Glitches in Superfluid Neutron Stars

GLITCHES IN SUPERFLUID NEUTRON STARS

Marco Antonelli

[email protected]

Centrum Astronomiczne im. Mikołaja Kopernika Polskiej Akademii Nauk

Quantum Turbulence: Cold Atoms, Heavy Ions ans Neutron Stars INT, Seattle (WA) – April 16, 2019 Outline

Summary:

Why glitches (in radio pulsars) tell us something about rotating neutron stars?

- A bit of hystory: why superfluidity is needed to explain glitches.

Intrinsic difficulty: model the exchange of angular momentum that causes the glitch (mutual friction)

- Many-scales (coherence length → stellar radius) - Possible memory effects (also observed in He-II experiments)

Which microscopic input do we need? I will focus on two quantities:

- Pinning forces (or better, the critical current for unpinning) - Entrainment

Is it possible to use glitches to obtain “model-independent” statements about neutron stars interiors?

SPOILER: yes, we have at the moment 2 models:

1 – Activity test → “entrainment” 2 – Largest glitch test → “pinning forces”

Question: is it possible to go beyond these two tests? What can be done? Neutron stars – RPPs

What we observe, since: What we think it is, since: Hewish, Bell et al., Observation of a rapidly pulsating Pacini, Energy emission from a neutron star (1967) radio source (1968) Gold, Rotating neutron stars as the origin of the pulsating radio sources (1968)

Magnetic field lines

Radiation beam

Coordinated observations with three telescopes: 22-s data slice of the pulsed radiation at four different radio Open issue: precise description bands obtained of the 1.2 s pulsar B1113+16. of beamed emission mechanism

Why?

Coherent (i.e. non-thermal) emission + brightness + small period: only possible for very compact objects

A vibrating WD or NS? excluded by pulsar-timing data: P increases with time. BH accretion? No regular pulses...

SOLUTION: pulsars are strongly magnetized rotating neutrons stars.

Neutron stars - structure

Sketch of nuclear matter phase diagram What we can not observe: internal structure

Main idea: compressing matter liberates degrees of freedom.

Gravity, holds the star together

Electromagnetism, makes pulsars pulse and magnetars flare

Strong interaction, determines the internal composition and prevents gravitational collapse

Weak interaction, determines the internal composition and affects reaction rates (chemical equilibrium and neutrino cooling) Neutron stars – M-R relation

TOV equations: hydrostatic equilibrium in GR

TOV inversion: Lindblom, ApJ 398, 569 (1992)

EOS lines not intersecting the J1614-2230 band are ruled out.

Rotation increases the maximum possible mass for each EOS: ≲2% correction for P~3 ms. Demorest et al. Nature 467 (2010) Shapiro delay for PSR J1614-2230

PB Demorest et al. Nature 467, 1081-1083 (2010)

For nearly edge-on millisecond radio

y pulsar systems, Shapiro delay allows to m r a l o infer the masses of both the neutron star f e

l d

and the companion. a o n r i o i p t a c h n u S f ” R G

t u o h t i w “

s l a u d i s e R Millisecond pulsar

” (1.97±0.04) M⊙ R G

h t i w “

s l a

u Companion: d i

s helium-carbon-oxygen WD e

R (0.500±0.006) M⊙

In contrast with X-ray-based mass/radius measurements, Shapiro delay provides no information about the NS radius. Shapiro delay: high-mass MSP J0740-6620

Cromartie et al. arXiv:1904.06759 A very massive neutron star: relativistic Shapiro delay measurements of PSR J0740+6620

12-year data set yielded a pulsar mass of 2.17+/-0.10 MSun at 68.3% credibility. Modeling the thermal pulse profile of this MSP at X-ray energies will aid in constraining both M and R. Pulsar timing

Period P and spin-down rate (period derivative) are precisely determined.

Inferred magnetic Different classes populate different feld regions (inferred age and magnetic field B).

)

(

e Sanity check from the braking index, but

t the second derivative of P is needed.

Stable clocks with predictable spin- down... except for random timing irregularities (glitches).

To date ~500 glitch events in ~170 objects reported in the Jodrell Bank catalogue (including few glitches in Period ( s ) magnetars and millisecond pulsars and a very small glitch in a binary system). Pulsar glitches

“Lack” of radiative/pulse profile changes (in RPPs):

→ Evidence for internal origin Vela: almost no recovery Long recoveries:

→ Impossible to explain if viscosity is present

Diverse phenomenology in RPPs:

→ probably due to different age (temperature), mass, rotational parameters...

Crab: nearly complete recovery Key point: to describe glitches we need that a NS is comprised of (at least) two components that exchange angular momentum.

Can we identify the (two?) components ?

Which part of the NS provides the angular momentum needed to spin-up the “observable component” ? Starquake model Ruderman, Nature 223, 597 (1969)

Liquid interior

The quake causes a sudden rearrangement of the moment of inertia and ultimately a glitch.

With this model it is possible to explain only the “jump”, not the subsequent relaxation.

Further problem: according to this theory, glitches should be rare: Baym&Pines, Ann. Phys. 66 (1971) ( in the Vela every ~3 yrs, not enough time to develop stresses, Giliberti et al. arXiv:1902.06345 )

Glitch recovery

The recovery from glitches in not uniform:

- Sometimes an “exponential” relaxation is observed, sometimes a “step-like” increase of the frequency.

- Both behaviours can be seen in the same object!

The instantaneous jump may be described in terms of a sudden rearrangement of the star crust, but not the recover.

We need another ingredient (the superfluid) to describe glitches.

Two-component model Baym et al. “Spin Up in Neutron Stars : The Future of the Vela Pulsar”, Nature (1969)

The long recovery time-scales (of order months) observed after the first Vela glitch were considered to be evidence for a weakly coupled superfluid component in the stellar interior.

→ Superfluidity → “no viscosity” → long relaxation timescale p i r d

n o r t u e braking N

” torque t s u → r c ? “ e r o c

r e ” n s n n I o r t

n “

Internal torque (“mutual friction”)

Two-component model Baym et al. “Spin Up in Neutron Stars : The Future of the Vela Pulsar”, Nature (1969)

P o s t-

) g

s l it / ch d a sp The long recovery time-scales (of order months) r i

n - 5 d - o observed after the first Vela glitch were considered to be w 0 n 1 m ~

( o gl

evidence for a weakly coupled superfluid component in d Q ΔΩ el gl y ΔΩ t the stellar interior. i c o l e v

gl r (1-Q) ΔΩ

l l u a w g n o n o d Superfluidity → “no viscosity” → long relaxation timescale i n A t i a w

v P

r r e-

k g e li gl s c tch (1-Q) ΔΩ a b s l pin O b - dow n mo del braking Time (~weeks) ” torque t s u Spin-up is given by the settling of the crust (starquake): r c “ the change in the moment of inertia sets the amplitude and the “healing parameter” Q.

” s Coupling timescale: fitted from post-glitch relaxation. n

o r t u → Relaxation is always exponential e

n “ → Still problems with very large glitches Internal torque (too much oblateness and building stress rate needed) (“mutual friction”)

Glitch mechanism (vortex mediated) Anderson & Itoh Pulsar glitches and restlessness as a hard superfluidity phenomenon (1975)

- The charged component steadily looses angular momentum

- Vortices are pinned the superfluid cannot spin-down a

p r

l → vortex line carried by the charged component a

F → a velocity lag builds up → neutron current in the frame of the normal component Magnus force

- Magnus force ≃ pinning force: the vortex line unpins

→ analogy between unpinning lag and critical current in superconductors → vortices can move: mutual friction between the components

Expulsion of vortex lines from bulk superfluid

Local: vortex creep Global: vortex avalanche (thermally activated) (trigger?) Analogy: “noisiness” in superconductors

velocity of flux-lines

S. Field et al. Superconducting Vortex Avalanches (1995)

velocity of flux-lines

“Strong pinning regimes by spherical inclusions in anisotropic type-II superconductors” R. Willa et al. (2018) In the complete phase diagram of the dynamical phases also the variable J should be considered! Hydrodynamics in neutron stars

The dynamics of vortices is hidden inside this term. Pinning modifies the mutual friction (when all vortices are perfectly pinned the mutual friction is zero) (x = n or p)

Assumption: the charged component Widely used approximation: (and possibly the superfluid core) circular flow (no meridional circulation) rotates as a rigid body

On the other hand, the superfluid in the crust can rotate non-uniformly. Entrainment coupling: crust and core

- In the crust:

Chamel N. Neutron conduction in the inner crust of a neutron star in the framework of the band theory of solids, Phys Rev C 85 (2012)

- Bragg scattering by crustal lattice, non-local effect: m* > 1

→ Consequence: the crustal superfluid is entrained by the normal component: reduced mobility of free neutrons is a potential problem for pulsar glitch theory.

- In the core:

Chamel N., Haensel P. Entrainment parameters in a cold superfluid neutron l

star core, Phys. Rev. C 73 (2006). x

t r Ind o uc

d e

v d l v th elo - Entrainment is due to the strong interaction between protons and neutrons e e c

e i ne ity f u f h ie tr l t on d

in c fl f u i id

- Very different mechanism: actually more similar to the original A&B idea e e

t n

r g

o a - Local effect: m*<1 C d V an m s

n roto s d p n ed tro e c → Consequence #1: Scattering of electrons off vortex cores: the core is rain ele

h t En ged

c g

dra

coupled to the crust on the timescale of a second. a t

Alpar et al. Rapid postglitch spin-up of the superfluid core in pulsars(1984) t A

→ Consequence #2: Dipole-dipole interaction with flux-tubes (core pinning?) Test #1: entrainment Entrainment parameters: Chamel, N. (2012). Neutron conduction in the inner crust of a neutron star in the framework of the band theory of solids.

Cumulated glitch amplitude of Vela

If the domain of the superfluid density is restricted to the crust, the inequality can be satisfied only if the Vela has a very low mass.

Entrainment correction is encoded into the moment of inertia. Note that the integration domain is defined by the superfluid density! Physical scales

Different scales are involved in glitch modelling:

Core → “Abrikosov lattice” spacing between flux-tubes ~ 1000 fm

Crust → crustal lattice spacing ~ 100 – 20 fm Inter-vortex spacing

Vortex-nucleus interction → coherence length ~ 10 – 100 fm

Vortex dynamics and vortex-lattice interaction → “mesoscale” (inter-vortex spacing)

Wigner-Seitz radius Mean inter-neutron spacing

Negele & Vautherin (1973) Chamel & Haensel, Living Rev.Rel. 11 (2008) Pinning energies P. Donati & P. Pizzochero, Nuclear Physics A, 724 (2004)

Pinning energy

Semiclassical approach: static LDA calculation (i.e. the local Fermi momentum is a function of the neutron number density). The pinning depends also on the vortex dynamics: TDLDA, Wlazlowski (2016) Vortex pinning and dynamics in the neutron star crust

Energy contributions: 2 → negative condensation energy ~ Δ / EF Alpar (1977) Pinnning and threading of quantized vortices in the pulsar crust superfluid → kinetic energy of the irrotational vortex-induced flow

→ Fermi energy EF of neutrons → nuclear cluster energy (Woods-Saxon potential)

Uncertain pairing gap Δ: modifies the strength and location of the pinning energies. Maximum pinning energies < 3.5 MeV.

Significant pinning occurs only in a restricted range: 0.07 n0 < nB < 0.2 n0 Donati & Pizzochero, Phys Lett B, 640 (2006) Pinning energies P. Donati & P. Pizzochero, Nuclear Physics A, 724 (2004)

Pinning energy

Negele & Vautherin (1973) Pinning forces in the inner crust ( S. Seveso et al., MNRAS, 2016 )

Lattice spacing ~ 50 – 10 fm

Qualitatively:

ξ ~ 10 – 100 fm Coherence length ξ ~ vortex core radius. Epin ~ 3 – 0.02 Mev Strong pinning when ξ < lattice spacing. Inner crust: Pinning to single defects VS “collective pinning”: Problem: how to calculate the “vortex-lattice” Rigid (straight) vortices are “less pinned”. interaction from the “vortex-nucleus” interaction ?

IDEA: consider a segment of vortex line (the length L is given by the tension) and average over translations Coherence length estimates: Mendell, ApJ 380 (1991) and rotations of the total pinning force divided by L.

Pinning forces (S. Seveso et al., MNRAS, 2016)

Lattice spacing ~ 50 – 10 fm The pinning force per unit length of a vortex depends on the mesoscopic interaction between the vortex and many pinning sites. Thus we to account for:

- rigidity of the vortex, tension T) - vortex radius (coherence length ξ) - the lattice spacing

ξ ~ 10 – 100 fm

Epin ~ 3 – 0.02 Mev

Inner crust:

Problem: how to calculate the “vortex-lattice” interaction from the “vortex-nucleus” interaction ?

IDEA: consider a segment of vortex line (the length L is given by the tension) and average over translations and rotations of the total pinning force divided by L. Pinning forces (S. Seveso et al., MNRAS, 2016)

Input parameters: Lattice spacing ~ 50 – 10 fm

Resulting pinning forces per unit length:

ξ ~ 10 – 100 fm

Epin ~ 3 – 0.02 Mev

Inner crust:

Problem: how to calculate the “vortex-lattice” interaction from the “vortex-nucleus” interaction ?

IDEA: consider a segment of vortex line (the length L is given by the tension) and average over translations and rotations of the total pinning force divided by L. Pinning forces - core (S. Seveso et al.,in preparation)

Core:

Vortex-flux tube interaction → Vortex-array interaction Pinning to flux-tubes negligible for normal pulsars

Coherence length estimates: Mendell, Astrophys. J., 380:515, 1991

Overlap of a vortex line and a flux tube is energetically favored because the volume of uncondensed fluid is minimized by such overlap (Srinivasan et al. 1990)

A larger contribution to the interaction energy is the magnetic interaction between the vortex and a flux-tube. The magnetic field in a flux tube is B ∼ 1015 G

(Alpar et al 1984, Jones 1991, Link 2012) Pinning forces - core (S. Seveso et al.,in preparation)

Non entangled fluxtube array

Entangled fluxtube array Dynamical simulation of maximal glitches

(Momentum per particle)

(Particle conservation: kinematic velocity) (x = n or p)

Simplified model: Antonelli & Pizzochero, MNRAS 2017

Assuming straight vortices, we project the 3D problem becomes 1D.

→ the vorticity of the superfluid is parallel to the rotation axis of the star → the normal component is rigid on the timescales of interest Dynamical simulation of maximal glitches

Vela largest glitch

Overshoot: transient event in which the normal component (“p”) rotates faster than the superfluid component (“n”).

This implies that the average velocity lag is reversed (before the glitch the neutrons must spin faster, otherwise there is no angular momentum reservoir). Test #2: pinning forces

Simulation of a Vela-like glitch: Δf / f ~ 10 −6 → Δf ~ 10 −5 Hz based on the model in Antonelli & Pizzochero Axially symmetric equations for differential pulsar rotation (2017)

Time evolution 4.0⋅10-4 protons v

3.5⋅10-4 Overshoot? The “corotation” point sets an upper -4 ) 3.0⋅10 limit to the observed glitch amplitude s / d a

r -4

( 2.5⋅10

] e s d / 1

u -4 [

t 2.0⋅10

i l Ω

Δ Observations p

m -4 a 1.5⋅10

h c t i l 1.0⋅10-4 G Instantaneous corotation of the two components 5.0⋅10-5 First minute: black window

0.0⋅100 1.0⋅10-1 1.0⋅100 1.0⋅101 1.0⋅102 1.0⋅103 1.0⋅104 1.0⋅105 Time since thet-t0 triggering[s] event (s)

Entrainment → cannot change the total moment of inertia → does not change the unpinning threshold (at least for circular flows)

Hence, the theoretical maximum glitch amplitude depends only on the unpinning threshold and the total moment of inertia: Test #2: results

Maximum glitch amplitude at corotation:

→ Only dependent on pinning forces and on the mass of the star

→ Entrainment independent

→ No need to consider straight vortex lines

→ As long as pinning is crust-confined the maximum glitch amplitude does not depend on the extension of vortices in the outer core

Pizzochero P.M., Antonelli M., Haskell B., Seveso S. (2017). Constraints on pulsar masses from the maximum observed glitch. Relativistic corrections in a nutshell

Antonelli M., Montoli A., Pizzochero P.M. Effects of general relativity on glitch amplitudes and pulsar mass upper bounds (2018)

Each component has 4-velocity (p is rigid, n can rotate non-uniformly)

The idea is to select the contribution of the superfluid to the total angular momentum

ZAMO observer Namely

… but the momenta contain the entrainment coupling and a Lorentz factor

In principle the angular momentum is not a linear functional of the lag → Hartle slow rotation: Test #2: relativistic corrections

Antonelli M., Montoli A., Pizzochero P.M. Effects of general relativity on glitch amplitudes and pulsar mass upper bounds (2018)

The unpinning condition for slack vortices is:

The corresponding maximum glitch amplitude is: Summary

- Lack of radiative/pulse profile changes:

→ evidence for internal origin

- Long recoveries:

→ thought to be due to superfluid component in the star

- Diverse phenomenology:

→ probably due to different age, mass, rotational parameters…

- Cross contamination between different fields is necessary:

→ condensed matter, astrophysics, nuclear physics

Key point: to describe glitches we need that a NS is comprised of (at least) two components that exchange angular momentum. Starquakes alone cannot describe Vela large glitches.

Which part of the NS provides the angular momentum needed to spin-up the “observable component” ?

How to describe the complex dynamical phases of superlfuid vortices in the crust/outer core ?

Better understanding of pulsar dynamics will probably lead to reliable mass constraints of isolated NS.

Two independent tests

Two different methods can be used to test our understanding of mesoscopic physics in the inner crust of a neutron star with glitch observations:

1 - Observed activity: provides a method to test the strength of entrainment coupling

Datta & Alpar Implications of the crustal moment of inertia for neutron stars EOSs (1993). [GR, no entrainment effect] Link et al. Pulsar Constraints on Neutron Star Structure and Equation of State (1999). [GR, no entrainment effect] Chamel, Crustal Entrainment and Pulsar Glitches (2013). [GR, +entrainment effect] Ho et al, Pinning down the superfluid and measuring masses using pulsar glitches (2015). [GR, cooling]

2 - Observed largest glitch amplitude: can be used to test the strength of pinning forces

Pizzochero et al. Constraints on pulsar masses from the maximum observed glitch (2017). [Newtonian, +pinning] Antonelli et al. Effects of general relativity on glitch amplitudes and pulsar mass upper bounds (2018) [GR, +pinning] Andersson N. et al. “Pulsar Glitches: The Crust is not Enough”. Phys. Rev. Lett. (2012) Chamel N. “Crustal Entrainment and Pulsar Glitches”, Phys. Rev. Lett. (2013)

For pulsars that have glitched many times it is possible to estimate the moment of inertia of the Cumulated glitch amplitude of Vela superfluid component (at least ~2% of the total).

Moment of inertia of the crustal superfluid would be sufficient to explain the observations, as long as entrainment is ignored.

Entrainment: Bragg scattering in the crust lowers the effective moment of inertia by a factor ~ 5.

Possible solution: as the star cools down the superfluid region extends into the core. Neutron star cooling simulations using “minimal” cooling scenario to find the interior temperature at various ages. (Ho et al. Sci. Adv. 2015)

Entrainment correction on the moment of inertia A simple but robust of the superfluid model assures that:

By using values for the Vela Activity parameter (this is problematic as implies very low masses or very stiff EOS) Macroscopic equations

Simple construction based on angular momentum equation for the whole star and conservation of vortex lines. It provides a rigorous and consistent realization of the early vortex-creep model (Alpar,1984) under the hypothesis that the Taylor-Proudman theorem is valid for spinning-down neutron stars: Ruderman & Sutherland, Rotating superfluid in neutron stars (1974).

Every variable must be expressed in terms of ΩP(t) and ΩV(t,x). The vortex velocity can always be expressed as

Vortex velocity can be found by solving vortex dynamics:

Message: by using ΩV instead of ΩN there is no need to introduce extra terms into the hydrodynamical equations (i.e. “entrainment” torques). Macroscopic equations: mutual friction

The vortex velocity can always be expressed as

Vortex velocity can be found by solving vortex dynamics:

This is the definition of infinitely rigid vortex lines: a vortex is a massless and rigid rod

We obtain where the Y term modulates the radial velocity of vortex lines. Perfect pinning prescription: if the actual velocity lag is below the critical lag, vortices are pinned ( Y = 0 ) Details: Mesoscopic Superfluid Entrainment pinning forces fraction parameters AntonelliPizzochero & Microscopic quantities: Microscopic functions of density functions , Axially symmetric equations for differential pulsar rotation, MNRAS (2017) 464 (1) 464 (2017) MNRAS differentialrotation, for pulsar Axiallysymmetricequations profiles needed profiles to simulate the simulate to the spherical spherical the and obtain obtain and Solve TOV Solve cylindrical cylindrical Obtain the Obtain dynamical dynamical equations + EOS + profiles

Critical lag for Moment of unpinning inertia weight functions of cylindrical radius cylindrical functions of Macroscopic Macroscopic quantities: Relativistic corrections: frame drag

Frame drag in the surrounding of a spinning NS has deep impact on the pulsar rotational dynamics. Slow rotation approximation (Hartle & Sharp, 1967): equatorial velocity much less than c (non-millisecond).

Total moment of inertia is “reduced”

Moment of inertia associated with the reservoir (corrected by entrainment, Antonelli et al. 2018) Pulsar parameters

Pizzochero, Antonelli, Haskell, Seveso, Constraints on pulsar masses from the maximum observed glitch, Nature Astronomy 1 (2017) Unified EOSs

Astrophysical imlications: Fantina et al. (2013) Neutron star properties with unified equations of state of dense matter

Unified EOSs of catalysed matter for application to non-accreting and non-magnetised cold Nss:

→ Outer crust: based on the seminal BPS model (Baym,1971). Assumption: BCC and full ionization.

→ based on effective density-dependent NN force with parameters fitted on nuclei properties

→ Semiclassical approach: BSk20, BSk21: extended Thomas Fermi + eff. Skyrme force Goriely, et al. Phys. Rev. C88 (2013)

→ Classical approach (compressible liquid drop model): SLy: based on the effective NN interaction SLy4 Douchin & Haensel, A&A 380 (2001) Inner crust structure

HF calculation of the ground state in the inner crust with effective NN interaction: density profiles of neutron and protons, at several average densities, along a line joining the centers of two adjacent unit cells. No pairing correlations are considered. Negele & Vautherin, Neutron star matter at sub-nuclear densities (1973)

Include pairing correlations: Baldo et al. The role of superfluidity in the structure of the neutron star inner crust (2005)

Band theory of solids: Carter et al. Entrainment Coefficient and Effective Mass for Conduction Neutrons in Neutron Star Crust: Macroscopic Treatment (2006) Negele & Vautherin (1973) zones

Negele & Vautherin, Neutron star matter at sub-nuclear densities (1973)

Parameters used by: Seveso et al., Mesoscopic pinning forces in neutron star crusts (2016)

Last two columns, pinning energies from: Donati & Pizzochero, Realistic energies for vortex pinning in intermediate-density neutron star matter (2006) References

Andersson, N. and Comer, G. L. (2006). A flux-conservative formalism for convective and dissipative multi-fluid systems, with application to Newtonian superfluid neutron stars. Andersson, N., Glampedakis, K., Ho, W. C. G., and Espinoza, C. M. (2012). Pulsar Glitches: The Crust is not Enough. Andersson, N., Sidery, T., and Comer, G. L. (2006). Mutual friction in superfluid neutron stars. Chamel, N. (2012). Neutron conduction in the inner crust of a neutron star in the framework of the band theory of solids. Chamel, N. (2013). Crustal Entrainment and Pulsar Glitches. Donati, P. and Pizzochero, P. M. (2004). Fully consistent semi-classical treatment of vortex–nucleus interaction in rotating neutron stars. Donati, P. and Pizzochero, P. M. (2006). Realistic energies for vortex pinning in intermediate-density neutron star matter. Jones, P. B. (1991). Rotation of the neutron-drip superfluid in pulsars - The interaction and pinning of vortices. Jones, P. B. (1992). Rotation of the neutron-drip superfluid in pulsars: the Kelvin phonon contribution to dissipation. Lombardo, U. and Schulze, H. J. (2001). Physics of Neutron Star Interiors, Lecture Notes in Physics. Springer Berlin. Mendell G. (1991) Superfluid hydrodynamics in rotating neutron stars. I - Nondissipative equations. II - Dissipative effects. Negele, J. and Vautherin, D. (1973). Neutron star matter at sub-nuclear densities. Prix, R. (2004). Variational description of multifluid hydrodynamics: Uncharged fluids. Seveso S., Pizzochero P.M., Grill F., Haskell B. (2016). Mesoscopic pinning forces in neutron star crusts.

Antonelli M., Pizzochero P.M. (2017). Axially symmetric equations for differential pulsar rotation with superfluid entrainment. Pizzochero P.M., Antonelli M., Haskell B., Seveso S. (2017). Constraints on pulsar masses from the maximum observed glitch. Antonelli M., Montoli A., Pizzochero P.M. (2018). Effects of general relativity on glitch amplitudes and pulsar mass upper bounds. Seveso S., Antonelli M., Pizzochero P.M., Haskell B., Mesoscopic pinning forces in magnetized neutron star core (in preparation). Montoli A., Anontelli M., Inferred neutron star masses with dynamical models of pulsar glitches.