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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
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Representative terms from entire chapter:
black hole
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/~3 ~ 10-3 (here
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/~2 ~ 10-~° where ~ 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 , 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
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