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The PSR 2127+12 as an Indicator of a Massive Black Hole in the Core of Globular Cluster M 15 K.A. POSTNOV, M.E. PROKHOROV, AND N.I. SHAKURA Sternberg Astronomical Institute ABSTRACT The 110-millicsecond pulsar PSR 2127+12 in the core of the globular cluster M15 is distinguished by having a negative period derivative P ~ -2 10-~7. As value cannot be provided by acceleration in the mean gravitational potential of the core. A flyby of a star ~ 300 AU away could explain the P observed, but the probability of such an event is small (~ 10-3~. We suggest that the pulsar motion Is governed by the presence of a moderately massive (A 2 - 3 104 Mel black hole in the cluster's center. The idea is further supported by an observed post-collapse morphology of the M15 core. INTRODUCTION Pro pulsars in globular clusters out of the seven known so far exhibit anomalous values of P. The first one, the pulsar PSR 1821-24 in M28 with a period of 3 ms, has unusually large for millisecond pulsars dP/dt = 1.6 · 10-~8 (Foster et al. 1988), and the second one, PSR 2127~12 in M15 with a period of 110 ms has negative penod derivative dP/dt ~ -2 10-17 (Wolszczan et al. 1989). The observed P in the PSR 1821-24 can be connected with the magnetic braldog, while the arrival time analysis in the latter pulsar strongly evidences the Doppler shift due to a motion with acceleration toward the observer. In this note we concentrate on PSR 2127~12 and show that its acceleration can be due lo the presence of a massive black hole in the core of the globular cluster M15. Other possible 316
HIGH-ENERGY ASTROPHYSICS 317 reasons for acceleration of this pulsar fail in explaining the observational data or are very tentative. DYNAMICAL EFFECTS OF TlIE: MEAN CORE POTENTL\L Dynamical effects in the globular cluster should exert influence upon the pulsar timing because any change in pulsar radial velocity vie results in changing of the observed pulsar period according to the relation - r ~ r do /aft dP/dt _ _o (1) with c as speed of light. This effect may mimic the proper pulsar dP/dt. The pulsar radial velocity can vary due to (i) the motion in the potential of the cluster's core and (ii) the motion in the potential of a nearby star. By measuring P one can discriminate between these two possibilities (Blandford et al. 1987~. The knowledge of the globular cluster parameters and the angular position of the pulsar in the cluster's core make it possible to estimate the mammal radial acceleration produced by the potential of the core which can evidently be both positive or negative. If the pulsar acceleration were produced solely by the core potential and not exceeded a maximal value, one could find in principle two possible spatial positions of the pulsar inside the core. The mammal value of the acceleration In the framework of standard King model of M15 cluster structure (see Webbink 1985) is about 10-6 cm2/s, whereas go ~ 6 10-6 cm2/s is required to explain properties of PSR 2127+12. ACCELERATION IN A NEARBY STAR POTENTIAL Let's turn to the possibility of pulsar accelerating in the potential of a nearby star. For the potential required to be produced, the second star with a mass of 0.8 Me should be closer than a ~ 5 10~5 cm, which corresponds to a characteristic time of stellar encounter ~ 300 yrs. The probability of finding a star such as this In the dense core of a globular cluster is of the order of (a/<a>~3 ~ 10-3 (here <an = n*~~/3 denote me mean distance between stars in the core) for a Apical stellar density 105 pC-3. Note that whether or not this second star forms a binary system with the pulsar is of no importance. The existing data for PSR 2127+12 ~1olszczan et al. 1989) about the constancy of dP/dt during 9 months of observation rule out all binary periods less than 100 yrs. A group of nearby stars may produce a fluctuation of the potential 6
318 AMERICAW AND SOVIET PERSPECTIVES times the mean core value, but the probability of such situation is extremely low. OTHER SOURCES OF ABNORMAL TIMING i) Gravitational Lensing Gravitational leasing effect in the core of a globular cluster causes the brightness variations (see Sazhin 1987~. The same effect should be seen in pulsar timing. The variations In pulsar period can be produced by a flyby of a star near the line of sight between the pulsar and the observer. Such a star will cause a delay in the time of arrival of radio pulses (so-called Shapiro delay). This results in a period denotative of order of dP/dis ~ 2(r,/c)(v/b)2P (2) where rg is the star's gravitational radius, b is the impact parameter and v is the transversal velocity of the star. For a typical stellar mass 0.8 Me and velocity 10 km/s the impact parameter required to produce the observed in PSR 2127+12 P is ~ 3 10~i cm. The probability of such event in the core is (b/<b>~2 ~ 10-~° where <b> ~ Rc/~ ~ 3 · 10~6 cm is the mean impact parameter, Rc and N are the cluster core radius and the number of stars inside it. But the typical event duration is of the order of ~ ~ by ~ 3 · 1Os s and is too short Note that the effect of the Shapiro delay produced by the core itself is estimated quite analogically by substituting r9 for the core and v for the pulsar velocity, and is as large as the effect from the single star with the mean impact parameter <b>, he., is extremely small amplitude although large enough in duration. ii) The Nelltron Star Precession Pulsar period variations could also be caused by some internal irreg- ularities in the NS rotation. For example, neutron star free precession is known to change pulsar period as P ~ (Pp~r/Ppr)2. For P ~ 10-~7 and PA 0.1 s precession period required proves to be very small, about 3 · 105 s. So this possibility Is ruled out by the observations. Geodetic precession of a NS in the cluster's core has, instead, a very large period ~ T/(v/c32 (where T denotes characteristic traveling time for a star with velocity v across the core; T ~ 104 yrs) and carrot produce any notable P. Another reason could be connected with the NS internal spinup/spin- down lee in the rotational pendulum. But it remains unclear why such universal effects are not seen in other pulsars.
HIGH-ENERGY ASTROPHYSICS PULSAR MOTION IN TlIE POTENTIAL OF COLLAPSED CORE 319 Dynamical evolution of a globular cluster at the late stages can lead to a core collapse (see Spitzer 1985 for a review). As a result, a massive black hole can be formed. This observationally should be expressed by a cusp in the core stellar density and surface brightness. The globular cluster M15 is one of the clusters whose surface brightness can be interpreted as showing such a feature (Djorgovski and Penner 1985~. For an estimation of the acceleration produced by the collapsed core potential let's use again expression (1) in the for GMy 9 = (x2 ~ y2~312 (~3) where x and y are Cartesian coordinates as shown in Figure 1. The function has a maximum at x = Any. x is observed to be 2", corresponding to ~ 0.1 pc at a distance 10 kpc. Thus the central mass M capable of producing the acceleration required must be ~ 3 104 My, or ~ 10% of the total mass of the core. Such mass in the center of the cluster begins to significantly affect the cluster dynamics only in the innermost parts of the core. To other millisecond pulsars recently discovered in M15 (Anderson et al. 1989) are situated 2' away from the center and thus should not be subjected to the central mass influence. A black hole of similar mass will tidally disrupt the stars inside me loss cone at a rate (HID 1975) Non I = in* rat rh ~ ~ 710 ~ 7m`3 yr (here r' and rh denote tidal disruption radius of the star and the black hole's capture radius, respectively, and ~ is the velocity dispersion). This rate agrees well with the simple estimate of the growing rate of the black hole during the Hubble time. So we conclude this hole could really be formed in the course of the cluster's evolution. CONCLUSIONS lathe negative period derivative observed in the PSR 2127+12 must be due to an accelerated pulsar motion in the core of the globular cluster M15. The mean core potential under standard assumptions about the core structure can provide the acceleration an order of magnitude lower than the required 6 · 10-6 ~l',2/s, but a nearby star ~ 300 AU away from the pulsar would do. However the probability of this event is a rather small one, ~ 10-3. Measurements of P in the pulsar timing analysis would confirms or discard the latter possibility in the nearest future.
320 AMERICAN AND SOVIET PERSPECTIVES y 1 / / k PSR 2" / V To the observer FIGURE 1 Geometry of pulsar position in the globular cluster core. x Other reasons for possible period changing, such as the neutron star precession or rotational pendulum-like motions seem inappropriate. The Shapiro delay in pulsar timing induced by a flyby of a field star across the line of sight fails as well. We suggest that a black hole with a mass of 2 . 3 · 104 M:> governs the pulsar motion. This black hole could be a remnant of the core collapse which occured in the course of the dynamical evolution of the cluster. Its mass would exert influence upon only the innermost regions of the core and would not contradict the cluster's age. This idea seems to be further supported by the obsen ations of a steep brightness gradient in the core of M15 the phenomenon predicted by theories of the globular cluster cores having massive post-collapse remnants. The black hole in the cluster's center would also cause an anomalous (= 30 binds) velocity dispersion to be observed in the M15 core. So its measurements would be strong desired.
HIGH-ENERGY ASTROPHYSICS 321 Thus the pulsar timing analysis being a probe of the core gravitational potential can be an indicator of the core having a massive black hole in its center. cessions. ACKNOWLEDGEMENTS We thank the staff of the relativistic astrophysics department for dis REFERENCES Anderson, S., P. Gorham, S. Kulkarni, T. Prince, and A. Wolszczan. 1989. IAU Circ 4762, 477Z Blandford, R.D., R.W Romani, and J.H. Applegate. 1987. Mon. Notic. Roy. Astron. Soc. 225: 51P. Djorgovski, S., and H. Penner. 1985. In: Goodman, J., and P. Hut (eds.~. Dynamics of star clusters. IAU Symp. 113: 73. Foster, R.S., D.C. Backer, J.H Taylor, and W.M. Goss. 1988. Astrophys. J. Lett. 326: L13. Hills, J.C. 1975. Nature 254: 295. Sazhin, M.V. 1987. In: 11th International Conference on General Relativity and Gravitation. Stokholm II: 519. Spitzer, In, Jr. 1985. In: Goodman, J., and P. Hut (eds.~. Dynamics of star clusters IAU Symp 113: 109. Webbink, R.F. 1985. In: Goodman, J., and P. Hut (eds.~. Dynamics of star clusters. IAU Symp. 113: 541. Wolsz~an, A, S.R. Kulkan~i, J. Middleditch, D.C. Backer, AS. F~uchter, and RJ. Dewey. 1989. Nature 337: 531.