbit. mids natural candidates for providing matter to BHs at These define the base of the pyramid. Defining (ex0,ey0) the centers of galaxies. to be the values of (ex,ey) when this occurs, vthe Hamil- Pyramid orbits can be treated analytically if the fol- tonian has numerical value lowing two additional aproximations are made: (1) the 2 5 5 2 2 angular momentum is assumed to be small, ℓ 1; (2) H = ǫc ǫcex0 + (ǫc ǫb)ey0 . (20) the triaxial component of the potential is assumed≪ to 2 − 2 − be small compared with the spherical component, i.e. Equations (19) have the form of coupled, nonlinear os- ǫb,ǫc 1. As shown below, these two conditions are cillators. Given solutions to these equations, the time ≪ 2 consistent, in the sense that ℓmax ǫb,c for pyramid or- dependence of the additional variables (ℓ,ℓz, ̟, Ω) fol- bits. ∼ lows immediately from equations (16) and (18): 2 Removing the second-order terms in ǫb,ǫc and in ℓ 2 2 2 from the orbit-averaged Hamiltonian (13a), we find e˙x +e ˙y (e ˙xey exe˙y) 1 2 − − 2 2 2 ℓ = 2 2 = (e ˙x +e ˙y +e ˙z), 9(1 ex ey) 9 3 2 5 2 2 2 − − H = ℓ + ǫc(1 ci )s̟ + ǫb(c̟sΩ + cis̟cΩ) ℓ = (e ˙ e e e˙ )/3, −2 2 − z x y − x y 2 2 2 (15) 1 ex ey e 2 − − z where again ci cos i = ℓz/ℓ. sin ̟ = 2 2 = 2 2 , Because an orbit≡ described by (15) is essentially a pre- 1 ℓz/ℓ 1 ℓz/ℓ − 2 − cessing rod, one expects the important variables to be (exe˙x + eye˙y) e˙2 = , e2 =1 e2 e2, the two that describe the orientation of the rod, and z 2 2 z x y 1 ex ey − − its eccentricity. This argument led us to search for ex- − − act solutions to the equations of motion in terms of the and a quite lengthy expression for Ω which we choose not Laplace-Runge-Lenz vector, or its dimensionless counter- to reproduce here. part, the eccentricity vector, which point in the direction In the limit of small amplitudes, (ex,ey) 1 (which of orbital periapse. 5 ≪ corresponds to H 2 ǫc), the oscillations are harmonic We therefore introduced new variables ex, ey and ez: and uncoupled, with≈ dimensionless frequencies
ex =cos ̟ cosΩ sin ̟ cos i sin Ω (16a) ν √15ǫ , ν 15(ǫ ǫ ). (22) − x0 c y0 c b e = sin ̟ cos i cosΩ+ cos ̟ sin Ω (16b) ≡ ≡ − y The apoapse traces out a 2d Lissajousp figure in the plane ez = sin ̟ sin i (16c) perpendicular to the short axis of the triaxial figure. This is the base of the pyramid (e.g. Merritt & Valluri (1999), which correspond to components of a unit vector in the Figure 11). The solutions in this limiting case are direction of the eccentricity vector. Of these, only two 2 2 2 are independent, since ex +ey +ez = 1. In terms of these ex(τ)= ex0 cos(νx0τ + φx), (23a) variables, the Hamiltonian (15) takes on a particularly ey(τ)= ey0 cos(νy0τ + φy), (23b) simple form: 2 2 2 2 2 ℓ (τ)= ℓx0 sin (νx0τ + φx)+ ℓy0 sin (νy0τ + φy) 3 2 5 2 2 H = ℓ + ǫ ǫ e (ǫ ǫ )e . (17) where φx, φy are arbitraty constants and −2 2 c − c x − c − b y ℓx0 = νx0ex0/3, ℓy0 = νy0ey0/3. (24) As expected, the Hamiltonian depends on only three vari- ables: ex and ey, which describe the orientation of the Figure 3a plots an example. orbit’s major axis, and the eccentricity ℓ. Equations (23) describe integrable motion. Remark- To find the equations of motion, we must switch to a ably, it turns out that the more general (anharmonic, Lagrangian formalism. Taking the first time derivatives coupled) equations of motion (19) are integrable as well. 6 Merritt and Vasiliev
Fig. 3.— A pyramid orbit, in three approximations. Each orbit has the same (ex0,ey0) = (0.5, 0.35). (a) The simple harmonic oscillator 2 (SHO) approximation, equations (23), valid for small ℓ , (ǫb,ǫc) and (ex0,ey0). (b) From equation (19), which does not assume small 2 (ex0,ey0). (c) From the full orbit-averaged equations (13), which does not assume small ℓ , ǫ or (ex0,ey0). Aside from the fact that the latter orbit is fairly close to a 5 : 2 resonance, the correspondence between the physically important properties of the approximate orbits is good. The triaxiality parameters are (ǫb,ǫc) = (0.0578, 0.168), corresponding to a pyramid orbit with a = 0.1r0 in a nucleus with triaxial axis ratios (0.5, 0.75), density ratio ρt(r0)/ρs(r0) = 0.1, and γ = 1. The frequencies for the SHO case are νx0 = 1.59, νy0 = 1.28 (equation 22); frequencies for planar orbits with the same ex and ey amplitudes are 1.48 and 1.24 respectively (equation 27).
The first integral is H; an equivalent, but nonnegative, of oscillation of the planar orbits (ex = 0 or ey = 0) integral is U where described by these equations are easily shown to be U 15ǫ 6H = ν2 e2 + ν2 e2 + (e ˙2 +e ˙2 +e ˙2). (25) ν P (e )=4K e2 (e = 0), (27) ≡ c − x0 x y0 y x y z x0 x0 x0 y 2 The second integral is obtained after multiplying the first νy0P (ey0)=4K ey0 (ex = 0) of equations (19) by 15ǫce˙x, the second by 15(ǫc ǫb)e ˙y, and adding them to obtain a complete differential.− The where K(α) is the complete elliptic integral: integral W is then π/2 2 −1/2 2 2 2 2 2 4 2 2 2 2 2 4 K(α)= 1 α sin x dx. W = νx0(e ˙x + νxex νx0ex)+ νy0(e ˙y + νy ey νy0ey) − − − Z0 2ν2 ν2 e2 e2, (26a) − x0 y0 x y For small α, K π/2 and P 2π/ν0. As α 1, 2 2 2 2 K ; this corresponds≈ to a pyramid≈ that precesses→ νx U + νx0 , νy U + νy0. (26b) ≡ ≡ from→ the ∞ z axis all the way to the (x, y) plane. The The existence of two integrals (U, W ), for a system with oscillator is “soft”: increasing the amplitude increases two degrees of freedom, demonstrates regularity of the also the period. Figure 3 shows comparison of orbits with motion. the same initial conditions, calculated in three different Regular motion can always be expressed in terms of approximations. action-angle variables. The period of the motion, in each Pyramid orbits can be seen as analogs of regular box degree of freedom, is then given simply by the time for orbits in triaxial potentials (Schwarzschild 1979), with the corresponding angle variable to increase by 2π. We three independent oscillations in each coordinate. Like were unable to derive analytic expressions for the action- box orbits, they do not conserve the magnitude of the angle variables corresponding to the two-dimensional mo- sign of the angular momentum about any axis. The dif- tion described by equations (19). However the periods ference is that a BH in the center serves as a kind of “re- Orbital structure of triaxial nuclei 7
W Short-axis tubes and saucers resemble distorted rect- 0.03 angular regions in the ex,ez plane. Again, the corner points (with subscript 0) are defined to havee ˙x =e ˙z =0 2 and ey = 0, withe ˙y > 0, and therefore d SAT U = ν2 e2 +e ˙2 , (33a) 0.02 x0 x0 y0 Saucer LAT c W = ν2 U + (ν2 e2 + ν2 ν2 )e ˙2 . (33b) x0 x0 x0 y0 − x0 y0 a Both these families have W ǫcU, i.e. lie above the line b ≥ 0.01 (30). SAT orbits intersect the plane ez = 0, so we can set ex0 = 1 in (33). (Alternatively, for SATs, both angles Pyramid circulate, so we can set ̟ = Ω = 0, which again gives ex = 1). We then find that SATs lie below the line (31). U On the other hand, saucers never reach e = 0 (since 0 z 0.05 0.1 0.15 0.2 for them sin2 ̟> 0), so that they lie above the line (31). Fig. 4.— Regions in the U − W plane occupied by the different To obtain the upper limit for W at fixed U, we substitute orbit families. Lower (dark blue) line (a), equation (29); upper e˙2 from the first equation in (33) in the second, and (red) curve (d), equation (34); green line (c), equation (30); light y0 blue line (b), equation (31); red points, equation (35); blue points, then seek a maximum of W with respect to ex0 at fixed equation (32). Plotted for ǫb = 0.002,ǫc = 0.008. U. This gives flecting boundary”, so that a pyramid orbit is reflected 2 2 2 2 ◦ W = ν U + (U + ν ν ) /4. (34) by 180 near periapsis, instead of continuing its way to x0 y0 − x0 the other side of x y plane as a box orbit would do. − This curve intersects (30) and (31) in the points 4.3. The complete phase space of eccentric orbits U = ν2 ν2 , W = ν2 U, (35a) x0 − y0 x0 While our focus is on the pyramid orbits, the low- U = ν2 + ν2 , W = ν2 U + ν4 , (35b) angular-momentum Hamiltonian (15) also supports or- x0 y0 x0 y0 bits from other families. In this section, we complete the which define the left- and rightmost bounds for the saucer discussion of the phase space described by equation (15), region. by delineating the regions in the U W plane that are All these criteria are summarized in Figure 4. In par- − occupied by each of the four orbit families (Figure 4). ticular, pyramid orbits exist in the following cases: Pyramids and LATs both resemble distorted rectan- gles in the e ,e plane. The corner points of this region for 0 H 5 ǫ they coexist with LATs; x y • ≤ ≤ 2 b correspond toe ˙x =e ˙y = 0. Evaluating the two integrals at a corner (denoted by the subscript 0) gives for 0 H 5 (ǫ ǫ ) they coexist with SAT • ≤ ≤ 2 c − b 2 2 2 2 2 saucers; U = νx0 ex0 + νy0 ey0 +e ˙z,0 , (28a) 4 2 4 2 2 2 above these values they are the only population for W = νx0 ex0 + νy0 ey0 + (U e˙z,0)e ˙z,0 . (28b) • 5 − H 2 ǫc, which is the maximum allowed value of The difference between pyramids and LATs arises from H.≤ the last term: corner points of pyramid orbits correspond 2 to ℓ = 0 and hence (from (21a)) toe ˙z,0 = 0. For LATs below H < 0 pyramids do not exist (this is easily 2 • seen from equation (15): since the term in square the condition is, conversely,e ˙z,0 > 0, and ez,0 = 0 (hence 2 2 brackets is always non-negative, it is impossible to ey0 =1 ex0). Analyzing these expressions, we find that have ℓ2 = 0 when H < 0). for pyramids− and LATs the lower and upper boundaries for W given U are Figure 5 shows Poincar´esurfaces of section for Ω = 5 2 π/2 and 0 tion, what can we conclude about the orbit class? It is clear that without knowledge of the derivatives of ex,y the answer will only be probabilistic. It turns out that the probability p for an orbit with “sufficiently high” ec- 2 2 centricity (i.e. with ℓ ℓmax) to be a pyramid depends mostly on the z component≤ of the eccentricity vector: p 0.7 4 ǫb (1 ǫb ) e1.5 (here the normalization comes ≈ ǫc − ǫc z from theq total number of pyramids among low-ℓ orbits). That is, an orbit lying in the plane defined by the long and intermediate axes of the potential is certainly not a pyramid, and the highest probabilitiy occurs for orbits directed toward the short axis. Fig. 5.— Poincar´esection for ℓ,̟ plotted at Ω = π/2 for en- −2 −2 4.4. Large ℓ limit ergy H = 0.02 (ǫb = 0.99 − 1,ǫc = 0.96 − 1) showing the three possible types of orbit: LATs, pyramids and SATs/saucers. In the previous sections we considered the case ǫb,c 1 This figure adopts the same potential parameters as Figure 5b in and ℓ2 ǫ , which allowed a simplification leading≪ to Sambhus & Sridhar (2000), but those authors chose H = −0.02 ∼ c which precludes pyramid orbits. Boundaries are marked by the integrable equations. same letters as in Figure 4. In the opposite case, when ǫ ℓ2 . 1, the frequency b,c ≪ of in-plane precession, ν̟, is much greater than the rates We now return from the simplified Hamiltonian (15) to change of Ω and i (Figure 2). In this limit we can carry the full Hamiltonian (13), i.e. we no longer require ǫ to out a second averaging of the Hamiltonian (13a), this be small. The full Hamiltonian retains all the qualitative time over ̟. Thus properties of the simplified system but requires numerical integration of the equations of motion (13) to determine 1 2π 3 H = Hd̟ = ℓ2 (38) orbit classes. h i 2π −2 To quantify the overall fraction of pyramid orbits in a Z0 5 3ℓ2 given potential, one should uniformly sample the phase + − ǫ s2 + c2c2 + ǫ 5 3ℓ2 1 c2 . space for all four variables and determine the orbit class 4 b Ω i Ω c − − i for each initial condition. From equation (36), we can −1 restrict ourselves to values of ℓ2 ℓ2 (0) = 5ǫc On timescales T & νp the orbit resembles an annulus ≤ max 3+4ǫc−ǫb that lies in the plane defined by the angles i and Ω. The (but we must take care not to filter out initial conditions only remaining equations of motion are those that de- corresponding to H < 0). 2 scribe the change in orientation of the orbital plane: We calculated the proportions of the ℓmax-restricted fraction of phase space occupied by each family of orbits. dℓ ǫ 4 z b 2 2 Initial conditions were drawn randomly for 10 points = 5 3ℓ 1 ci s2Ω, (39) 2 2 dτ − 4 − − (with uniform distribution in ℓ [0..ℓmax], in ℓz [0..ℓ], π ∈ ∈ dΩ ǫb 2 2 ǫc 2 and in ̟, Ω [0, 2 ]). The proportions were found to ℓ = 5 3ℓ cΩci 5 3ℓ ci. depend very∈ weakly on ǫ if ǫ 1. To elucidate the dτ 2 − − 2 − c c ≪ dependence on ǫb/ǫc (the degree of triaxiality) we took One expects the natural variables in this case to be the 15 values in the range (0.001 0.999). components of the angular momentum: We found that the relative− fraction η of pyramids low- among ℓ orbits is almost independent of ǫc (Figure 6): ℓx = ℓ sin i sin Ω, (40) ℓy = ℓ sin i cosΩ, ǫb ǫb η 0.28 4 1 . (37) ℓ = ℓ cos i ≈ ǫ − ǫ z s c c and ℓ2 + ℓ2 + ℓ2 = ℓ2 = constant. In terms of these The fraction of pyramids among all orbits isη ˜ = x y z 2 5 variables, the Hamiltonian is ηℓmax(0) = 3 ǫc η. For comparison, the left panel of Figure 6 shows the results obtained using the simpli- 2 3 2 (5 3ℓ ) 2 2 2 2 fied Hamiltonian (15) and the analytical classification H = ℓ + − ǫb ℓ ℓy + ǫc ℓ ℓz . scheme described above, while the middle panel, made h i −2 4ℓ2 − − (41) for ǫc = 0.1, shows almost the same behavior, with the addition of a small number of chaotic orbits. We note After some algebra, one finds the equations of motion: that for ǫc 1 the phase space becomes largely chaotic. 1 ℓ ℓ These estimates∼ of the relative fraction of pyramid or- ˙ 2 y z ℓx = (ǫc ǫb) 5 3ℓ 2 , (42) bits are directly applicable to a galaxy with an isotropic −2 − − ℓ ǫc ℓxℓz distribution of stars at any energy. This assumption may ℓ˙ = 5 3ℓ2 , not be valid, for example, in the case of induced tangen- y 2 − ℓ2 tial anisotropy following the merger of supermassive BHs ǫ b ℓxℓy ℓ˙ = 5 3ℓ2 (Merritt & Milosavljevic 2005). z − 2 − ℓ2 One can ask a different question: if we know the in- stantaneous value of an orbit’s eccentricity and orienta- (only two of which are independent). These can be writ- Orbital structure of triaxial nuclei 9 2 2 Fig. 6.— Proportions of the restricted part of phase space (defined by ℓ <ℓmax(0)) that are occupied by the major orbit families: LATs, pyramids, SATs, and SAT/saucers, as a function of the ratio ǫb/ǫc. Left: analytic estimates from the simplified orbit-averaged Hamiltonian (15) for ǫc → 0; middle: orbit-averaged Hamiltonian (13) for ǫc = 0.1; right: real-space integration for orbits with semimajor axis a = rinfl (equal to the BH influence radius) and ǫc = 0.1. ten We note the following property of the pyramid orbits: as long as the frequencies of e and e oscillation are in- d~ℓ x y = T ℓ,~ (43) commensurate, the vector (ex,ey) fills densely the whole dτ × available area, which has the form of distorted rectangle. 5 3ℓ2 0 The corner points correspond to zero angular momen- T = − ǫbℓy . tum, and the “drainage area” is similar to four holes in 2ℓ2 ǫcℓz ! the corners of a billiard table. Unless otherwise noted, in this section we adopt the Conservation of the Hamiltonian (41) implies simple harmonic oscillator (SHO) approximation to the 2 2 (ex,ey) motion, that is, we use the simplified Hamil- ǫbℓy + ǫcℓz = constant = C. tonian (15) and its solutions (23); these orbits have 2 2 This is an elliptic cylinder; the axis is parallel to the ℓ - ex + ey 1 and they form a rectangle in the ex ey x ≪ − axis, and the ellipse is elongated in the direction of the plane, with sides 2ex, 2ey. As long as the motion is inte- ℓy-axis. In addition, we know that grable, the results for arbitrary pyramids with ex,ey . 1 will be qualitatively similar. Quantitative results may be 2 2 2 2 ℓx + ℓy + ℓz = constant = ℓ obtained by numerical analysis and are presented near the end of this section. which is a sphere. So, the motion lies on the intersec- Figure 7 shows a two-torus describing oscillations in tion of a sphere with an elliptic cylinder. There are two (e ,e ) for a pyramid orbit. In the SHO approximation, possibilities. x y 2 solutions are given by (23). If the two frequencies νx0,νy0 1. ℓ > C/ǫb. In this case, the cylinder intersects the are incommensurate, the motion will fill the torus. In sphere in a deformed ring that circles the ℓx-axis. This this case, we are free to shift the time coordinate so as corresponds to a LAT orbit. 2 to make both phase angles (φ1, φ2) zero, yielding 2. ℓ < C/ǫb. In this case, the locus of intersection is a deformed ring about the ℓz-axis. This orbit is a SAT. 2 2 2 2 2 In other words, precession of the angular momentum vec- ℓ (τ)= ℓx0 sin (νx0τ)+ ℓy0 sin (νy0τ) (45a) 2 2 2 2 tor can be either about the short or long (not intermedi- = ℓ sin θ1 + ℓ sin θ2 (45b) ate) axes of the triaxial ellipsoid. x0 y0 5. CAPTURE OF PYRAMID ORBITS BY THE BH where θ1 = νx0τ,θ2 = νy0τ. (In the case of exact com- mensurability, i.e. m1νx0 + m2νy0 = 0 with (m1,m2) As we have seen, pyramid orbits can attain arbitrar- integers, the trajectory will avoid certain regions of the ily low values of the dimensionless angular momentum torus and such a shift may not be possible.) In the ℓ. The BH tidally disrupts or captures stars with angu- SHO approximation, ν = √15ǫ ,ν = 15(ǫ ǫ ) lar momentum less than a certain critical value L•, or – x0 c y0 c b (22). More generally, integrable motion will still be− rep- in dimensionless variables – ℓ• L•/I(a). We can ex- ≡ resentable as uniform motion on the torusp but the fre- press ℓ• in terms of the capture radius rt, the radius at which a star is either tidally disrupted or swallowed. For quencies and the relations between ℓ and the angles will 8 be different. BH masses greater than 10 M⊙, main sequence stars 2 Stars are lost when ℓ(θ ,θ ) ℓ•. Consider the loss avoid disruption and r ∼r 2GM•/c ; for smaller 1 2 t Schw region centered at (θ ,θ ) = (0≤, 0). This is one of four M•, tidal disruption occurs≈ outside≡ the Schwarzschild ra- 1 2 dius; e.g. at the center of the Milky Way, r 10r such regions, of equal size and shape, that correspond to t ≈ Schw the four corners of the base of the pyramid. For small for solar-type stars. Defining rt = ΘrSchw and writing 2 ℓ•, the loss region is approximately an ellipse, L GM•r , then gives • ≈ t −1 2 2 2 rSchw −5 M• a ℓx0 2 ℓy0 2 ℓ• =Θ 10 Θ . (44) θ1 + θ2 . 1. (46) a ≈ 108M⊙ 1 pc ℓ2 ℓ2 • • 10 Merritt and Vasiliev plane defined by 2 2 νx0ℓ θ1 + νy0ℓ θ2 ψ = x0 y0 , (50) 2 2 2 2 νx0ℓx0 + νy0ℓy0 qν ℓ ℓ θ + ν ℓ ℓ θ ϑ = − y0 x0 y0 1 x0 x0 y0 2 . (51) 2 2 2 2 νx0ℓx0 + νy0ℓy0 With this transformation,q the phase velocity becomes ˙ 2 2 2 2 1/2 ˙ ψ = (νx0ℓx0 + νy0ℓy0) , ϑ = 0 (52) and the loss regions become circles of radius ℓ•. The angular displacement in one radial period is 2 2 2 2 1/2 ∆ψ =2π(νp/νr) νx0ℓx0 + νy0ℓy0 . (53) 2 −1 The density of stars is (4π ℓx0ℓy0) . At any point in the (ψ, ϑ) plane, stars have a range of radial phases. Assuming that the initial distribution satisfies Jeans’ theorem, stars far from the loss regions are uniformly distributed in χ where Fig. 7.— Two-torus describing oscillations of (ex,ey) for a pyra- r dr mid orbit. The ellipses correspond to regions near the four corners χ = P −1 ; (54) • of the pyramid’s base where ℓ ≤ ℓ . In the orbit-averaged approx- rp vr imation, trajectories proceed smoothly along lines parallel to the Z solid lines, with slope tan α = νy/νx. In reality, successive periapse here P 2π/νr is the radial period, rp is the periapse passages occur at discrete intervals, once per radial period. ≡ distance and vr is the radial velocity. The integral is performed along the orbit, hence χ ranges between 0 and 1 as r varies from rp to apoapse and back to rp. (χ = w mod 2π, where w is mean anomaly). The area enclosed by this “loss ellipse” is Figure 8 shows how stars move in the (χ, ψ) plane at fixed ϑ. The loss region extends in ψ a distance 2 2 2 ℓ 2 ℓ• ϑ , from ψin to ψout. Stars are lost to the BH if π • . (47) − ℓ ℓ they reach periapse while in this region. x0 y0 pTwo regimes must be considered, depending on whether ∆ψ is less than or greater than ψ ψ . There are four such regions on the torus; together, they out − in constitute a fraction 1. ∆ψ < ψout ψin (Figure 8a). In one radial period, stars in the orange− region are lost. One-half of this region 1 ℓ2 lies within the loss ellipse; these are stars with ℓ<ℓ but µ = • (48) 0 π ℓx0ℓy0 which have not yet attained periapse. The persistence of stars inside the “loss cone” is similar to what occurs in of the torus. the case of diffusional loss cone repopulation, where there Stars move in the (θ1,θ2) plane along lines with slope is also a “boundary layer”, the width of which depends 2 2 on the ratio of the relaxation time to the radial period tan α = νy0/νx0, at an angular rate of νx0 + νy0. Since (e.g. Cohn & Kulsrud (1978)). The other one-half con- periapse passages occur only once perq radial period, a sists of stars that have not yet entered the loss region. star will move a finite step in the phase plane between The area of the orange region is equal to the area of a encounters with the BH. The dimensionless time between rectangle of unit height and width ∆ψ; since stars are successive periapse passages is ∆t =2πνp/νr. The angle distributed uniformly on the (χ, ψ) plane, the number of traversed during this time is stars lost per radial period is equal to the total number of stars, of any radial phase, contained within ∆ψ. 2 2 2. ∆ψ > ψ ψ (Figure 8b). In this case, some ∆θ =2π(νp/νr) νx0 + νy0. (49) out in stars manage to cross− the loss region without being cap- q The rate at which stars move into one the four loss el- tured. The area of the orange region is equal to that of a rectangle of unit height and width ψ ψ . The num- lipses is given roughly by the number of stars that lie out − in an angular distance ∆θ from one side of a loss ellipse, ber of stars lost per radial period is therefore equal to divided by ∆t. the number of stars, of arbitrary radial phase, contained 2 2 This is not quite correct however, since a star may within ψout ψin =2 ℓ• ϑ . − − precess past the loss ellipse before it has had time to To compute the total loss rate, we integrate the loss reach periapse. We carry out a more exact calculation per radial period overpϑ. It is convenient to express the by assuming that the torus is uniformly populated at results in terms of q where some initial time, with unit total number of stars. To ∆ψ νp −1 2 2 2 2 simplify the calculation, we transform to a new phase q = π ℓ• νx0ℓx0 + νy0ℓy0. (55) ≡ 2ℓ• νr q Orbital structure of triaxial nuclei 11 Fig. 9.— Illustrating the area of the (ψ,ϑ) phase plane that is lost into the BH each radial period. The circle centered at (0, 0) is the loss region corresponding to one corner of the pyramid orbit; its radius is ℓ•. Regions marked in bold denote the area of the torus that is lost in one radial period, for q < 1 (∆ψ1) and q > 1 (∆ψ2). While the number of stars lost per radial period is proportional to the marked areas, the region on the torus from which those stars come is more complicated since it depends also on an orbit’s radial phase (Figure 8). ϑ = 0. For q 1, the area integral becomes c ≤ ϑc ℓ• 4qℓ• dϑ +4 ℓ2 ϑ2dϑ • − Z0 Zϑc p1 2 2 2 2 =4qℓ• 1 q +4ℓ• dx 1 x − √1−q2 − p Z p 2 2 2 Fig. 8.— Trajectories of stars in the (ψ,χ) plane as they en- = ℓ• π +2q 1 q 2 arcsin 1 q counter a loss region from left to right, defined as ψin ≤ ψ ≤ ψout. − − − 2 χ increases from 0 at periapse, to 1/2 at apoapse, to 1 at subsequent =4qℓ•f(q), p p periapse. Trajectories are indicated by dashed lines. Stars are lost 1 1 if they reach periapse while inside the loss region. Stars within the f(q)= 1 q2 + arcsin(q) (59) orange region are lost in one radial period. (a) ∆ψ < ψin − ψout; 2 − 2q (b) ∆ψ > ψin − ψout. p 2 and for q > 1 it is πℓ•. The function f(q) varies from f(0)= 1 to f(1) = π/4 0.785. The area on the phase≈ plane that is lost each radial period can be interpreted in a very simple way geomet- rically, as shown in Figure 9. Considering that there are four loss regions, the in- q 1 corresponds to an “empty loss cone” and q 1 stantaneous total loss rate , in dimensionless units, is to≪ a “full loss cone”. However we note that – for≫ any F q < 1 – there are values of ϑ such that the width of the 2ℓ• µ 4qf(q) loss region, ψout ψin, is less than ∆ψ. In terms of the 2 2 2 2 = f(q) 2 νx0 ℓx0 + νy0 ℓy0 = integral W defined− above (28), q becomes simply F π ℓx0ℓy0 Pνp π q for 0 q 1, (60a) Pν ≤ ≤ p √ 2 q = W. (56) −1 ℓ• 1 ℓ• νr 6ℓ• 2 2 2 2 = q νx0 ℓx0 + νy0 ℓy0 = 2 F 2πℓx0ℓy0 2π ℓx0ℓy0 νp Unlike the case of collisional loss cone refilling, where µ q = for q> 1. (60b) q = q(E) is only a function of energy, here q is also a Pνp function of a second integral W . Pyramid orbits with small opening angles will have small W and small q. The second expression for the loss rate, equation (60b), The area on the (ψ, ϑ) plane that is lost, in one radial can be called the “full-loss-cone” loss rate, since it corre- period, into one of the four loss regions is sponds to completely filling and empyting the loss regions in each radial step (Figure 9). Note that the loss rate for ϑc ℓ• q < 1 is q times the full-loss-cone loss rate. A similar 2 ∆ψdϑ +2 (ψout ψin)dϑ (57) relation holds∼ in the case of collisionally repopulated loss − Z0 Zϑc cones (Cohn & Kulsrud 1978). The inverse of the loss rate gives an estimate of where F the time tdrain required to drain an orbit, or equivalently 2 ϑc ℓ• 1 q (58) the time for a single star, of unknown initial phase, to ≡ − go into the BH. In this approximation, the loss rate re- is the value of ϑ where ∆ψp= ψ ψ ; for q 1, mains constant until t = t at which time the torus out − in ≥ drain 12 Merritt and Vasiliev focusing (see equation 7 of Merritt & Poon (2004)). The coefficient µ for each orbit is calculated as (ℓ2 < 2 P ℓ•). As seen from equation (48), the smaller the extent of a pyramid in any direction, the greater µ – this is true even for orbits with large ex0 or ey0. While µ varies greatly from orbit to orbit, its overall distribution over the entire ensemble of pyramid orbits follows a power law: ℓ2 (µ > Y ) (Y/µ )−2 , µ • ; (62) Pµ ≈ min min ≈ 2˜η µ is the probability of having µ greater than a certain Pvalue andη ˜ is the fraction of pyramids among all orbits (37). The average µ for all pyramid orbits is therefore Fig. 10.— 2 2 Proportion of phase space (ℓ <ℓmax(0)) that is occu- µ =2µmin, and the average fraction of time that a ran- pied by the major orbit families: LATs, pyramids, SATs, saucers, dom orbit of any type and any ℓ spends inside the loss and chaotic orbits, as a function of semimajor axis a based on 2 real-space integrations. Triaxiality parameters are ǫb = 0.5ǫc and cone is µη˜ ℓ• (almost independent of the potential pa- −1 ≃ ǫc = 0.12/[0.2+(a/rinfl) ] (equation 11), corresponding to a den- rameters ǫb and ǫc) – the same number that would result sity cusp with γ = 1 and ǫc = 0.1 at a = rinfl. For a & rinfl most from an isotropic distribution of orbits in a spherically- low-ℓ orbits are chaotic (Poon & Merritt 2001). symmetric potential. is completely empty. In reality the draining time will 6. always be longer than this, since after 1 precessional COMPARISON WITH REAL-SPACE INTEGRATIONS periods, some parts of the torus that are∼ entering the loss We tested the applicability of the orbit-averaged ap- regions will be empty and the loss rate will drop below proach by comparing the orbit-averaged equations of mo- equation (60). For ∆ψ ψin ψout, the downstream tion, equation (13b), with real-space integrations of or- density in Figure 8, integrated≥ over− radial phase, is eas- bits having the same initial conditions (and arbitrary ra- ily shown to be 1 q−1 1 ϑ2/ℓ2 times the upstream dial phases). The agreement was found to be fairly good − − • density while for ∆ψ < ψin ψout the downstream den- for orbits with semimajor axes a . 0.1rinfl: about 90% sity is zero. Integrated overp −ϑ, the downstream depletion of the orbits were found to belong to the same orbital factor becomes class, and the correspondence between values of µ and 2 π 1 ℓmin was also quite good for individual orbits. Averaged 1 1 q2(1 + q)+ sin−1 1 q2 (61) over the ensemble, the proportion of phase space occu- − 4q − − 2q − pied by the different orbital families, as well as the net p p for q 1 and 1 π/4q for q > 1; it is 0 for q = 0, flux of pyramids into the BH, is almost the same for the 0.215≤ for q = 1− and 1 for q . For small q, the two methods. However, at larger radii, the relative frac- ∼torus will become striated, containing→ ∞ strips of nearly- tion of pyramids and saucers decreases (Figure 6 (right), zero density interlaced with undepleted regions; the loss 10). Since the maximum possible angular momentum for rate will exhibit discontinous jumps whenever a depleted orbits with a given semimajor axis a grows faster than region encounters a new loss ellipse and the time to to- √GM•a, this means that the fraction of pyramid orbits tally empty the torus will depend in a complicated way among all (not just low-ℓ) orbits is even smaller. For orbits with semimajor axis a & rinfl the frequency of ra- on the frequency ratio νx/νy and on ℓ•. For large q, the loss rate will drop more smoothly with time, roughly as dial oscillation becomes comparable to the frequencies of 3 precession, and when these overlap, orbits tend to be- an exponential law with time constant tdrain. We postpone a more complete discussion∼ of loss cones come chaotic. (Weakly chaotic behavior starts earlier). in the triaxial geometry to a future paper. Here we make So low-ℓ orbits with a > 1.5rinfl are mostly chaotic, as a few remarks about pyramids with arbitrary opening seen from Figure 10, confirming that regular pyramid or- angles, i.e. for which e ,e are not required to be small. bits (along with saucers) exist only within BH sphere of x0 y0 4 For each orbit one can compute µ, the fraction of the influence (Poon & Merritt 2001). torus occupied by the loss cone (equation 48), by nu- merically integrating the equations of motion (13) and 7. EFFECTS OF GENERAL RELATIVITY analyzing the probability distribution for instantaneous In the previous sections we considered the BH as a values of ℓ2: (ℓ2 6π GM• ∆̟ = (63) c2 (1 e2)a regular LAT − 0.02 per radial period, with c the speed of light (Weinberg min max 1972), making the orbit-averaged precession frequency n2 chaotic LAT or pyramid Qmax = y0/6 3GM• ν = ν . (64) GR r c2 aℓ2 0.01 We can approximate the effects of this precession by crit adding an extra term to the orbit-averaged Hamiltonian (13): κ 3 2 H = ℓ + ǫ H + ǫ H (65a) 0 −2 b b c c − ℓ 0.05 0.1 0.15 2 Fig. 11.— νGRℓ 3GM• νr rSchw M• Illustrating the allowed variations in angular momen- κ tum ℓ for orbits in the presence of general relativistic precession. = 2 . (65b) ≡ νp c a νp ∼ a M(a) The solid (red) curve represents the function P (ℓ) (equation 67); the dashed (blue) parabola is the same function in the Newtonian 2 This is equivalent to adding the term κ/l to the equa- case (κ = 0). If κ =6 0, P (ℓ) has a minimum at ℓcrit (equation 66). tion of motion for ̟, i.e. to the right hand side of Orbits make excursions along the curve P (ℓ) in the range from a certain value Pmax to Pmax −Qmax (here Qmax is given for the case d̟/dτ = ∂H/∂ℓ. When ℓ = ℓcrit, where 5 2 of planar orbits of § 7.1 and equals 2 (ǫc − ǫb) ≡ νy0/6). If during κ 1/3 such an excursion ℓ does not cross ℓcrit, then the orbit resides on ℓcrit = , (66) one branch of P (ℓ), typically remaining regular. Otherwise it flips 3 to the other branch, reaching lower values of ℓmin (equation 79), becoming a (typically chaotic) LAT or pyramid orbit. the precession due to GR exactly cancels the preces- sion due to the spherical component of the distributed and ℓ = ℓ0. In the absence of GR, such an orbit would 5 mass. Since the angular momentum of a pyramid or- be a LAT for ℓ0 > 3 ǫb and a pyramid otherwise. bit approaches arbitrarily close to zero in the absence of The Hamiltonian and the equations of motion are GR, there will always come a time when its precession κ κ 2 is dominated by the effects of GR, no matter how small 5 3 2 3 2 νy0 2 κ ǫc H = ℓ0 + = ℓ + + cos ̟ , (69) the value of the dimensionless coefficient . 2 − 2 ℓ0 2 ℓ 6 We again restrict consideration to the simplified Hamil- ν2 κ tonian (15), valid for ǫ ,c 1, ℓ2 . ǫ , now with the ˙ y0 b c ℓ = sin 2̟ , ̟˙ = 3ℓ + 2 . (70) added term due to GR. This≪ Hamiltonian may be rewrit- − 6 − ℓ ten as The orbit in the course of its evolution may or may not 5 3 κ 5 5 attain ̟ = 0 (mod π). If it does, then the angle ̟ circu- ǫ H = ℓ2 + + ǫ e2 + (ǫ ǫ )e2 lates monotonically, with̟ ˙ = 0. In Figure 11, the con- 2 c − 2 ℓ 2 c x 2 c − b y 6 dition̟ ˙ = 0 corresponds to reaching the lowest point in P (ℓ)+ Q(e ,e ) , (67) the P (ℓ) curve, ℓ = ℓcrit. Whether this happens depends ≡ x y on the value of ℓ0: since the orbit starts from Q = 0 and where P and Q denote the expressions in the first and P = P (ℓ0), it can “descend” the P (ℓ) curve at most by second sets of square brackets. The minimum of P (ℓ) 2 Qmax = νy0/6. If this condition is consistent with reach- occurs at ℓ = ℓcrit: ing P (ℓcrit), the orbit will flip to the other branch of the 3 2 Pmin = 81κ /8. (68) P (ℓ) curve. The condition for this to happen is κ κ 2 The function Q can vary fromp 0 to some maximum value 3 2 3 2 νy0 2 2 ℓ0± + = ℓcrit + + ; (71) Qmax due to the limitation that ex + ey 1. 2 ℓ0± 2 ℓcrit 6 Two differences from the Newtonian case≤ are apparent. ℓ0+ and ℓ0− are the upper and lower positive roots of 1) For each value of (ex,ey) (and therefore Q), there are now two allowed values of ℓ. One of these is smaller than this equation. If ℓ0 > ℓ0+, the orbit behaves like a Newtonian LAT ℓcrit while the other is greater (Figure 11). 2) Both the minimum and maximum values of ℓ – both of (Figure 12, case a): it has̟ ˙ < 0 and ℓ>ℓcrit. If which correspond to the maximum value of P (Figure 11) ℓ0 <ℓ0−, the orbit is again a LAT, but now it precesses in the opposite direction (̟ ˙ > 0) due to the dominance – are attained when Q = 0, i.e. when ex = ey = 0. of GR, and ℓ never climbs above ℓcrit (Figure 12, case e). The maximum of Q corresponds to ℓ = ℓcrit. In the Newtonian case, the minimum of ℓ corresponds to the In these cases the condition ̟ = 0 gives the extremum maximum of Q. of ℓ, which is found from equation (69): 3 κ 3 κ ν2 7.1. Planar orbits ℓ2 + = ℓ2 + y0 . (72) 2 extr,L ℓ 2 0 ℓ − 6 We first consider orbits confined to the y z plane extr,L 0 − (ex = 0, hence Ω = π/2,ℓz = 0 throughout the evolu- This extremum appears to be a minimum (ℓextr,L < ℓ0) tion). Namely, we start an orbit from ̟ = π/2 (ey = 0) if ℓ0 >ℓ0+ and a maximum (ℓextr,L >ℓ0) if ℓ0 <ℓ0−. 14 Merritt and Vasiliev Pyramid orbits are those that reach ℓ = ℓcrit.̟ ˙ 0.1 changes sign exactly at ℓcrit, but the angular momentum continues to decrease beyond the point of turnaround, a reaching its minimum value only when ̟ returns again 0.08 to π/2, i.e the z-axis. The two semiperiods of oscillation b are not equal: the first (ℓ>ℓcrit and̟ ˙ < 0) is slower, the other is more abrupt (Figure 12, cases b, d). 5 In 0.06 d effect, the orbit is “reflected” by striking the GR angular momentum barrier. After the orbit precesses past the z axis in the oposite sense, the angular momentum be- 0.04 c gins to increase again, reaching its original value after the precession in ̟ has gone a full cycle and the orbit 0.02 e has returned to the z axis from the other side. If ℓ0 = ℓcrit, there is no oscillation at all – the GR t and extended mass precession balance each other exactly 0 5 10 15 20 25 (Figure 12, case c). For ℓ0 . ℓcrit the orbit precesses in the opposite sense to the Newtonian precession. w We can find the extreme values of ℓ by setting ℓ˙ = 0 in 6 equation (70). This occurs for ̟ = π/2, i.e. for Q = 0 or P (ℓ)= P (ℓ0). This gives 4 e ℓ0 b ℓ = 1+8ℓ3 /ℓ3 1 . (73) extr,P 2 crit 0 − d q 2 c If ℓ0 > ℓcrit, this root corresponds to the minimum ℓ, with ℓ0 the maximum value; in the opposite case they 3 t exchange places. For κ 3ℓ this additional root is 0 ≪ 0 5 10 15 20 25 3 κ 2ℓcrit 2 a ℓmin 2 = 2 . (74) ≈ ℓ0 3 ℓ0 –2 Thus the minimum angular momentum attained by a Fig. 12.— Planar y − z orbits, solutions of equation (69) in −2 −4 pyramid orbit in the presence of GR is approximately a potential with ǫc = 10 ,ǫb = ǫc/2, κ = 10 , started with proportional to κ. Note the counter-intuitive result that ̟ = π/2 and different ℓ0: (a) 0.104, (b) 0.103, (c) 0.0322, (d) 0.006, (e) 0.0058. The first two orbits lie close to the separatrix the pyramid orbit with the widest base (largest ℓ0) comes between LATs and pyramids, ℓ0+ = 0.10392 (71); the third is closest to the BH. the stationary orbit with ℓ0 = ℓcrit; and the last two lie near the Figure 13 shows the dependence of the maximum and separatrix between pyramids and GR-precession-dominated LATs, minimum values of ℓ on ℓ0 for the various orbit families. ℓ0− = 0.005845. Top panel shows the evolution of ℓ(τ), bottom panel shows ̟(τ). For pyramid orbits (b–d), the angle ̟ librates 7.2. around π/2, and ℓ crosses the critical value ℓcrit; tube orbits (a,e) Three-dimensional pyramids have ̟ monotonically circulating, and ℓ is always above or below In the case of pyramid orbits that are not restricted to ℓcrit. a principal plane, numerical solution of the equations of motion derived from the Hamiltonian (65a) are observed This ellipse defines the base of the “pyramid” (which to be generally chaotic, increasingly so as κ is increased now rather resembles a cone). As in the planar case, (Figure 14). This may be attributed to the “scattering” the maximum and minimum values of ℓ are attained not effect of the GR term κ/l in the Hamiltonian, which on the boundary of this ellipse (i.e. the corners in the Newtonian case), but at e = e = 0, where Q = 0 and causes the vector (ex,ey) to be deflected by an almost x y random angle whenever ℓ approaches zero. In the limit P attains its maximum. These values are given by the roots of the equation P (ℓ)= 5 ǫ H, or that the motion is fully chaotic, H remains the only inte- 2 c − gral of the motion. The following argument suggests that 3ℓ3 (5ǫ 2H)ℓ +6ℓ3 =0 . (77) the minimum value of the angular momentum attained − c − c in this case should be the same as in equation (73). The two positive roots of this cubic equation are given Suppose that the Hamiltonian (67) is the only integral by that remains. Then the vector (e ,e ) can lie anywhere x y 2 π inside an ellipse ℓ = 5ǫ 2H sin φ , (78) min,max 3 c − 6 ± 5 2 2 κ Q(ex,ey) ǫcex + (ǫc ǫb)ey Qmax , (75) 1p 9 ≡ 2 − ≤ φ = arccos 3/2 . 3 (5ǫc 2H) whose boundary is given by − The plus sign in the argument of the sine function gives 5 ℓ while the minus sign gives ℓ . These two values Qmax = ǫc H Pmin. (76) max min 2 − − are linked by a simple relation: 5 T. Alexander has suggested that these be called “windshield- ℓ 2 κ wiper orbits.” max κ 3 ℓmin = 1+8 ℓmax/3 1 2 , (79) 2 − ≈ 3 ℓmax p Orbital structure of triaxial nuclei 15 1 for a chaotic LAT: some orbits from this range do not at- 0– 0+ tain ℓ<ℓcrit because of the existence of another integral of motion besides H (that is, they are regular). GR precession LAT Pyramid a LAT Finally, we consider the character of the motion when d the precession is dominated by GR, as would be the case 10–1 very near the BH. This is equivalent to staying on the left branch of P (ℓ), with ℓmin,max ℓ0− <ℓcrit (71). In c ≪ crit this limit there is a second short time scale in addition e to the radial period, the time for GR precession. This situation is similar to the high-ℓ case ( 4.4), in the sense b § 10–2 that we can carry out a second averaging over ̟ and nw µ obtain the equations that describe the precession of an annulus due to the triaxial torques. The orbits in this case are again short- or long-axis tubes. The only differ- ence from 4.4 is that we have to add the term κ/ℓ to 0 the averaged§ Hamiltonian (41), but since ℓ is constant− in 10–3 10–3 10–2 10–1 1 this approximation, the equations of motion for ℓz, Ω do not change. These very-low-ℓ regular tube orbits can be Fig. 13.— Minimum and maximum values of ℓ for a series of easily captured by the BH, however their number is very orbits with initial conditions ℓ = ℓ0, ω = π/2, ℓz =0, Ω= π/2. −2 −4 small and we do not consider them when computing the Potential parameters are ǫc = 10 ,ǫb = ǫc/2, κ = 10 . The straight line is ℓ = ℓ0; dashed line is the extremum for pyramids, total capture rate. equation (73). These two curves intersect at ℓcrit (equation 66), We argue in 8 that the conservation of ℓ for orbits in where they exchange roles. For ℓ>ℓ0+ and ℓ<ℓ0− (equation 71) this limit can have§ important consequences for resonant the orbit is a tube, and the minimum (or maximum) is given by equation (72). Dotted grey line shows the leading frequency of relaxation. −2 ̟ oscillations, ν̟ × 10 ; for high-ℓ orbits ν̟ ≈ 3ℓ, for orbits 2 dominated by GR precession ν̟ ≈ 2κ/ℓ . Letters denote the 7.3. Capture of orbits by the BH in the case of GR position of orbits shown in Figure 12. The inclusion of general relativistic precession has the where the latter approximate equality holds for κ effect of limiting the maximum eccentricity achievable by 3 ≪ ℓmax. In the same approximation a pyramid orbit. However, if ℓmin ℓ•, an orbit can still come close enough to the BH to be≤ disrupted or captured. 2κ 5ǫ 2H 2 c Introducing the quantity w ℓmin/ℓ•, we can write ℓmin , ℓmax − . (80) ≡ ≈ 5ǫc 2H ≈ 3 − ℓmin 1 2κ 3 νr ℓ• Equation (73) for planar pyramids is a special case of w & , (82) ≡ ℓ• ≃ ℓ• 5ǫc 2H 5ǫc νp Θ this relation where ℓ0 = ℓmax. − The ellipse (75) serves as a “reflection boundary” for where we used (64, 65b) and set H = 0 as a lower limit trajectories that come below ℓ ℓcrit. If this happens, for pyramid orbits (orbits with the smallest H have the the vector (e ,e ) is observed to≈ be quickly “scattered” x y largest ℓmax and the smallest ℓmin). Comparison with by an almost random angle (Figure 14, right, denoted by equation (56), with W (15ǫ )2, shows that the red segments), similar to the rapid change in ̟ that ≤ c occurs in the planar case (Figure 12). Roughly speaking, 3π w q−1. (83) all pyramid orbits and some tube orbits (those that may ≈ Θ 6 attain ℓ ℓcrit) will be chaotic. The distinction≤ between pyramids and chaotic tubes is Roughly speaking, the condition that stars be captured in the radius of this ellipse: pyramids by definition have (w< 1) is equivalent to the statement that the loss cone 2 2 is full (q > 1). This is not a simple coincidence: a full a fixed sign of ez, or ex + ey < 1, which means that the 2 2 loss cone implies that for low-ℓ orbits the mean change ellipse (75) should not touch the circle ex +ey = 1. Hence of ℓ during one radial period ( ν0ℓ0 νp/νr) is of order pyramids have Q 5 (ǫ ǫ ), and ∼ ≤ 2 c − b ℓ•, while the condition w = 1 requires that for the lowest 5 5 allowable ℓ, the GR precession rate (64) is comparable to ǫb Pmin H ǫc Pmin. (81) the radial frequency. These two conditions are roughly 2 − ≤ ≤ 2 − equivalent. This condition is different from the one described in 4 We can express this necessary condition for capture § even in the case κ =0= Pmin, since now pyramids do in terms of more physically relevant quantities. Writing not coexist with LAT orbits. equation (14a) as The condition for LATs to be chaotic (i.e. to pass 5 νp 1 M(a) through ℓcrit) is Pmin + Qmax ǫc H. For LATs the , (84) ≤ 2 − ellipse (75) always intersects the unit circle, so this con- νr ≈ 2 M• 5 dition can be satisfied for Pmin H 2 ǫb Pmin. and approximating rinfl of equation (2) as However, this is a necessary− but not≤ sufficient≤ condition− GM• 6 r r (85) A small fraction of the “flipping” orbits, especially those that infl ≈ σ2 ≡ 0 oscillate near ℓcrit (close to the lowest point on the P (ℓ) curve of Figure 11), may retain regularity by virtue of being resonant. with σ the one-dimensional stellar velocity dispersion at 16 Merritt and Vasiliev ey ey ey ex ex ex Fig. 14.— Three pyramid orbits with the same initial conditions (ℓ = 0.05,ℓz = 0.02,̟ =Ω= π/2; ǫc = 0.01,ǫb = 0.005) and three values of the GR coefficient κ (equation 65b). Left: κ = 0 (regular); middle: κ = 10−6 (weakly chaotic); right: κ = 10−5 (strongly chaotic). The green ellipse marks the maximal extent of the (ex,ey) vector, equation (75), i.e. ℓ = ℓcrit, equation (66); red segments correspond to ℓ<ℓcrit, blue to ℓ>ℓcrit and to the nonrelativistic case. r = r , the condition w 1 becomes do not cancel exactly, and there is a residual torque that infl ≤ − produces a change in the angular momentum: 1 σ a γ 7/2 . 1. (86) L d Gm √Nm −1 ǫc√Θ c r0 √N = Lc 2πP (90) dt ≈ a M• The Milky Way BH constitutes one extreme of the BH mass distribution. Writing Θ 10 (solar-mass main (here m is the stellar mass, P = 2π/µr is the radial 2 −1≈ sequence stars), σ 10 km s , and γ =3/2 gives period, Lc √GM•a is the angular momentum of a cir- ≈ cular orbit≡ with radius a, and N is roughly the number 1/2 a −2 ǫc & 10 (87) of stars within a sphere of radius a). In a non-resonant r0 system this net torque changes the direction randomly af- −3 ter each radial period, but in the case of near-Keplerian e.g. for a =0.1r0 0.3 pc, ǫc & 10 is required for stars to be captured. This≈ is a reasonable degree of triaxiality motion, for example, orbits remain almost the same for for the Galactic center. many radial periods, so the change of angular momentum At the other mass extreme, we consider the galaxy produced by this torque continues in the same direction M87, for which σ 350 km s−1 and Θ 3. Setting for a much longer time, the so-called coherence time tcoh, γ =0.5, corresponding≈ to a low-density core,≈ we find until the orientation of either the test star’s orbit or the other stars’ orbits change significantly. If this decoher- 1/3 a −1 ence is due to precession of stars in their mean field, then ǫc & 10 . (88) r0 M• P t t ν−1 (91) A dimensionless triaxiality of order unity is reasonable coh ≈ M ≡ p ≈ m N for a giant elliptical galaxy. In 9 we estimate the loss rate using the expression where the relevant precession time is that for an orbit of (60b)§ for the full loss cone rate, with the modification average eccentricity. L that the fraction of time µ an orbit spends inside the The total change of during tcoh is loss cone is now given not by (48), but by Gm L (∆L) √N t c . (92) 2 2 coh coh ℓ• ℓmin ≈ a ≈ √N µ 2 − 2 . (89) ≈ ℓ0 ℓ On timescales longer than t the angular momentum − min coh experiences a random walk with step size (∆L) and This relation implies that the instantaneous value of ℓ2 coh 2 2 time step tcoh. The relaxation time is defined as the time is distributed uniformly in the range [ℓmin..ℓmax]. This required for an orbit to change its angular momentum by is a good approximation for chaotic pyramid orbits and L , and hence it is given by a reasonable (within a factor of few) approximation for c 2 chaotic LATs (and also for regular orbits). Lc M• In the next section we point out the importance of tRR,s tcoh P . (93) ≈ ∆L ≈ m the angular momentum limit for the rate of gravitational wave events due to inspiral of compact stellar remnants. The above argument describes “scalar” resonant relax- L 8. ation, in which both the magnitude and direction of CONNECTION WITH “RESONANT RELAXATION” can change. On longer timescales, precessing orbits fill Resonant relaxation (RR) is a phenomenon that arises annuli, which also exert mutual torques; however, since in stellar systems exhibiting certain regularities in the these torques are perpendicular to L, they may change motion (Rauch & Tremaine 1996; Hopman & Alexander only the direction, not the magnitude of L. This effect is 2006a). Due to the discreteness of the stellar distribu- dubbed vector resonant relaxation (VRR), and its coher- tion, torques acting on a test star from all other stars ence time is given by the time required for orbital planes Orbital structure of triaxial nuclei 17 to change. In a spherically symmetric system the only √N torques that drive RR with the torques due to the mechanism that changes orbital planes is the relaxation triaxial distortion. The justification is as follows: Dur- itself. 7 Hence for VRR the coherence time is given by ing the coherent RR phase, the gravitational potential setting dL/dt = Lc/tcoh in equation (90): from N orbit-averaged stars can be represented in terms | | of a multipole expansion. If the lowest-order nonspheri- M• P tcoh tΩ,VRR √NtM (94) cal terms in that expansion happen to coincide with the ≡ ≈ m √N ≈ potential generated by a uniform-density triaxial cluster, and the relaxation time, equation (93), becomes the behavior of orbits in the coherent RR regime would be identical to what was derived above for orbits in a tri- M• axial nucleus. We stress that this is a contrived model; in tRR,v tΩ,VRR P (95) ≈ ≈ m√N general, an expansion of the orbit-averaged potential of N stars will contain nonzero dipole, octupole etc. terms which is √N times shorter than the scalar relaxation that depend in some complicated way on radius. Nev- ∼ time. ertheless the comparison seems worth making since (as We begin by comparing RR timescales with timescales we argue below) there is one important feature of the for orbital change due to a triaxial background potential. motion that should depend only weakly on the details of Consider a star on a (regular) pyramid orbit confined to the potential decomposition. the x z plane. It experiences periodic changes of angular Equating the torques due to RR − momentum ℓ L/Lc with frequency . νx0νp (22) and ≡ Gm amplitude ℓx0 . νx0/3 (24). Hence, the typical rate of T √N (98) change of angular momentum is RR ≈ r dL 2 Nm −1 with those due to a triaxial cluster, Lc νx0νp/3 Lc 5ǫc 2πP . (96) dt ≈ ≈ M• GNm T ǫ (99) Comparison with (90) shows that the rate of change of triax ≈ r angular momentum due to unbalanced torques from the our ansatz becomes other stars (RR) is greater than the rate of regular pre- − √ ǫ N 1/2. (100) cession if ǫ N . 1. However, the coherence time for ≈ RR is a typical precession time of stars in the cluster, −1 As shown above ( 7), GR sets a lower limit to the νp , whereas pyramids change angular momentum on a angular momentum of§ a pyramid orbit (equation 74): √ −1 longer timescale (νp 15ǫ) . On the other hand, in the κ case of RR the angular momentum continues to change κ rSchw M• √ ℓmin 2 N; (101) in a random-walk manner on timescale longer than tcoh, ≈ ℓ0 ≈ ǫ ≈ a M(a) while in the case of precession in triaxial potential its variation is bounded. the third term comes from setting ℓ0 ℓmax √ǫ, the maximum value for a pyramid orbit,≈ while the≈ fourth Next we consider VRR, which corresponds to changes κ in orbital planes defined by the angles Ω and i = term uses our ansatz (100) and the definition (65b) of . Expressed in terms of eccentricity, arccos(ℓz/ℓ). The frequency of orbital plane precession in a triaxial potential, νΩ, is νp√ǫ for low-ℓ orbits 2 2 ∼ rSchw M• 1 (pyramids and saucers) and even lower for other orbits 1 emax . (102) (Figure 2). The corresponding timescale may be written − ≈ a m N(a) as There is another way to motivate this result that does M• P tΩ,triax & . (97) not depend on a detailed knowledge of the behavior of m N√ǫ pyramid orbits. If we require that the GR precession Comparison with the VRR timescale (95) shows that time: ℓ2 t /t . √Nǫ. For the Milky Way, these two −1 −1 RR,v Ω,triax νGR νp (103) timescales are roughly equal at a 0.5 pc (Figure 15). ≈ κ ∼ For sufficiently large N the regular precession due to tri- (eq. 64) be shorter than the time axial torques goes on faster than the relaxation, so the − coherence time for VRR is now defined by orbit preces- dℓ 1 ℓ ℓ ν−1 (104) sion, and the relaxation time itself becomes even longer. dt ≈ ǫ p On the other hand, for small enough N the VRR destroys orientation of orbital planes before they are substantially for torques to change ℓ by of order itself, then affected by triaxial torques. It seems that VRR in triax- κ r M• ial (or even axisymmetric) systems can be suppressed by ℓ . √Nκ Schw √N (105) regular orbit precession; we defer the detailed analysis of ǫ ≈ ≈ a M(a) relaxation for a future study. as above. In other words, when ℓ . ℓ , GR precession So far we have considered the torques arising under min RR as being independent of the torques due to the elon- is so rapid that the √N torques are unable to change the gated star cluster. Suppose instead that we identify the angular momentum significantly over one precessional period. 7 If the BH is spinning, precession due to the Lense-Thirring In order for this limiting angular momentum to be rele- effect also destroys coherence (Merritt et al. 2010). vant to RR, the timescale for changes in the background 18 Merritt and Vasiliev potential should be long compared with the time over 9. ESTIMATES FOR REAL GALAXIES which an orbit with ℓ ℓ appreciably changes its an- ≈ min In this section we estimate the fraction and lifetime gular momentum. As just shown, the latter timescale of pyramid orbits to be expected in the nuclei of real is galaxies. 2 −1 ℓ −1 κ −1 We restrict calculations to the case of “maximal triax- tGR νGR νp Nνp . (106) ≡ ≈ κ ≈ iality,” ǫb = ǫc/2, although we leave the amplitudes of ǫb,ǫc free parameters. We also limit the discussion to or- The former timescale is the coherence time for VRR, bits within the BH influence radius, r . rinfl, where our equation (94): analysis is valid and where orbits are typically regular8. M• P Beyond rinfl, centrophilic (mostly chaotic) orbits still t . (107) ∼ Ω ≈ m √ exist and could dominate, e.g., the rate of feeding of a N central BH (Merritt & Poon 2004). The first set of parameters is chosen to describe the The condition tΩ tGR is then ≫ center of the Milky Way. The BH mass is set to 6 M• P M• = 4 10 M⊙ (Ghez et al. 2008; Gillessen et al. κNν−1 (108) × m √ ≫ p 2009a,b). The density of the spherically symmetric stel- N 5 −3 −γ lar cusp is taken to be ρs =1.5 10 M⊙pc (r/1 pc) or (Sch¨odel et al. 2007), with γ =1×.5 (Sch¨odel et al. 2008); a M• the corresponding BH influence radius is r 3 pc.9 √N . (109) infl ≈ rSchw ≫ m The triaxial component of the potential is highly un- certain; one source would be the nuclear bar with den- −3 Applying this to the center of the Milky Way, the condi- sity ρt 150 M⊙pc (Rodriguez-Fernandez & Combes tion becomes 2008),≈ yielding a triaxiality coefficient at r = 1 pc of a −3 −2 N(a) 102 (110) ǫc 10 . We also considered a larger value, ǫc = 10 , mpc ≫ which≈ may be justified by some kind of asymmetry on p spatial scales closer to rinfl than the bar. In this model, which is likely to be satisfied beyond a few mpc from the precession time due to the spherical component of the SgrA∗. potential, 2π/(3νp) for a circular orbit, is independent of On timescales longer than t , the torques driv- 5 ∼ coh radius and equals 1.7 10 yr; the two-body relax- ing RR will change direction. This is roughly equiva- ation time is also constant∼ × (5 109 yr), and timescales lent in our simple model to changing the orientation of for scalar and vector resonant× relaxation are given by the triaxial ellipsoid, or to changing ℓ0 at fixed ℓ. Such equation (93), (95) with stellar mass m =1 M⊙. changes might induce an orbit to evolve to values of ℓ The second set of parameters is intended to describe lower than ℓmin, by advancing down the narrow “neck” the case of galaxies with more massive BHs, using the in the lower left portion of Figure 13. However such evo- so-called M• σ relation in the form lution would be disfavored, for two reasons: (1) it would − require a series of correlated changes in the background 4.86 8 σ potential, increasingly so as ℓ became small; (2) as ℓ de- M• 1.7 10 M⊙ (113) ≈ × 200 km s−1 creased and νGR increased, changes in the background potential would occur on timescales progressively longer (Ferrarese & Ford 2005). Combined with the definition than the GR precession time, and adiabatic invariance 2 of r GM•/σ , we get would tend to preserve ℓ ( 4.4). These predictions can infl ≈ in principle be tested via direct§ N-body integration of 0.59 M• small-N systems including post-Newtonian accelerations rinfl 13 pc . (114) ≈ 108 M⊙ (e.g Merritt et al. 2010). A lower limit to the angular momentum for orbits near a massive BH could have important implications for the The two-body relaxation time evaluated at rinfl (as- rate of gravitational wave events due to extreme-mass- suming a mean-square stellar mass m⋆ = 1 M⊙ and a ratio inspirals, or EMRIs (Hils & Bender 1995). The Coulomb logarithm ln Λ = 15) is critical eccentricity at which the orbital evolution of a σ 7.5 ⊙ 13 10M compact object begins to be dominated by gravi- t2br(rinfl) 2.1 10 yr (115a) tational wave emission is ≈ × 200 km s−1 1.54 − 2/3 4/3 12 M• −5 tr M• 9.6 10 yr (115b) 1 eEMRI 10 (111) ≈ × 108 M⊙ − ≈ 109yr 106M⊙ 8 Excepting for the effects of GR, which as noted above may in- with tr the relaxation time (e.g. Amaro-Seoane et al. troduce chaotic behavior even for orbit-averaged parameters. The chaos that sets in at r & rinfl (§ 6) arises from the coupling of the 2007, eq. (6)). By comparison, equation (102), after sub- orbit-averaged and radial motions. stitution of N( (the fraction of phase space occupied by the loss cone) T, yr 2 5 is given by equation (89) with ℓ ℓ•, ℓ ǫ min ≪ max ≈ 3 c 10 (equation 80): 10 2-body relaxation T drain 1 5π c2 –2 t = a5/2ǫ (a) (117) 9 e=10 drain 3/2 c 10 νp ≈ 3Θ (GM•) Scalar RR –3 F e=10 −3/2 5/2 9 ǫc(a) M• a 8 10 yr 8 . 10 Relativistic ≈ × Θ 10 M⊙ 1 pc precession –3 A more exact calculation of tdrain(a) for the Milky Way, 7 e=10 10 W based on numerical analysis of properties of orbits sam- precession e=10–2 pled from the entire phase space, is shown in Figure 15. Finally, we estimate the total capture rate for all pyra- 106 Vector RR mids inside rinfl, using tpyr tdrain(rinfl) as a typical Newtonian precession timescale and applying (113,≡ 114, 117): 5 10 r −0.025 bh 11 ǫc(rinfl) M• tpyr 6 10 yr . (118) ≈ × × Θ 108 M⊙ 104 Dynamical time The capture rate from pyramids is then 3 r, pc 1.025 10 –3 –2 1– ˙ ǫc(rinfl)M• −4 −1 M• 10 10 10 1 10 Mpyr 1.6 10 M⊙ yr Θ ≈ t ≈ × 108 M⊙ pyr Fig. 15.— Various timescales for Milky Way center. Green and (119) −5 −1 blue solid curves are the pyramid draining times (more exact cal- For the Milky Way we find 4 10 M⊙yr for ǫc = −2 −3 −3 −4 −1 ∼ × −2 culation than equation 117) for ǫc = 10 and 10 ; dotted green 10 and 10 M⊙yr for ǫc = 10 . and blue curves denote typical orbital plane precession timescales ∼ −1 2 This capture rate should be compared with that due to νΩ for low-ℓ orbits (ℓ ≈ ǫc(a)); Newtonian and relativistic pre- cession times are given for circular orbits; other timescales are two-body relaxation, which is estimated to be (Merritt marked on the plot. Vector resonant relaxation is suppressed when 2009) −1 tRR,v & νΩ (marked by conversion of line into dashed). −0.54 ˙ M• −6 −1 M• (Merritt et al. 2007). M2br 0.1 10 M⊙yr 8 ≈ t2br ≈ × 10 M⊙ We first estimate the radius rcrit that separates the empty (q < 1) and full loss cone regimes. As noted in (120) the previous section, GR precession prevents a pyramid Thus even for a Milky Way-sized galaxy, the capture orbit from reaching arbitrarily low angular momenta; the rate of pyramids could be comparable with or greater radius beyond which capture becomes possible is roughly than that due to two-body relaxation. For more massive r . Using equation (56) with W = (15ǫ )2 (the maxi- galaxies this inequality becomes even stronger. However, crit c this is only the initial capture rate – after t , all stars mum value for pyramids) and equations (14a), (44), the ∼ pyr condition q = 1 translates to on pyramid orbits would have been consumed, at least in the absence of other mechanisms for repopulating the 3 3−γ ′ 4π ρsr0 rcrit 2α 5πǫc(rcrit) small-ℓ parts of phase space (not necessarily ℓ . ℓ•, but 1= the much broader region ℓ . √ǫc from which draining is 3 γ M• r0 3(2 γ) Θr /r − − Schw crit effective). If we take r0 to be rinfl, and ρs and σ asp the density and In the most luminous galaxies, like M87, standard velocity dispersion at this radius, we obtain mechanisms for relaxation are expected to be ineffec- tive even over Gyr timescales and pyramid orbits once 2/7 r σ √Θ depleted are likely to stay depleted. Setting ǫc = 0.1, crit 9 0.5 . (116) Θ = 3 and M• = 4 10 M⊙ gives for M87 tpyr 5 rinfl ≈ c ǫc(r0)! ×8 ≈ Gyr and Mpyr 4 10 M⊙. This could be an effective mechanism for≈ creating× a low-density core at the centers The radius rcrit typically lies in the range (0.2 0.7)rinfl, weakly dependent on the parameters. Since regular− pyra- of giant elliptical galaxies. mid orbits exist only for a . rinfl (Figure 10), there is 10. CONCLUSIONS evidently a fairly narrow range of radii for which cap- ture of stars from pyramid orbits is possible.10 However We discussed the character of orbits within the radius pyramid-like, centrophilic can exist at much larger radii of influence rinfl of a supermassive BH at the center of (Poon & Merritt 2001). a triaxial star cluster. The motion can be described as Next we make a rough estimate of the pyramid drain- a perturbation of Keplerian motion; we derive the orbit- averaged equations and explore their solutions both ana- ing time at a > rcrit, using the expression (60b) for the flux in the full loss cone regime, = µ/(Pν ); µ lytically (when the triaxiality is small) and numerically. F p Orbits are found to be mainly regular in this region. 10 This is also roughly the radial range from which extreme- There exist three families of tube orbits; a fourth or- mass-ratio inspiral events are believed to originate; (e.g. Ivanov bital family, the pyramids, can be described as eccentric 2002). Keplerian ellipses that librate in two directions about 20 Merritt and Vasiliev the short axis of the triaxial figure. At the “corners” of rate due to two-body relaxation in the most luminous the pyramid, the angular momentum reaches zero, which galaxies, although in the absence of mechanisms for or- means that stars on these orbits can be captured by the bital repopulation, these high consumption rates would BH. We derive expressions for the rate at which stars on only be maintained until such a time as the pyramid or- pyramid orbits would be lost to the BH; there are many bits have been drained; however the latter time can be similarities with the more standard case of diffusional measured in billions of years. loss cone refilling, but also some important differences, due to the fact that the approach to the loss cone is deterministic for the pyramids, rather than statistical. The inclusion of general relativistic precession is shown We thank T. Alexander, S. Hughes, A. Rasskazov and to impose a lower bound on the angular momentum. We B. Kocsis for fruitful discussions. DM was supported by argue that a similar lower bound should apply to orbital grants AST-0807910 (NSF) and NNX07AH15G (NASA). evolution in the case that the torques are due to resonant EV acknowledges support from Russian Ministry of relaxation. The rate of consumption of stars from pyra- science and education (grants No.2009-1.1-126-056 and mid orbits is likely to be substantially greater than the P1336). REFERENCES Amaro-Seoane, P., Gair, J. R., Freitag, M., Miller, M. C., Merritt, D., & Valluri, M. 1999, AJ, 118, 1177 Mandel, I., Cutler, C. J., & Babak, S. 2007, Classical and Polyachenko, E. V., Polyachenko, V. L. & Shukhman, I. G. 2007, Quantum Gravity, 24, 113 MNRAS, 379, 573 Bartko, H., et al. 2010, ApJ, 708, 834 Poon, M. & Merritt, D. 2001, ApJ, 549, 192 Buchholz, R. M., Sch¨odel, R., & Eckart, A. 2009, A&A, 499, 483 Poon, M. Y., & Merritt, D. 2004, ApJ, 606, 774 Cappellari, M., et al. 2007, MNRAS, 379, 418 Rauch, K. & Tremaine, S. 1996, New Astron., 1, 149 Chandrasekhar, S. 1969, The Silliman Foundation Lectures, New Richstone, D. O. 1982, ApJ, 252, 496 Haven: Yale University Press, 1969 Rodriguez-Fernandez, N. J., Combes, F. 2008, A&A, 489, 115 Cohn, H.& Kulsrud, R. 1978, ApJ, 226, 1087 Sambhus, N. & Sridhar, S. 2000, ApJ, 542, 143 Do, T., Ghez, A. M., Morris, M. R., Lu, J. R., Matthews, K., Sanders, J. A., & Verhulst, F. 1985, Averaging methods in Yelda, S., & Larkin, J. 2009, ApJ, 703, 1323 nonlinear dynamical systems. Applied Mathematical Sciences, Eilon, E., Kupi, G. & Alexander, T. 2009, ApJ, 698, 641 Vol. 59. Springer-Verlag, New York - Berlin - Heidelberg - Erwin, P., & Sparke, L. S. 2002, AJ, 124, 65 Tokyo Ferrarese, L., & Ford, H. 2005, Space Science Reviews, 116, 523 Sch¨odel, R. et al. 2007, A&A, 469, 125 Franx, M., Illingworth, G., & de Zeeuw, T. 1991, ApJ, 383, 112 Sch¨odel, R., Merritt, D., Eckart, A. 2008, Journal of Physics: Ghez, A. et al., ApJ, 689, 1044 Conference Series, 131, 012044 Gillessen, S. et al. 2009a, ApJ, 692, 1075 Schwarzschild, M. 1979, ApJ, 232, 236 Gillessen, S. et al. 2009b, ApJ, 707, 114 Schwarzschild, M. 1982, ApJ, 263, 599 Goldstein, H., Poole, C., & Safko, J. 2002, Classical mechanics Seth, A. C., Blum, R. D., Bastian, N., Caldwell, N., & (3rd ed.) San Francisco: Addison-Wesley Debattista, V. P. 2008, ApJ, 687, 997 Hils, D., & Bender, P. L. 1995, ApJ, 445, L7 Shaw, M. A., Combes, F., Axon, D. J., & Wright, G. S. 1993, Hopman, C., & Alexander, T. 2006, ApJ, 645, 1152 A&A, 273, 31 Hopman, C., & Alexander, T. 2006b, ApJ, 645, L133 Sridhar, S. & Touma, J. 1997, MNRAS, 287, L1 Ivanov, P. B. 2002, MNRAS, 336, 373 Sridhar, S. & Touma, J. 1999, MNRAS, 303, 483 Ivanov, P. B, Polnarev, A. G. & Saha, P. 2005, MNRAS, 358, 1361 Statler, T. S., Emsellem, E., Peletier, R. F., & Bacon, R. 2004, Lees, J. F., & Schwarzschild, M. 1992, ApJ, 384, 491 MNRAS, 353, 1 Merritt, D. 2009, ApJ, 694, 959 Valluri, M., & Merritt, D. 1998, ApJ, 506, 686 Merritt, D., Alexander, T., Mikkola, S., & Will, C. M. 2010, Wang, J., & Merritt, D. 2004, ApJ, 600, 149 Phys. Rev. D, 81, 062002 Weinberg, S. 1972, Gravitation and Cosmology: Principles and Merritt, D., Mikkola, S., & Szell, A. 2007, ApJ, 671, 53 Applications of the General Theory of Relativity (Wiley) Merritt, D. & Milosavljevic, M. 2005, Living Rev. Rel., 8, 8 de Zeeuw, T. 1985, MNRAS, 216, 273 Merritt, D., & Poon, M. Y. 2004, ApJ, 606, 788