The influence of the hydrodynamic drag from an accretion torus on extreme mass-ratio inspirals

# The influence of the hydrodynamic drag from an accretion torus on extreme mass-ratio inspirals

Enrico Barausse SISSA, International School for Advanced Studies and INFN, Via Beirut 2, 34014 Trieste, Italy    Luciano Rezzolla Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Potsdam, Germany INFN, Sezione di Trieste, Via A. Valerio 2, I-34127 Trieste, Italy
August 21, 2019
###### Abstract

We have studied extreme mass-ratio inspirals (EMRIs) in spacetimes containing a rotating black hole and a non self-gravitating torus with a constant distribution of specific angular momentum. We have found that the dissipative effect of the hydrodynamic drag exerted by the torus on the satellite is much smaller than the corresponding one due to radiation reaction, for systems such as those generically expected in Active Galactic Nuclei and at distances from the central supermassive black hole (SMBH) which can be probed with LISA. However, given the uncertainty on the parameters of these systems, namely on the masses of the SMBH and of the torus, as well as on its size, there exist configurations in which the effect of the hydrodynamic drag on the orbital evolution can be comparable to the radiation-reaction one in phases of the inspiral which are detectable by LISA. This is the case, for instance, for a SMBH surrounded by a corotating torus of comparable mass and with radius of gravitational radii, or for a SMBH surrounded by a corotating torus with radius of gravitational radii. Should these conditions be met in astrophysical systems, EMRI-gravitational waves could provide a characteristic signature of the presence of the torus. In fact, while radiation reaction always increases the inclination of the orbit with respect to the equatorial plane (i.e., orbits evolve towards the equatorial retrograde configuration), the hydrodynamic drag from a torus corotating with the SMBH always decreases it (i.e., orbits evolve towards the equatorial prograde configuration). However, even when initially dominating over radiation reaction, the influence of the hydrodynamic drag decays very rapidly as the satellite moves into the very strong-field region of the SMBH (i.e., ), thus allowing one to use pure-Kerr templates for the last part of the inspiral. Although our results have been obtained for a specific class of tori, we argue that they will be qualitatively valid also for more generic distributions of the specific angular momentum.

###### pacs:
04.30.-w,04.70.-s,98.35.Jk,98.62.Js

## I Introduction

One of the most exciting prospects opened up by the scheduled launch of the space-based gravitational-wave detector LISA LISA will be the possibility of mapping accurately the spacetime of the supermassive black holes (SMBHs) which are believed to reside in the center of galaxies SMBH . Among the best candidate sources for this detector there are Extreme Mass Ratio Inspirals (EMRIs), i.e. stellar-mass black holes () or compact objects orbiting around the SMBH and slowly inspiraling due to the loss of energy and angular momentum via gravitational waves (radiation reaction). In order for the signal to fall within the sensitivity band of LISA, the SMBH must have a mass , i.e., the low end of the SMBH mass function.

It is currently expected that a number of such events ranging from tens to perhaps one thousand could be measured every year gair_event_rates , but since they will have small signal-to-noise ratios, their detection and subsequent parameter extraction will require the use of matched-filtering techniques. These basically consist of cross-correlating the incoming gravitational-wave signal with a bank of theoretical templates representing the expected signal as a function of the parameters of the source.

This will not only allow one to detect the source, but also to extract its properties. For instance, the accurate modelling of the motion of a satellite in a Kerr spacetime will allow one to measure the spin and the mass of the SMBH. Although producing these pure-Kerr templates has proved to be a formidable task, particularly because of the difficulty of treating rigorously the effect of radiation reaction (see Ref. Poisson_review for a detailed review), considerable effort has gone into trying to include the effects of a deviation from the Kerr geometry. These attempts are motivated by the fact that possible “exotic” alternatives to SMBHs have been proposed (e.g., boson stars monica , fermion balls fermion_balls and gravastars gravastars ), although the presence of these objects would require to modify radically the mechanism with which galaxies are expected to form. On the other hand, non-pure Kerr templates might allow one to really map the spacetimes of SMBHs and to test, experimentally, the Kerr solution.

Different approaches to this problem have been considered in the literature. EMRIs in a spacetime having arbitrary gravitational multipoles should be considered in order to maintain full generality ryan , but this method does not work very well in practice and would only apply to vacuum spacetimes. For this reason, alternative approaches have been proposed and range from EMRIs around non-rotating boson stars gair_boson , to EMRIs in bumpy black-hole spacetimes bumpyBH (i.e., spacetimes which are almost Schwarzschild and require naked singularities or exotic matter) or in quasi-Kerr spacetimes quasi_kerr (i.e., spacetimes, consisting of Kerr plus a small quadrupole moment).

Interestingly, none of these methods is suitable for taking into account the effect of the matter which is certainly present in galactic centers. SMBHs can indeed be surrounded by stellar disks (as in the case of the Galactic center stellar_disks ) or, as in the case of Active Galactic Nuclei (AGNs) agn , in which we are most interested, by accretion disks of gas and dust which can be even as massive as the SMBH Hure . While the gravitational attraction of a disk can have important effects on EMRIs if this disk is very massive and close to the SMBH BHTorus , an astrophysically realistic accretion disk can influence an EMRI only if the satellite crosses it, thus experiencing a “hydrodynamic” drag force.

This drag consists of two parts. The first one is due to the accretion of matter onto the satellite black hole (this was studied analytically by Bondi & Hoyle bondi and subsequently confirmed through numerical calculations petrich ; font1 ; font2 ). This transfers energy and momentum from the disk to the satellite, giving rise to a short-range interaction. The second one is instead due the gravitational deflection of the material which is not accreted, which is therefore far from the satellite, but which can nevertheless transfer momentum to it. This long-range interaction can also be thought of as arising from the gravitational pull of the satellite by its own gravitationally-induced wake (i.e., the density perturbations that the satellite excites, by gravitational interaction, in the medium), and is often referred to as “dynamical friction”. This effect was first studied in a collisionless medium by Chandrasekar chandra , but acts also for a satellite moving in a collisional fluid rephaeli ; petrich ; ostriker ; sanchez ; kim ; my_dyn_frict .

All of these studies have been been carried out within a Newtonian or pseudo-Newtonian description of gravity (with the partial exception of Ref. karas_subr0 , in which the orbits are Kerr geodesics, but the disk model and the hydrodynamic drag is not relativistic). In this paper, instead, we provide a first relativistic treatment for satellite black holes moving on generic orbits around a rotating SMBH surrounded by a thick disk (i.e., a torus). We consider the torus to have constant specific angular momentum and neglect its self-gravity (i.e., we consider the metric to be pure Kerr). Under these assumptions, an analytical solution exists for this system constant_l_disks1 ; constant_l_disks2 . This configuration can be proved to be marginally stable with respect to axisymmetric perturbations marginally_stable (i.e., if perturbed, such a torus can accrete onto the SMBH), and is expected to be a good approximation at least for the inner parts of the accretion flow constant_l_disks1 ; constant_l_disks2 .

We have found that for a system composed of a SMBH with mass and a torus with mass and outer radius , the effect of the hydrodynamic drag on the motion of the satellite black hole is much smaller than radiation reaction at those distances from the SMBH which can be probed with LISA (i.e., for ). Although these values for , and are plausible for AGNs, an overall uncertainty is still present and has motivated an investigation also for different masses and sizes of the torus. In this way we have found that the effect of the torus can be important in the early part of the inspiral and that it could leave an observable imprint in the gravitational waveforms detected by LISA, if the radius of the torus is decreased to or, even for and , if . In this latter case, in fact, LISA could detect an EMRI event at distances as large as from the SMBH, although the event needs to be sufficiently close to us because the amplitude of the gravitational-wave signal decreases as .

In addition, if non-negligible, the effect of the hydrodynamic drag would have a distinctive signature on the waveforms. Radiation reaction, in fact, always increases the inclination of the orbit with respect to the equatorial plane (i.e., orbits evolve towards the equatorial retrograde configuration) NHSO . The hydrodynamic drag from a torus corotating with the SMBH, on the other hand, always decreases this angle (i.e., orbits evolve towards the equatorial prograde configuration). Should such a behavior be observed in the data, it would provide a strong qualitative signature of the presence of the torus. However, it is important to point out that even for those configurations in which the hydrodynamic drag plays a major role, this is restricted to the initial part of the inspiral detectable by LISA, whereas its effect rapidly vanishes in the very strong-field region of the SMBH (i.e., ). As a result, the pure-Kerr templates would provide a faithful description of the last part of the inspiral even in these cases.

The rest of the paper is organized as follows. In Sec. II we review the equilibrium solutions that we used for the orbiting torus. In Sec. III.1 we present the equations governing the interaction between the satellite black hole and the torus, while in Sect. III.2 we apply the adiabatic approximation to the hydrodynamic drag. Results are then discussed in Sec. IV.1 for equatorial circular orbits and in Sec. IV.2 for generic (inclined and eccentric) orbits. Finally, the conclusions are drawn in Sec. V. Throughout this paper we use units in which .

## Ii Modelling the torus

The properties of non self-gravitating, stationary, axisymmetric and plane-symmetric toroidal fluid configurations in Kerr spacetimes are well-known in astrophysics but are less well known within the community working on EMRIs. Because of this, in this Section we briefly review the basic facts, referring the interested reader to Refs. constant_l_disks1 ; constant_l_disks2 ; font_daigne ; zanotti_etal:03 ; zanotti_etal:05 for additional information.

Let us consider a perfect fluid with 4-velocity , which is described by the stress-energy tensor

 Tμν =(ρ+p)uμfluiduνfluid+pgμν =ρ0huμfluiduνfluid+pgμν, (1)

where , , and are the pressure, rest-mass density, energy density and specific enthalpy of the fluid. In what follows we will model the fluid with a polytropic equation of state , where is the internal energy per unit rest-mass, and and are the polytropic constant and index, respectively. Because we are neglecting the self-gravity of the fluid, we can also consider as given by the Kerr metric in Boyer-Lindquist coordinates, which reads MTW

 ds2 = −(1−2MrΣ) dt2+ΣΔ dr2+Σ dθ2 (2) +(r2+a2+2Ma2rΣsin2θ)sin2θ dϕ2 −4MarΣsin2θ dt dϕ,

where

 Σ≡r2+a2cos2θ, Δ≡r2−2Mr+a2. (3)

The fluid is assumed to be in circular non-geodesic motion with 4-velocity

 ufluid=A(r,θ)[∂∂t+Ω(r,θ)∂∂ϕ] =U(r,θ)[−dt+ℓ(r,θ)dϕ], (4)

where the second equals sign underlines that the vector and the 1-form are each the dual of the other. Here, is the angular velocity, is called the redshift factor, is the energy per unit mass as measured at infinity and is the specific angular momentum as measured at infinity (i.e., the angular momentum per unit energy as measured at infinity). Note that is conserved for stationary axisymmetric flows, as can be easily shown using Euler’s equation. The specific angular momentum and the angular velocity are trivially related by

 Ω=−gtϕ+gttℓgϕϕ+gtϕℓ,ℓ=−gtϕ+gϕϕΩgtt+gtϕΩ, (5)

while the normalization condition gives

 U=√ϖ2gttℓ2+2gtϕℓ+gϕϕ, (6) A=√−1gtt+2gtϕΩ+gϕϕΩ2, (7) AU=11−Ωℓ, (8)

where . Note that in this paper we will always consider (torus rotating in the positive -direction), while we will allow the spin parameter of the black hole to be either positive (black hole corotating with the torus) or negative (black hole counter-rotating with respect to the torus).

To calculate the structure of the torus, we need to use Euler’s equation, which in its general form reads

 aμfluid=−(gμν+uμfluiduνfluid)∂νpp+ρ, (9)

where is the 4-acceleration of the fluid. In particular, if the pressure is assumed to depend only on and and if the equation of state is barotropic [i.e., if ]222This is of course the case for a polytropic equation of state, because ., from Eq. (9) one easily gets that the 4-acceleration can be expressed as the gradient of a scalar potential :

 afluidμ=∂μW,W(p)=−∫pdp′p′+ρ(p′). (10)

On the other hand, from the definition of 4-acceleration (), Eqs. (4), (7) and (8), and the Killing equation for and , one easily gets

 afluidμ=∂μW=−∂μpp+ρ=∂μlnU−Ω1−Ωℓ∂μℓ. (11)

In particular, taking the derivative of this equation, anti-symmetrizing and using the trivial fact that , we obtain that . This implies and thus that and have the same contour levels [i.e., ]. Using this fact, we can then write Eq. (11) in an integral form:

 W−Wout=−∫p0dp′p′+ρ(p′) =lnU−lnUout−∫ℓℓoutΩ(ℓ′)dℓ′1−Ω(ℓ′)ℓ′, (12)

where and are the potential and specific angular momentum at the outer edge of the torus.333Of course, and can be replaced by the values of and at the inner edge of the torus if this is present.

In the case of a torus with constant specific angular momentum [i.e., constant], Eq. (12) provides an analytical solution, because once has been fixed the integral on the right-hand side is zero and Eq. (6) gives an analytical expression for :

 W−Wout=−∫p0dp′p′+ρ(p′)=lnU−lnUout. (13)

Note that if one requires that when (i.e., for an equipotential surface closing at infinity), this equation gives : then corresponds to open equipotential surfaces, while corresponds to closed equipotential surfaces. Interestingly, the potential well can present a minimum and a saddle point. Because of the plane-symmetry, these points are located in the equatorial plane, thus corresponding to local extremes of , and mark two important positions: respectively, the center of the torus (i.e., the point where the density reaches its maximum) and its cusp (i.e., the mass-shedding point). Noticeably, these points are located at the radii where the specific angular momentum of the torus, , coincides with that of the geodesic circular equatorial orbit (the “Keplerian” orbit) corotating with the torus,

 ℓK(r,a)=r2−2a√Mr+a2(r−2M)√r/M+a. (14)

This immediately follows from the fact that at the extremes of the function one has , which leads, through Eq. (10), to (in other words, at the center and at the cusp the pressure gradients are zero and only gravitational forces act).

In this paper we will indeed consider constant- tori. A detailed classification of these models depending upon the values of and of can be found in Refs. constant_l_disks1 ; constant_l_disks2 ; font_daigne . Here we simply recall that in order to have a closed equipotential surface with a cusp, one needs to have a value of between the specific angular momenta and of the marginally stable and marginally bound equatorial geodesic (i.e., “Keplerian”) orbits corotating with the torus. This can be easily understood by noting, from Eq. (13), that the potential is simply the effective potential describing the equatorial motion of a test particle around a Kerr black hole. As such, and can be calculated easily using Eq. (14) and the formulas for the radii of the marginally stable and marginally bound circular equatorial orbits in Kerr rotating in the positive -direction (i.e., corotating with the torus):

 ℓms=ℓK(rms),ℓmb=ℓK(rmb), (15) rms/M=3+Z2−sign(~a)√(3−Z1)(3+Z1+2Z2), (16) rmb/M=2−~a+2√1−~a, (17) Z1=1+(1−~a2)1/3[(1+~a)1/3+(1−~a)1/3], (18) Z2=√3~a2+Z21, (19)

where .

In order to pick up a particular solution having both an inner and an outer radius, one needs also to choose a negative value for the “potential barrier” at the inner edge of the torus,

 ΔW=Win−Wcusp=Wout−Wcusp≤0. (20)

If , the inner radius of the torus is larger than the radius at which the cusp occurs (), while if the potential barrier reduces to zero, the torus exactly fills its outermost closed equipotential surface and . Note that because of the considerations that we have made above about the value of , for constant- tori we have (with only if ) and (with only if ). If instead , the fluid overflows the outermost closed equipotential surface and mass transfer is possible at the cusp: for a polytropic equation of state, the accretion rate can be shown to be .

The integral Euler equation for constant- tori [Eq. (13)] further simplifies if the equation of state is polytropic, because in this case

 ∫p0dp′p′+ρ(p′)=lnhhout, (21)

where is the specific enthalpy at the outer edge of the torus. Since for a polytropic equation of state the enthalpy is given by

 h=1+ΓΓ−1κρΓ−10, (22)

it is clear that (because at the outer edge of the torus), and Eqs. (13) and (21) give

 ρ0(r,θ)={Γ−1Γ[eWout−W(r,θ)−1]κ}1/(Γ−1). (23)

Once the rest-mass distribution is known, the total rest mass of the torus is given by

 Mt,0=∫ρ0√−gutd3x , (24)

where and is the coordinate 3-volume element, while the mass-energy reads

 Mt=∫(Trr+Tϕϕ+Tθθ−Ttt)√−gd3x= 2π∫ρ0>0(gϕϕ−gttℓ2gϕϕ+2gtϕℓ+gttℓ2ρ0h+2P) ×(r2+a2cos2θ)sinθ drdθ. (25)

Clearly, the smaller the ratio between the mass of the torus and that of the SMBH, the better the approximation of neglecting the self-gravity of the torus.

## Iii Modelling the orbital motion

This Section is dedicated to the discussion of the hydrodynamic drag on the satellite black hole. Although the two aspects are closely inter-related, we first discuss the equations governing the interaction between the satellite black hole and the torus and then describe their use in the calculation of the changes of the orbital parameters within the adiabatic approximation.

### iii.1 The hydrodynamic drag

As already mentioned in Sec. I, the hydrodynamic drag acting on the satellite black hole can be written as the sum of a short-range part, due to accretion, and a long-range part, due to the deflection of the matter which is not accreted or, equivalently, to the gravitational interaction of the satellite with the density perturbations gravitationally induced by its own presence:

 dpμsatdτ=dpμdτ∣∣accr+dpμdτ∣∣defl, (26)

where is the proper time of the satellite.

Accretion onto a moving black hole was studied analytically in a Newtonian framework by Bondi & Hoyle bondi , who found the rest-mass accretion rate to be

 dm0dτ=4πλm2ρ0(v2+v2s)3/2, (27)

where is the mass of the black hole, and are respectively the velocity of the black hole with respect to the fluid and the sound velocity, and is a dimensionless constant of the order of unity, which for a fluid with polytropic equation of state and polytropic index has the value BHWDNSbook

 λ=(12)(Γ+1)/[2(Γ−1)](5−3Γ4)−(5−3Γ)/[2(Γ−1)]. (28)

Subsequent numerical work petrich ; font1 ; font2 treated instead the problem of accretion in full General Relativity, and showed that Eq. (27), with given by Eq. (28), is correct provided that it is multiplied by a factor when and become relativistic (cf. Table 3 of Ref. font1 ). However, because a fit for this correction factor is, to the best of our knowledge, not yet available, and the published data is not sufficient for producing one, we use the Bondi accretion rate [Eqs. (27) and (28)], bearing in mind that it could slightly underestimate the drag at relativistic velocities and .444As we will see in section IV, and can become relativistic only for orbits counter-rotating with respect the torus and very close to the SMBH. For these orbits the dominant part of the hydrodynamic drag is the long-range one, and the relativistic correction factor to the Bondi accretion rate (which is roughly for these orbits, as can be seen comparing the middle panel of Fig. 1 with Table 3 of Ref. font1 ) does not change this conclusion. Once the accretion rate is known, the short-range part of the drag reads petrich

 dpμdτ∣∣accr=hdm0dτuμfluid, (29)

where we recall that is the specific enthalpy of the fluid. Note that this equation basically follows from the conservation of the total 4-momentum of the satellite and the fluid.

The long-range drag is instead more complicated. The gravitational interaction of a body with the density perturbations that it excites gravitationally in the surrounding medium was first studied by Chandrasekhar chandra in the case of a collisionless fluid, and is also known as “dynamical friction”. Although less well recognized, dynamical friction acts also for a body moving in a collisional medium rephaeli ; petrich ; ostriker ; sanchez ; kim ; my_dyn_frict . In particular, a satellite moving on a circular planar orbit (e.g., a circular orbit around a Schwarzschild black hole or a circular equatorial orbit around a Kerr black hole) experiences a drag in the tangential direction rephaeli ; petrich ; ostriker ; sanchez and one in the radial direction kim :

where is a unit spacelike vector orthogonal to and pointing in the direction of the motion of the fluid,

 σ=ufluid−γusat√γ2−1 (31)

(the Lorentz factor encodes the relative motion of the satellite with respect to the fluid of the torus), and

 χ=−usatrusat−σrσ+∂/∂r[grr−(usatr)2/(γ2−1)]1/2, (32)

is a unit spacelike vector, orthogonal to both and and poiting in the radial direction. In particular, the tangential and radial drags are given by kim ; my_dyn_frict

where and are complicated integrals. Fits to the numerically-computed steady-state555Fortunately, the steady-state values for these integrals are reached over timescales which are comparable with either the sound crossing-time , being the radius of the circular orbit, or with the orbital period. values for these integrals are given in Ref. kim :

 Itang=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩0.7706ln(1+M1.0004−0.9185M)−1.4703M,for M<1.0,ln[330(r/rmin)(M−0.71)5.72M−9.58],for 1.0≤M<4.4,ln[(r/rmin)/(0.11M+1.65)],for M≥4.4, (35)

and

 Irad=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩M2 10 3.51M−4.22,for M<1.1,0.5 ln[9.33M2(M2−0.95)],for 1.1≤M<4.4,0.3 M2,for M≥4.4, (36)

where is the radius of the circular orbit, is the capture impact parameter of the satellite black-hole, while is the Mach number.

These fits are valid for and are accurate within 4% for and within 16% for . However, the fit for does not go to zero when goes to zero, while goes to zero only as in this limit: these behaviors would give a non-zero radial drag and a diverging tangential drag for [cf. Eqs. (33) and (34)]. This is clearly a spurious behavior: dynamical friction must vanish for , since in this case the pattern of the density perturbations is spherically symmetric around the body (as there is no preferred direction). However, as we will see in Sec. III.2, the effect of the radial drag vanishes if one uses the adiabatic approximation (as it is usually done in EMRI-studies drasco_scalar ; drasco_GW ; adiabatic ; mino ), and therefore this artifact of the fit (36) cannot cause any harm in our numerical code. This is instead not the case for the tangential drag: in order to eliminate its spurious divergence, we have approximated with its straight-line functional form at low Mach numbers. Since the dynamical friction drag for straight-line subsonic motion is given by Eq. (33) with , we can assume that is given, for , by

 Itang=0.9563(M33+M55), (37)

where the factor is introduced to match the above fit at .

Note that although Eq. (30) is strictly valid only for circular planar motion (i.e., in the case of a Kerr spacetime, for circular equatorial orbits), we expect it to be a good approximation also for generic orbits around a Kerr black hole. Indeed, thanks to the choice of the unit vectors and , Eq. (30) gives a tangential drag parallel the direction of the flow and a drag in the radial direction perpendicular to the direction of the flow. Both of these components are expected to be present also for generic orbits. In particular, the tangential drag should be given approximately by Eqs. (33) and (37) if the radius appearing in Eq. (35) is replaced by the semi-latus rectum of the orbit [see Eq. (III.2) for the definition of this quantity].666Note that the tangential drag given by Eqs. (33), (35) and (37) is approximately correct also for straight-line motion, if replaced in Eq. (35) by being the time for which the satellite has been active ostriker ; my_dyn_frict – as long as is smaller than the size of the medium, and by a cutoff-length of order of the size of the medium at later times. To see this, compare Eqs. (35) and (37) to the functional form of for straight-line motion, which is for subsonic motion and for supersonic motion ostriker ; my_dyn_frict . Although this prescription is not exact, the results of Ref. kim suggest that the relevant lengthscale in the Coulomb logarithm appearing in the second and third lines of eq. (35) should be one characterizing the orbit, rather than the size of the medium, as commonly assumed in most of the works on dynamical friction predating Refs. ostriker ; kim (see the introduction of Ref. kim and references therein for more details about this point). Of course, this lengthscale could be different from the semi-latus rectum of the orbit, but different choices for it would have only a slight impact on the results because of the logarithmic dependence.

The extrapolation of the radial drag given by Eqs. (34) and (36) from circular planar to generic orbits is instead a bit more problematic, although one expects it to be a good approximation at least for orbits with small eccentricities and small inclinations with respect to the equatorial plane. At any rate, as we have mentioned earlier, in Sec. III.2 we will show that the effect of this radial drag on the orbital evolution averages to zero when adopting the adiabatic approximation. (Note that this agrees with Ref. kim , which found that the effect of the radial drag on the orbital evolution was subdominant with respect to that of the tangential drag.) Nevertheless, a non-zero effect may still be present in cases in which the adiabatic approximation is not valid (i.e. if the hydrodynamic drag acts on a timescale comparable to the orbital period), or possibly even in the adiabatic approximation if more rigorous expressions for the radial drag should be derived in the future.

The rate of change of the mass of the satellite with respect to the coordinate time follows immediately from : denoting the derivative with respect to with an overdot, we have

 ˙m=−usatμutsatdpμsatdτ=−usatμutsatdpμaccrdτ=hγutsatdm0dτ. (38)

It is well-known carter that Kerr geodesics can be labeled, up to initial conditions, by three constants of motion, the dimensionless energy and the angular momentum as measured by an observer at infinity,

 ~E=−usatt,~Lz=usatϕ/M, (39)

and the dimensionless Carter constant carter ,

 ~Q=(usatθM)2+~a2cos2θ(1−~E2)+cot2θ~L2z (40)

where . We will now derive expression for the rates of change of these quantities.

To this purpose, let us first introduce the tetrad based in the position of the satellite and write the change in the 4-velocity due to accretion and deflection of the flow as

 δuμsat=δu(t)satuμsat+δu(i)sateμ(i), (41)

where and are the components with respect to the tetrad. In particular, perturbing to first order one easily gets , and using then the fact that to zeroth order, one obtains . Using now , and (), Eq. (41) becomes

Using now Eqs. (4), (39) and (42), we immediately obtain

In order to calculate instead the rate of change of the dimensionless Carter constant , let us note that from Eq. (42) it follows that the variation of in a short time interval due to accretion and deflection of the flow is

We can then write as the sum of a term coming from the gravitational evolution (i.e., the geodesic equation) and one coming from collisions with the surrounding gas:

The evolution of therefore follows from Eq. (40):

where the partial derivatives are meant to be calculated with Eq. (40). Note that the first and the second term of the first line cancel out because is conserved for geodesic motion.

A useful alternative form for the evolution rate of can be obtained by rewriting Eq. (40) using the normalization condition :

 ~Q=~Δ−1[~E(~r2+~a2)−~a~L]2−(~L−~a~E)2−~r2−~Δ(usatr)2, (48)

where and . Proceeding as above and in particular using the fact that

[from Eqs. (4) and (42)], one easily gets

where the partial derivatives are now calculated with Eq. (48). Note that for circular orbits Eq. (50) becomes

 ˙~Q=∂~Q∂~E˙~E+∂~Q∂~Lz˙~Lz (51)

[use Eq. (48) and the fact that for circular orbits]. This condition ensures777Note in particular that the proof presented in Ref. kennefick_ori , which was concerned mainly with radiation reaction, applies also to the case of the hydrodynamic drag. Note also that the resonance condition which was found in Ref. kennefick_ori as the only possible case that could give rise to a non-circular evolution for an initially circular orbit is never satisfied in a Kerr spacetime ryan2 . that circular orbits keep circular under the hydrodynamic drag and in the adiabatic approximation, as it happens for radiation reaction.

Finally, let us note that the rates of change of , and [Eqs. (43), (44), (47) and (50)] go smoothly to zero as the velocity of the satellite relative to the fluid goes to zero. This is easy to check using the fact that, when approaches zero, [cf. Eqs. (33) and (37)], , , , , and , and using the fact that keeps finite in this limit [in particular, from Eqs. (31) and (32) it follows , , and ]. Note that this is indeed the result that one would expect. First of all, a body comoving with the fluid clearly does not experience any dynamical friction and the only active mechanism is accretion. The body then accretes mass and consequently energy and angular momentum (because the fluid carries a specific energy and a specific angular momentum). However, the dimensionless constants of motion , and entering the geodesic equation cannot change because of the weak equivalence principle. Pictorially, one may think of a satellite comoving with a gaseous medium. Consider a sphere centered in the satellite, with radius small enough to ensure that the gas contained in the sphere has approximately the same velocity as the satellite. Suppose now that all the gas in this sphere is accreted by the satellite. The velocity of the satellite will clearly be unaffected, because of the conservation of momentum: for the weak equivalence principle this is enough to ensure that the orbit of the satellite will be unaffected, in spite of its increased mass.

At the heart of our approach is the calculation of the changes of the orbital parameters experienced by Kerr geodesics as a result of the hydrodynamic drag, and their comparison with the corresponding changes introduced by radiation reaction. To this purpose, let us recall that up to initial conditions Kerr geodesics can be labeled by a set of three parameters, the semi-latus rectum , the eccentricity and the inclination angle . These are just a remapping of the energy, angular momentum and Carter constant introduced in Sec. III.1, and are defined as schmidt

 p=2rarpra+rp,e=ra−rpra+rp, θinc=π2−Dθmin, (52)

where and are the apastron and periastron coordinate radii, is the minimum polar angle reached during the orbital motion and for orbits corotating with the SMBH whereas for orbits counter-rotating with respect to it. Note that in the weak-field limit and correspond exactly to the semi-latus rectum and eccentricity used to describe orbits in Newtonian gravity, and that goes from for equatorial orbits corotating with the black hole to degrees for equatorial orbits counter-rotating with respect to the black hole, passing through degrees for polar orbits.

In order to fix the initial conditions of a geodesic, let us first parametrize it with the Carter time , which is related to the proper time by carter

 dτdλ=Σ. (53)

This is a very useful choice because it makes the geodesic equation separable carter :

 (drdλ)2=Vr(r), dtdλ=Vt(r,θ), (dθdλ)2=Vθ(θ), dϕdλ=Vϕ(r,θ), (54)

with

 Vt(r,θ)/M2= ~E[(~r2+~a2)2~Δ−~a2sin2θ]+~a~Lz(1−~r2+~a2~Δ), Vr(r)/M4= [~E(~r2+~