Pulsar glitches and neutron star masses

Alessandro Montoli

First-year PhD student workshop 2018

Milano, 10th October 2018

Neutron stars and pulsars

Extreme features of neutron stars:

• R ≈ 10 km

• M ≈ 1− 2M

• ρc & 1014 g / cm3

• P ≈ 10−3 − 101 s

• B ≈ 108 − 1015 G

Lighthouse model:

• Magnetic field axis misaligned with respectto the rotational axis

• Conversion of rotational energy intoelectromagnetic radiation

• Pulse frequency = rotational frequency

Pulsar glitches

-400 -200 0 200 400Days from Glitch

-374

9-3

744

ν (1

0-13 H

z s-1

)

•-400 -200 0 200 400

Days from Glitch

-374

7-3

742

00.

20.

40.

6 B0531+2151740.656

Δν

(μHz

)(a)

-400 -200 0 200 400

00.

6

B0531+2152084.072

(b)

-400 -200 0 200 400Days from Glitch

-376

0-3

740

-400-200 0 200 400

-40

48 B0531+21

53067.0780(c)

-400 -200 0 200 400Days from Glitch

-373

4-3

726 -400-200 0 200 400

00.

20.

40.

60.

8 B0531+2153970.1900

(d)

Four glitches of the Crab pulsar, Espinoza et al. (2011)

Two components: a normal charged one (visible) and a superfluid one.

Long recoveries ⇒ Effect due to superfluid components.

Diverse phenomenology:

• Mostly radiopulsars, but also magnetars and millisecond.• Periodic glitchers vs single glitchers.• Similar size vs. different size.

Neutron star structure

Glitch mechanism

• The normal component loses angular momentum.

• The superfluid rotates by forming an array ofquantised vortices.

• These vortices may pin to impurities of the crust.

The relative motion between the vortices and thesuperfluid itself causes a Magnus force. If the Magnusforce is strong enough to overcome the pinning force,vortices detach and a glitch occurs.

F

v

Critical lag

10-5

10-4

10-3

10-2

10-1

100

0 0.2 0.4 0.6 0.8 1

ωcr

[rad/

s]

x / Rd

M=1.1 MSunM=1.4 MSunM=1.9 MSun

∫vortex

fP dl =

∫vortex

fM dl

Critical lag for vortices extended to the whole star (Antonelli & Pizzochero,2017)

Maximal glitch

Angular momentum of the star:

L = IΩp + ∆L[Ωnp]

Time derivative of the angular momentum:

I Ωp + ∆L[Ωnp] = −I |Ω∞|

Integration between a time before and one after the glitch:

⇒ ∆Ωmax(t) =∆L[Ωnp(t)]

I

In the Newtonian framework:

∆L[Ωnp] =

∫d3x x2ρn(r) Ωnp

Lag as a function of time

Our aim is that of calculating the time dependence of the lag.

The equations that allow us to calculate this value are complicated and theydepend on the rotational parameters of the star (see Antonelli & Pizzochero2017).

We can use a simplified (and unified)prescription that allow us to study thequantitative trend of the lag:

Ωvp(x , ω∗) = min [Ωcrvp, ω

∗],

where Ωvp ≡ (1− εn)Ωnp and ω∗ ≡ |Ω∞|t.

Pulsar activity and ω∗act

The absolute pulsar activity is defined as:

Aa =1

Tobs

N∑i=1

∆Ωi .

We can also define a timescale for a pulsar that defines the mean waiting timefor a fictitious pulsar that exhibits only glitches of size ∆Ωobs and has activityAa:

tact =∆Ωobs

Aa.

Finally, we can find the nominal lag associated to this timescale:

ω∗act = tact|Ω∞|.

Mass estimate Mact

The maximal glitch ∆Ω(ω∗,M) depends only on the mass once themicrophysical parameters are fixed.

Measuring a maximum glitch ∆Ωobs and calculating the activity, we canestimate a mass for the pulsar inverting the relation:

∆Ω(ω∗act,Mact) = ∆Ωobs.

This mass estimate depends on the extension of the superfluid reservoir.

Critical lag

0.0 0.2 0.4 0.6 0.8 1.0r/R

10 4

10 3

10 2

10 1

100

101

Criti

cal l

ag (r

ad/s

)

Whole star1 n0

0.75 n00.68 n0

0.6 n0Crust

The critical lag depends on the cutoff considered.

Mass dependence on cuts

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Mac

t (M

sun)

CrustWhole star

0.60 n0Whole star

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Largest observed glitch (1e-4 rad/s)

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Mac

t (M

sun)

0.68 n0Whole star

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Largest observed glitch (1e-4 rad/s)

0.75 n0Whole star

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Mas

s [M

sun]

J0537-6910J0537-6910J0537-6910 J0835-4510J0835-4510J0835-4510

0.45 0.50 0.55 0.60 0.65 0.70 0.75Cutoff [n0]

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Mas

s [M

sun]

J1341-6220J1341-6220J1341-62200.45 0.50 0.55 0.60 0.65 0.70 0.75

Cutoff [n0]

J1801-2304J1801-2304J1801-2304

BSk20SLy4DDME2

Montoli, Antonelli & Pizzochero, arXiv:1809.07834Bayesian two-sample test: Antonelli, Gherardi, Montoli & Pizzochero, work in

progress.

Conclusions

• The mass estimate Mact depends on the extension of the superfluid region.

• Mact is generally lower with a smaller superfluid reservoir.

• Like in other models (Andersson et al. 2012, Chamel 2013, Delsate et al.2016), also here there is too little angular momentum stored in the crustfor glitches.

• A little bit more of superfluid reservoir is enough to describe glitches.

Open question: Where actually resides the superfluid involved in the glitch?

Thank you!

Mass upper limit Mmax

Maximum glitch = maximal glitch in the case of the critical lag.

∆Ωmax =π2

Iκ

∫ Rd

0dr r3 fP(r).

As for Mact, we can find a mass corresponding to the maximum glitch, Mmax.This is analytically independent on the extent of the superfluid region - as longas this extends at least in the pinning region - and entrainment paramters.

Already discussed in Pizzochero,Antonelli, Haskell & Seveso (2017) forthe Newtonian case.

Slow rotation approximation

R3Ω2

GM 1 Slow rotation condition, Hartle (1967)

Extreme cases (millisecond pulsars, M = 1.4M, R = 10km):

• J1748-2446ad, Ω = 4501 rad s−1 (not seen glitching):

R3Ω2/(GM) ≈ 0.11

• J1824-2452A, Ω = 2057 rad s−1 (seen glitching, Cognard & Backer 2004):

R3Ω2/(GM) ≈ 0.023

Mass upper limit Mmax, relativistic corrections

The calculation of the maximum glitch has been remade in the slow rotationframework in Antonelli, Montoli & Pizzochero (2018).

The dependence of Mmax on relativistic corrections is weak.

0.8 1.0 1.2 1.4 1.6 1.8 2.0M(MSun)

0

2

4

6

8

Maxim

um

glit

ch a

mplit

ude (

1e-4

rad

/s)

EOS: SLy

Newtonian (N)

Rigid vortices (R)

Slack vortices (S)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Maximum glitch amplitude (1e-4 rad/s)

0.00

0.01

0.02

0.03

0.04

0.05

Mm

ax(R

,S)

/ M

max(N

) -

1

Rigid vortices (R)

SLy4

BSk20

BSk21

SLy4

BSk20

BSk21

Slack vortices (S)

Maximum glitch amplitude

Maximal glitch

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0M

axim

al g

litch

(1e-

4 ra

d/s)

Crust Nmax < 3Nmax < 4Nmax > 4

0.60 n0 0.68 n0

10 3 10 2 10 1 100

Nominal lag (rad/s)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Max

imal

glit

ch (1

e-4

rad/

s)

0.75 n0

10 3 10 2 10 1 100

Nominal lag (rad/s)

1 n0

10 3 10 2 10 1 100

Nominal lag (rad/s)

Whole star

Pulsars have been chosen with Ngl > 2, Nmax > 1.1, Tobs |Ω∞|> 10−3 rad/s.

Metric of a slowly rotating star

ds2 = −e2Φ(r)dt2 + e2Λ(r)dr2 + r2 [dϑ2 + sin2 ϑ(dϕ− ω(r)dt)2]The radial profiles are still given by the TOV equations + EoS. ω(r) is given byan additional differential equation.

Drag of the local inertial frames

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 2 4 6 8 10 12 14

ω‾ (

r)/Ω

r [km]

1.4 MSun SLy41.4 MSun BSk211.4 MSun BSk20

The relevant quantity is ω(r) = Ω− ω(r).

Due to the linearity of the equation giving ω(r), ω(r)/Ω does not depend on Ω.

Moments of inertia

Total moment of inertia:

I =8π3

∫ R

0dr r4e−Φ(r)+Λ(r)

(ρ(r) +

P(r)

c2

)ω(r)

Ω

Moment of inertia of the superfluid component:

Iv =8π3

∫ R

0dr r4e−Φ(r)+Λ(r)xn(r)

(ρ(r) +

P(r)

c2

)mn

m∗n (r)

Moments of inertia

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

1 1.5 2

SLy

Mom

ent

of

inert

ia (

10

45

g c

m2)

M (MSun)

Itot (N)Itot (GR)In (N)In (GR)

1 1.5 2

BSk21

M (MSun)

1 1.5 2 2.5

BSk20

M (MSun)

• The total moment of inertia varies up to 50%.

• In GR the moment of inertia of the superfluid reservoir exceeds the totalone.