BLACK HOLE CANDIDATE SAGITTARIUS A*. J. M. HOLLYWOOD,1,2 FULVIO MELIA,1,3,4 LAIRD M. CLOSE,3,5 DONALD W. MCCARTHY, JR.,3. AND TATIA ...
THE ASTROPHYSICAL JOURNAL, 448 : L21–L24, 1995 July 20 1995. The American Astronomical Society. All rights reserved. Printed in U.S.A.
GENERAL RELATIVISTIC FLUX MODULATIONS IN THE GALACTIC CENTER BLACK HOLE CANDIDATE SAGITTARIUS A* J. M. HOLLYWOOD, 1,2 FULVIO MELIA, 1,3,4 LAIRD M. CLOSE, 3,5 DONALD W. MCCARTHY, JR., 3
AND
TATIA A. DEKEYSER 1
Received 1995 January 27; accepted 1995 May 12
ABSTRACT The proximity of the unique radio source Sgr A* at the Galactic center is allowing us to make unprecedented observations of the spectral formation region near the event horizon of a massive black hole. Near-infrared flux measurements hint at a possible 10.4 minute modulation period in this source, which suggests an emission surface with a Keplerian velocity near the last stable orbit of a 10 6 M J Schwarzschild object. In this Letter, we set up the fully general relativistic framework for calculating such modulations, which we suggest might arise from orbiting disk inhomogeneities. The application of our method to the currently available data shows excellent agreement between the calculated light curve and the observed temporal profile, promising exciting future observations of Sgr A* with NICMOS on Hubble Space Telescope. Subject headings: accretion, accretion disks — black hole physics — Galaxy: center — relativity small, radiating disk that forms from angular momentum fluctuations associated with the shifting accretion patterns (Ruffert & Melia 1994). If this picture is correct, general relativistic effects may significantly influence Sgr A*’s infrared spectrum (Hollywood & Melia 1994), which would provide an observational signature that may be tested with ongoing ground-based studies of the Galactic center (Close et al. 1995) and more conclusively with the upcoming deployment of the near-infrared camera NICMOS on the Hubble Space Telescope (HST). Chief among these effects is the expected amplification of nonaxisymmetric modulations in the disk flux due to strong light bending and Doppler shift corrections within 14 –5 Schwarzschild radii of the black hole. The groundwork for interpreting and understanding a system such as this was laid out by Bardeen et al. (1972), Cunningham & Bardeen (1973), and Cunningham (1975). Subsequent adaptations of these early developments to a study of variability data for AGNs have been carried out by several authors, most notably Abramowicz et al. (1989), Bao & Stuchlik (1992), and Chakrabarti & Wiita (1994). These workers have already considered the role played by relativistic corrections in determining specific AGN characteristics, such as the shape of their light curves. Our goal in this Letter is to provide a framework for predicting and ultimately testing the short timescale IR variability of Sgr A* as a function of the (inferred) disk inhomogeneity, while we take into account the effects of Doppler and gravitational redshifts, the light-travel time factor, and the strong lensing. In the adaptation described here (see also Hollywood & Melia 1994), we solve the equations for the photon trajectories without invoking the weak field limit, and we include both the Kerr and the Schwarzschild metrics. As an immediate application of these results, we compare our predicted modulation amplitudes and temporal profiles with the currently available data.
1. INTRODUCTION
The dynamical center of the Galaxy appears to be coincident (within 110) with Sgr A*, a unique, nonthermal radio source (e.g., Lacy et al. 1994), whose properties point to a close kinship with the nuclei of such nearby galaxies as M31, M81, and M104 (e.g., Melia 1992a, b). As do those of the more powerful active galactic nuclei (AGNs), Sgr A*’s spectral characteristics suggest that it may be a massive (110 6 M J ) black hole, as first anticipated by Lynden-Bell & Rees (1971). Sgr A* is luminous across more than 10 decades in frequency. Numerous observations with the VLA and radio VLBI have provided a wealth of spectral and temporal variability data (e.g., Lo 1989; Backer 1993), and several highenergy instruments have detected X-ray (and possibly g-ray) emission from its direction (Watson et al. 1981; Skinner et al. 1987; Predehl & Tru ¨mper 1994), although it is interesting and at first somewhat puzzling that its X-ray luminosity is not so large as one might naively expect from X-ray observations of other (usually much smaller) black hole candidates. From recent Granat observations, Sunyaev et al. (1990) report a variable 4 –20 keV source of luminosity 10 36 ergs s 21 located within 19 of Sgr A*, and of potential interest is the more recent detection, by EGRET onboard the Compton GRO, of a central (,18) ?100 MeV continuum source with luminosity 210 36 ergs s 21 (Mattox 1995). At infrared wavelengths, Sgr A* appears to be an m K 2 13 source (Eckart et al. 1992; Close et al. 1992), although a measurement of its flux over several years indicates that the emission at this wavelength may also be variable on a timescale 11 yr (cf. Herbst, Beckwith, & Shure 1993). There is even a hint of more rapid variability (on a timescale 110 minutes; Close, McCarthy, & Melia 1994). Whereas the radio and high-energy radiation appears to originate from within a large-scale (=10 16 cm) Bondi-Hoyle inflow via bremsstrahlung and cyclotron/synchrotron emission (Melia 1992a; 1994), the infrared luminosity may be due to a
2. DISK GEOMETRY AND CALCULATIONAL PROCEDURE 1 2 3 4 5
Physics Department, University of Arizona, Tucson, AZ 85721. NASA GSRP Fellow. Steward Observatory, University of Arizona, Tucson, AZ 85721. Presidential Young Investigator. NSERC (Canada) Graduate Scholar.
We assume here that the accretion disk is optically thick, geometrically thin, and axisymmetric in structure (though not necessarily in its temperature profile), that it resides in the equatorial plane of the black hole, and that it radiates as a sum L21
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of blackbodies. The justification for these assumptions may be found in Hollywood & Melia (1994), along with a prescription for calculating the observed specific power flux once the optical appearance of the accretion disk at infinity has been determined. For calculational purposes we regard the disk as consisting of a relatively large number of concentric rings of emitting particles at source radii r s and source polar angle u s 5 p/ 2. The radial temperature function used here is that of Page & Thorne (1974), but we now incorporate into our accretion disk model a nonaxisymmetric ‘‘hot’’ patch, propagating as a fixed pattern around the accretion disk with an angular velocity v 1 and a temperature T 1 (r) enhanced over its normal radial Page-Thorne value T(r) by a constant factor f = 3. For simplicity, we here take this patch to be a wedge, although any shape can be incorporated into the calculation when more precise fitting is required. An inhomogeneity such as the one considered here might result from a standing shock rotating around the black hole with a fixed pattern speed. A lack of azimuthal symmetry in the disk temperature might also arise from the manner in which gas fluctuations in the larger scale inflow settle down into a disklike configuration, not unlike the creation of a ‘‘hot’’ spot in which the inflowing stream intercepts the disk in a binary system. A partial description of the numerical techniques used here to determine the apparent image of the accretion disk at infinity has been provided elsewhere (Hollywood & Melia 1994). The validity of these procedures rests ultimately on the fundamental equations governing photon propagation in the Kerr metric first set forth by Carter (1968) and amplified and applied in a variety of contexts by, most notably, Cunningham & Bardeen (1973), Bardeen et al. (1972), Cunningham (1975), Abramowicz et al. (1989), Fu & Taam (1990), Bao & Stuchlik (1992), and Chakrabarti & Wiita (1994). Briefly, a photon trajectory originating at source coordinates (t s , r s , u s , and f s ) and terminating at observation coordinates (t o , r o , u o , and f o ) may be characterized by the azimuthal component of angular momentum r f , the energy (at infinity) E, and the polar component of angular momentum p u 5 (Q 1 a 2 E 2 cos 2 u 2 p f2 cot 2 u ) 1/ 2 . Here Q is a constant of the motion that arises when the Hamilton-Jacobi equation is solved by separation of variables, and a [ J/Mc is a parameter specifying the magnitude of the black hole’s spin angular momentum. Taking the unit of length to be the Schwarzschild radius r g [ 2GM/c 2 5 1, setting G 5 c 5 1 in accordance with the convention of ‘‘geometrized units’’ (so that the black hole mass is M 5 1/ 2), and introducing the dimensionless parameters l [ p f /E and q [ Q 1/ 2 /E, one may write the propagation equations as
E
ro
rs
Df 5
dr @R~r!# 21/ 2 5
E
ro
1l
E
d u @Q~ u !# 21/ 2 ,
dr @R~r!# 21/ 2 ˜ D 21 $r@a 1 l ~r 2 1!#%
E
uo
d u @Q~ u !# 21/ 2 cos 2 u ,
us
ro
rs
(1)
us
rs
Dt 5
E
uo
dr @R~r!# 21/ 2 ˜ D 21 @~r 2 1 a 2 ! 2 2 a 2 ˜ D 2 a l r#
(2)
1 a2
Vol. 448
E
uo
d u @Q~ u !# 21/ 2 cos 2 u ,
(3)
us
where R(r) 5 r 4 1 (a 2 2 l 2 2 q 2 )r 2 1 [(a 2 l) 2 1 q 2 ]r 2 a 2 q 2 , D 5 r 2 1 a 2 2 r. Q(u ) 5 q 2 1 a 2 cos 2 u 2 l 2 cot 2 u, and ˜ For each ring of emitting particles in the accretion disk and S o consisteach polar observation angle u o , there is a superset ˜ ing of an infinite number of sets S on of (l, q), each of which satisfies equation (1), with the index n (50, 1, 2, · · ·) indicating the number of times the photon trajectories in the set transit the equatorial plane as they propagate from source to observer. Each set S on corresponds in turn to a set of image coordinates (‘‘impact parameters’’) (a, b), where a 5 2l/sin u o , b 5 H(q 2 2 a 2 cos u o2 2 l 2 cot u o2 ) 1/ 2 , and the sign of b is given by (u/r) o /u(u/r) o u. When plotted on the plane of the celestial sphere at infinity (with the origin of coordinates at the center of the black hole and the positive z-axis defined by the hole’s spin angular momentum vector), the set (a, b) determines the nth-orbit image of a ring of emitting particles as observed at (r o , u o ). In general only the direct-orbit (n 5 0) image contributes significantly to the specific power flux observed at infinity, partly because the higher orbit rays are occluded by the accretion disk, but also because gravitational focusing ‘‘squeezes’’ the higher orbit images so that their apparent areas at infinity are small compared to that of the direct-orbit image. For simplicity, therefore, we consider only the direct-orbit image for the calculations reported here. Since the relations given above for a and b are invertible, it follows that for every ring image in the image plane (a, b) there is a corresponding complementar y ring image in a complementar y image plane (l, q). It is possible to deduce a set of rules that, for each ring of emitting particles, restricts the set of (l, q) satisfying equation (1) to a definite region in the lq-plane. One may then partition this region into a grid with intersection points (l i , q i ). With r s fixed and r o 3 E, one evaluates the left-hand side of equation (1) along the appropriate radial path for each (l i , q i ); then, with u s 5 p/ 2, one evaluates the right-hand side of equation (1), along the polar path, stepwise, in small increments Du, until equality holds; the final value of u is then recorded as the polar angle of observation u o . By this means one obtains a table of values giving u o numerically as a function of (l i , q i ). A numerical inversion then gives the set S oo of (l, q) that satisfies equation (1) for a specified u o . In the case of a homogeneous, axisymmetric, steady state disk, one needs no additional information to calculate the time-independent specific power flux observed at infinity. On the other hand, if one assumes a temporal variation in the emitted power flux and desires to determine the time-dependent flux observed at infinity, one must then go on to insert each member of S oo into equations (2) and (3) to obtain the corresponding azimuthal angular displacements Df [ f o 2 f s and photon travel times Dt [ t o 2 t s , respectively. Given these quantities and some model that describes the time evolution of the disk in its local reference frame, one may then easily determine the corresponding time evolution of the apparent image of the disk as observed at infinity. By integrating the source intensity function (along with its frequency-shift weighting factors) over a series of ‘‘snapshots’’ of the apparent disk image, one thus obtains the time-dependent flux.
No. 1, 1995
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FIG. 1.—Projected image in the plane of the sky of a small accretion disk and a Schwarzschild black hole for a polar observation cosine m 5 0.4 at two different observational phases. (a) Appearance at the zero of observational time. The ‘‘normal’’ image of a 508 hot wedge (large dark patch at bottom of disk image) is distorted here primarily by light-travel time effects. The two auxiliary wedge images (small dark patches at top of disk image) are formed by rays which circulate completely about the z-axis of the black hole, here oriented in the 1b-direction. (b) Appearance at observational phase 0.7, which corresponds approximately to a maximum in the observed frequency-integrated flux. Here the distortion and splitting of the wedge image arise primarily from general relativistic time delays and light bending. 3. PRELIMINARY RESULTS
The two panels in Figure 1 show the optical appearance of the disk as observed in asymptotically flat space for a polar observation (i.e., inclination) angle cosine m 5 0.4 and for two phases of observation. The zero of source time is chosen so that at that instant, according to an observer fixed in the reference frame of the black hole, the hot wedge is centered on f s 5 0. The ray emitted with zero angular momentum at (t s 5 0, r s 5 r max , u s 5 p/ 2, f s 5 0), where r max is the radius of the outermost disk ring, is then taken as a reference ray. Arrival of this ray at (t o , r o , u o and f o ) defines the zero of the observer’s time. Then the ‘‘observational phase’’ is defined to be the time elapsed, according to the observer, since t o 5 0, divided by the period t 1 [ 2p/v 1 , modulo unity. The asymmetry in the appearance of the wedge at zero observational phase (Fig. 1a) arises from light-travel time delays of rays from other regions of the wedge relative to the reference ray emitted at f s 5 0. For comparison, Figure 1b exhibits the appearance of the same disk at an observational phase of 0.7, which corresponds approximately to the observed maximum in the frequency-integrated flux.
The effects of the various factors which determine the character of the observed flux are illustrated by the curves plotted in Figure 2a, which show the fractional change in K-band flux (i.e., the flux density integrated over the effective K-band wavelength limits from 2.00 to 2.40 mm) arising from (1) differential light-travel times for rays originating at different points in a Newtonian disk, (2) a combination of these and Doppler frequency shifts, and (3) the foregoing plus the effects due to gravitational redshift (GR) and light bending. Although a modulation is expected even in a nonrelativistic context, the general relativistic modifications greatly enhance the amplitude of the ‘‘pulse’’ and shift it to an earlier phase. The observer’s inclination angle has an additional effect on the amplitude of the observed modulation, as demonstrated by the three curves in Figure 2b. 4. APPLICATION TO CURRENTLY AVAILABLE DATA AND DISCUSSION
As an illustration of the possible future application of this model, we have calculated the time profile of the K-band flux from Sgr A* to compare with the best currently available data
FIG. 2.—Fractional change (from the mean value) in K-band flux plotted as a function of the observational phase, for a 10 6 M J black hole accreting at 0.4 3 10 22 g s 21 (Melia 1994). The disk size is 5r g . (a) Newtonian values (1), Newtonian values incorporating the classical Doppler and special relativistic frequency shifts (2), and the fully GR flux (3). All three curves are for a ‘‘hot’’ [T 1 (r) 5 3T(r)], 508 wedge with an orbital period approximately equal to the observed short timescale IR variability period of the black hole candidate Sgr A*. The polar observation cosine is again m 5 0.4. (b) Three fully GR flux curves for polar observation cosines m 5 0.1, 0.4, and 0.7. The wedge size and temperature enhancement factor are the same as in (a).
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FIG. 3.—Data show the observed modulated K-band flux from Sgr A* folded onto a period t 1 5 10.4 minutes (Close et al. 1995). The curve represents the fully GR flux calculated assuming a black hole plus disk system with the same parameter values as those of Fig. 2b (m 5 0.4). The x 2 value for this fit is 0.956 (see text).
(Close et al. 1995). We have assumed a 10 6 M J black hole accreting at a rate of 0.4 3 10 22 g s 21 . These values, and the adopted disk size of 5r g , appear to be consistent with the broadband spectrum of Sgr A* across roughly 10 decades in frequency (Melia 1994). These values of the black hole mass and disk size are also consistent with the time-averaged flux of K 1 12.1 mag (Close et al. 1995). The remaining parameters, which we use for the fit shown in Figure 3, are the size of the inhomogeneity (here assumed to be a wedge), the temperature enhancement factor f , and the disk inclination angle cosine m 5 cos i. In addition, we have taken the pattern speed to be v 1 5 0.01 rad s 21 , so that the corresponding period t 1 is approximately equal to the observed short timescale IR variability period, t obs 2 10.4 minutes. It is noted that this angular speed corresponds to that of particles orbiting a 110 6 M J black hole in a circular orbit of radius r s 2 3.7R g , which for a Schwarzschild metric is close to the innermost stable orbit. Treating the zero point phase in the data as an additional degree of freedom, we find that x 2 is minimized to less than 0.96 for a disk inclination angle i (5cos 21 m) of about 658,
f 2 3, and a wedge size of about 508. For this data set, the actual shape of the disk inhomogeneity does not produce a significant effect on the light profile. Of course, the close agreement between these data and the calculated light profile is at best only suggestive because the detected modulation is not significant beyond the 2 s level (Close et al. 1995). The inferred disk characteristics should not yet be taken too seriously. However, NICMOS when installed on HST should do considerably better with its stable diffraction-limited point-spread function, offering us the promise to ‘‘map’’ out the disk structure using a far more accurate light profile. We expect that a direct comparison between our fits and the data could not only confirm the expectations of this model—and perhaps provide considerable evidence for the black hole nature of Sgr A*— but also may allow us to distinguish between various possible disk inhomogeneities. This research was supported in part by NSF grant PHY 88-57218, NASA grant NAGW-2822, and a NASA GSRP Fellowship.
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