14'13 BLACK HOLES

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14.13 BLACK HOLES
Neutron stars are the closest objects to black
holes which have been found to date on the
stellar scale. The upper mass limit of 2M0 is
rather small and we know that the collapsing core
in more massive stars can readily exceed this
value. No known physics is capable of stopping
the collapse (if there were a quantum theory of
gravity then a singularity should not occur) and
it is probable that a black hole is created. This
singularity will be causually disconnected with
the outside world. The Schwartzschild radius RS
2GM/c2 determines the point at which not even
photons can escape since the observed frequency
n0 of light becomes
where ne is the emitted frequency
A useful numerical value for astronomers is
When the motion of test particles are considered
then the angular momentum has to be taken into
account. One important result is the existence of
a last stable orbit about the point mass which
has radius R 3RS. Circular orbits with radii
less than this cannot exist, the particles spiral
rapidly to r 0.
Observable Properties of Black Holes
Given the likely existence of black holes, how
and by what means can we recognise them. Let us
try and identify their usable properties. The
stationary solutions to the Einstein field
equations depend on three parameters the mass
M, the angular momentum J, and the charge Q. All
other information about the initial state is
radiated away in the form of electromagnetic and
gravitational waves during the collapse..
Mass
The Schwartzschild radius confines the mass of
the object to a given volume of space. Whereas
photons cannot communicate to the outside world
gravitational forces can. Thus we will be able to
measure the mass in e.g. a binary system.
Rotation - Kerr Black Holes
Since all astronomical objects rotate, so that we
can expect black holes formed by gravitational
collapse from a rotating parent to be rotating,
probably quite rapidly. Observationally we have
already seen this for the case of Pulsars.
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Rotating black holes are often referred to as
Kerr black holes, from the case of the Kerr
metric (Q 0) ( As opposed to the Schwarzschild
metric, Q 0, J 0). They are of particular
interest in the context of binary X-ray sources.
Spin axis
A key parameter in the Kerr metric is the angular
momentum of the BH per unit mass, a.
Event Horizon
Some key factors for Kerr black holes are

• No BH can be formed for J gt GM2/c
• For a maximally rotating BH the horizon R
  GM/c2
• i.e. half the Static value of RS
• The rotation causes the dragging of inertial
  frames i.e. the rotation drags all objects near
  it into orbital motion in the same direction as
  the hole rotates. A surface exists within which
  no observer can remain at rest, and must rotate
  in the same direction as the hole.
• The region of space-time between this surface
  and R is called the ergosphere. See figure. It
  is possible that the Penrose process takes place,
  in which a particle enters the ergosphere and
  splits into two sub-particles. If one of them
  falls down the hole then the other escapes to
  infinity with greater energy than the original
  particle had when it fell in. The source of
  energy comes from the rotation of the BH.
• The last stable orbit depends on the direction
  it is R GM/c2 for co-rotation and is much
  larger, R 9GM/c2 for counter-rotation and
  outside the ergosphere.
• The maximum amount of energy loss (i.e. the
  binding energy) in order that the material
  reaches the innermost bound orbit is 40 of the
  mass energy. This is the process which provides
  the energy source which makes BH astronomical
  objects visible.
• It is also possible to tap the rotational energy
  from Kerr BHs. About 30 of the rest mass energy
  can be made available to power astrophysical
  phenomena.

Ergosphere
The Electrodynamics of Black Holes
It is unlikely that charged black holes will be
important astrophysically since it is expected
that a charged astrophysical object is rapidly
neutralised by the surrounding.
plasma. A magnetic field can be associated with
the BH provided it is tied to the surrounding
medium and not the BH itself. The field lines can
for example be linked to the accretion disk and
the surrounding medium and create a geometrical
field structure which is determined by the
presence of the black hole.
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15. CLOSE BINARY SYSTEMS
Since 50 of all stars are originally found in
binary systems, and since compact objects (WD,
NS, BH) are derived from main sequence stars, we
may expect to find these objects in binary
systems. We should be able to estimate their
masses.
Close Binary Systems are defined as systems in
which pairs of stars are so close that mass
transfer takes place between the two objects.
Clearly this will effect their evolution which
will be different to normal (i.e. free) stars of
the same masses.
15.1 THE GRAVITATIONAL POTENTIAL - ROCHE LOBES
W
The gravitational potential is modified for the
case of two stars rotating about each other in
close proximity.
v
M1
M2
The equipotentials for the Newtonian gravitation
centrifugal potential in the orbit plane of a
binary star system with a circular orbit is shown
below
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• In the figure the ratio of the masses of the
  normal star to the compact object is 10 1,
  and the normal star is on the left. The values of
  the equipotentials are labelled in units of G(M1
  M2)/a, where a is the separation of the centres
  of mass of the two stars.
• The equipotentials close to the stars are
  approximately spherical and dominated by the 1/r
  dependence, as if the other star is not there
• At very large distances the equipotentials again
  are spherical for a mass of M1 M2
• The potentials have local stationary points (DF
  0), called Lagrangian points. These are at the
  locations marked L. The inner Lagrangian L1, for
  example, represents the easiest point for
  material from the normal star to escape to the
  other object.
• The two volumes enclosed by the equipotentials
  passing through L1 are called the Roche lobes,
  they can be used to define the enclosed material
  as belonging to one star or the other.

15.2 MASS TRANSFER - ACCRETION ONTO COMPACT
OBJECTS
At particular stages in the lifetime of stars we
have seen that they can suffer considerable mass
loss. The two most notable cases in the immediate
context are when they suffer extensive stellar
wind losses (e.g. OB supergiants) and when (e.g.
Red Giants) they expand outside the limits of
their Roche lobes
Stellar Wind Losses
High Mass X-Ray Binaries (HMXRB) The adjacent
figure illustrates the situation in which stellar
wind losses dominate. The primary star lies
inside the Roche lobe but loses mass via a
stellar wind. The orbiting compact star is an
obstacle in the wind and a bow-shaped shock front
is formed around it by the action of its
gravitational field. Some of the shocked material
is captured by the compact object.
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Roche Lobe Overflow Low Mass X-Ray Binaries
(LMXRB)
If the primary star expands to overfill its Roche
lobe material the material will flow through the
L1 saddle point between the two stars and much of
it is quickly captured by the compact object.
This gas has a considerable amount of angular
momentum and will spiral down to the secondary
object in the form of a disk structure - the
so-called Accretion Disk. A fraction will find an
easy escape route via the L2 point.
The material in the accretion disk moves in
approximately Keplerian orbits as it spirals down
the gravitational potential well. The velocities
will be
As the material swirls in there will be large
shear forces between adjacent parts of the disk
which are dissipated in the form of viscous
heating as the gas flow has a differential
velocity and collisions/turbulence is created.
Note that the total energy available from the
gravitational field is very high for compact
objects such as neutron stars, and can amount to
0.1 mp c2 100 MeV for each proton.
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Luminosity from Accretion
Consider a proton of mass mp falling onto a star
with a mass M and a radius R. The energy gain
from the gravitational field is
The emergent luminosity is
If we write the efficiency of the radiative
emission as e RS/R
Then
where RS GM/c2 is the Schwarzschild radius
15.3 THE EDDINGTON LUMINOSITY LIMIT
Now the luminosity cannot be made arbitrarily
high since the outflowing photons will drive the
infalling matter back and hence cut off the prime
source of the radiation. We may estimate the
maximum luminosity under these conditions
The downward force on the protons due to gravity
is
Thompson cross section
The upward force due to Compton scattering by the
outgoing photons is
Equating we obtain
Leading to a critical luminosity of
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Rough Estimate of the Luminosity and Energy of
the Emitted Photons
The material at the surface of a normal star
emits photons at hn 1 eV, so that we may expect
the temperature of the inner reaches of an
accretion disk to achieve higher temperatures and
higher energy photons to emerge
where R is the inner radius of the accretion disk
Now
so that with a reasonable accretion rate for
close binary systems of dM/dt 10-9 M0 y-1 we
can make approximate estimates as follows
NOTE

• The spectral range of emission from the various
  objects
• Neutron stars and black holes are X/g-ray
  sources which should emit at the Eddington
  luminosity limit

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15.4 ACCRETION ONTO WHITE DWARFS
A wide range of available geometries exist for
accretion to take place onto WD, as a result a
number of different types of phenomena occur.
NOTE all Cataclysmic Variables(CVs) are strong
uv sources. Classical Novae are the most
dramatic, exhibiting typically 10 magnitudes of
brightening over a few days and remaining
luminous for many days. The luminosity is
typically Eddington for a WD with a total energy
release of 1038 J. In the 1960s Novae were
finally observed to be related to WD in binary
systems. They are thought to be related to the
accretion of a critical mass of H-rich material
onto the WD, followed by degenerate thermonuclear
runaway which eventually stabilises itself. The
light curve could well be sustained by the 22Na
g-ray emitting radioactive isotope. About 30
Novae occur in the Galaxy per year, although only
a small fraction are observed due to optical
extinction.
Recurrent Novae are less luminous, but as the
name suggests the outbursts take place on the
timescales of decades to months and with a great
deal of variety between the various objects.
These dwarf novae have been found to contain
accretion disks. The binary periods of
cataclysmic variables are typically of a few
hours and all less than 1/2 day, a fact which
dictates that the companion is a low mass star
which is undergoing Roche lobe overflow.
Accretion stream
White dwarf
Polars Here the magnetic field is very strong (
103 T) and the rotation of the WD star is phase
locked to the orbit of the binary. AM Herculis is
the archetypal example. The accreting matter
flows directly along the magnetic field lines
from the primary star onto the poles of the WD,
no accretion disk is formed.
Companion star
Intermediate Polars These objects have weaker
magnetic fields and since the rotation can not be
synchronised with the orbit the rotation period
of the white dwarf is shorter than the orbital
period. The magnetic field is not strong enough
at large distances to force the accreting
material directly down the field lines and an
accretion disk is formed. Close to the star the
magnetic field will disrupt the accretion disk
and matter will accrete onto the star only at the
poles. Both of these classes of objects exhibit
high and low states of activity which are related
to the different states of mass accretion.
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15.5 ACCRETION ONTO NEUTRON STARS AND BLACK
HOLES
Since both NS and BH will emit in the X-ray
region of the spectrum, what differences may we
expect to find to distinguish them from one
another?
The mass is clearly the main factor to look for,
however there may be more subtle observational
differences based indirectly on such aspects as
the ergosphere of magnetic field configuration.
Regularly Pulsating X-ray Binary Sources.
The discovery of regularly pulsating X-ray binary
sources by the UHURU satellite was a major
milestone in the understanding of neutron star
systems. The two archetypal objects are Her X-1
and Cen X-3.
Counts per 0.096 s BIN
BINS
The presence of 4.8 second X-ray pulsations in
Cen X-3 as revealed by the UHURU satellite, thus
confirming that it includes a rapidly spinning
neutron star.
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Period P
Delay Dt
Doppler t value
IX
Detailed Study of the Pulsations

• Whereas the X-ray pulsations may, at a first
  glance, look to be extremely regular, in fact
  they are not completely stable as for the case of
  a free neutron star in the form of a pulsar.
  However their variability is not random and is
  linked to the fact that the neutron star, whilst
  pulsating regularly, is in fact rotating around
  the primary star. Thus we have an accurate clock
  on one of the members of the binary system, and
  this may be used to gain more information about
  the overall system. The pulses experience three
  kinds of modulation (for an eclipsing binary
  system) which are related to the phase of the
  orbit as illustrated in the above figure for Cen
  X-3
• The modulation of the X-ray intensity IX as the
  compact object passes behind the primary (Cen X-3
  has a period of 2.087 d)

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• The arrival time delay varies sinusoidally with
  the period P due to the extra distance the
  radiation has to travel when the secondary is
  furthest away from the Earth.

This enables us to evaluate the diameter D of the
secondary orbit i.e. for CenX-3 Dt 39.7 s so
that D 2 x 39.7c 2.38 1010 m which
means the NS orbit is very close to the surface
of the primary star.

• The x-ray pulse period fluctuates sinusoidally
  throughout the orbit due to a periodic Doppler
  effect

The period decreases as the neutron star
approaches the Earth and increases as it recedes.
For Cen X-3 t0 4.84 s which is consitent with a
projected velocity of
i
Mx
MP
The Estimated Masses of Neutron Stars
There are a number of X-ray binaries for which
enough observational information is available,
including an estimate of the angle i, to permit
the mass of the neutron star to be evaluated with
some accuracy.
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Neutron Star Equation of State
Note that the masses lie within the range
discussed earlier. However if we are to obtain an
equation of state for neutron stars then we have
to evaluate their sizes. Measurement of the red
shift by fine spectroscopy of any nuclear
emission lines (including the electron-positron
annihilation line at 511 keV) emitted from the
surface may enable this to be done.
Orbital Geometry
Many of the neutron stars are very close to the
primary star. the diagramme on the left gives
some idea of how close they are, and makes it
easy to understand why such a large fraction of
them are eclipsing. Below is a sketch of an
accreting binary system.
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Pulsation Mechanism
The masses of the pulsating X-ray binaries
indicate that the compact object is a neutron
star. The pulsations are thought to be due to the
strong dipole magnetic field.
Accretion Column
Accretion Disk
Emergent X-ray beam
The accreting material is constrained to flow
along the magnetic field lines towards the polar
caps. As the material hits the neutron star
surface a hot shock is formed in which X-rays are
produced. The column above the emitting region
means that the emission will not be uniform but
shadowed into a fan beam. If the polar caps are
displaced from the rotation axis then modulation
of the emission will be synchronised with the
rotation period of the neutron star. Again, as
for pulsars, a kind of lighthouse effect causes
the pulsations.
Pulse Period Changes
The periods of pulsating X-ray binaries are
found to decrease as a function of time. This
spin-up must be related to the transfer of
angular momentum from the material of the
accretion disk to the neutron star during the
infall.