Pulsars -...

Pulsars

D.Maino

Physics Dept., University of Milano

Radio Astronomy II

D.Maino — Pulsars 1/43

Pulsar Properties

Pulsars are magnetized neutron stars emitting periodic, shortpulses at radio wavelenghts with periods P between 1.4msand 8.5s

Pulsar=pulse and star. They show a lighthouse effect: theyemit rotating beams of radiation flashing when beam sweepsline-of-sight

Pulse periods are quite stable and Ppulse = Prot

Radio emission mechanism is still not completely clear. Butpulsars are astrophysical tools for

Neutron stars extreme conditions: deep grav. potentialGM/(rc2) ∼ 1, ρ ∼ 1014g/cm

−3,B ' 1014÷15Gauss

Pulse periods fractional errors ' 10−6 ⇒ power emitted inGW, NS masses, GR in strong fields, long-w GW from BHmergers or primordial.


Discovery

Serendipitously in 1967 by graduate student Jocelyn Bell andAnthony Hewish

Look for the un-expected!


Discovery

Serendipitously in 1967 by graduate student Jocelyn Bell andAnthony Hewish

Look for the un-expected!

JB observed the dips with P ≈ 1.3s, different from anyinterference signal and reappear exactly once per sideral day⇒ origin outside the Solar System


A brief history

1926∼1932: Fowler, Anderson & Stoner derived theChandrasekhar limit for White Dwarf (WD) mass

1932: Chadwick discovers neutron

1934: Baade & Zwicky predict NSs (unobservable)

1967: “LGM” discovery by Jocelyn Bell (Nobel Prize toHewish!)

1974: Hulse-Taylor binary pulsar - GW emission (Nobel Prize)

today: ∼ 2500 known pulsars


Neutron Stars: Formation

Neutron Stars: final stage of stellar evolution. Result from acore-collaspe SN

8M <∼ progenitor mass <∼ 20÷ 30MIron Core collapses when M > MCh → electron capture andphoto-disintegration of iron nucleiT 109K - e− + p → n + ν: neutrinos emissionCore radius decreases from 3000 km down to ∼ 30 km; B fluxand ω almost conservedρ > 4× 1017g/cm3: neutron degeneracy pressure halts thecollapse: shock on outer envolepose → supernovaGM2/R ' 1053ergs liberated

See Fogliazzo et al. PASA, 2015, 32, 9


Why neutron stars?

Short pulsation period → very compact objects (WD, BH orNS)

Stability of pulsation rule out BH due to their strong activity

In normal pulsating stars: relation between P and ρ (but orderof days not seconds)

Consider a spherical star with M and R with angular velocityΩ = 2π/P: limit on P by centrifugal accel. at equator notexceeds gravitational accel.

Ω2R <GM

R2⇒ P2 >

(4πR3

3

)3π

GM=

(3π

Gρ

)⇒ ρ >

3π

GP2

CP1919 has P = 1.3s⇒ ρ ≈ 108 g/cm−3 consistent with WD.


Why neutron stars?

Soon a pulsar with P = 0.033s discovered in the Crab Nebula⇒ ρ any stable WD (electron-degeneracy pressure)

Crab Nebula is a SN remnant: observed by Chineseastronomers in 1054

This confirms Baade & Zwicky: NS are compact remnants ofSN


NS & SuperNova Remnant

The number of association between pulsar and its SNR isdifficult:

SNR are visible for upto ∼ 0.1Myrpulsars are 10 to 100 times longer livedHigh pulsar velocities (' 100km/s) compared to progenitors(∼ 1÷ 10 km/s). Origin still debated: depends on SN Typeand neutrinos also play a role.

Best found in young SNR


Magnetic Field

Most of normal stars possess a dipolar magnetic field ∼ 100G

Star inner parts are fully ionized ⇒ good conductor

Remember:charges move along field lines and field lines aretied to charged particles

Collapse: R ∼ 106km → 10km, flux Φ

Φ =

∫daB · n

is conserved (Alfven’s Theo) ⇒ field boosted by ∼ 1010

reaching (typically) 1012G

Additional dynamo effect can create larger fields (1014−15G)observed in magnetars - very young neutron stars


Alfven’s Theorem

Consider a fluid (NS surface) and Maxwell Eqs. together withOhm’s Law

∇ · B = 0∂B

∂t+∇× E = 0

E + u× B =J

σ

In the limit σ →∞ (matter is degenerate) we get

∂B

∂t= ∇× (u× B)

Consider B flux and its time derivative

dΦB

dt≡ d

dt

∫SB · dS


Alfven’s Theorem

Either B or dS are changing

dΦB

dt=

∫S

∂B

∂t· dS +

∮CB · u× d l

=

∫S

∂B

∂t· dS−

∮Cu× B · d l

=

∫S

[∂B

∂t−∇× (u× B)

]· dS = 0

Remember that

a · (b× c) = b · (c× a) = c · (a× b)


Magnetic Dipole Pulsar Model

Traditional magnetic dipolemodel (Pacini 67)

Angle α between rotationand magn. axes

Gaps where particles areaccelerated

Light-Cylinder: whereco-roteting speed ΩRL = c



Inclined Magnetic Dipole emits e/m radiation with Ω = Ωrot

extracting kinetic-energy from NS

Larmor formula for a rotating electric dipole:

Prad =2q2v2

3c3=

2

3

(qr sinα)2

c3=

2

3

p2⊥c3

where p = qr is electric dipole moment and p⊥ = p sinα

This is the same for the magnetic dipole m so that

Prad =2

3

m2⊥

c3



For a sphere of radius R with surface magnetic field B themoment m = BR3

If dipole is rotating with velocity Ω it can be written asm = m0exp(−iΩ t) and hence we get

m = −iΩm0exp(−iΩ t)

m = Ω2m0exp(−iΩ t) = Ω2m

Emitted power from dipole is therefore:

Prad =2

3

m2⊥Ω4

c3=

2

3c3(B R3 sinα

)2(2π

P

)4

where P is the pulsating period (Ω = 2π/P).


Spin-Down Luminosity

Rotational energy E of a spinning object is related to itsmoment of inertia I with E = IΩ2/2 = 2π2I/P2

Moment of inertia of a sphere with radius R and mass M isI = 2/5M R2 and for a “canonical” NS it is around1045g cm2. For the Crab Nebula pulsar with P = 0.033 s therotational energy E ' 1.8× 1049 erg.

The magnetic dipole extracts energy from NS and thusincresing the rotation period P > 0

Combining information on P and P we derive the rate E atwhich energy is changing

We define spin-down luminosity as

−E ≡ −dErot

dt= − d

dt

(1

2IΩ2

)= −IΩ Ω


Spin-Down Luminosity

The Crab pulsar has P = 0.033 s and P = 10−12.4. AssumingI = 1045g cm2 we get:

−E =4π2 I P

P3=

4π2 1045 10−12.4

(0.033)3≈ 4×1038erg s−1 ≈ 105×L

If Prad ≈ −E the Crab lumonosity at low-frequencies(ν = P−1 = 30Hz) is the entire radio output of our Galaxy!

It is even large than the Eddington limit for a NS: ok sinceenergy source is not accretion

Most of this energy heats up the surrounding Crab Nebula:Megawave oven

The observed bolometric luminosity is consisten with thissimple model and with the following conversion andre-emission from radio to X-ray


Minimum Magnetic Field Strenght

If Prad = −E we can infer the minimum B strenght on the NSsurface

2

3c3(BR3sinα

)2(2π

P

)4

=4π2I P

P3⇒ B2 =

3c3I PP

2 · 4π2R6sin2α

Since sin2α ∈ [0, 1] we can arrange terms to obtain

B >

(3c3I

8π2R6

)1/2 (PP)1/2

The first term can be computed for a canonical pulsar

(B

gauss

)> 3.2× 109

(PP

s

)1/2


Minimum Magnetic Field Strenght

Let’s again consider the Crab pulsar (P = 0.033s andP = 10−12.4)(

B

gauss

)> 3.2× 109

(0.033× 10−12.4

)1/2= 4× 1012

Extremely large B: the associated energy (UB = B2/8π2) of 1cm3 of this B is ≈ 6× 1023erg ' 2 GW/year


Pulsar Age

Suppose Prad = −E and pulsar geometry almost constant(Bsinα ≈ constant) we can derive pulsar age τ from PP(initial P0 actual period)

PP =8π2R6 B2sin2α

3c3I

PP ≈ constant

Rewrite PP = PP as PdP = PPdt. Integrating over pulsarage: ∫ P

P0

PdP =

∫ τ

0(PP)dt = PP

∫ τ

0dt

since PP ≈ constant


Pulsar Age

Integration givesP2 − P2

0

2= PPτ

In the (reasonable) limit of P0 P pulsar age is given by

τ =P

2P

Note that τ depends on the measured P and P but not onother pulsar characteristics e.g. radius R, momentum ofinertia I or perperdicular magnetic field Bsinα

Very young pulsars are very oblate (high rotational velocity)→ emits quadrupole GW radiation and hence slow down →τ >∼ true age for young pulsars


Pulsar Age

Crab Pulsar: P = 0.033s and P = 10−12.4

τCrab =0.033s

2× 10−12.4≈ 4.1× 1010s ≈ 1300yr

somewhat larger since Crab Supernova dates 1054 AD

Vela Pulsar: P = 0.0893s and P = 10−12.9

τVela =0.0893

2× 10−12.9≈ 7.1× 1011s ≈ 11300yr

in agreement with other age estimates


PP diagram

Plays a similar role of theHR diagram for normal stars

Lines of constant τ , −E ,B

Newly formed pulsarsup-middle (SNR), movedown-right to populate the1 s pulsar. After that tooslow to power radio emission

Millisec pulsar down-left aremainly binary pulsarsrecycled via accretion


Braking Index

If magnetic dipole radiation is the only source of spin-downthen Prad = −E that becomes

2

3

m2⊥Ω4

c3= −IΩΩ⇒ Ω3 ∝ Ω

In general the braking index n is defined as

Ω ∝ Ωn = CΩn

and for pure magnetic dipole we have n = 3

n is determined by the observed P,P and P


Braking Index

Take time derivative of Ω

Ω = CnΩn−1Ω = n

(CΩn

Ω

)Ω = n

(Ω

Ω

)Ω = n

Ω2

Ω⇒ n =

ΩΩ

Ω2

Convert from Ω to P

Ω = 2πP−1 Ω = −2πP−2P

Ω = −2πP−2P + 4πP−3P2 = 2π

(P2

P3− P

P2

)


Braking Index

Combining into n gives

n =

(2π

P

)(P4

4π2P2

)(−2πP

P2+

4πP2

P3

)

n = 2− PP

P2

n = 5 is quadrupole radiation (both e/m and GW)

For different n other emission mechanisms

accretion from debris disk after explosioninstabilities on internal structure (glitches)a relativistic stellar wind


Glitches


Glitches

Slow-down process is not constant but:

almost periodic speed-up are presentmore common/regular in young pulsars and rare/spaced inolder onesbraking index during glitches n ∼ 1.4 → no magnetic dipole

Explanation involves internal NS structure

Coupling of angular momenta of solid crust and superfluidinterior

Transfer of (quantized) angular momentum from superfluid tosolid region → impeded by crystal structure up to a certainpoint


Pulsar Binary Systems

Almost all pulsars with P < 0.1s and B < 1011G are in binarysystems (variations in the observed pulse period)

Left:nearly circular orbit (e = 4.6× 10−5) with a companion mass ≈ 0.47M.

Right: Largest eccentricity (e = 0.888) and massive WD or NS companion

Recycled Pulsars: they are restored despite their low B byaccretion (mass and angular momentum) from companion

NS B move hot ionized gas to polar gaps → hot to emit inX-rays


Pulsars and ISM

Pulsars short duration pulses, small sizes and high TB probesthe ionized ISM


Pulsars and ISM

Pulsars short duration pulses, small sizes and high TB probesthe ionized ISM

Electron in ISM form a cold plasma with refractive index

µ =

[1−

(νpν

)2]1/2where ν is radio frequency and νp is the plasma frequency

vp =

(e2neπme

)1/2

≈ 8.97 kHz( necm−3

)1/2where for typical ISM (ne ≈ 0.03 cm−3) → vp ∼ 1.5 kHz


Pulsars and ISM

If µ is imaginary: no propagation through the ISM

For propagating waves: µ < 1 and group velocity vg = µc

For observations at ν νp we get

vg ≈ c

(1−

ν2p2ν2

)

If the source is at distance d a delay due to dispersion isobserved

t =

∫ d

0

dl

vg− d

c'∫ d

0

(1 +

v2p2ν2

)dl − d

c


Pulsars and ISM

t =

(e2

2πmec

)ν−2

∫ d

0nedl

We define the dispersion measure - DM (units of pc cm−3)

DM ≡∫

nedl

Final delay( t

sec

)≈ 4.149× 103

(DM

pc cm−3

)( ν

MHz

)−2


Pulsars and ISM

Gray-band: uncorrected ν−2

dispersion delay over thefrequency band

Data are folded into phaseperiod

With known DM (or afterseveral trayes) we get thebinned dispersion correctedpulse


Pulsars and ISM

Inhomogeneities in turbolent ISM can cause diffractive andrefractive scintillations similar to Earth atmosphere

Diffractive: from minutes to hours time-scales and from kHzto MHz frequency range with ≈ 1 order of magnitude fluxvariationsRefractive: timescale of weeks and a <∼ 2 factor on flux

Scintillation is related to scattering and hence pulsebroadering

Inhomogeneities in ISM cause multiple scattering resulting indifferent waves path with strong (∝ ν−4) frequencydependence with a long exponential pulse tail


Pulsars and ISM


Pulsar Timing

Pulsar Timing: the regular monitoring of rotation of the NSby measurements of the arrival of radio pulse times

Once you have timing you can:

probe the internal structure of NSprecise astrometrytest of theories of gravity in strong field regimepossible detection of GWs

Pulses are “folded” on the phase period → increase S/N

TOA (Time Of Arrival) precision can be very high ≈ µsMeasured TOAs have to be corrected for seveal effects:

t = tt−t0+∆clock−∆DM+∆R+∆E+∆S+∆R+∆E +∆S

determined from accurete position/orbital meas.


Pulsar Timing

For binary pulsars: time delay across NS orbit allows for highprecision meas. of orbital parameters

Keplerian parameters: projected semi-major axis x , longitudeof periastron ω, time of periastron passage T0, orbital periodPb and eccentricity e

Relativistic binaries are those involving a NS and anothercompact object i.e. WD, NS or even a BH

Additional 5 post-Kepleriar parametersrate of periastron advance ω (ellipt. orbits do not close in GR);orbital period decay Pb (emission of GW)γ for time-dilation and grav. redshift and two other Shapiroterms (r and s called range and shape)


Pulsar Timing

ω = 3

(Pb

2π

)−5/3

(TM)2/3(1− e2)−1

γ = e

(Pb

2π

)1/3

T2/3 M−4/3m2(m1 +m2)

Pb = −192π

5

(Pb

2π

)−5/3 (1 +

73

24e2 +

37

96e4)(1− e2)−7/2T

5/3 m1m2M

−1/3

r = Tm2

s = x

(Pb

2π

)−2/3

T−1/3 M2/3m−1

2

Here T ≡ GM/c3 = 4.925 . . . µs is solar mass in time units,

M = m1 + m2 total mass system in solar masses

Given 2 of these we can derive system individual masses


Pulsar Timing

Hulse and Taylor followed a binary pulsar B1916+13 made bytwo NSs

Accurate measurements of ω and γ allowed for the estation ofthe two masses m1 and m2

They compare the observed orbital period decay with GRpredictions


Pulsar Timing


Pulsar Timing Array (PTA)

Take a number of pulsars distributed across the sky

GW background would manifest as a local (at Earth)distortions of TOA common to all pulsars

The single i pulsar fraction frequency change δνi is

δνiνi

= αiA(t) + Ni (t)

Cross-correlate fractional changes from pulsar i and pulsar j

〈δiδj〉 = αiαj〈A2(t)〉+αi 〈A(t)Nj(t)〉+αj〈A(t)Ni (t)〉+〈Ni (t)Nj(t)〉

All blue terms are 0 since GW amplitude is not correlated withintrinsic noise; same for individual noises


Overall GW detectors


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