References:
1. A. G. Lyne & F. Graham-Smith, Pulsar Astronomy
Cambridge University Press, 1998
2. Shapiro & Teukolsky, WD, NS & BHs, Chapters 9 & 10
3. Lorimer: astro-ph/0104388 & 0301327
4. Camilo: astro-ph/0210620
1967: Hewish, Bell et al.
discover radio pulsars.
1974: Nobel prize to Ryle
(aperture synthesis)
and Hewish (pulsars).
1968: Gold proposes rotating NS model for Pulsars
Why neutron stars?
• Pulsation timescale for WD is: (R3/GM)1/2/2pi ~ 1 s
(The period of the closest orbit is similar;
moreover, these timescales decrease
with time - not increase as for pulsars).
• The break-up rotation period for WDs is also ~ 1 s.
• Not possible to get highly stable periodic signal
from BHs.
• The break-up rotation period, pulsation or dynamical
time for a NS is ~ milli-sec; rotation can explain the
observed period range and stability.
Observational Properties of Pulsars
•
Period range: 1.5 milli-sec --- 8 sec.
• Luminosity in the radio band ~ 1025 -- 1028 erg/s
• Radio luminosity distribution: N(L) dL  L-1
dL
(This holds over 3-decades in L. The total number
of active pulsars for L> 1 mJy kpc2 is ~ 150,000;
pulsars we observe are more luminous than
average for the Galaxy by a factor 10-100, the
Typical flux is of order 100 mJy).
• The spectrum index is ~ 1.5 I.e. f  --1.5
for  < 1 GHz.
Period Derivative
Some elementary considerations:
Collapse of a star -- conserving angular momentum &
magnetic flux -- to NS gives rise to msec P and B~1012 G
• M R2  = M R2n n  Pn = P (Rn/R)2
Pn ~ 1 ms
(P ~ 1 month; R/ Rn ~ 1010)
• R2 B = R2n Bn  Bn = B (R/Rn)2
Bn ~ 1012 Gauss
Pulsar Distance Determination
1. Parallax
2. Neutral H absorption at 21 cm:
The Doppler shift of the 21cm absorption line
together with the dynamical model of the Galaxy
can be used to identify the location of the H-cloud
and determine the distance to the pulsar.
3. Dispersion measure:
(pulses at different  arrive at different times)
2 = 2p + k2 c2
2p = 4 ne e 2 /me = 3x109 ne (rad/s)2
DM =
 dl n
e
Pulse Dispersion
(Lyne & Graham-Smith in “Pulsar Astronomy)
Lyne & Graham-Smith in “Pulsar Astronomy)
Magnetic dipole Radiation formula
Magnetic dipole rad. energy loss rate:
dE/dt = -2(d 2 m/dt 2)2 /3c3 ; m = Bn R3n/2
m: the magnetic moment of the NS
Or dE/dt = - Bn2 R6n n4 sin2 /6c3
dE/dt ~ 1035 erg/s for Bn ~ 1012 & P=0.1s
Solution of this equation and breaking index
E = I  n2
dn/dt = - K na; a: breaking index
For the dipole model a=3. Observations give a between 1.4 & 2.8
B determined from the dipole radiation formula
(Lyne & Graham-Smith in “Pulsar Astronomy”; Cambridge U. press 1998)
Pulsar magnetosphere
• NS surrounding is completely dominated by Electro-dynamics.
The pressure scale height on a NS for 108 K plasma is ~ 100 cm.
Thus, the number density 100 m above the NS surface < 10-5/cc
(provided that EM forces are unimportant).
• Goldreich-Julian model (aligned rotator)
• Charge density (pulled from the surface of NS)
• Electric potential drop along open B-field lines
• Poynting flux at the light-cylinder & NS slowdown rate

Summary of Axisymmetric NS magnetosphere results
E r (R) 
FE
FG
BR
2C
 4x10 B12
7
10 B12 Statvolt cm-1
7
n   10 B12
10
cm-3
(Goldreich-Julian density)
Poyinting flux:


dE BR2 R 64

dt
8 c 4
(same as the dipole
radiation formula)
Summary of last lecture
Crab nebula
The nebula is powered
by poynting outflow
from the pulsar.
(Plerion)
e-s with energy > 1014 ev
are accelerated by the
electric field in the polar
region; these e-s are needed
for emission at 10 kev.
Rotational energy of the
NS Is the energy source for
The Luminosity ~ 1038 erg/s
(mostly x-ray & gamma)
Synchrotron radiation
Blue: x-ray
Red: optic
Green:rad
Pulsar radio-emission must be coherent radiation
• Pulsar radio luminosity, assuming conical geometry,
is found to be in the range of 1025
--
1028 erg/s.
• The source area ~ (c t)2 ; where t is the pulse width
(t ~ a few milli-sec)
This implies
The brightness temperature Tb ~ 1023
--
1026 K!
This is clearly not possible --- as it will lead to enormous luminosity.
ms pulsars
Milli-sec pulsars have low magnetic field
(Lyne & Graham-Smith in “Pulsar Astronomy”; Cambridge U. press 1998)
Spin-up of a NS in a binary system
(Spherical accretion)
Ram pressure of in-falling gas balances the magnetic pressure:
Ýv
Ý
M
M
2
v 

2
4 r
4 r 2
Or
Req 
where
1
(8G )1/ 7
6


2GM B (r) B R*


 
r
8
8  r 
2
2
*
12
 27
 37  27
8 47
7
Ý
Ý cm
B* R* M M  1.25x10 B12R6 m m
4
7
12
7
 17
M  mMo, MÝ 1.39x1018 m mÝ
g s-1
(For disk accretion the viscous torque in the disk is equated to the
magnetic torque in from the star; Req turns out to have the same
form as above and the numerical coefficient is also similar.)


Spin-up Equilibrium
eq 
GM
3
R eq
or
Peq 
2
 eq
6
7
18
7
 87
 37
Ý
 2.2 B9 R6 m m
ms
Propeller effect: If the period of the NS is smaller than Peq
then matter is not accreted onto the NS. Click here to find details.
Spin-up Line:
The fastest spin rate for a NS corresponds to dm/dt =1.
Substitute for B in terms of P & dP/dt in the above equation
18
 37
 87
7
Ý
Pmin P15  0.8 R6 m
4
7
2 3
Ý

B


All binary radio pulsars lie below the spin-up line.

Many single ms pulsars are seen, and they too lie below
this line. It is believed that
these too were spun-up in a
binary system, and either the companion was evaporated
by the pulsar or was lost in a binary collision.
Spin-up Time
A crude model describing the time evolution of NS spin is:
d
Ý(  eq )Req2
I
 M
dt
or

2
 MÝR eq
t /I
(t)  eq 1 e

 The spin-up time:

tspinup 
I
2
MÝR eq
 87
9
 24
7
6
 17
 37
Ý I45 yr
 3.6x10 B R m m
6

Anomalous x-ray pulsars (AXPs)
References:
1.
2.
3.
4.
5.
Mereghetti et al., 2002, Astro-ph/0205122
Thompson, astro-ph/0010016 & 0110679
Pavlov et al., 2001, Astro-ph/0112322
Hurley, 1999, astro-ph/9912061
Gaensler, 2002, astro-ph/0212086
Summary of observational properties
Five confirmed cases of AXPs as of 2004.
Pulsation period: 5--12 s.
x-ray luminosity: Lx ~ 1034 --1036 erg s-1.
PÝmeasured gives P/PÝ ~ 103.5 -- 105.5 yr.
Black-body kT < 0.5 kev + steep power-law spectrum
No radio emission.
No binarycompanion detected.
2 or 3 are associated with supernovae remnants.
Energy source for AXPs?
•
P ~ 6s &  ~ 1 s-1  EKE ~ 5x1044 erg (insufficient to explain Lx).
(So unlike normal pulsars the energy source is NOT rotational)
•
Accretion is also ruled out since AXPs are not in binary systems.
The most likely source is the dissipation of magnetic field
P & dP/dt give B ~ 1014 -- 1015 Gauss.
(click here for the P-B diagram)
 Energy in B-field ~ 1045 -- 1047 erg
This is sufficient to explain Lx as resulting
from a steady decay of B-field inside NS!
Soft gamma-ray repeaters
References:
•
•
•
Thompson, astro-ph/0010016 & 0110679
Kaspi, V., 2004, Astro-ph/0402175
Woods, P.M., 2003, astro-ph/0304372
Summary of observational properties
• 4-6 objects are known.
(Rare events)
• All but one SGRs are in the Galactic plane (one in LMC).
(bursts are associated with young stellar population)
• The one in the LMC is in a supernova remnant.
(associated with NS or a BH)
• Three SGRs have been seen to pulse with period in
the range 5--8 s. Two of these 3 have pulsations in
x-rays during quiescence as well & are spinning down.
(almost certainly SGRs are associated with NS)
• Soft -ray and x-ray bursts with typical energy ~ 1041 erg.
Rise time ~ 10 ms & duration ~ 100 ms. Occasionally energy
Greater than 4x1044 erg. But no binary companion detected.
(Not accretion powered! KE of NS rotation too little as well)
15 Gauss.
Ý
In
2
cases
the
measured
P
and
gives
B
~10
P
•
(The energy in magnetic field ~ 1047 erg; sufficient to
power these bursts).
could be inactive for years and
• Bursts repeat episodically;

then hundreds of bursts could appear in a week.
• Generally thermal Bremsstrahlung spectrum with
kT ~ 20 - 50 kev.
Taken from:
Exploring the x-ray
Universe -- Charles &
Seward, 1995,
Cambridge U. Press
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Manchester, 2000, astro-ph/0009405
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