Towards relativistic orbit fitting of Galactic center stars and pulsars
aa r X i v : . [ a s t r o - ph . GA ] A ug Towards relativistic orbit fitting of Galactic center stars and pulsars
Raymond Ang´elil and Prasenjit Saha
Institute for Theoretical Physics, University of Z¨urich,Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland andDavid Merritt
Department of Physics and Center for Computational Relativity and Gravitation,Rochester Institute of Technology,Rochester, NY 14623
ABSTRACT
The S stars orbiting the Galactic center black hole reach speeds of up to a few percent the speed oflight during pericenter passage. This makes, for example, S2 at pericenter much more relativistic thanknown binary pulsars, and opens up new possibilities for testing general relativity. This paper developsa technique for fitting nearly-Keplerian orbits with perturbations from Schwarzschild curvature, framedragging, and the black hole spin-induced quadrupole moment, to redshift measurements distributedalong the orbit but concentrated around pericenter. Both orbital and light-path effects are taken intoaccount. It turns out that absolute calibration of rest-frame frequency is not required. Hence, if pulsarson orbits similar to the S stars are discovered, the technique described here can be applied withoutchange, allowing the much greater accuracies of pulsar timing to be taken advantage of. For example,pulse timing of 3 µ s over one hour amounts to an effective redshift precision of 30 cm s − , enough tomeasure frame dragging and the quadrupole moment from an S2-like orbit, provided problems like theNewtonian “foreground” due to other masses can be overcome. On the other hand, if stars with orbitalperiods of order a month are discovered, the same could be accomplished with stellar spectroscopy fromthe E-ELT at the level of 1 km s − .
1. Introduction
Tracing the orbits of the S stars in the Galactic center region reveals a central mass of ∼ × M ⊙ (Ghez et al.2008; Gillessen et al. 2009a), presumably a supermassive black hole. The roughly 20 known S stars are mostly main-sequence B stars, and it is likely that they share their environment with many fainter stars. Hence discovery bythe next generation of telescopes of stars far closer to the black hole is anticipated. The dynamics of the orbit andlight trajectories provide an opportunity to test hitherto unobserved predictions of general relativity in the blackhole neighborhood, and in doing so, confirm or revise our understanding of the nature of gravity (Gillessen et al.2010). Meanwhile, searches for pulsars in the Galactic-center region are also underway (e.g., Macquart et al. 2010).If pulsars on orbits similar to S stars are discovered, their time-of-arrival measurements may provide accuraciesorders of magnitude beyond those available from optical spectroscopy. 2 –It is interesting to compare S stars with other relativistic orbits. General relativistic perturbations of a nearlyKeplerian orbit typically scale as an inverse power of the classical angular momentum p a (1 − e ). So a simplemeasure of the strength of relativistic effects in a nearly Keplerian orbit is the dimensionless pericenter distance. ∗ Figure 1 plots pericenter distance against orbital period for a sample of S stars and binary pulsars, together withMercury and artificial satellites. At present, binary pulsars are the leading laboratories for general relativity. Theprinciples are summarized in Taylor (1994). Taylor’s Figure 11, depicting five intersecting curves, indicating fourdistinct tests of general relativity, is especially memorable, and descendants of this figure with newer data appear inrecent work (e.g., Figure 3 in Kramer & Wex 2009). So it is remarkable that the star S2 has a pericenter distance ∼ × , an order of magnitude lower than the most relativistic binary-pulsar systems, and four orders of magnitudelower than Mercury. This suggests that Galactic center orbits may show relativistic effects not measured in binarypulsars, such as frame dragging. On the other hand, S2 has an orbital period of ∼ . Time dilation at the source is a consequence of the equivalence principle, and is the strongest relativisticperturbation. Its measurability in the context of the Global Positioning System (GPS) is well known (Ashby2003). The manifestation of time dilation in pulsar timing is known as the Einstein time delay. For the Sstars, a spectroscopic detection of time dilation is expected around pericenter passage (Zucker et al. 2006).2.
Orbit perturbations are a test of space curvature, and at higher order, of black hole spin. • The leading order precession of orbital periapse is well measured in binary pulsars. (For a discussion ofhow pulsar timing effects scale to the Galactic-center region, see Pfahl & Loeb 2004, .) Astrometric mea-surement of precession in the orbit of S2 is considered feasible with current instruments (Eisenhauer et al.2009), but the long orbital period, and the likelihood of Newtonian precession due to the distributed mass,both present serious difficulties. If stars much further in are discovered, however, precession of orbitalplanes due to higher-order spin and quadrupole effects become accessible (Will 2008; Merritt et al. 2010). • Orbital decay due to gravitational radiation is not considered measurable because of the long time scalesand the extreme mass ratio between the star or pulsar and the supermassive black hole. • In contrast to the above secular effects, there are also GR induced velocity perturbations, which varyalong the orbit (Kannan & Saha 2009; Preto & Saha 2009). In binary pulsars, velocity perturbations ∗ In this paper the semi-major axis a and the pericenter distance a (1 − e ) are always expressed in units of GM/c and are thereforedimensionless. Light paths and travel times are affected by the black hole mass, and again at higher order, by spin. • Strong deflection of starlight (Bozza & Mancini 2009) or a pulsar beam (Wang et al. 2009b,a) would bea spectacular though rare event. • Small perturbations of photon trajectories, too small to measure astrometrically, can nevertheless producedetectable changes in the light travel time. In the pulsar literature this is known as the Shapiro delay.The analogous effect for S stars is a time-dependent redshift contribution (Ang´elil & Saha 2010), whichfor a star like S2 is comparable to the velocity perturbations. Hence, in the Galactic center, relativisticperturbations on both orbits and light paths must be considered.Later in this paper, we will group time dilation with the orbital effects.Motivated by the above considerations, in this paper we develop a method to fit a relativistic model to ob-servables. This strategy (i) takes relativistic perturbations on both the orbits and the light into account, (ii) treatsstellar spectroscopy and pulse timing in a unified way, and (iii) is well suited to analyzing data obtained from asingle orbital period or even less. We include the effects of time dilation, space curvature, frame dragging, andtorquing-like effects induced by the quadrupole moment from the black hole spin. Some further issues remain; mostimportantly, how to include Newtonian perturbations (from mass other than the black hole’s) in the fit — hencethe “towards” in the title.The observable we consider is the apparent frequency ν . As remarked above, this can refer to spectral lines orpulses. The source frequency ν is in principle known for stars, but unknown for pulsars. Note that c ln( ν /ν ) = c ln(1 + z ) ≃ cz (1)where z is redshift in the usual definition. For this paper, however, we will use “redshift” to mean c ln( ν /ν ). Thepossibly unknown source frequency now appears as a harmless additive constant in the redshift. For the S stars,the current spectroscopic accuracy is ≈
10 km s − under optimal conditions (Gillessen et al. 2009a). If a pulsar on asimilar orbit is discovered, the accuracy would likely be much higher. Taking, as an example, a one-hour observationwith pulses timed to 3 µ s (cf. Janssen et al. 2010) implies an accuracy of one part in 10 , equivalent to a redshiftaccuracy of 30 cm s − . The same level of accuracy is not inconceivable from spectroscopy of S stars, since planetsearches routinely achieve ∆ v < − (e.g. Lovis et al. 2006), but would require some technical breakthroughs toachieve for faint infrared sources like the S stars.For a given orbit and redshift accuracy, the principal quantities we will calculate are the signal-to-noise ratio S/N for different relativistic effects. For each of four relativistic effects — time dilation, space curvature, framedragging, and quadrupole moment effects — we will present the results in two ways: (i) the redshift accuracy neededto reach some
S/N for a given orbit, and (ii) the orbital parameters needed to reach some
S/N for a given redshiftaccuracy. 4 –
2. Model
The main calculations in this paper will assume that the trajectories of both S stars (or pulsars) and photonsare geodesic in a pure Kerr metric. Naturally, geodesics of the former are timelike, and the latter null. Even ifEinstein gravity is correct, the pure Kerr metric is naturally an approximation because this solution to the fieldequations neglects all mass in the vicinity of the black hole (more on this later), and the mass of the star itself. Ingeneral relativity, geodesic motion is fully described by the super Hamiltonian H = g µν p µ p ν , (2)with H = 0 for null geodesics.Rather than carry out all computations with the Hamiltonian resulting from the full Kerr metric, it is usefulto consider two different approximations for the cases of orbits and light paths. These we will call H star and H null .(Word labels in subscripts indicate orbits, in superscripts light paths.) We express these approximate Hamiltoniansperturbatively as H star = H static + ǫ H Kep + ǫ H Schw + ǫ H FD + ǫ H q , (3)and H null = H Mink + ǫ H Schw + ǫ (cid:0) H FD + H q (cid:1) , (4)where the meanings of the various component Hamiltonians will be explained shortly. Note that ǫ is just a label toindicate orders: if ǫ n appears, the term is of order v n (where v is the stellar velocity in light units), but numerically ǫ = 1. The series in (3) and (4) actually consist of the same terms, but because H star is to be applied only totrajectories that are strongly timelike, and H null only to compute null geodesics, the orders of same terms are oftendifferent.Expressions for all the Hamiltonian terms are derived perturbatively from the Kerr metric, using the basicmethod explained in Ang´elil & Saha (2010). We will not repeat the details here, but simply list the terms and theirphysical meanings.First we consider the orbit (Equation 3). • To leading order, the star feels no gravity and the clock simply ticks: H static = − p t . (5) • At next-to-leading order, gravity first appears, with H Kep = p r p θ r + p φ r sin θ − p t r , (6)which is the classical Hamiltonian modified by 1 /r → p t /r . This modification leaves the problem spatiallyunchanged, but introduces a temporal stretching causing a gravitational time dilation. This is originally aconsequence of the Einstein Equivalence Principle, and affects the redshift at Ø( v ). 5 – • Appearing next are the leading order Schwarzschild contributions, H Schw = − p t r − p r r , (7)of which the first term effects a further time dilation, and the second curves space, leading to orbital precession. • Next, the first spin term emerges with H FD = − sp t p φ r . (8)producing frame dragging, an r -dependent precession around the spin axis. Here s is the spin parameter,which points in the θ = 0 or + z direction. Maximal spin is s = 1. • The last set of orbital terms we will consider are H q = s sin θp r r − p φ r sin θ + cos θp t r − cos θp θ r ! , (9)which are the leading-order quadrupole moment terms. These result in the orbit being torqued towards thespin-plane, as well as other less intuitive effects.Then we consider light paths (Equation 4). • At leading order, we have H Mink = − p t p r p θ r + p φ r sin θ , (10)which is to say, the spacetime is Minkowski and the photon trajectories are straight lines. • The leading order Schwarzschild contribution H Schw = − p t r − p r r (11)(which notice is not identical to H Schw ) introduces lensing and time delays. • The frame-dragging term for photons H FD = − sp t p φ r (12)debuts at one higher order than H FD . Frame dragging lenses photons around the black hole in a twistedfashion. A φ -dependent time-dilation also occurs. • Finally, we consider higher order Schwarzschild terms, and quadrupole moment terms H q = − p t r + s sin θp r r − s cos θp θ r − s p φ r sin θ (13)which, as before, produce a torque towards the spin plane and as well as some more subtle effects.The well-known separability properties of the Kerr metric (cf. Chandrasekhar 1983) provide solutions up toquadratures, but not explicit solutions. Analytic solutions for geodesics and null geodesics are available for variousspecial cases, for example for the leading-order Schwarzschild case (D’Orazio & Saha 2010). But the present workuses numerical integration of Hamilton’s equations. 6 –
3. Calculating redshifts
Given a set of orbital parameters for the star, we calculate a redshift curve, meaning ln ( ν /ν ) against observertime or pulse arrival time, by the following method.First the orbit is calculated by numerically solving Hamilton’s equations for H star (Equation 3). If desired,particular orbital relativistic effects can be isolated by omitting other relativistic terms from H star . The independentvariable is, of course, not time but the affine parameter (say λ ). Along this orbit, from pairs of points separated by∆ λ , photons are sent to the observer. The difference in proper time between the emission of two photons is∆ τ = ∆ λ √− H star . (14)The photons themselves travel along paths determined by H null (Equation 4) with the additional condition H null =0. Again, one is free to isolate particular relativistic effects on the photons by discarding terms from H null . If ∆ t is the difference in the photons’ times of arrival at the observer, then ν ν = ∆ t ∆ τ . (15)This calculation is carried out along the orbital path. Some snapshots of this procedure are illustrated in Figure 2.Now, photons are emitted from the star in all directions, but we need to find exactly those photons which reachthe observer. Solving this boundary value problem is the computationally intensive part. Our algorithm for doingso is explained in Ang´elil & Saha (2010). In stronger fields, the redshift curve takes longer to compute than inweaker ones, both because the orbit integration needs smaller stepsizes over more relativistic regions, and becausethe boundary value problem requires more iterations before satisfactory convergence is reached.The observable redshift is of course a combination of all the relativistic effects atop the classical redshift. Onecan, however, estimate the strengths of each effect in isolation, by toggling each term in Equations (3) and (4) onand off, and then taking the difference in redshift. Figures 3 and 4 show relativistic redshift contributions measuredin this way. These orbits refer to the pericenters of a range of orbits, varying in a but with e and the other orbitalparameters fixed at the values of S2. These effects have simple scalings with the orbital period, readily deducedfrom the scaling properties of the Hamiltonians, and summarized in Table 1. Numerical results in Ang´elil & Saha(2010) verify that such scalings apply not only at pericenter but all along the orbit.The extended mass distribution in the Galactic center region, due to all the other stars, stellar remnants anddark matter particles that are present, introduces Newtonian perturbations. The distribution and normalization ofthis mass are poorly constrained on the scales of interest (Sch¨odel et al. 2009), although some upper limits exist(Gillessen et al. 2009b) and some numerical experiments have been done (Merritt et al. 2010). Assuming sphericalsymmetry for this distributed mass, models of the form ρ ( r ) ∝ r − γ (16)are sometimes adopted. Such a model ignores the torques due a nonspherical or discrete mass distribution, and theonly change it implies in the orbital dynamics is an additional (prograde) pericenter advance. Figure 5 comparesthe orbital and photon signals strengths for each effect, including an extended mass distribution of the type (16).Note however, that in the analysis which follows, we do not attempt to include the effects of the extended massdistribution. 7 –
4. Fitting
The computational demands for the present work are far greater than calculations of a few redshift curvesfor given parameters (as in Ang´elil & Saha 2010), because fitting redshift curves in a multidimensional parameterspace requires evaluations of up to tens of thousands of such curves. To this end, parallel functionality was addedto the implementation. The work is distributed evenly among the available processors. A workload is the chargeof calculating the redshift from a single position on the star’s orbit, i.e., a single evaluation of (15), calling for thefinding of two photons travelling from star to observer. The program is available as an online supplement.
The redshift curve of an S star or pulsar depends on eight essential parameters, but there can be any numberof additional parameters for secondary effects. In this paper we consider nine parameters in all, as follows. • The black hole mass M BH sets the overall time scale, and accordingly we express it as a time. Changing M BH simply stretches or shrinks the redshift curve in the horizontal direction. • The intrinsic frequency ν for pulsars is the pulse frequency in proper time, whereas for spectroscopy, it hasthe interpretation of absolute calibration. Altering ν simply shifts the redshift curve vertically. • Then there are the Keplerian elements, referring to the instantaneous pure Keplerian orbit with the initialcoordinates and momenta. Our orbit integrations all start at apocenter.1. The semi-major axis a (or equivalently the period P = 2 πa / M BH ) is single most important parameterdictating the strength of the relativistic signals.2. The eccentricity e sets how strongly peaked the redshift curve is at pericenter.3. The argument of pericenter ω sets the level of asymmetry of the redshift curve.4. The orbital inclination I changes the amplitude of the redshift. Classically, the inclination enter the red-shift depends only on M BH sin I (on M BH /M sin I for finite mass ratio). In relativity, the degeneracyis broken, because time dilation is independent of I .5. The epoch ǫ basically the zero of the observer’s clock.The longitude of the ascending node Ω does not appear, because, in our chosen coordinate system, Ω rotatesthe whole system about the line of sight, which has no effect on the redshift. Formally, we simply fix Ω = 0. • For the pulsar case, we include a simple spin-down model, with a constant spin-down rate ν → ν − ˙ ν ( t a ).Here ˙ ν is an additional parameter.We take the black hole spin as maximal, and to point perpendicular to the line of sight. Proper motion of theblack hole is not considered.To fit, we use a quasi-Newton limited-memory Broyden-Fletcher-Goldfarb-Shanno optimization routine withbounds (Zhu et al. 1997; Byrd et al. 1994). 8 – With an orbit fitting algorithm in hand, we now need to quantify the notion of detectability of particularrelativistic effects. One way would be to attach a coefficient to each term in the relativistic Hamiltonians and thensee how accurately that coefficient can be recovered. For this paper, however, we take a simpler approach. Since,for now, we only aim to identify which effects and which regimes are promising, we will simply attempt to estimatethe threshold for detecting the presence of each term in the Hamiltonians (3) and (4). Accordingly, we proceed asfollows.1. For some chosen parameters, we generate a redshift curve including all Hamiltonian terms up to some Ø( ǫ n ).These redshift curves are sampled at 200 points, with dense sampling near pericenter.2. We add Gaussian noise to the redshifts, at some chosen level.3. We then fit the parameters, with a redshift curve lacking a particular Hamiltonian contribution. If the noiselevel is too high, a reduced χ ≃ χ will be higher. We define S/N ≡ q χ (17)as the signal to noise.Naturally, it is necessary to check that a large χ is not the result of some algorithmic problem, by verifying thata good fit is obtained if the Hamiltonian term in question is consistently included. Figure 6 shows the
S/N of time dilation, space curvature, frame dragging, and quadrupole terms, all as afunction of redshift accuracy, assuming the orbit of S2. Time dilation is well above the current spectroscopicthreshold of ≈
10 km s − . Space curvature is somewhat below, while frame dragging and quadrupole are far below.The detectability thresholds are summarized in the first part of Table 2. In Figures 7, 8 and 9, the redshift accuracy is set to 10 km s − , − and 30 cm s − respectively, while theorbital period varies along the horizontal axis.Table 2 summarizes the detectability thresholds.
5. Summary and Outlook
This paper addresses the problem shown schematically in Figure 2, which is to infer orbital parameters anddetect relativistic effects from redshifts or pulsar timings along an orbit. The specific observable of interest is the 9 –ratio of rest-frame to observed frequency ν /ν and how it varies along an orbit, especially around pericenter. It isnot necessary to measure ν separately, it can be treated as a parameter to be inferred. As a result, spectroscopicredshifts and pulsar timings can be treated in a unified way. We argue that redshift variation over one or a few orbits,with special attention given to pericenter passage, provides a possible route to testing relativity using Galactic-centerstars, or (if discovered) pulsars on similar orbits. There are two reasons why a different strategy is called for herethan in the binary pulsar case. First, pericenter speeds of S stars are typically much higher than those of binarypulsars, and second, the orbital periods are too long for cumulative effects to build up.The observable redshift contains several different relativistic contributions affecting the orbit of the star orpulsar and the light traveling to the observer. For calculations, we use a four-dimensional perturbative Hamiltonianformulation derived from the Kerr metric. Each relativistic effect appears conveniently as a Hamiltonian term thatcan be toggled on and off to examine its import. Computation of redshift curves using this Hamiltonian approachwas demonstrated in a previous paper (Ang´elil & Saha 2010). In this paper we have developed a pipeline forsolving the inverse problem of inferring the orbital parameters from a redshift curve. We then compute the redshiftresolution needed to distinguish between redshift curves with and without each relativistic perturbation included,thus simulating the analysis of future observations. The redshift resolution needed to uncover each effect is a fewtimes finer than the maximum contribution of that effect.In our treatment, we have assumed that frequency data is the only observable at hand. The prospects fordetection would be improved if astrometric information were included in the fitting procedure. Information providedby astrometry is particularly potent in constraining the angular Keplerian elements, and would alleviate the burdenplaced on spectroscopy or pulse timing.A further way in which the detection prospects could be improved would be to obtain data from multipleorbits. The analysis of frequency data from stars or pulsars with periods < ∼
10 km s − accuracies, detecting time dilation appears comfortably feasible. Detecting space curvature appears feasible with amodest improvement in redshift resolution. On the other hand, the discovery of a pulsar on an orbit comparableto S2 could push the effective redshift accuracy to < − and, in principle, bring frame dragging and evenquadrupole effects within reach, as shown in Figure 9.This paper, however, assumes a source of negligible mass in pure Kerr spacetime. The finite mass of S2 (being < − M BH ) would perturb the redshift by a similar factor, via its effect on the motion of the supermassive blackhole. This would be much smaller than the space-curvature effect, but more than the frame-dragging contribution. Apotentially much more serious problem, however, is the Newtonian perturbations due to other mass in the Galacticcentre region. This Newtonian “foreground” is not necessarily fatal – the very specific time-dependence of therelativistic effects may enable them to be extracted from under a larger Newtonian perturbation (Merritt et al.2010) – but further research is needed to assess this.If sources inwards of S2 are discovered — something we may hope for from the E-ELT and the Square KilometreArray (SKA) — the prospects for relativity improve. Pfahl & Loeb (2004) argue that there may be 100 pulsars withorbital periods less than 10 years, although the number may be much smaller (e.g. Merritt 2009). A large fraction 10 –of these are expected to be found by the SKA. Not only do the relativistic effects get stronger as we move closer tothe black hole, but the Newtonian perturbations are likely to weaken. Figures 7 and 8 shows the orbital periods atwhich one would need to find stars or pulsars given redshift accuracies of 10 km s − and 1 km s − respectively. Ifstars with a period of less than one year are discovered, the prospects become very exciting indeed. Acknowlegements
We thank Antoine Klein and Daniel D’Orazio for discussion and comments. DM was supported by grantsAST-0807910 (NSF) and NNX07AH15G (NASA). 11 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
13 –Effect Orbit Light pathClassical P − / Time dilation P − / Schwarzschild P − P − Frame-dragging P − / P − / Quadrupole P − / P − / Table 1: Relativistic effects considered in this paper, and how they scale with the orbital period.Effect Required redshift accuracy Required orbital period (yr)for S2 for given redshift accuracy10 km s − −
30 cm s − Time dilation ∼
60 km s − ∼ ≫ P S ≫ P S Schwarzschild ∼ − ∼ ∼ ≫ P S Frame dragging ∼
10 m s − ∼ . ∼ . > P S Quadrupole ∼
50 cm s − ∼ . ∼ . ∼ P S Table 2: Summary of the observational thresholds at which different relativistic effects are exposed, assuming asource of negligible mass in a pure Kerr spacetime. 14 – -3 -2 -1 Period (years)10 P e r i c e n t e r d i s t a n c e ( G r a v i t a t i o n a l r a d ii ) Known S StarsGPS satelliteMercuryBinary PulsarsS2-like StarsS Stars, e=0S Stars, e=0.99
Fig. 1.— Pericenter distance (in units of
GM/c ) against orbital period for a variety of systems. The known S stars,having smaller pericenter distances, are more relativistic than known binary pulsars. But the long orbital periodsof the S stars render it infeasible to measure cumulative effects over many orbits. Hence other techniques must bedevised. The pulsar examples are taken from Lorimer (2008). For the S stars, the orbital elements in Gillessen et al.(2009a) have been used. In this paper we also treat fictitious stars that lie along the dashed line, that is, having arange of semi-major axis a values but with the same eccentricity and angular elements as S2. 15 – −50050100150200250 −100−50050100150200250050100150200250300 −50050100150200250 −100−50050100150200250050100150200250300−50050100150200250 −100−50050100150200250050100150200250300 −50050100150200250 −100−50050100150200250050100150200250300 Fig. 2.— Depiction of the system we deal with, shown at four slices of coordinate time. The curve ending in astar symbol traces a stellar orbit, starting from apocenter, through its second pericenter passage. The star emitsphotons in all directions at equal intervals of proper time. Dots show photons which will reach the observer (marked × ). Note that the dots represent a sequence of photons emitted at different places, hence joining the dots does notrepresent the path of any particular photon. Time dilation and Schwarzschild perturbations are included in thisexample, but not frame dragging or quadrupole. This means that the star’s orbit precesses, and the photons arelensed. In order to make the relativistic effects visually discernible, the star has a very low value of a = 100. Sucha star may, however, yet be discoverable by the E-ELT (Lyubenova & Kissler-Patig 2009). Also, the observer hasbeen placed at an unrealistically close distance of 300. 16 – -3 -2 -1 Period (years)10 -4 -3 -2 -1 R e d s h i f t c o n t r i b u t i o n ( k m / s ) (cid:0) to strong field to Kepler (cid:1) S2 Classical velocityTime dilation signalSchwarzschild signalFrame-dragging signalQuadrupole signal
Fig. 3.— Redshift contributions of different orbital effects. Shown here is the contribution of each term in Equation(3) at pericenter, with the orbital period being varied and the other orbital parameters fixed at the S2 values. Thepericenter distance varies from 3600 (approximately the value for S2) down to 6. Note that for the last two signals -the frame-dragging and the quadropole - we have taken the black hole spin to be maximal and to point perpendicularto the line of sight. The signal scalings are listed in Table 1. The curves begin to intersect as we move towards thestrong field regime - attributable to the the breakdown in our perturbative approximation for small r . 17 – -3 -2 -1 Period (years)10 -3 -2 -1 R e d s h i f t c o n t r i b u t i o n ( k m / s ) (cid:2) to strong field to Minkowski (cid:3) S2 Minkowski redshiftSchwarzschild signalFrame-dragging signalQuadrupole signal
Fig. 4.— Like Figure 3 but for photon propagation delays (Equation 4). The Schwarzschild propagation signal is forthe most part slightly smaller than the Schwarzschild orbital signal, and scales in the same way. The frame-draggingpropagation signal is considerably smaller than the corresponding orbital signal, and scales like z FD ∝ P − / , asopposed to z FD ∝ P − / . The frame-dragging signal remains approximately an order of magnitude weaker on thephotons than on the star. This suggests that in attempting to measure the spin of the black hole in the post-Newtonian regime, its manifestation on the orbit is what matters. Note however, that this is not the case for theSchwarzschild effects - for which neither the photon nor orbit perturbations may be neglected. 18 – -3 -2 -1 Period (years)10 -2 -1 R e d s h i f t c o n t r i b u t i o n ( k m / s ) (cid:4) to strong field to Kepler (cid:5) S2 Classical velocityTime dilationSchwarzschildFrame-draggingQuadrupoleExtended mass
Fig. 5.— Relativistic redshift effects, with orbital and light-path contributions summed. Two estimates for thesignal due to the extended mass distribution are also shown, using the crude model (16) normalized so that thecircular velocity at the r = 10 GM BH /c (or ∼ . ∼
100 km s − (cf. Gillessen et al. 2009a). The flat andsloping dashed curves correspond to γ = 2 . -4 -3 -2 -1 Spectral dispersion km/s10 S / N Time dilation testSchwarzschild testFrame-dragging testQuadrupole moment test
Fig. 6.— The signal-to-noise ratio (as defined by equation 17) for different relativistic effects in the orbit of S2as a function of redshift accuracy. Each point on this figure corresponds to a simulated data set of 200 redshifts,distributed over the orbit but with highest density around pericenter. 20 – -2 -1 Period (years)10 S / N S2 Time dilation testSchwarzschild testFrame-dragging testQuadrupole test
Fig. 7.— Similar to Figure 6, except that the redshift accuracy is fixed at 10 km s − and the orbits vary, beingscaled-down and speeded-up versions of the orbit S2. For large S/N these curves scale with the powers in Table 1.Current instrumentation, operating under optimum conditions, should manage an accuracy of 10 km s − indicatingthat gravitational time dilation should be able to be detected on some of the currently known S Stars. 21 – -1 Period (years)10 S / N S2 Schwarzschild testFrame-dragging testQuadrupole test
Fig. 8.— Like Figure 7 but for a redshift accuracy of 1 km s − - matching the capabilities of the E-ELT(Lyubenova & Kissler-Patig 2009). 22 – Period (years)10 S / N S2 Frame-dragging testQuadrupole test
Fig. 9.— Like Figures 7 and 8 but for redshift accuracy 30 cm s −1