14' NEUTRON STARS AND BLACK HOLES

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14. NEUTRON STARS AND BLACK HOLES
14.1 PULSARS
The existence of stars supported by neutron
degeneracy had been predicted in the 1930s,
however it was generally thought that the only
way to observe them would be via thermal X-ray
emission from their surfaces. The discovery by
the Cambridge radio astronomers of astronomical
objects which emit periodic pulses of radio waves
in 1968 was unexpected although not occurring in
a complete theoretical vacuum. Pulsars are a
remarkable class of galactic objects, they
radiate short pulses of electromagnetic radiation
at extremely well maintained intervals in the
region of 1 s. The fastest are typically a few ms
and the slowest about 5 s. Although most radiate
only in the radio range a few are also X/g-ray
emitters. Some are associated with supernovae
remnants, which indicates production at the end
point of stellar evolution. Several hundred
have been discovered to date, all are found to be
slowing down.
THE CRAB PULSAR
The preceding sections have shown

• There is a central source of electrons within
  the Crab nebula
• The electrons have energies up to 1014 eV
• There is a central source of energy 2 - 3 x
  1031 J s-1

The discovery of a pulsar at this central point
may well provide an explanation.

• If the pulsar can supply the electrons and the
  energy, then we can explain the dynamics of the
  Crab nebula. The pulsar has the following
  observational characteristics
• A period of 33 ms
• An emission spectrum as shown
• We will return to the Crab when we understand
  more about pulsars

n (Hz)
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14.2 OBSERVATIONAL PROPERTIES OF PULSARS
Pulsar Periods.
The fastest pulsars have periods which are less
than 10 ms. Using light travel times we can
estimate the maximum size of the underlying
object to be
where P is the period
and recognise that they are probably considerably
smaller. What causes the clock mechanism? We know
that stars undergo radial pulsations, so what
about the radial pulsations of white dwarfs
With a density of 1010 kg m-3 one second is just
about feasible, but for periods less than 0.1s
radial pulsations are not possible.
The only other small object known is the neutron
star, with R 104 m and r 1012 kg m-3.
Whereas 10-3 s radial pulsational periods are
possible, these object will have problems in
having their made pulsational mode at periods as
long as one second. We are left with the
possibility of rotating neutron stars as the
basic clock mechanism, white dwarfs cannot rotate
at the required speed..
The Galactic Distribution and Pulsar Distances
Number of Pulsars
SUN
Schematic representation to show that pulsars are
found to be associated with younger stars within
the galaxy, and that most pulsars detected to
date lie within a few kpc of the sun. Only a
small fraction of galactic pulsars have been
detected.
0
500
1000
-500
Height above Galactic Plane (pc)
Distribution of pulsars with respect to height
above the galactic plane. The scale height is
typically 230 pc.
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Since there are many free electrons (density ne)
in the interstellar medium they create a
refractive index for the radio waves which is
frequency (w) dependent.
The resultant variation of the group velocity of
the pulses means that there will be frequency
dependent dispersion which enables the distance
to be measured, and the spatial distributions
illustrated above to be obtained. The dispersion
measure DM is
and enables the distance to be estimated from the
measurement of the delay Dt between pulses
measured over a frequency interval Dw.
Velocities and Kinematic Ages
The proper motions of a number of pulsars have
been measured and revealed that their velocities
are much larger that those of typical stars in
the galactic disk. Typical values of their speed
s are a few hundred kilometers per second. Since
their observed distribution in z above the
galactic plane corresponds to typically 200 - 300
pc then it automatically follows that if they
were created from OB stars situated in the
galactic plane, then they have a lifetime of
not more than about 107 years. After this time
they must cease to pulsate.
Birthrate
This is extremely difficult to estimate, since we
will see that the pulsed emission may well be
beamed. However from the measurements of the
distribution of the pulsars near to the solar
neighbourhood, and assuming that the beaming
factor is typically 2, one may estimate the
birthrate to be of the order of one pulsar birth
per 30 to 100 years. There must be a large
(108-9) and invisible number of ex-pulsars
associated with our galaxy.
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Pulsar Period Distribution
(Above) Observed period distribution of 400
pulsars.. (Right) Distribution of the periods and
period derivatives. Six binary pulsars are
circled. The death line and spin-up line are
discussed later
The range of pulsar periods extends from
typically a few milli-seconds to a few seconds
with a median period a little below 1 s. The
periods of pulsars can be measured with a
precision of 1 part in 109. All pulsars are found
to be slowing down at a rate 10-14 s s-1. No
pulsars are seen with periods greater than about
5 seconds. Since neutron stars will not suddenly
stop rotating we can only conclude that the
mechanism for creating pulsations has to stop.
14.3 THE APPLICATION OF CONSERVATION LAWS TO
STELLAR COLLAPSE
Rotation Axis
If the rotating neutron star is indeed the result
of the core collapse discussed in the context of
supernovae explosions, since the core is likely
to be rotating along with the rest of the
pre-supernova star which also will be permeated
by a magnetic field, then we may expect the
conservation of these parameters as the collapse
takes place.
Collapsing Core
Expanding Supernova Remnant
Magnetic Field
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Conservation of Magnetic Flux
The higher mass stars which are likely to create
neutron stars have typical magnetic fields of the
order 10-2 Tesla. Thus conservation yields
Since the radii of neutron stars are typically
104 m.
Conservation of Angular Momentum
Angular momentum
For a uniform density spherical core of mass MC,
radius RC and initial angular velocity W rad s-1
Thus
For a progenitor star with a typical starting
rotational period of one day. We can easily
understand that pulsars obtain their timing from
rotational motion and that periods of 10-3
seconds are reasonable for young objects.
Note There will be a minimum period possible
for pulsars which corresponds to the point at
which the surface velocity of the star at the
equator approaches the speed of light.
Pulsars are likely to be created with rotational
velocities close to the practical limit
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14.4 SECULAR PERIOD CHANGES AND THE AGE OF
PULSARS
All pulsars are found to be slowing down. If
Period
has been constant then the age is
Now
For the case of the Crab this gives
Time
t1
t0
Since we know the crab pulsar is 1000 years
old, it follows that there must have been a
stronger braking force at an earlier epoch.
Let us look at the physical mechanisms which can
slow down a rotating neutron star.
Secular Decrease in the Angular frequency of
Pulsars
W
The basic picture of a pulsar which emerges from
the above considerations is a rapidly rotating
neutron star which has a very strong dipole
magnetic field. Note that the dipole field will
not in general be aligned with the spin axis of
the neutron star, since this does not generally
occur in main sequence progenitors. Given this
picture we can evaluate how the object can lose
energy.
m
m?
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Stellar Wind Losses
-ve sign means loss
Magnetic Dipole Losses
Where m is the magnetic dipole moment
perpendicular to the rotation axis.
Gravitational Quadrupole Losses
When a neutron star is formed it will be rotating
very rapidly. The centrifugal forces on the
material of the star will be immense. If we
evaluate the value of W for a spherical object
which corresponds to the speed at which the
centrifugal forces at the equator are equal to
the Newtonian gravitational forces we get
i.e.
For a neutron star this value is close to the
initial value so that we may expect a
considerable deviation from a sphere. Such an
object can radiate as a quadrupole.
Where D is the gravitational quadrupole moment
perpendicular to the spin axis.
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The Braking Index
The angular momentum is defined as J IW and is
related to the rotational energy as follows
The values of the constants n and k depend upon
the physical processes involved.
Thus
and
Substitute for k
Now
So that the braking index n is defined as
n 1 is characteristic of stellar wind losses
(i.e. similar to mud off a wheel) n 3 is
characteristic of dipole radiation n 5 is
characteristic of quadrupole radiation
Generally the measurement of W is not possible,
combining the results from many pulsars we can
obtain a general picture
Now
The compiled slope of pulsars shows reasonable
agreement with n 3, indicating that magnetic
dipole losses are generally important.
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THE AGE OF PULSARS ASSUMING ONLY MAGNETIC DIPOLE
LOSSES
We have shown that in general
where T is the age corresponding to W
Integrating we obtain
Since
for n 3
This gives
Thus taking only magnetic dipole losses into
account we obtain an age of 1250 years for the
Crab. Any slight correction for Doppler motions
is not sufficient to obtain the correct age. Some
other physics must be involved, and losses by
gravitational quadrupole radiation at an earlier
epoch is a possibility.
14.5 THE MAGNETIC FIELD STRENGTH AT THE NEUTRON
STAR SURFACE
For the case of magnetic dipole losses we have
and obtain
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If the magnetic braking is responsible for the
slow down of a neutron star then the measurement
of the change in period can be used to estimate
the magnetic field intensity at the surface of
the neutron star. If we assume that the field is
a dipole then classically we have for the
magnetic field at distance r
where m0 is the magnetic dipole moment
so that at r R the surface magnetic field
strength is BS m0 m0 /4pR3
substitution into the energy loss equation gives
For the case of a uniform density sphere the
moment of inertia I 2MR2/5
we get
For the case of the Crab we obtain a value of
typically BS 3 108 T, in general the values for
free pulsars lie in the range 106 to 109 T.
Acceleration of the Filaments in the Crab Nebula
Substitution of the expression for BS in to the
magnetic dipole energy loss equation yields
Now if this is the source of energy which is
responsible for the acceleration of the
filamentary structure of the Crab, estimated at
Ein 3 1031 J s-1 , then we may make an estimate
of the magnetic field strength by assuming dE/dt
Ein, this yields a value for the surface
magnetic field on the Crab pulsar to be
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PERIOD EVOLUTION AND MAGNETIC FIELDS
No pulsars are found with a period more than
about 5 seconds. since neutron stars will not
suddenly stop spinning, and since a longer period
means an older object then a possible explanation
is that the underlying pulsational mechanism
ceases at about this time. a possible scenario is
one in which the magnetic field dies away,
although co-alignment of the spin and magnetic
axis could provide an alternative explanation.
The age of a pulsar undergoing MD losses is
Lines of constant age according to this formula
are shown dashed on the adjacent figure. It can
be seen that the typical lifetime of a pulsar is
about 107 years indicating that this is the
characteristic time scale for magnetic field
changes. As pulsars such as the Crab and Vela
age they will move from their present locations
on the figure downwards and to the right
following lines of slope -1. (Curve a)
(a)
104 years
(b)
106 years
(c)
1010 years
However if the magnetic field were to decay on a
time constant T, such that
Then the evolutionary tracks will steepen
eventually becoming nearly vertical as the pulsar
approaches its terminal period. Curves b and
c illustrate the trajectory for values of 107
and 106 years respectively. Since the luminosity
of a pulsar is likely to be dependent on the
magnitude of the magnetic field strength, then it
is likely that at this time the luminosity will
decrease and the rotating neutron star cease to
radiate as a pulsar.
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14.6 MAGNETOSPHERIC STRUCTURE OF PULSARS -
ALIGNED ROTATOR
The Light Cylinder
The dipole field can only co-rotate with the star
up to a radius corresponding to the distance at
which the field lines would have to travel with
the speed of light.
i.e.
and since
we obtain
This is known as the light cylinder and has a
value Rc 1.5 106 m for the Crab.
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Inside the Light Cylinder
The outermost field line leaves the neutron star
at an angle qP which is defined by
For the case of the Crab this has a value of
about 60
The polar cap region has a radius RP RNS Sin
qP 1.6 km (for the Crab)
The critical field line in the above figure is at
the same electrical potential as the interstellar
medium and divides the regions of positive and
negative current flow from the star. Charged
particles stream along the field lines since the
electric field outside the star lies parallel to
the magnetic field lines and can be shown to have
an approximate magnitude of
For 0 lt q lt qC the field lines are at a lower
electrostatic potential than the surrounding
medium and electrons stream out along the
magnetic field lines. For qC lt q lt qp the
reverse is true and positive ions will move
outwards. Such a strong field imparts a
tremendously strong force to the particles For
protons we have
Particles will be readily torn off the surface of
the neutron star and provide a plasma around the
star - particles that can be readily accelerated
to energies 1015 electron volts or more.
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Outside the Light Cylinder
Outside the light cylinder we see an oscillating
magnetic dipole which creates electromagnetic
waves with frequency n 1/P 30 Hz (for the
Crab). Close to RC there will be immensely
powerful electromagnetic waves which drive the
plasma and any other material away by radiation
pressure (wind zone). These can further
accelerate charged particles and provide an
explanation of the wisp motions.
14.7 THE CRAB NEBULA REVISITED

• A number of mysteries surrounding the Crab
  supernova remnant can be explained by the
  existence of the pulsar
• The magnetic dipole losses balance the power
  requirements for the acceleration of the
  filamentary structure
• The age of the pulsar is in reasonable agreement
  with the date of the original supernova explosion
  if magnetic dipole losses are considered. A
  perfect fit requires some other energy loss
  mechanism to contribute
• The pulsar can explain the production and
  replenishment of the high energy electrons (E
  1015 eV) required to generate synchrotron photons
  up to g-ray energies
• The dependence of the synchrotron size of the
  nebula versus photon frequency is explained by
  the pulsar as the central source of electrons
• The observed motions of the wisps is explained by
  the immensely powerful 30 Hz electromagnetic
  waves which are generated just outside the light
  cylinder

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14.8 EMISSION CHARACTERISTICS OF PULSARS
Pulse Shapes
The observed wave forms have varied shapes but
generally have the common feature of a small duty
cycle (pulse width divided by period), typically
only a few per cent. Frequently a strong
interpulse is seen, the interpulse is a region of
emission separated from the main pulse by roughly
half a period.
Spectra
Log In
n Hz
108
109
1010
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The few radio pulsars which do emit outside the
radio range have short ( 0.1 s) periods,
indicating that they are young pulsars. PSR 0531
21, 0833 - 45, and 1509 - 58, are still
surrounded by the remnant of the supernova
explosion in which they were born. It is
interesting to note that there is an emission dip
at intermediate frequencies (around the optical),
and that they are all powerful g-ray emitters.
Although this spectral gap is evident, and
indicates some possible change in the emission
mechanism, it is interesting to note that the
both the primary and interpulse is found to be
found in phase across the full electromagnetic
emission spectrum. The g-rays are responsible for
nearly all the emission luminosity of the
pulsars. One of them, Geminga, emits only at the
high frequency end of the spectrum.
Polarization
Pulsar emission is highly polarized, 50 linear
polarization is not uncommon, sometimes with a
weaker circular component as well. The position
angle of the polarization vector generally
rotates smoothly throughout the pulse as
indicated in the adjacent figure. The
instantaneous plane of polarization is found to
be independent of frequency, after due allowance
is made for Faraday rotation in the interstellar
medium.
Pulse Profile
Flux
Radio Polarization Vector
Time
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THE PULSAR EMISSION MECHANISM
Probably pulsars dont pulsate !

• There are a number of requirements which are
  apparent from the observational evidence
• The radiation must be coherent (from bunches of
  particles rather than single independent
  particles) in the radio band. The brightness
  temperatures are much higher than is possible
  from a thermal point of view. Coherence is not
  required at high energies.
• The radiation must be emitted in a narrow (10o)
  beam, fixed in orientation with respect to the
  neutron star with a stable beam shape which is
  maintained in both time and across the em
  spectrum.
• The polarization indicates that the emission
  mechanism is highly ordered

• The system must remain stable for long time
  intervals
• The radiation process must produce broad-band
  emission and and generate the observed
  luminosities at different wavelengths

Whereas the underlying mechanism(s) for the
pulsar emission process is still poorly
understood, it is likely that the basic scenario
is one in which the rotating star accelerates
beams of particles along the field lines and that
these particles subsequently radiate a beam of
photons into space. The pulsations are an
artefact of the rotation. Rather like a
lighthouse we see regular flashes of radiation as
the beam sweeps across our path.. The synchrotron
and curvature mechanisms involving the high
energy electrons in the dipole field are strong
candidates for the radiation.
The Origin of Cosmic Rays has been a mystery
since their discovery. Pulsars can accelerate to
extremely high energies. Energetically the rate
of pulsar generating supernovae within the Galaxy
should be sufficient to provide the observed
energy density contained with the cosmic ray
fluxes throughout the Galaxy. However there may
be some problems in explaining the highest energy
(1020 eV) cosmic ray particles. Furthermore
these particles have radii of curvature within
the galactic magnetic field which is greater that
the characteristic size of the Galaxy, and either
must be observed to come from the direction of
known pulsars or have been generated
extra-galactically.