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Pulsars
High Energy Astrophysics
[email protected]
http://www.mssl.ucl.ac.uk/
Introduction
• Pulsars - isolated neutron stars
- radiate energy via slowing down of rapid
spinning motion (P usually < 1sec, dP/dt>0)
• Pulsating X-ray sources / X-ray pulsators
compact objects (generally neutron stars) in
binary systems. Accrete matter from normal
star companion. (P ~ 10s secs, dP/dt<0)
Pulsars cont.
• Discovered in radio
Averaging over many pulses we see:
Period
pulse
~P/10
interpulse
• Measuring pulse complicated by Doppler
motion of Earth and frequency dispersion
in pulse arrival times.
• Individual pulses:
av. very constant, individual pulses variable
Pulsar period stability
• Period extremely stable: 1 part in 10 12
indicates some mechanical clock
mechanism - although this mechanism must
be able to accommodate pulse variablity.
• Pulsations of white dwarf??? (but Crab
pulsar period (P~1/30 sec) too short).
• Rotation of neutron star???
Rotation of a neutron star
Gravitational force > centrifugal force
GMm mv

2
r
r
2
2r
where v 
and P is the period
P
Reducing:
M

GM 4 r
=>


3
2
2
2
4r
PG
r
P
2
M
but  
4 3
r
3
G = 6.67x10
-11
so
3
3
 2
PG
-1 -2
m kg s
-3
; PCrab = 33x10 s
Substituting numbers for Crab then:
3

11
6
6.67 10 1100 10
kg m
-3
so  > 1.3 x 10 14 kg m -3
This is too high for a white dwarf (which has
a density of ~ 10 9 kg m-3 ), so it must be a
neutron star.
Pulsar energetics
• Pulsars slow down => lose rotational energy
- can this account for observed emission?
• Rotational energy:
1 2 I  4  2 I
E  I   2   2
2
2 P 
P
2
2
2
2


dE
d
2
I

4
I

dP
so
  2    3
dt dt  P 
P dt
Energetics - Crab pulsar
Crab pulsar
- M = 1 solar mass
- P = 0.033 seconds
- R = 10 4 m
2
2
2
30
8
2
I  MR   2 10 10 kg m
5
5
= 0.8 x 10 38 kg m 2
dE  4  0.8 1038
 1 dP 

10
 watts
and
2
dt
0.033
 P dt 
 1 dP 
 3 10 
 watts
 P dt 
42
from observations: 1 dP
P dt
~ 10
11 1
s
thus energy lost
dE
31
 310 watts
by the pulsar 
dt
This rate of energy loss is comparable to that
inferred from the observed emission, for
example in the 2-20keV range, the observed
luminosity in the Crab Nebula is approx.
1.5 x 10 30 watts.
Thus the pulsar can power the nebula.
Irregularities in pulsar emission
• Short timescales - pulsar slow-down rate is
remarkably uniform
• Longer timescales - irregularities apparent
- in particular, ‘glitches’
P
glitch
t
A glitch is a
discontinuous
change of
period
Glitches
Glitches are caused by stresses and fractures
in the external layers, the so-called ‘crust’ of
the neutron star.
P
10
For example,
~ 10
P
is the observed value for the Crab pulsar.
Pulse profiles
• Average pulse profile very uniform
• Individual pulses/sub-pulses very different
in shape, intensity and phase
t
Sub-pulses show high
degree of polarization
which changes throughout
pulse envelope
average envelope
Neutron Stars
• General parameters:
- R ~ 10 km (104 m)
- inner ~ 1018 kg m-3 = 1015g cm-3
- M ~ 0.2 - 3.2 solar masses
- surface gravity ~ 1012 m s-2
• We are going to find magnetic induction, B,
of a neutron star.
Magnetic induction
Magnetic flux,
 BdS 
constant
surface
radius Sun
8
4
Radius collapses from 7 x 10 m to 10 m
Surface
change
gives
Bns  7 10
 
4
BSun  10
8
2

9
  5 10

The general field of Sun is uncertain but
should be ~ 0.01 Tesla.
Thus the field for the neutron star,
Bns ~ 5 x 107 Tesla = 5 x 10 11 Gauss
Next - how long does B ns last?
Decay time of magnetic field
Decay time of
magnetic field:
3000m
  D s 0
10km
2
Polar cap
D - typical dimension over which field varies
significantly (for n.s., D ~ 3 x 10 3 m)
s - conductivity
Thus,
t ~ (3 x 103 ) (1010 ) (410 -7)
11
3
~ 10 seconds ~ 3 x 10 years
But magnetic field Crab pulsar still intense
after 1000 years => interior must be
superconducting (s and  both very large)
Neutron stars very dense and zero-T energy
supports star and prevents collapse.
Neutron star structure
crust
Neutron star segment
neutron
1.
liquid
solid
Superfluid
core?
neutrons, 2.
superconducting
p+ and e1km
crystallization
of neutron
9km
matter
10km
1018 kg m -3
Heavy nuclei (Fe)
find a minimum
energy when
arranged in a
crystalline lattice
2x1017 kg m -3
4.3x1014 kg m -3
109 kg m -3
1. Between densities of 4.3 x 10 14 kg m -3 and
2 x10 17 kg m -3, the lowest energy state is
reached when nuclei are embedded in an
electron and neutron fluid.
2. Above 2x1017 kg m -3, there is a continuous
neutron fluid with electrons and protons as
minor constituents.
Pulsar Magnetosphere
First, defining scale height
p
p  h
h
h
h
p
h
The pressure
difference
supports the
element of
atmosphere
The pressure difference is given by:
p
h  h    g
h
where  is the density

But p   kT
m
thus
(where m is the mass of
constituent particles)
p
mg
pm
and
 p

h
kT
kT
Formula for scale height
Integrating:
 mg 
p  p0 exp  
h
 kT 
=> pressure falls off exponentially with height
in atmosphere with uniform temperature.
 kT  has the dimensions of distance

h0  
 mg  and is called the ‘scale height’.
Neutron star scale height
For a neutron star,
g ~ 1012 m s-2
T ~ 1 million K
thus h0 ~ 0.01m
Thus the atmosphere of a neutron star is
only the order of 10cm!
Forces exerted on particles
Particle distribution determined by
gravity
etemperature
Fg n s
electromagnetism
FB
Gravity:
Fgns  me gns  9 10
31
10  10
12
18
Newton
Magnetic force:
FB  evB  1.6 10
5
19


2 10 m
8
10 T
3
33 10 s
4
 3  10 Newton
This is a factor of 1013 larger than the
gravitational force and thus dominates
the particle distribution.

Neutron star magnetosphere
Neutron star rotating in vacuum:

B
Electric field induced
immediately outside n.s. surface.
E  Bv  10  2 10 Vm
14
1
 2 10 Vm
8
6
1
pd on scale of neutron star radius:
  ER  1018V
Electron/proton expulsion
Neutron star particle emission

B
electrons
protons
Cosmic
rays
In reality...
• In reality, the charged particles will
distribute themselves around the star to
neutralize the electric field.
=> extensive magnetosphere forms
• Number difference +ve and -ve charges:
n  n ~ 7 10
8
B
Pperiod
m
3
(B in
Tesla)
Crab pulsar particle density
• This relationship gives an indication of the
particle density n:
• take, for example, the Crab pulsar 8
10
3
18 3
n  7 10
m  10 m
2
3 10
8
Pulsar models
Magnetic and rotation axes co-aligned:
eCo-rotating plasma,
mag field lines are
closed inside light
cylinder
Radius of light cylinder
must satisfy:
p
light cylinder, R L
2RL
c
P
A more realistic model...
• Note that if radiation pulses are to be
predicted, magnetic axis and rotation axis
cannot be co-aligned.
• => plasma distribution and magnetic field
configuration around a neutron star is much
more complicated.
Radio beam
Open
magnetosphere
B
A better picture
r=c/
Light cylinder
Closed magnetosphere
Neutron star
mass = 1.4 solar masses
radius = 10 km
B = 10 4 to 109 Tesla
The dipole aerial
Even if a plasma is absent, a spinning neutron
star will radiate if the magnetic and rotation
axes do not coincide.
This is the case of a
a
‘dipole aerial’
dE
4 6 2
2
  R B sin a
dt
Quick revision of pulsar structure
1. Pulsar can be thought of as a non-aligned
rotating magnet.
2. Electromagnetic forces dominate over
gravitational in magnetosphere.
3. Field lines which extend beyond the light
cylinder are open.
4. Particles escape along open field lines,
accelerated by strong electric fields.
Radiation mechanisms in pulsars
Emission mechanisms
Total radiation
intensity
exceeds
does not
exceed
Summed intensity of
spontaneous radiation
of individual particles
coherent
incoherent
Incoherent emission - example
eg. Radiating particles in thermodynamical
equilibrium ie thermal emission.
blackbody => max emissivity
So is pulsar emission thermal?
consider radio: n~10 8 Hz; 100MHz; 3m
Use Rayleigh-Jean approximation to find T:
2kTn
I n  
2
c
2
Watts m -2 Hz -1ster -1
Flux density at Earth, F~10-25 watts m -2 Hz -1
Source radius, R~10km at distance D~1kpc
then:


 
 D  10 3 10
F
I     F  2  
2
4

10
R 
(1)
2
 25
19 2
6
= 10 watts m -2 Hz -1 ster -1
From equation (1):


 
I n c
10 3 10
T
K
2
 23
8
2kn
2 1.4 10 10
2
 310 K
29
6
8 2
2
this is much higher
than a radio blackbody
temperature
K
Incoherent X-ray emission?
• In some pulsars, eg. Crab, there are also
pulses at IR, optical, X-rays and g-rays.
• - Are these also coherent?
• Probably not – brightness temperature of Xrays is about 100 billion K, equivalent to
electron energies 10MeV, so consistent with
incoherent emission.
radio
coherent
IR, optical, X-rays, g-rays
incoherent
Models of Coherent Emission
high-B sets up large pd => high-E particles
e-
ep+
electron-positron
pair cascade
B=1e8Tesla
1e16V
cascades results in bunches
of particles which can radiate
coherently in sheets
Emission processes in pulsars
• Important processes in magnetic fields :
- cyclotron
Optical & X-ray
emission in pulsars
- synchrotron
• Curvature radiation => radio emission
B
V. high mag fields; efollow field lines very
closely, pitch angle ~ 0
Curvature Radiation
• This is similar to synchrotron radiation.
If ve- ~ c and  = radius of curvature,
radiation v. similar to e- in circular orbit
with:
c
where nL is the
L 
2g
gyrofrequency
‘effective frequency’ of
emission is given by:
 m   Lg
3
Curvature vs Synchrotron
Synchrotron
Curvature
B
B
• Spectrum of curvature radiation
- similar to synchrotron radiation,
Flux
n 1/3
e -n
n
nm
• e- intensity c.r. << cyclotron or synchrotron
=> radio produced this way, need coherence
X-rays from curvature radiation?
• At frequency 1018Hz
luminosity ~ 10 29 J/s
requires g ~ 105
and no. particles radiating nV ~ 10 40- 1041
depending on density.
• This is too many for such energetic particles
=> X-rays emitted by normal synchrotron
Beaming of pulsar radiation
• Beaming => radiation highly directional
• Take into account
- radio coherent, X-rays incoherent
- location radiation source dep on frequency
• Model:
- radio from magnetic poles
- X-rays from light cylinder
Radio beam
Open
magnetosphere
B
A better picture
r=c/
Light cylinder
Closed magnetosphere
Neutron star
mass = 1.4 solar masses
radius = 10 km
B = 10 4 to 109 Tesla
Magnetic poles
Radiation source localized near mag poles.
b
(simple, axisymmetric case)
Rad source localized near
poles, narrow beam
produced along mag field.
Polar caps defined by
field lines tangential to
light cylinder.
2 b ~ 1  10
light cylinder
Important observed properties
• Pulses observed only when beam points at
Earth.
• Rad source probably localized within light
cylinder close to neutron star surface…
- no ‘wandering’ and directionality
• Problem: ALL radiation mechanisms at
different frequencies (coherent or not) must
have same orientation along magnetic field.
Origin of subpulses
subpulses
Brightening on
boundaries between
closed and open
lines may produce
subpulses
boundary
co-rotating plasma
Light Cylinder
Radiation source close to surface of light
cylinder.
P
P’
simplified case
Light cylinder – realistic but
complex!
Top view
aligned
 and B
Cross-section
torus
aligned
 and B
r=c/
B
What we see?
• Relativistic beaming may be caused by ~c
motion of source near light cylinder radiation concentrated into beam width :
g ,
1
g
1
1  b 
2
(the
Lorentz
factor)
• Also effect due to time compression (2g 2 ),
so beam sweeps across observer in time:
 P
 
 2
P
 1
 2 
3
 2g g 4g
Long Period Pulsars
• Not generally seen in optical or X-rays
- is this emission produced at light cylinder?
power radiated by
synchrotron
E B
For dipole magnetic field :
Also :
cP
RL 
2
2
2
Br
3
• So if particles of the necessary energy E
exist in all pulsars and emission occurs at
RL , we expect:
6
- radiated power  P
• and thus long period pulsars are weak
emitters.
In summary...
• Radio emission
- coherent
- curvature radiation at polar caps
• X-ray emission
- incoherent
- synchrotron radiation at light cylinder
Magnetic energy & nebula
• Neutron star slows down
=> energy sufficient to feed nebula
• What about the magnetic energy?
Consider energy
released at light
2RL
cylinder.
RL
Area =
2
4RL
• Magnetic field at RL is stretched out to v~c.
• Magnetic energy density =
B2
2 0
• Mag energy crossing light cylinder per sec:
3
B2
R0
2
But
for
PB 
4RL c
BRL  B0 3
2 0
RL
mag dipole
6
so
2


R
2

cR
2
L
PB  B0  0 
 RL   0
Substituting values for the Crab pulsar:
 
PB  10
8 2
 10 Js
31
6

 
 10  2 3 10 10
 6 
7
4 10
 10 
4
8
6 2
Js
1
1
like rotational energy release, this is also
comparable to observed emission from Crab
Nebula
Age of Pulsars
.
Ratio P / P (time) is known as ‘age’ of pulsar
In reality, may be longer than the real age.
Pulsar characteristic lifetime ~ 107 years
Total no observable pulsars ~ 5 x 10 4
Pulsar Population
• To sustain this population then, 1 pulsar
must form every 50 years.
• cf SN rate of 1 every 50-100 years
• only 8 pulsars associated with visible SNRs
(pulsar lifetime 1-10million years, SNRs
10-100 thousand... so consistent)
• but not all SN may produce pulsars!!!