The Astrophysics of Resonant Orbits in the Kerr Metric

2.2 MB·English

by  Jeandrew Brink

The Astrophysics of Resonant Orbits in the Kerr Metric
Jeandrew Brink,1,2 Marisa Geyer,3,4 and Tanja Hinderer5
1National Institute for Theoretical Physics (NITheP), Bag X1 Matieland, Stellenbosch, 7602, South Africa
2Department of Applied Mathematics, Stellenbosch University, 7602, South Africa
3Physics Department, Stellenbosch University, 7602, South Africa
4Department of Astrophysics, University of Oxford, UK
5Maryland Center for Fundamental Physics & Joint Space-Science Institute,
Department of Physics, University of Maryland, College Park, MD 20742, USA
(Dated: February 2, 2015)
ThispapergivesacompletecharacterizationofthelocationofresonantorbitsinaKerrspacetime
for all possible black hole spins and orbital parameter values. A resonant orbit in this work is
defined as a geodesic for which the longitudinal and radial orbital frequencies are commensurate.
5 Our analysis is based on expressing the resonance condition in its most transparent form using
1 Carlson’s elliptic integrals, which enable us to provide exact results together with a number of
0
conciseformulascharacterizingtheexplicitdependenceonthesystemparameters. Thelocations of
2
resonantorbitsidentifyregionswhereintriguingobservablephenomenacouldoccurinastrophysical

n situations where various sources of perturbation act on the binary system . Resonant effects may
a have observable implications for the in-spirals of compact objects into a super-massive black hole.
J During a generic in-spiral the slowly evolving orbital frequencies will pass through a series of low-

0 orderresonanceswheretheratiooforbitalfrequenciesisequaltotheratiooftwosmallintegers. At
3 these locations rapid changes in the orbital parameters could produce a measurable phase shift in
the emitted gravitational and electromagnetic radiation. Resonant orbits may also capture gas or

] larger objects leading to further observable characteristic electromagnetic emission. According to
c
theKAMtheorem,loworderresonantorbitsdemarcatetheregionswheretheonsetofgeodesicchaos
q
could occur when theKerrHamiltonian isperturbed. Perturbationsare induced for exampleif the
-
r spacetime of the central object is non-Kerr, if gravity is modified, if the orbiting particle has large
g
multipole moments, or if additional masses are nearby. We find that the 1/2 and 2/3 resonances
[
occur at approximately 4 and 5.4 Schwarzschild radii (Rs) from the black hole’s event horizon.
1 For compact object in-spirals into super-massive black holes ( 106M⊙) this region lies within
∼
v the sensitivity band of space-based gravitational wave detectors such as eLISA. When interpreted
8 within the context of the super-massive black hole at the galactic center, Sgr A*, this implies that
2 characteristic length scales of 41µas and 55µas and timescales of 50min and 79min respectively
7 should be associated with resonant effects if Sgr A* is non-spinning, while spin decreases these
7 values by up to 32% and 28%. These length-scales are potentially resolvable with radio VLBI
∼ ∼
0 measurementsusingtheEventHorizonTelescope. Wefindthatalllow-orderresonancesarelocalized
. to the strong field region. In particular, for distances r > 50R from the black hole, the order of
1 s
theresonancesissufficientlylargethatresonanteffectsofgenericperturbationsarenotexpectedto
0
lead to drastic changes in the dynamics. This fact guarantees the validity of using approximations
5
1 based on averaging to model the orbital trajectory and frequency evolution of a test object in this
: region. Observing orbital motion in the intermediate region 50Rs < r < 1000Rs is thus a “sweet
v spot” for systematically extracting the multipole moments of the central object by observing the
i
X orbit of a pulsar – since the object is close enough to be sensitive to the quadruple moment of the
central object but far enough away not tobe subjected to resonant effects.
r
a
PACSnumbers: 98.35.Jk,98.62.Js,97.60.Lf,04.20.Dw
I. INTRODUCTION during such a perturbation. Resonance phenomena are
ubiquitous in any multi-frequency system. In celestial
mechanicsthey stronglyinfluence satellite dynamicsand
Super-massiveblackholessuchasSgrA*atthecenter ring formation. Examples include the gaps in the aster-
of our galaxy are at zeroth order mathematically ide- oidbeltbetweenMarsandJupiter[1]andthegapsinthe
alized as Kerr black holes. In practice this description rings of Saturn [2–4]. Resonances are further intimately
is not complete due to a plethora of small perturbing ef- connected with dynamical chaos [5].
fectswhichslightlyalterthespacetimegeometry. Ingen-
eralthese perturbationsaresmallandwellaccountedfor As radio telescopes increase in sensitivity and collect-
with canonical perturbation theory. In the special case ingareawewillbeabletoresolvelength-scalestypicalof
that the perturbation excites one of the intrinsic reso- resonantphenomenainthespacetimeoftheblackholeat
nant structures of the spacetime’s orbits, the effect may the center of our galaxy. The Event Horizon Telescope
be larger than normally expected due to an anomalous is one such observational tool currently under develop-
transfer of energy and angular momentum that occurs ment [6]. Space based gravitational wave detectors such
2
as eLISA will be sensitive to in-spiral frequencies tra- ical resonance time and length-scales accessible to the
verse the resonant regime and may observe shifts in the larger astrophysics community by means of easily evalu-
phasing of the gravitational waves emitted during the ated formulas and tabulated results.
in-spiral of a compact object as it passes through the To explorethe resonanceeffects we describethe orbits
various resonant bands. X-ray, optical and infrared tele- in the Kerr metric using a set of variables adapted to
scopes do not have the resolving power to image Sgr A* the orbital geometry [28] that reduce to the Keplerian
directly, but can potentially record flux variations from orbital parameters in the Newtonian limit rather than
this region that may display timescales characteristic of the constants of motion associated with the spacetime’s
resonant events. Killing fields. The properties of the Keplerian constants
This paper investigates resonant orbits in the Kerr will be reviewed in Sec. IIIA. Plotting the location of
metric expanding on the discussion in [7]. The aim is resonancesintermsofthesevariablesimmediatelyallows
to provide a complete characterization of the parameter us to interpret the result as a physical location in the
space where resonant orbits occur as a function of black actual spacetime.
hole spin and the orbital parameters. Since geodesic or- Aresonantorbitoccursiftheratioofthecharacteristic
bitalmotioninKerriscompletelyintegrable,itisakinto radial, ω , and longitudinal, ω , frequencies is a rational
r θ
geodesic flow on a two-dimensionaltorus in phase space. fraction, ω /ω =n/m where n, m N. For a more ex-
r θ
∈
Generic orbits are ergodic and sample the entire surface tendeddiscussionoftheorbitalgeometryandfrequencies
ofthetorusafterasufficientlylongtime. Loworderreso- seeSec.IIIA.Mostofthe technicalaspectsofthis paper
nantorbitshoweveronlytraceoutasimple,co-dimension deal with how to efficiently examine this expression and
one, curve on the torus. Some of the features of reso- extractthe physics. Closedformanalyticexpressionsfor
nant orbital trajectories are illustrated in [8–12]. By the the frequencies in terms of elliptic integrals have been
Kolmogorov-Arnold-Moser(KAM)theoremwhichisdis- presented by [28, 29] which serve as companions to this
cussedinSec.II,loworderresonantorbitsaremostlikely work. Here, however, we opt in Sec. IV to take advan-
toexhibitthenonsmoothanomalousbehaviorassociated tage of a more symmetric representation of the elliptic
with a rapid change in the constants of motion and the functionsappearingintheresonanceconditionandwrite
breaking of the resonant torus. Test particles entering a them in terms of Carlson’s integrals [30–32]. This al-
low order resonance often display subsequent dynamics lowsustoidentifytheimportantparametersintheprob-
with a sensitive dependence on initial conditions. lem and exploit the identities associated with Carlson’s
To date a number ofauthorshavestudied resonantef- integrals to manipulate the expressions. In Sec. V we
fects in Kerr-like metrics in the context of various forms consider solutions to the resonance condition. We first
of perturbations. The effect of perturbations originat- specialize to the weak field limit where we introduce the
ing from adding a quadruple moment to the Kerrmetric key properties of a “resonant surface” in the parameter
has been quantified by exploring orbital motion in the space. Wethengiveanumberofexactanalyticsolutions
Manko-Novikov metric [13–17]. Perturbations from the to the resonance condition that can be used to describe
presence of a disk were considered in Ref. [18], and the resonances in the strong field region near the black hole.
effects ofthe smallmass’spinin[19–21]. Thefeaturesof Finally, several low order (small n+m value) resonant
traversingaresonanceduringanin-spiral,wheretheper- surfaces such as the 1/2, 2/3 and 3/4 are evaluated nu-
turbation arises from the small mass’ gravitational self- merically and compared to the analytic results and ap-
force, have been explored by [22, 23], and the possibility proximate formulae.
of sustained resonance has been considered in [24]. Res- The breakdown of integrability around a resonance in
onances involving one of the fundamental frequencies of “almost”-Kerr spacetimes is often quantified by numeri-
the motion on the torus and the orbit’s rotational fre- cally generating Poincar´emaps for a fixed energy E and
quency were studied in the context of enhanced gravi- angularmomentumcomponentL . Associatedwitheach
z
tational recoil [25, 26], and isofrequency orbits were dis- Poincar´eplot is a rotation curve which characterizes the
cussed in [27]. frequency ratio as a function of initial condition given
Most of these studies have focused on a particular or- a fixed E and Lz. In Sec. VII we give a representative
bital trajectoryor a small subsetof parametersin a spe- example of orbital breakdown around the 2/3 resonance
cificperturbedsetting. Theideaofthispaperistorefrain and analytically compute Kerr’s rotation curve. We fur-
fromspecializingtoaparticularperturbationandinstead ther provide expressions for finding the E and Lz values
provide insights that apply to all types of resonant be- associated with a particular resonance.
havior. We willuse tools suchasthe results ofthe KAM The exact nature of a perturbed system’s response in
theoremthatholdtrueregardlessofthesourceofpertur- the region of a resonance depends on the source of per-
bation. The results obtained here are thus robust in the turbation. In Sec.VIII we heuristically discuss how one
sensethatthe time andlength-scalesofresonanceeffects wouldestimate the size of a perturbationrequiredto see
for astrophysical applications are to be associated with a dramatic change in dynamics. It is important to note
properties of the underlying Kerr metric and resonance that the KAM theorem does not guarantee the break-
location rather than the details of the effect causing the down of integrability at any particular resonance. It
perturbation. The aim of this paper is to make the typ- merely states that if integrability breaksdown it will oc-
3
metric functions, the third is due to the conservation
of rest mass and the fourth integral is known as the
Carter constant [33]. Integrability implies that action-
angle variables can be defined. The phase space is foli-
ated by invariant level surfaces of the actions with the
compact dimensions of these surfaces diffeomorphic to a
torus. Geodesic motion in an integrable system is thus
θ1
akin to geodesic flow on a torus.
θ
2 ToillustratethisideafortheKerrmetric,considerthe
reduced Hamiltonian, which is constructed by replacing
ω the conjugatemomentaassociatedwiththe time andaz-
Δθ2 = 2πω21 imuthal symmetries by their constant values to obtain
an integrable two degree of freedom system with an ef-
FIG.1. Theorbitsinanintegrablesystemwithtwodegrees fective potential[16]. The main features of geodesic flow
of freedom can bevisualized as trajectories wrapping around for such a system are sketched in Fig. 1. The trajectory
atwodimensionaltorusinphasespacewithcharacteristicfre- on the torus is described by two characteristic frequen-
quenciesω1andω2,relatingtotheangularadvancesinθ1and cies,associatedwiththeanglesθ andθ ,labeledω and
1 2 1
θ2. Forrationalvaluesofω2/ω1 =m/ntheorbitaltrajectory ω , which in the Kerr metric correspond to the radial
will trace out a distinct path, wrapping n times around the 2
and longitudinal motions. For rational values of ω /ω
θ1 axis and m times about the θ2 axis. For irrational values 2 1
the orbit will sample only a finite regionof the torus be-
ofω2/ω1 ontheotherhandatrajectorywillfillthesurfaceof
fore retracing its own path, while for irrational values of
thetorus densely.
ω /ω atrajectorywillfillthetorusdensely. Orbitswith
2 1
rationalfrequencyratiosinvolvinglargeintegersarevery
cur first at the location of a low order resonant orbit ∗. similartoirrationalones,itisonlythosewithsmallinte-
Since this is true of all possible sources of perturbation, gerratiosthataresubstantiallydistinctfromtheergodic
the cumulative effect of many sources of perturbation case.
could result in a Saturn ringlike structure (see Fig. 16) When describing the astrophysical environment
being established around the black hole. This and other around a black hole such as Sgr A* we need to take into
potentially observable effects due to resonances are dis- account a number of corrections to the mathematically
cussed in Sec. IX. We focus in particular on the galactic idealized vacuum Kerr metric. In this case we are inter-
center, Sgr A*, as a possible observational realization of ested in the Hamiltonian
an extreme mass ratio in-spiral (EMRI). We note which
= +ǫ , (1)
detectorswill be sensitive to chaoticorbitsas wellasthe H HK H1
implicationsofregionswerewecanguaranteetheabsence
where is the perturbing Hamiltonian and ǫ is a di-
of low-order resonances and in which we expect orbits H1
mensionlessparametercharacterizingthe strengthofthe
to be approximately integrable. Regions that only con-
perturbation. contains information about a possi-
tain high order resonances we consider to be the “sweet H1
ble accretion disk [34–36], other sources of matter [37]
spot” for observationally determining the higher order
or dark matter [38, 39], structural deviations of the cen-
multipole moments of the super massive black hole in
tral black hole away from the Kerr metric (i.e. Bumpy
the Milky way.
Blackholeeffects[40–44]),theinfluenceofmodifiedgrav-
ity [45, 46], or effects of the multipoles ofthe smallmass
[19, 47, 48]. The exact nature of the perturbation does
II. KAM THEOREM AND IMPORTANCE OF
not concern us here. In what follows we simply assume
RESONANT ORBITS
thesemodificationstobesmallandrepresentthisbycon-
sidering the case where ǫ 1.
Bound geodesic motion in the Kerr spacetimes is in- ≪
To quantify the effect of an arbitrary perturbation on
tegrable [33] since the Hamiltonian = 1/2gµνp p ,
wheregµν is the inverseKerrmetric aHnKdp the tKestµpaνr- the orbital motion and to find the regions where the
K µ impact of the perturbation will be greatest, we make
ticle’s four momentum, admits a full set of isolating in-
use of the Kolmogorov–Arnold–Moser (KAM) theorem
tegrals. Two of these integrals result from the absence
[49, 50]. The KAM theorem investigates the stability of
of explicit time and azimuthal dependence in the Kerr
near-integrablesystemsandsuggeststhatatorusassoci-
atedwitharationalratiooffrequencieswillbedestroyed
inthepresenceofperturbations. However,providedthat
theperturbationissmallenough,toriforwhichtheratio
∗ Integrability could also break down at a homoclinic orbit, e.g.
ofassociatedcharacteristicfrequenciesaresufficientlyir-
thelaststableorbitdiscussedintheAppendixA.However,this
is of less observational interest than the resonances because it rational will remain stable and persist, although slightly
marksthetransitiontotheplunge,wherethenatureofthemo- deformed, in the perturbed Hamiltonian [51, 52]. More
tionchanges drastically. specifically, consider the vector of frequencies ω in the
4
unperturbedHamiltonianandavectorofintegersk,and
let d denote the dimension of these vectors. The condi-
tion for resonance is ω k = 0, which can generally be
·
satisfied to arbitraryaccuracy by choosing large integers
for k. When sufficiently large integers are necessary to
satisfytheresonanceconditionthetoriwillbepreserved, p p
where the definition of sufficiently is such that Arnold’s rp= 1 + e ra= 1- e
criterion holds [49, 50]
p
d −(d+1) 1 - e2
ω k >K(ǫ) k . (2)
i
| · | | |!
i=1
X
We will henceforth call O = d k the order of the
k i=1| i|
resonance. The factor K(ǫ) in Eq. (2) approaches zero
P
as the perturbation vanishes, i.e. lim K(ǫ) 0, but
ǫ→0 spin axis
→
its functional form depends on the nature of the pertur-
bation. In a non-integrable Hamiltonian system, when
ǫ . 1, Eq.(2) suggests a hierarchy of resonant orbits
of increasing order whose stability cannot be guaran-
teed. These are the low-order resonances 1, 1/2, 1/3,
*
2/3, 1/4, 1/5, 3/4, 2/5, 1/6. We expect these tori to
be destroyed first if the Hamiltonian is perturbed, how-
ever, from Eq.(2) we cannot guarantee their destruction
either. Changing the Kerr metric’s spin parameter is an
p
example of a Hamiltonian perturbation to an integrable rp= 1+ e
Hamiltonian for which none of the lower order resonant p
tori are broken. horizon ra=1- e
The destruction of resonant tori corresponds to the
physical idea that energy transfer takes place most FIG. 2. Top Keplerianorbital parameters. Theeccentricity, e, is
a measure of how ellipticthe orbit it is. When e=0, the orbit is
rapidlyifthefrequencyofthedrivingforcecoincideswith
circular e = 1 the trajectory becomes parabolic. The semi-latus
multiples of the internal frequencies of the system. Sim-
rectum,p,canbedefinedintermsoftheeccentricityandthesemi-
ilarly, even without a direct input of energy, if a system major axis of the ellipse as is show in green in the figure. The
is deformed the modes that could potentially be altered pointofclosestapproachrpiscalledtheperiastron,whilethemost
mostarethosewhosefrequenciesarerationallyrelatedto outlying point the orbit reaches is the apastron denoted by ra.
Bottom Theorbitaltrajectory asshowninthreedimensions. The
other modes and which thus have the greatest potential
third orbital parameter, namely the maximum inclination angle
to exchange energy and interact among themselves. ι=π/2−θ∗,istheanglewithrespecttotheblackhole’sequatorial
The study of torus destruction is not the subject of planeandθ∗ istheminimumBoyer-Lindquistθ valueattained.
this paper. We do howevergive a heuristic discussionon
how to estimate the size of the perturbation requiredfor
the onsetof stronglychaotic dynamics in Sec. VIII. The
detailed calculation will differ depending on the charac- ωφ. A schematic representationof a typical elliptic orbit
teristics of the perturbation. The main focus in the fol- andtheKeplerianvariablesusedtodescribeitisgivenin
lowing sections is to identify the regions in parameter Fig. 2. By contrast in the Kerr metric bound orbits are
and physical space where resonant dynamics are likely notrestrictedtoaplanebutareconfinedtoatoroidalre-
to occur. If they do occur the KAM theorem limits the gionwhoseshapeischaracterizedbytheconstantsofmo-
impact to low-order resonances. tion,theenergyE,thez componentofangularmomen-
−
tumL andCarterconstantQ. ForgeodesicsinKerr,the
z
rotationalfrequency ω describing the rotationalmotion
φ
III. GEODESIC MOTION IN THE KERR intheazimuthaldirectionisaugmentedbytwolibration-
METRIC type frequencies ωr and ωθ which characterizemotion in
the radial and longitudinal directions respectively. The
bottom panel in Fig. 2 gives a schematic representation
A. Physically motivated constants of motion
oftheoriginoftheω andω frequenciesassociatedwith
r θ
the orbit.
The orbital motion of a bound trajectory of two bod-
ies in Newtonian gravity is described completely by an In the subsequent sections we will explore the loca-
ellipse restricted to a plane. The manner in which this tion of the resonances for the ω and ω frequencies
r θ
ellipse is traversedis characterizedby a single frequency, in Boyer-Lindquist coordinates. Instead of using the
5
constants of motion E/µ, L /µ, Q/µ2 † we will de- generalized Keplerian variables are defined in terms of
z
{ }
scribe the orbits using variables analogous to the Keple- the roots of the potential functions as:
rianvariablesofclassicalcelestialmechanics,namelythe
p p
eccentricity (e), sine of the maximum inclination angle r1 = , r2 = , z− =cos(θ∗) (8)
1 e 1+e
(sinι=cosθ∗) and semi-latus rectum (p) as illustrated −
in Fig. 2. These are defined by writing the periastron When quantifying the resonance behavior in the subse-
or point of closest approach to the central object as quent sections we would like to express the results en-
rp =p/(1+e), the apastron or furthest point the tra- tirely in terms of the variables {p, e, z−} rather than
jectory reaches as r =p/(1 e) and the turning point using E, L , Q . The fact that the roots r , r , and
ofthelongitudinalmaotionasθ−. Thetypicalfrequencyof z2 can{notbezview}edasindependentfunctions3but4rather
∗ +
oscillationsbetweenr andr isdescribedbyω ,whereas mustbeinterpretedasfunctionsofthesetofindependent
a p r
the longitudinal oscillations about the equatorial plane, variables p, e, z− complicatesthecalculation. Bycom-
{ }
(π/2 θ ) ι π/2 θ , are described by ω . paringEqs. (5)and(7)wecanfindz2 isgivenexplicitly
∗ ∗ θ ±
− − ≤ ≤ −
in terms of E, L , Q as follows
z
{ }
B. Equations of motion z2 = (L2z +Q+β2)± (L2z +Q+β2)2−4Qβ2 .
± h p2β2 i
For a test mass in orbit around a Kerr black-hole the (9)
equations governing the radial and longitudinal motion,
Equating the coefficients of r in the two expressions
expressed in Boyer-Lindquist coordinates (t,r,θ,φ), are
for the radial equation, Eqs. (4) and (6) allows us to
[33]
obtain the following expressions relating the constants
dr 2 dz 2 {E, Lz, Q} to the roots of the factorization
=R(r), =Θ(z), (3)
dλ dλ E2 2(1 e2)
(cid:18) (cid:19) (cid:18) (cid:19) =1 − ,
µ2 − 2p+(1 e2)̟
+
where z = cos(θ) and we have chosen to parameterize −
the orbit in terms of a non-affine evolution parameter L2z = 2p(p+2̟+)−2a2 1−e2
λ = dτ/(r2 +a2cos2θ), rather than the proper time µ2 2p+(1−e2)̟(cid:0)+ (cid:1)
τ, sothatthe radialandlongitudinalequationsdecouple 2 a2 1 e2 p2 ̟
(thisRin fact just corresponds to workingin the extended + − − ×,
a2(2p+(1 e2)̟ )
phase space). The radialand longitudinalpotentials can (cid:0) (cid:0) −(cid:1) (cid:1)+
respectively be expressed as Q 2p2̟×
= (10)
µ2 a2(2p+(1 e2)̟ )
+
R= (r2+a2)E aL 2 ∆ µ2r2+(L aE)2+Q , −
− z − z− where we have set ̟+ = r3 +r4 and ̟× = r3r4. In
(4)
(cid:2) (cid:3) (cid:2) (cid:3) addition the condition
Θ=Q(1 z2) (µ2 E2)a2(1 z2)+L2 z2. (5)
− − − − z 2aELz =a2+ 2 ̟×−2a2 (1−e2)−2p̟×
(cid:2) (cid:3) µ2 (2p+(1 e2)̟ )
where ∆ = r2 2Mr+a2 and a = S/M is the spin per (cid:0) (cid:1) − +
unit mass. (We−will henceforth use units where M =1.) p2(̟+ 2)+4p̟+
− − (11)
The R and Θ potentials are quartic polynomials of their (2p+(1 e2)̟ )
+
−
respective arguments and can equivalently be expressed
mustalsohold. SquaringEq. (11) andthen substituting
in factored form as
in the expressions for E2 and L2, Eq. (10) results in a
z
β2 quadratic equation for ̟ and ̟ in terms of p and e:
+ −
R= (r r )(r r )(r r )(r r ) (6)
−a2 − 1 − 2 − 3 − 4
Θ=β2(z2 z2)(z2 z2) (7) ̟+̟× p(1 e2)(p+4 a2)+p2(p 4) a2 1 e2 2
− − − + − − − − −
+̟2 hp+e2 1 2+2p̟ p a2 p+e2 (cid:0)1 +(cid:1) i
whereβ2 =(µ2 E2)a2. InEq. (6)welabeltherootsso × − × − −
that r1 r2 r−3 > r4 and in Eq. (7) so that z+ z−. 1̟2 a(cid:2)4 1 e2 2(cid:3) 2a2 1 (cid:2)(cid:0)e2 p(p(cid:1)+(cid:0)4)+(p 4(cid:1)(cid:3))2p2
Forboun≥dorb≥its,Eq. (3)dictatesthatther(λ)and≥z(λ) 4 + − − − −
functions describing the orbitalmotion oscillatebetween +p̟h a(cid:0)2 1 (cid:1)e2 (a2+(cid:0)p) a2(cid:1)p(p+4) (p 4)p2 i
+
− − − −
two of the roots of Eqs. (6) and (7) respectively. The +p2 a2(cid:2) (cid:0)p 2 =0(cid:1). ((cid:3)12)
−
If z2 (cid:0)=0 we(cid:1)can use Eqs. (9) and (10) to rewrite z2 as
− 6 +
† The rest mass of the probe, µ, is introduced here to ensure the z2 = p2̟× . (13)
constants ofmotionaredimensionless. + a4(1 e2)z2
− −
6
We will always treat the z2 =0 or Q=0 limit of orbits Substituting Eqs. (6) and (7) we find that this is equiv-
−
restricted to the equatorial plane separately. Using Eqs. alent to the condition
(9), (10) and (13) we can further show that
r2 dr
an =
2a2p̟+z−2 = a2 1−e2 z−2 −p2 a2z−2 −̟× . (14) Zr1 (r1−r)(r−r2)(r−r3)(r−r4)
Eq. (14) is a li(cid:0)nea(cid:0)r condi(cid:1)tion in ̟(cid:1)+(cid:0)and ̟× w(cid:1)hich, in p m z− dz . (17)
caonndju̟nct(ipo,ne,wzi2th)(a1n2d),tihmupslitchitelyrodoettserrmiannesd̟r+(inp,tee,rzm−s) − Z−z− (z2−z−2)(z2−z+2)
× − 3 4
q
of p,e,z . Substituting Eq. (14) into (12) eliminates
{ −} The subject of the rest of the paper is to characterize
̟ and yields a quadratic equation in ̟ . As a result
+ × the solutions to this equation. The strategyis to express
a closed form expressioncan easily be found for ̟ and
× both the radial and longitudinal integrals in their most
subsequently ̟ . We will not give the expressions here
+ symmetric form using Carlson’s integrals [30–32] . Carl-
and continue to work with the implicit quantities ̟
+ son’s integral of the first kind is defined to be
and ̟ , substituting their actual values only at the end
×
of the calculations. The two solutions that result from 1 ∞ dt
the quadratic equation canbe interpretedas test masses RF(α,β,γ)= . (18)
2 (t+α)(t+β)(t+γ)
thateitherco-rotateorcounter-rotatewithrespecttothe Z0
spin, a, of the black-hole. Orbits that co-rotate with the InAppendix Bwelistanpumberofidentities andrapidly
black-hole(L hasthesamesignasa)arecalledprograde
z converging approximation techniques that make Carl-
and those that counter-rotate (L has the opposite sign
z son’s integralsa valuable analytic tool for characterizing
toa)areretrograde orbits. Foragiven p,e,z thepro-
− the resonantsurfaces. UsingEq. (B2)ofAppendix B we
{ }
grade orbit’s angular momentum is higher than that of
can rewrite Eq. (17) as
the retrograde orbit. On the other hand prograde orbits
have lower orbital energy than their retrograde counter- anR (0,(r r )(r r ),(r r )(r r ))=
F 2 3 1 4 2 4 1 3
parts [28]. − − − −
mR (0,(z +z )2,(z z )2). (19)
One special set of orbital parameters is the case when − F − + −− +
the roots satisfy r = r , which corresponds to the in-
3 2 This expressioncan be further simplified using the iden-
nermost stable orbit (ISO) separating stable bound or-
tity (B8)to rewritethe righthandside andthe factthat
bits from those that plunge into the black-hole. For
the equations are homogeneous (B6) to absorb the con-
a given eccentricity and longitudinal parameter z the
− stant factor. We shall refer to the resulting equation,
semi-major axis satisfying this condition demarcates the
smallestvalue of p atwhich a stable bound orbitcanex-
R (0,(r r )(r r ),(r r )(r r ))=
F 2 3 1 4 2 4 1 3
ist. We shall explicitly solve for the ISO for all values − − − −
of a, e and z− in Appendix A and use it as a compara- RF(0,κa2(z+2 −z−2),κa2z+2), (20)
tive benchmark for the location of resonantorbits in the
as the resonance condition and explore its properties by
subsequent sections.
studying various limiting cases. In this expression we
have defined the parameter 0 < κ < 1 to indicate which
resonance we are considering,
IV. THE RESONANCE CONDITION
n2
Inthissectionwebegintocharacterizetheorbitswhich κ= . (21)
m2
will exhibit resonant behavior. We are interested in the
parametervaluesforwhichωr andωθ arecommensurate. In the subsequent section we will explore all the qual-
Givenrelativelyprime integersm andn weseek the sur- itative features of a resonance by examining an easily
face in the three dimensional parameter space spanned evaluated approximation to Eq. (20) for large p. We
by p, e and z− where then give a number of formulae valid in the region near
the blackhole in special cases.
mω =nω . (15)
r θ
Thisisequivalenttosayingthatthetimeittakesthelon-
V. SOLUTIONS TO THE RESONANCE
gitudinal motion to traverse exactly n times between its
CONDITION
turningpointsisequaltothetimeittakestheradialmo-
tion to traverse m times between its turning points. For
Kerrgeodesicsm nsincetheradialfrequencyisalways When seeking solutions of Eq. (20) it is convenient to
thesmallestofthe≥threefrequencies. Thistranslatesinto rewrite it in terms of a rapidly converging series. This
the following integral condition series allows us to identify the three important parame-
ters in the problem. The first sets the overall scale and
z− dz r1 dr a rough location of the resonance. The remaining two
m =n . (16)
Z−z− √Θ Zr2 √R are expansion parameters < 1 that determine the more
7
subtle structure of the resonance surface. We give ex- less than unity ensuring the convergence of the series in
plicit expressions for theses parameters in terms of the Eq. (24). The parameters δ /y and δ /y vanish when
1 1 2 2
variables introduced in Sec. IIIB. Next, we evaluate the e = 0 and z = 0 respectively. The special limiting
−
series in the large p limit to obtain a simple analytic case when both these conditions hold, allows us to find
modelwhichillustratestheimportantfeaturesofanyres- an exact analytic result that is valid in all regions of the
onance. We then turn to the astrophysically more inter- spacetime. WeshallexaminethisspecialcaseinSec.VC.
esting strongfield regionwhere the low-orderresonances Howeverbefore we do so,itis instructive to examine the
occur and give a number of exact analytic formulas for propertiesofresonancesthatoccuratlargepvalues. The
special cases. We conclude the section by numerically featuresweexploreinthislimitqualitativelycapturethe
computing the detailed behavior of the 2/3 resonance characteristics of resonances in general.
and compare our analytic results and approximations to
the numerical solutions.
B. Anatomy of a resonance in the weak field limit
p
→∞
A. General series expansion
In the weak field limit, when p , the dominant
→ ∞
The resonance condition (20) can be rewritten in the terms in the expansion of ̟× and ̟+ found by solving
form Eqs.(14) and (12) are,
RF(0,y1+δ1,y1 δ1)=RF(0,κ(y2+δ2),κ(y2 δ2)). 4 8a (1 z2) 1
− − (22) ̟× =a2z−2 1+ p ± pp3/−2 1 !+O(cid:18)p2(cid:19),
where 8+2a2(1 z2)
̟ =2+ − −
+
p2 p(r +r ) ep(r r ) p
3 4 4 3
y = − +r r , δ = − ,
1 1 e2 3 4 1 1 e2 1 z2 6 1
− − 4a − − 1+ +O . (26)
y2 = a22(2z+2 −z−2), δ2 =−a22z−2 . (23) ± s p (cid:18) p(cid:19) (cid:18)p2(cid:19)
Substituting Eq. (26) into Eq. (25) and simplifying the
It will be shown below Eq. (25) that δ /y 1 for all
i i ≪ result gives the dominant behavior of the three essential
physically interesting parameters. As a result, each side
parameters in the resonance condition
ofEq. (22)canbeexpandedinδ y ,usingtherapidly
i i
≪
converging seriesof Eq. (B10). Squaring the resulting
12a 1 z2
expansions, moving all the terms containing the small y1 6 − −
=1
parametersδi/yitooneside,andre-expandingtheresult, y2 − p ∓ qp3/2
we obtain the equation 3a2 e2 5 z2 +4 1
− − +O , (27)
y1 =1+ 3 δ12 δ22 9 δ12δ22 − (cid:0)(cid:0) 2p2(cid:1) (cid:1) (cid:18)p5/2(cid:19)
κy 8 y2 − y2 − 64y2y2
2 (cid:18) 1 2(cid:19) 1 2
123 δ4 δ4 δ6
+ 512 y14 − y14 +O(y6). (24) δ1 = 2e 1−a2z−2 4ae z−2 −1 +O 1
(cid:18) 1 1(cid:19) y1 − q p ∓ p3/2sa2z−2 −1 (cid:18)p2(cid:19)
In terms of the variables introduced in Sec. IIIB, the
δ a2 1 e2 z2 1
threequantitiesthatentertheexpansionoftheresonance 2 = − − +O . (28)
y − 2p2 p3
condition (24) are, 2 (cid:0) (cid:1) (cid:18) (cid:19)
y1 = 2a2z−2 (1−e2)̟×+p2−p̟+ , Tanhde1s/mp2alrlesδp1e/cyt1ivaenlydinδd2i/cya2tinpgartahmatettehresysbceacloemweitahlm1o/spt
y a4(e2 1)z4 +2p2̟
2 (cid:0) − − × (cid:1) negligiblefor largep andreaffirmingthe choice ofδi as a
(cid:0)ep ̟2 4̟ (cid:1) expansion parameter.
δ1 = − +− × , Toillustratethe basicpropertiesofthe resonancecon-
y1 (1 e2)q̟×+p2 p̟+ ditionwesubstituteEqs. (27)and(28)intoEq. (24)and
− −
δ 1 e2 a4z4 keeptermsuptoO(p−2). Theresultingapproximateres-
y2 = (1 e2)−a4z4 2−p2̟ . (25) onance condition,
2 −(cid:0) −(cid:1)− ×
The firsttermy /y ultimately setsthe overallscaleofp 24a p(1 z2)=2p2(κ 1)+12(p+a2)+3e2κ
1 2 ± − − −
at which a particular resonance occurs. Recall that the
q 3a2z2 e2(κ 1)+5 ,
parameters z−, e, a, κ [0,1]. ByexaminingEq.(25) − − −
{ }∈ (29)
one can further verify that the δi/yi terms are always (cid:0) (cid:1)
8
FIG.3. Graphicalrepresentationoftheapproximateresonancecondition(29)orequivalently(32)plottedfortheκ=(9/10)2 resonance.
Theeccentricitydependenceisshowninthethreeplotsontheleft. Thetopleftplotdisplaysthetypicalarchshapeseenforallresonances.
Here the arch is centered around p∗ ≈ 31.6 reaching a maximum value of z− = 1 at p = ppolar given in Eq. (33) and indicated by a
dark line on the plot. The lines where arch intersect the z− = 0 plane are indicated in magenta (right) and cyan (left) and show the
maximum (retrograde) and minimum (prograde) values p attains for a fixed e. Note that the resonance shape for a fixed spin is very
weakly dependent on the eccentricity of the orbits. The spin dependence of the approximate resonance condition is shown in the three
plotsontherightfore=1/10ande=9/10. Observethatasa→0thearchpinchesofftoalineatp=ppolar,Eq. (33). Themaximum
archwidthoccursforamaximallyspinningblackholea=1aspredictedbyEq. (34).
is valid for large p values only. However, this weak field tricity is
approximation demonstrates all the qualitative proper-
ties ofresonantsurfacesand givesa goodapproximation p∗ p∗ 6
enveerninrelwahtiivcehlyEcql.os(e29to) tbhreeabklsacdko-whonlef.oTr hloewproercdiseermreasno-- p(a=0,e,z−,κ)= 2 1+s1+e2(cid:18) p∗−2 (cid:19)!. (31)
nances is numerically explored in Sec. VE. Sincee 1andp∗ >6weseethattheeffectofeccentric-
To build our intuition of the typical features of res- ≤
ity on the resonance location is small. We also observe
onant surfaces and their dependence on the parameters
that in the case a 0 the location of the resonance be-
a,e,z ,κandpweanalyzeEq. (29)indetail. Forquasi- →
− comes independent of z .
−
circular orbits and vanishing black-hole spins (a,e) 0,
We now examine the general spinning case. Squaring
→
the resonances occur at
both sides of Eq. (29) results in a polynomial condition
that is quartic in p and quadratic in z2. We choose
6 −
p∗ =p(a=0,e=0,z−,κ)= 1 κ. (30) to analyze the solution surface by specifying the z−2 =
− z−2(a,e,p,κ) for a fixed κ rather than explicitly working
Foragivenκ,thisvalueofp∗setsthegeneralmeanradius withthequarticrootsassociatedwithp. Theappropriate
in physical space (measured in units of GM/c2) about expression for z−2 is
which all the interesting features of a resonance occur.
TsporhlouivsteioiisnnaSeverceo.nbVuinsCtt.rhFeesourrletagifitohxnaetdnerianermtetaghienersmhao,nrrizeeosxonan,caatnscawensaewlywittiilhcl z−2 = 4a2a+2 e52−−66pee∗22p+∗4p2 − 1a22p(cid:16)52−pe∗26p+e∗21(cid:17)2
κ=[(m 1)/m]2 correspondtothemaximumresonance (cid:16) (cid:17) (cid:16) (cid:17)
radiusgi−venbyp=6m2/(2m 1). Resonanceswithn< p a2 e2+ 6e2(1−a2)+4p2 4p2 1 6e2
m 1occurataradiuslesstha−nthatassociatedwithn= + 8 v − p∗ − p∗ .
m− 1. The maximum p associated with a denominator a2uu h 5 6e2 3 i − 5(cid:16) 6e2 4(cid:17)
m−thus scales linearly with m for large values of m. uu − p∗ − p∗
t (cid:16) (cid:17) (cid:16) (cid:17) (32)
In the limiting case a 0 the dependence on eccen-
→
9
ThisfunctionisdepictedinFig.3forafixedspinparam- p = p (Eq. (33)) when a 0. This indicates that
polar
eterofa=9/10andintegerratioκ=(9/10)2. Ithasthe resonancesin the nonspinning l→imit become independent
shapeofaparabolicarchcenteredaroundp∗ 31.6. Fur- of inclination because the longitudinal frequency degen-
≈
thermore, the qualitative features that will be discussed eratestotheφ-frequencyinthiscase. Astheblack-hole’s
here are characteristic for all resonances. The function spinincreasesfromzerotheopeningwidthofthearchbe-
given in Eq. (32) has a maximum value of z2 =1 which tween the pro- and retrograde branches increases until a
−
occurs when p =p(a,e,z =1,κ) has the value maximum arch-width is attained at a = 1. The result
polar −
is a ′V′-shaped footprint of the arch in the p a plane,
p 1 (e2 a2) 6(1 a2)e2 withthe ′V′ profile’svertexcorrespondingtoa−=0. The
polar
p∗ = 2 1+s1+ p−∗ − −p∗2 !. inclinationdependenceoftheresonancesurfacescansim-
ply be characterized as the monotonic closing off of the
(33) ′V′ profile’spro-andretrogradebrancheswith increasing
Since e, a [0,1] and p∗ 8, the maximum only de- inclinationuntiltheymergeintoasinglelineformingthe
viates{bya}fe∈w percentfrom≫the p∗ valueasthespinand arch’s spine at ppolar.
This completes our discussion of resonances in the
eccentricity deviate from zero. The analytic value of the
weak field limit. The features described here and the
maximum given by Eq. (33) is plotted as a dark line in
Fig. 3. When z2 < 1 there are two possible values of p ′U′ ′ V′ ′ I′ transitions are characteristic of all reso-
− − −
nances. The actual values of the resonant surface of the
that lie on the resonance sheet for a given eccentricity:
true resonanceconditionbegin to deviate fromour weak
the resonance for a retrograde orbit p >p and the
− polar
field model as the black hole is approached. The largest
resonancefora progradeorbitwhichoccurscloserto the
deviation occurs in the equatorial limit, where the effect
black hole p < p . The sign in the naming conven-
+ polar
of spin is most marked. In the polar regions, the weak
tionof retrogradeand progradeorbits relatesto the sign
field resonance condition remains a remarkably accurate
ofthe productofthe angularmomentumandthe spinof
approximationtothetrueresonancesurface. InSec.VE
theblackhole(aL ),andnottheorbit’srelativeposition
z
we numerically characterizeseverallow orderresonances
with respect to p .
polar
and provide a quantitative comparison with the approx-
As z decreasesandtheresonancesurfacemovesfrom
−
imate results obtained in this section.
the polar towards the equatorial region, the influence of
spinbecomesincreasinglyimportantandthedistancep As we shall see next, in the strong field region it is
−
to p monotonically increases. The expression for p possibletoobtainexactanalyticresultsforthe′V′ equa-
+ −
andp caneasilybefoundinclosedformbysubstituting torial footprint for e = 0. Since the resonant surface
+
z =0intoEq. (29)andsolvingtheresultingquarticfor depends very weakly on e this result is a good indicator
−
p. However,since the results are messy and add little to for all resonant behavior.
thediscussionwedonotgivethegeneralresultsexplicitly
and merely plot these curves in Fig. 3. To benchmark
the size of the arch we consider the limit of vanishing C. Exact solutions to the resonance condition in
eccentricity and inclination and obtain special cases
p (a,e=0,z =0,κ) 1 4a 2a
± − In this section we explore easily evaluated exact solu-
= 1+ 1 .
p∗ 2 s ∓ √p∗ ∓ √p∗! tions to the resonance condition of Eq. (20) that can be
(34) usedtocharacterizetheresonantbehaviorneartheblack
hole. Thecasewewillconsiderfirstisthelimitofcircular
The maximum span of the arch occurs for a maxi- equatorialorbits,i.e. e 0andz 0. Asremarkedin
−
→ →
mally spinning black hole, a = 1. For lower spin val- Sec.VA this casesets the parametersδ /y =δ /y =0
1 1 2 2
ues a good approximation of the span of the arch is in Eq. (22) and thus a valid solution to the resonance
(p− p+) 4a√p∗(1+a2/p∗). The lowest order cor- condition is found when
− ≈
rection to Eq. (34) with respect to eccentricity is
y
e2(p∗ 6)/(4p∗2) and is the same for both pro and ret- 1 =κ. (35)
− y
rograde orbits. 2
Havingthus exploredthe basic featuresofa resonance
Note that in evaluating this case we will not be resort-
for a given spin parameter a and observed the weak de-
ing to Eq. (25) that was derived using Eq. (13) which
pendence of these features on eccentricity, we will now
assumed that z2 = 0. Instead we return to Eq. (9) and
choose a representative eccentricity and then study the − 6
observethatz =0ifandonlyifQ=0. By Eq.(10)we
spin dependence. The right-hand three panels of Fig. 3 −
see that Q = 0 implies ̟ = 0. The simplified version
show the κ = (9/10)2 resonance surface for eccentricity ×
of Eq. (10) is
values of e = 9/10 and 1/10 as a function of black hole
spin and p. As predicted by Eq. (34) the arch-width
E2 2 L2 a2+3p2
exhibits a strong spin dependence. The arch’s inverted =1 , z =4p 2 . (36)
′U′ profile pinches off to a single column ′I′ profile at µ2 − 2p+̟+ µ2 − 2p+̟+
10
1
0.8
0.6
a
0.4
0.2
0 1/3 1/2 3/5 2/3 5/7 3/4 4/5 5/6 6/7
5 10 15 20 25
p
FIG. 4. The location of orbits with resonant frequencies in the limiting case of e=0 and z− =0 as a function of spin, a and
semi-latus rectum, p [7]. Resonances are labeled at their vertex by the rational ratio n/m. All resonances with m 7 are
shown. For a=0 the prograde and retrograde branches are degenerate at p=p∗, as the spin increases the retrograde≤branch
leans right (copper tinge) and the prograde branch leans left (blue tinge). In general lower order resonances are colored more
darkly than their higher order counterparts. Note theaccumulation of resonances as thestrong field region is approached
Resonance Location (cid:2)GM/c2(cid:3) Spin splitting Period (cid:2)GM/c3(cid:3) Galactic center: Sgr A*
√κ=n/m p∗ =6/(1 κ) Max[(p1 p+)/p ] T =2πp∗3/2 p [µas] T [min] f [10−4Hz]
− − ∗ ∗
ISCO 6 1.33 92.3 30.6 32.7 5.10
1/2 8 1.22 142.1 40.9 50.3 3.31
1/3 27/4 = 6.8 1.29 110.2 34.5 39.0 4.27
2/3 54/5 = 10.8 1.10 223.0 55.2 78.9 2.11
1/4 32/5 = 6.4 1.31 101.7 32.7 36.0 4.63
3/4 96/7 = 13.7 1.00 319.1 70.1 112.9 1.48
1/5 25/4 = 6.3 1.32 98.2 31.9 34.7 4.80
2/5 50/7 = 7.1 1.27 119.9 36.5 42.4 3.93
3/5 75/8 = 9.4 1.16 180.4 47.9 63.8 2.61
4/5 50/3 = 16.7 0.92 427.5 85.1 151.3 1.10
1/6 216/35 = 6.2 1.32 96.3 31.5 34.1 4.89
5/6 216/11 = 19.6 0.86 546.7 100.3 193.5 0.86
1/7 49/8 = 6.1 1.33 95.2 31.3 33.7 4.94
2/7 98/15 = 6.5 1.30 104.9 33.4 37.1 4.49
3/7 147/20 = 7.4 1.26 125.2 37.5 44.3 3.76
4/7 98/11 = 8.9 1.18 167.1 45.5 59.1 2.82
5/7 49/4 = 12.3 1.05 269.4 62.6 95.3 1.75
6/7 294/13 = 22.6 0.80 675.8 115.5 239.1 0.70
TABLE I.Time and length-scales associated with low-order resonances depicted in Fig. 4 . This table gives thevalues for the
e=0, a=0, z− =0 vertices seen in Fig. 4, first in dimensionless units and subsequently in physical units for the special case
of the Galactic center, Sgr A*. Lower order resonances, shown in bold in this table, are most likely to have observationally
detectable dynamics.
Substituting Eq.(36)intoEq.(9)givesanexpressionfor Setting ̟ = 0 in Eq. (11) results in a quadratic equa-
×
z , tion for ̟ which has the following roots,
+ +
2p(a √p)2
p(p+2̟ )
z+2 = a2 + . (37) ̟+ = p(p 4) ±a2 4a√p. (38)
− − ∓