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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, dP/dt < 0)
Pulsars
• Discovered through their pulsed radio emission
• Averaging over many pulses we see:
Period
pulse
~P/10
interpulse
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
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
For structural stability:
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 M
- P = 0.033 seconds
4
- R = 10 m
2
2
2
30
8
2
I  MR   2 10 10 kg m
5
5
38
= 0.8 x 10 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 - 20 keV range, the observed
luminosity in the Crab Nebula is
30
~ 1.5 x 10 watts.
Thus the pulsar can power the nebula.
Neutron Stars
• General parameters:
- R ~ 10 km (104 m)
- inner ~ 1018 kg m-3 = 1015g cm-3
- M ~ 0.2 - 3.2 M
12
-2
2
- surface gravity, g = GM/R ~ 10 m s
• We are going to find magnetic induction, B,
for a neutron star.
Magnetic induction
Magnetic flux,
 BdS 
constant
surface
RNS
R
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 and
varies with the solar cycle but should be
≈ 0.01 Tesla.
• Thus the field for the neutron star:
7
11
Bns ~ 5 x 10 Tesla = 5 x 10 Gauss
Neutron star structure
crust
inner
outer
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
Regions of NS Interior
Main Components:
(1) Crystalline solid crust
(2) Neutron liquid interior
- Boundary at  = 2.1017 kg/m3 – density of nuclear matter
Outer Crust:
- Solid; matter similar to that found in white dwarfs
- Heavy nuclei (mostly Fe) forming a Coulomb lattice embedded in a relativistic
degenerate gas of electrons.
- Lattice is minimum energy configuration for heavy nuclei.
Inner Crust (1):
- Lattice of neutron-rich nuclei (electrons penetrate nuclei to combine with protons and
form neutrons) with free degenerate neutrons and degenerate relativistic electron gas.
- For  > 4.3.1014 kg/m3 – the neutron drip point, massive nuclei are unstable and
release neutrons.
- Neutron fluid pressure increases with 
Regions of NS Interior (Cont.)
Neutron Fluid Interior (2):
- For 1 km < r < 9 km, ‘neutron fluid’ – superfluid of neutrons and superconducting
protons and electrons.
- Enables B field maintenance.
- Density is 2.1017 <  <1.1018 kg/m3.
- Near inner crust, some neutron fluid can penetrate into inner part of lattice and
rotate at a different rate – glitches?
Core:
- Extends out to ~ 1 km and has a density of 1.1018 kg/m3.
- Its substance is not well known.
- Could be a neutron solid, quark matter or neutrons squeezed to form a pion
concentrate.
Low Mass X-ray Binary provides
Observational Evidence of NS
Structure
Neutron star
primary
Accretion
disk
Roche
point
Evolved
red dwarf
secondary
Gravitationally Redshifted Neutron Star Absorption Lines
• XMM-Newton found red-shifted X-ray absorption features
• Cottam et al. (2002, Nature, 420, 51):
- observed 28 X-ray bursts from EXO 0748-676
• Fe XXVI & Fe XXV
z = 0.35
(n = 2 – 3) and O VIII
(n = 1 – 2) transitions
with z = 0.35
z = 0.35
z = 0.35
• Red plot shows:
- source continuum
- absorption features
from circumstellar gas
ISM
ISM
X-ray absorption lines
quiescence
low-ionization
circumstellar
absorber
Low T bursts
High T busts
Fe XXV & O VIII Fe XXVI
(T < 1.2 keV)
(T > 1.2 keV)
redshifted, highly
ionized gas
z = 0.35 due to NS
gravity suggests:
M = 1.4 – 1.8 M
R = 9 – 12 km
EXO0748-676
origin of X-ray bursts
circumstellar material
Pulsar Magnetospheres
Forces exerted on particles
Particle distribution determined by
- gravity
e- electromagnetism
FB
Fg n s
Gravity
Fgns  me gns  9 10
31
10  10
12
18
Newton
Magnetic force
FB  evB  1.6 10
5
19
 3  10 Newton

RNS

2 10 m
8
10 T
3
33 10 s
4
PNS
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
Potential difference on scale of
neutron star radius is:
  ER  1018V
1
Electron/proton expulsion
Neutron star particle emission

B
electrons
protons
Cosmic
rays?
In reality...
• Charged particles will distribute themselves
around the star to neutralize the electric field.
• => extensive magnetosphere forms
• Induced electric field cancelled by static field
arising from distributed charges or -
E + 1/c (W x r) x B = 0
Magnetosphere Charge Distribution
• Rotation and magnetic polar axes shown co-aligned
• Induced E field removes charge from the surface so charge and
currents must exist above the surface – the Magnetosphere
• Light cylinder is at the radial distance at which rotational velocity of
co-rotating particles equals velocity of light
• Open field lines pass through the
light cylinder and particles stream
out along them
• Feet of the critical field lines are at
the same electric potential as the
Interstellar Medium
• Critical field lines divide regions of
+ ve and – ve current flows from
Neutron Star magnetosphere
Pulsar models
Magnetic and rotation axes co-aligned:
e-
Co-rotating plasma is on
magnetic field lines that
are closed inside light
cylinder
Radius of light cylinder
must satisfy:
light cylinder, rc
2rc
c
P
A more realistic model...
• For pulses, magnetic and rotation axes
cannot be co- aligned.
• Plasma distribution and magnetic field
configuration complex for Neutron Star
• For r < rc, a charge-separated corotating magnetosphere
• Particles move only along field lines;
closed field region exists within field-lines
that touch the velocity-of-light cylinder
• Particles on open field lines can flow out of
the magnetosphere
• Radio emission confined to these open-field
polar cap regions
Radio
Emission
Radio
Emission
Velocity- of Light Cylinder
Radio beam
Open
magnetosphere
B
A better picture
r=c/
Light cylinder
Closed magnetosphere
Neutron star
mass = 1.4 M
radius = 10 km
B = 10 4 to 109 Tesla
The dipole aerial
Even if a plasma is absent, a spinning neutron
star will radiate – and loose energy, if the
magnetic and rotation axes do not coincide.
a
This is the case of a ‘dipole
aerial’ – magnetic analogue
of the varying electric dipole
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
For radiating particles in thermodynamic
equilibrium i.e. thermal emission.
Blackbody => max emissivity
So is pulsar emission thermal?
Consider radio: n~108 Hz or 100MHz; l~3m
Use Rayleigh-Jeans approximation to find T:
2kTn
I n  
2
c
2
Watts m -2 Hz -1ster -1
-25
-2
-1
Crab flux density at Earth, F~10 watts m Hz
Source radius, R~10km at distance D~1kpc
then:


 
 D  10 3 10
F
I     F  2  
2
4
W
10
R 
2
 25
19 2
(1)
So 6
In = 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 1011 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-
ee+
electron-positron
pair cascade
B = 1.108Tesla
R = 104 m
1.1018V
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
=>
- synchrotron
emission in pulsars
• Curvature radiation =>
Radio emission
B
High magnetic fields;
electrons follow field lines
very closely, pitch angle ~ 0o
Curvature Radiation
• This is similar to synchrotron radiation.
If ve- ~ c and  = radius of curvature, the
radiation very similar to e- in circular orbit
with:
c
where nL is the
L 
gyrofrequency
2g
‘effective frequency’ of
emission is given by:
 m   Lg
3
Curvature vs Synchrotron
Synchrotron
Curvature
B
B
• Spectrum of curvature radiation (c.r.)
- similar to synchrotron radiation,
Flux
n 1/3
exp(-n)
n
nm
• For electrons:
intensity from curvature radiation << cyclotron or
synchrotron
• If radio emission produced this way, need coherence
Beaming of pulsar radiation
• Beaming => radiation highly directional
• Take into account
- radio coherent, X-rays and Optical incoherent
- location of radiation source depends on frequency
- radiation is directed along the magnetic field lines
- pulses only observed when beam points at Earth
• Model:
- radio emission from magnetic poles
- X-ray and optical emission from light cylinder
Radio beam
Open
magnetosphere
B
The better picture
- again
r=c/
Light cylinder
Closed magnetosphere
Neutron star
mass = 1.4 solar masses
radius = 10 km
B = 10 4 to 109 Tesla
Light Cylinder
• Radiation sources close to surface of light cylinder
P
X-ray and
Optical beam
Outer gap region
- Incoherent emission
P`
Outer gap region
- Incoherent emission
Radio
Beam
Polar cap region
- Coherent emission
• Simplified case – rotation and magnetic axes
orthogonal
• Relativistic beaming may be caused by ~ c
motion of source near light cylinder radiation concentrated into beam width :
g ,
1
g
1
1   
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
In summary...
• Radio emission
- coherent
- curvature radiation at polar caps
• X-ray emission
- incoherent
- synchrotron radiation at light cylinder
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!!!
PULSARS
END OF TOPIC