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Pulsars. •. References:. •. 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. •. •. •.

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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

1. 1931: Chadwick --discovers neutrons.

2. 1934:Baade & Zwicky suggested neutron-stars, and

postulated their formation in supernovae.


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.

Derive the break-up rotation speed.

Argue that the higher harmonics of WD

cannot explain pulsars because of the very

high stability of the pulsar clock and because

mode periods decrease with age not increase

as seen for pulsar period.


Observational Properties of Pulsars

Period range: 1.5 milli-sec --- 8 sec.

Luminosity in the radio band ~ 1025 -- 1028erg/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.

1 Jy = 10^{-23} erg/cm^2/s/Hz


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)

R2B = R2nBn  Bn = B (R/Rn)2

Bn ~ 1012 Gauss


Lecture 4

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 ne


Pulse Dispersion

(Lyne & Graham-Smith in “Pulsar Astronomy)


Lyne & Graham-Smith in “Pulsar Astronomy)

Note: The derived <n_e> varies only by a factor of a few.


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

Larmor formula for electric dipole radiation: dE/dt = -2e^2 a^2/c^3 = -2(d d/dt)^2/c^3

The deviation of the breaking index from 3 could probably be due to torque on the

pulsar from outflow of particles.


B determined from the dipole radiation formula

(Lyne & Graham-Smith in “Pulsar Astronomy”; Cambridge U. press 1998)


Lecture 5 (detailed derivation of Goldreich-Julian results)

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

1969: Goldreich & Julian model published.


Summary of Axisymmetric NS magnetosphere results

Statvolt cm-1


(Goldreich-Julian density)

(same as the dipole

radiation formula)

Poyinting flux:


Lecture # 6

Summary of last lecture


Crab shows pulsed emission from radio to optical to >50 Mev! And moreover

The pulse shape is nearly the same over this entire EM spectrum, suggesting

A common origin for the radition which is believed to be synchrotron

(curvature radiation). The radio is produced not too far away from the

Neutron star (within 5-10 radaii) and high energy pulsed radiation is

Likely produced near the light cylinder.

The bolometric luminosity is pulsed radiation is about a factor 100 smaller

Than nebular radiation; pulsed radio is smaller than total pulsed radiation

By a factor of 10^4.

Crab nebula


The nebula is powered

by poynting outflow

from the pulsar.

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.

(linear polarization of 9%

averaged over nebula).

Rotational energy of the

NS Is the energy source for

The Luminosity ~ 1038 erg/s

(mostly x-ray & gamma)

Blue: x-ray

Red: optical

Synchrotron radiation


Plerion: is derived from the Greek word “pleres” which means “full”.

Crab nebula is the remnant of Sne explosion (perhaps type II) observed by the chinease

Astronomers in 1054 (July 4th). The pulsar at the center has a period of 33milli-sec.


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 temperatureTb ~ 1023 -- 1026 K!

This is clearly not possible --- as it will lead to enormous luminosity.

Laser pointers have power output in the optical

Light of ~ a few milli-watts, the bandwidth is

Less than an Angstrom, and the beamwidth is

About 1/2 degree. This gives the brightness

Temperature to be about 106 k!


Milli-sec pulsars have low magnetic field

(Lyne & Graham-Smith in “Pulsar Astronomy”; Cambridge U. press 1998)

Make sure to point out:

Most of the ms pulsars are in binary system.

Magnetic field for ms pulsars is low.

Ms pulsars lie below the spin-up line which

we will derive shortly.

ms pulsars are the most stable clock ---

dP/dt ~ 10-20 ; in other words it loses 10-13 s

in one year!


1. This is a low mass x-ray binary system (the companion star is low mass which

Supplying gas to the compact star via Roche-lobe overflow).

2. Milli-sec pulsars have been spun-up by the accreted gas.

3. Magnetic field must be low for the NS to be spun-up to milli-sec period.



g s-1

Spin-up of a NS in a binary system

(Spherical accretion)

Ram pressure of in-falling gas balances the magnetic pressure:



(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.)

The accretion is nearly spherical in that the accreting gas falls onto the star

roughly equally all around it, but the in falling gas is rotating at nearly the

Keplerian speed and carries angular momentum with it.


Spin-up equilibrium: accretion causes NS spin to Keplerian rotation

speed at Req. Milli-sec pulsars are formed in low-mass x-ray binaries

(LMXB) which have a NS with small magnetic field and a low-mass

Companion star. Such systems are old (compared to HMXB) and the

NS magnetic field might have decayed with time or burried by accretion.

Spin-up Equilibrium



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

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.

Example: SAX J1808.4-3658 is a LMXB with a 2.5ms x-ray pulsar

with magnetic field of 108--9 Gauss, and 2 hr orbital period.

Other LMXBs also have weak field but only 1 or 2 have pulsation.


Spin-up Time

A crude model describing the time evolution of NS spin is:


The spin-up time:



Anomalous x-ray pulsars (AXPs)

measured gives P/ ~ 103.5 -- 105.5 yr.


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.

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!


4-6 objects are known.

All but one SGRs are in the Galactic plane (one in LMC).

The one in the LMC is in a supernova remnant.

Soft gamma-ray repeaters


  • Thompson, astro-ph/0010016 & 0110679
  • Kaspi, V., 2004, Astro-ph/0402175
  • Woods, P.M., 2003, astro-ph/0304372

Summary of observational properties

(Rare events)

(bursts are associated with young stellar population)

(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.

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.

In 2 cases the measured P and gives B ~1015 Gauss.

Bursts repeat episodically; could be inactive for years and

then hundreds of bursts could appear in a week.

Generally thermal Bremsstrahlung spectrum with

kT ~ 20 - 50 kev.

(almost certainly SGRs are associated with NS)

(Not accretion powered! KE of NS rotation too little as well)

(The energy in magnetic field ~ 1047 erg; sufficient to

power these bursts).


Taken from:

Exploring the x-ray

Universe -- Charles &

Seward, 1995,

Cambridge U. Press

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