Pulsars

High Energy Astrophysics

jlc@mssl.ucl.ac.uk

http://www.mssl.ucl.ac.uk/

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4. Pulsars: Pulsed emission; Rotation and energetics; Magnetic field; Neutron star structure; Magnetosphere and pulsar models; Radiation mechanisms; Age and population [3]

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Introduction• Pulsars - isolated neutron stars

Radiate energy via slowing down of rapid spinning motion (P usually ≤ 1sec, dP/dt > 0)

• Neutron Stars – supported by degeneracy pressure; Fermi

exclusion principle restricts position hence Heisenberg uncertainty principle allows large momentum/high pressure

• 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)

4

Pulsars

• Discovered through their pulsed radio emission

• Averaging over many pulses we see:

Period

interpulse(~P/10)pulse

5

Pulse profiles

• Average pulse profile very uniform

• Individual pulses/sub-pulses very different in shape, intensity and phase

t

average envelope

Sub-pulses show high degree of polarization which changes throughout pulse envelope

6

Pulsar period stability

• Period extremely stable: 1 part in 10 indicates some mechanical clock mechanism

- this mechanism must be able to accommodate pulse-to-pulse variablity.

• Pulsations of white dwarf ??? (but Crab pulsar period (P~1/30 sec) too short)

• Rotation of neutron star ???

12

7

Rotation of a neutron star

Gravitational force > centrifugal force

where and P is the period

otherwise star would fly apart

r

mv

r

GMm 2

2

P

rv

2

For structural stability:

8

Reducing:

G = 6.67x10 m kg s ; P = 33x10 s

2

2

2

4

P

r

r

GM

GPr

M234

=>

3

34

r

M

but so

GP2

3

-11 3 -1 -2Crab

-3

9

Substituting numbers for Crab pulsar then:

so > 1.3 x 10 kg m

This is too high for a white dwarf (which has

a density of ~ 10 kg m ), so it must be a

neutron star.

611 1011001067.6

3

kg m -3

14 -3

9 -3

10

Pulsar energetics• Pulsars slow down => lose rotational

energy - can this account for observed emission?

• Rotational energy:2

2

2

22 24

22

1

P

I

P

IIE

dt

dP

P

I

P

I

dt

d

dt

dE3

2

2

2 42

so

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Energetics - Crab pulsar

Crab pulsar - M ~ 1 M

- P = 0.033 seconds - R = 10 m

= 0.8 x 10 kg m

4

8302 101025

2

5

2 MRI kg m2

38 2

12

and

from observations:

thus energy lost

by the pulsar

watts

dt

dP

Pdt

dE

110

033.0

108.042

38

wattsdt

dP

P

1103 42

11110~1 s

dt

dP

P

wattsdt

dE 31103

13

Rate of energy loss is greater than that inferredfrom the observed 2 - 20 keV emission, for which the observed luminosity in the Crab Nebula is

~ 1.5 x 10 watts.

Thus the pulsar can power the nebula.

Characteristic age for magnetic dipole energy loss

= P/2 P = 3.3.10-3/2 x 4.10-14 s ~ 1300 years

Crab Nebula exploded in 1054 AD

30

•

14

Neutron Stars

• General parameters: - R ~ 10 km (10 m) - ~ 10 kg m = 10 g cm - M ~ 1.4 - 3.2 M

- surface gravity, g = GM/R2 ~ 10 m s

• We are going to find magnetic induction, B, for a neutron star.

4

18 -3 15 -3

-2

inner

12

15

Magnetic induction

Magnetic flux,

Radius collapses from 7 x 10 m to 10 m

BdS constant

surface

8 4R

Surface change

gives

9

2

4

8

10510

107

Sun

ns

B

B

RNS

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• The Sun has magnetic fields of several different spatial scales and strengths but its general polar field varies with solar cycle and is ≈ 0.01 Tesla.

• Thus the field for the neutron star:

B ~ 5 x 10 Tesla = 5 x 10 Gauss

• If the main energy loss from rotation is through magnetic dipole radiation then:

B ~ 3.3 x 1015 (P P) ½ Tesla

or ~ 106 to 109 Tesla for most pulsars

ns7 11

•

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Neutron star structure

Neutron star segment

solid core?

crystallization of neutron

matter10 kg m 18 -3

neutron liquid

Superfluid neutrons, superconducting p+ and e-

crust

1km

9km10km

Heavy nuclei (Fe) find a minimum energy when arranged in a crystalline lattice

2x10 kg m

4.3x10 kg m

10 kg m

17

14

9 -3

-3

-3

1.

2.

outerinner

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Regions of NS InteriorMain 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

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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.

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White Dwarfs and Neutron Stars• In both cases, zero temperature energy – the Fermi energy, supports

the star and prevents further collapse

• From exclusion principle, each allowed energy state can be occupied by no more than two particles of opposite spin

• Electrons in a White Dwarf occupy a small volume and have very well defined positions – hence from uncertainty principle, they have large momentum/energy and generate a high pressure or electron degeneracy pressure

• Corresponding “classical” thermal KE would have T ~ 3.104 K and the related electron degeneracy pressure supports the star

• For a high mass stellar collapse, inert Fe core gives way to a Neutron Star and neutron degeneracy pressure supports the star

• NS has ~ 103 times smaller radius than WD so neutrons must occupy states of even higher Fermi energy (E ~ 1 MeV) and resulting degeneracy pressure supports NS

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Low Mass X-ray Binary providesObservational Evidence of NS

Structure

Neutron star primary

Evolved red dwarf secondary

Accretion disk

Roche point

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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

ISM

ISM

z = 0.35

z = 0.35

z = 0.35

• Fe XXVI & Fe XXV (n = 2 – 3) and O VIII (n = 1 – 2) transitions with z = 0.35

• Red plot shows: - source continuum - absorption features from circumstellar gas

• Note: z = (and = (1 – 2GM/c2r)-1/2

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X-ray absorption lines

quiescence

low-ionizationcircumstellar

absorber

redshifted, highly ionized gas

z = 0.35 due to NSgravity suggests:M = 1.4 – 1.8 M

R = 9 – 12 km

High T bustsFe XXVI(T > 1.2 keV)

Low T burstsFe XXV & O VIII(T < 1.2 keV)

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EXO0748-676

circumstellar material

origin of X-ray bursts

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Forces exerted on particles

Particle distribution determined by - gravity - electromagnetism

nsgFBFe-

181231 1010109 nseg gmFns

Newton

Gravity

Pulsar Magnetospheres

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Ts

mevBFB

83

419 10

1033

102106.1

Magnetic force

5103 Newton

This is a factor of 10 larger than the gravitational force and thus dominates the particle distribution.

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RNS

PNS

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Neutron star magnetosphereNeutron star rotating in vacuum:

B

Electric field induced immediately outside n.s. surface.

168 10210 VmBvE114102 Vm

VER 1810

Potential difference on scale of neutron star radius is:

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Electron/proton expulsion

B

protons

Neutron star particle emission

electrons Cosmic rays?

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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 ( x r) x B = 0

where E and B are electric and magnetic fields and is the vector angular velocity of the neutron star

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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

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Pulsar modelsHere magnetic and rotation axes co-aligned:

e-

light cylinder, r

Co-rotating plasma is on magnetic field lines that are closed inside light cylinder

cc

P

rc 2

Radius of light cylinder must satisfy:

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A more realistic model...

RadioEmission

RadioEmission

Velocity- of -Light Cylinder

• For r < rc, a charge-separated co-rotating 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

• For pulses, magnetic and rotation axes cannot be co- aligned.• Plasma distribution and magnetic field configuration complex for Neutron Star

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A better picture

Light cylinderOpenmagnetosphere

Radio beam

r=c/

BClosed magnetosphere

Neutron starmass = 1.4 M

radius = 10 kmB = 10 to 10 Tesla

4 9

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The dipole aerialEven if a plasma is absent, a spinning neutron

star will radiate – and loose energy, if the

magnetic and rotation axes do not coincide.

2264 sinBRdt

dE

This is the case of a ‘dipole aerial’ – magnetic analogue of the varying electric dipole

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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.

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Radiation Mechanisms in Pulsars

Emission mechanisms

Total radiation intensity

Summed intensity of spontaneous radiation of individual particles

exceeds

does not exceed

coherent

incoherent

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Incoherent emission - example

For radiating particles in thermodynamic

equilibrium i.e. thermal emission.

Blackbody => max emissivity

So is pulsar emission thermal?

Consider radio: ~108 Hz or 100MHz; ~3m

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Crab flux density at Earth, F~10 watts m Hz

Source radius, R~10km at distance D~1kpc

then:

2

22

c

kTI

Watts m Hz ster-2 -1 -1

-25 -2 -1

24

21925

2

2

10

10310

R

DF

FI (1)

Use Rayleigh-Jeans approximation to find T:

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I = 10 watts m Hz ster

From equation (1):

6 -2 -1-1

KK

k

cIT 2823

286

2

2

10104.12

10310

2

K29103 this is much higher than a radio blackbody temperature!

So -

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Incoherent X-ray emission?

• In some pulsars, eg. Crab, there are also pulses at IR, optical, X-rays and -rays.

• - Are these also coherent?

• Probably not – brightness temperature of X-rays is about 1011 K, equivalent to electron energies 10MeV, so consistent with incoherent emission.

radiocoherent

IR, optical, X-rays, -raysincoherent

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Models of Coherent Emissionhigh-B sets up large pd => high-E particles

1.1018V

B = 1.108TeslaR = 104 m

e- e-

e+

electron-positronpair cascade

cascades results in bunchesof particles which can radiatecoherently in sheets

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Emission processes in pulsars

• Important processes in magnetic fields : - cyclotron - synchrotron

• Curvature radiation => Radio emission

Optical & X-ray emission in pulsars

B High magnetic fields; electrons follow field lines very closely, pitch angle ~ 0o

=>

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Curvature Radiation

• This is similar to synchrotron radiation. If v ~ c and = radius of curvature, the radiation very similar to e- in circular orbit with:

2

cL where is the

gyrofrequencyL

e-

3 Lm ‘effective frequency’ of emission is given by:

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Curvature vs Synchrotron

Synchrotron Curvature

BB

45

• Spectrum of curvature radiation (c.r.) - similar to synchrotron radiation,

• For electrons: intensity from curvature radiation << cyclotron or

synchrotron • If radio emission produced this way, need coherence

Flux

1/3 exp(-)

m

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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

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Observational Evidence for Pulsar Emission Sites

• Radio pulses come from particles streaming away from the NS in the magnetic polar regions:

– Radio beam widths

– Polarized radio emission

– Intensity variability

• Optical and X-ray brightening occurs at the light cylinder– Radiation at higher energies only observed from young pulsars with short periods

– Only eight pulsar-SNR associations from more than 500 known pulsars

• Optical and X-radiation source located inside the light cylinder– Pulse stability shows radiation comes from a region where emission position does not vary

– High directionality suggests that emission is from a region where field lines are not dispersed in direction i.e. last closed field lines near light cylinder

– Regions near cylinder have low particle density so particles are accelerated to high energies between collisions

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The better picture - again

Light cylinderOpenmagnetosphere

Radio beam

r=c/

BClosed magnetosphere

Neutron starmass = 1.4 solar massesradius = 10 kmB = 10 to 10 Tesla4 9

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Light Cylinder

• Radiation sources close to surface of light cylinder

• Simplified case – rotation and magnetic axes orthogonal

Outer gap region- Incoherent emission

P

P`Outer gap region- Incoherent emission

X-ray andOptical beam

RadioBeam

Polar cap region- Coherent emission

Light Cylinder

50

• Relativistic beaming may be caused by motion of source with v ~ c near the light cylinder

- radiation concentrated into beam width

• Also effect due to time compression (2, so beam sweeps across observer in time:

2 – 3 needed to explainindividual pulse widths

,1 21

1

(the Lorentz factor)

32 42

1

2 PP

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In summary...

• Radio emission - coherent - curvature radiation at polar caps

• X-ray emission - incoherent - synchrotron radiation at light cylinder

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Age of Pulsars

Ratio (time) is known as ‘age’ of pulsar

In reality, may be longer than the real age.

Pulsar characteristic lifetime ~ 10 years

Total no observable pulsars ~ 5 x 10

.

2/ PP

7

4

•

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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!!!

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PULSARS

END OF TOPIC

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Neutron star segment

solid core?

crystallization of neutron matter

1018 kg m-3

neutron liquid

Superfluid neutrons, superconducting

p+ and e-

crust

1km

9km

10km

Heavy nuclei (Fe) find a minimum energy when arranged in a crystalline lattice

2x10 kg m

4.3x10 kg m

10 kg m

17

14

9 -3

-3

-3

outerinner

56

• Relativistic beaming may be caused by ~ c motion of source near light cylinder - radiation concentrated into beam width :

• Also effect due to time compression (2 ), so beam sweeps across observer in time:

,1 21

1

(the Lorentz factor)

32 42

1

2 PP

2

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Pulsar Model

• Radio emission from magnetic poles– Radio pulses due to particles streaming away from the

neutron star in polar regions along open field lines– Observed radio beam widths and polarized emission

support this model

• X-ray and optical emission from light cylinder– Radiation only seen from young short period pulsars

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Pulsars

Period

interpulse(~P/10)pulse

59

Pulse profiles

t

average envelope