pulsars -...
TRANSCRIPT
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.
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Discovery
Serendipitously in 1967 by graduate student Jocelyn Bell andAnthony Hewish
Look for the un-expected!
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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
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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
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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
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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.
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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
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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
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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
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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
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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)
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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
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Magnetic Dipole Pulsar Model
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
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Magnetic Dipole Pulsar Model
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).
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
)
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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
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Glitches
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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
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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
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Pulsars and ISM
Pulsars short duration pulses, small sizes and high TB probesthe ionized ISM
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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
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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
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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
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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
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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
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Pulsars and ISM
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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.
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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)
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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
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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
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Pulsar Timing
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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
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Overall GW detectors
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