The role of mass and superfluid reservoir on glitches

The role of mass, equation of state and superfluid reservoir in pulsar glitches

A. Montoli , M. Antonelli , P. M. Pizzochero

Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy
Istituto Nazionale di Fisica Nucleare, sezione di Milano, Via Celoria 16, 20133 Milano, Italy
Nicolaus Copernicus Astronomical Center of the Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland
E-mail: alessandro.montoli@unimi.itE-mail: mantonelli@camk.edu.pl
Preprint version, September 2018.
Abstract

Observations of pulsar glitches provide useful insights on the internal physics of a neutron stars: recent studies show how it is in principle possible to constrain pulsar masses from observations related to their timing properties. We present a generalisation of a previous model for the rotational dynamics of superfluid and rotating neutron stars. By examining the possibility of different extensions of the -wave superfluid domain, we study the dependence of the mass inferred within this model on the still uncertain extension of the region in which the neutron pairing gap is big enough to allow for superfluity. Hence, it is possible to quantify the general trend that to a smaller extension of the pairing channel’s region should correspond a smaller mass estimate of the glitching object. The employment of different equations of state for the star matter does not affect the general tendency described above: future independent estimates of masses for a couple of objects in our sample has the potential to calibrate the results and put indirect constraints on the microphysics of neutron stars.

keywords:
dense matter - stars:neutron - pulsars:general
pagerange: The role of mass, equation of state and superfluid reservoir in pulsar glitchesThe role of mass, equation of state and superfluid reservoir in pulsar glitchespubyear: 2018\AtBeginShipout

1 Introduction

In the current description of large pulsar glitches - a sudden decrease of the observed pulsation period - the neutron star is assumed to be divided into two components (Baym et al., 1969) that can rotate with slightly different angular velocities: a normal component spins down because of its strong coupling with the magnetic field of the star, while a superfluid component (comprised by Cooper pairs of neutrons) mimics macroscopic rotation by means of many quantised vortices that are present at the mescoscopic scale (for a recent review on superfluidity in neutron stars see Haskell & Sedrakian, 2017). The possibility of pinning between vortices and impurities in the inner crust makes the superfluid to lag behind the normal component (Anderson & Itoh, 1975). In this way a superfluid current develops in the frame of the crustal lattice, and an excess of angular momentum builds up during the observed inter-glitch period. A small fraction of this angular momentum reservoir is then released during a glitch. This set of ideas translates into a vast phenomenology of pulsars glitches: according to the data stored into the Jodrell Bank pulsar glitch catalogue (Espinoza et al., 2011), some stars have been observed glitching only once, others several times, and among these some pulsars have shown only glitches of approximately the same size, while others do not seem to have a preferred amplitude (Melatos et al., 2008a). In some cases, a pulsar can show a big glitch and several glitches of orders of magnitude smaller, a fact that it is tempting to be justified by invoking different types of glitch mechanisms. However, for the largest ones this vortex-mediated description is still the most accepted (see e.g. the review of Haskell & Melatos, 2015).

One of the many still open problems in the two-component scenario is where the superfluid reservoir involved in the glitch is located. In numerous works the neutron superfluid has been considered restricted only to the crust (in particular for what concerns the present work see Datta & Alpar, 1993; Link et al., 1999, for examples of models with crust-confined superfluidity). More recently, it has been shown that the presence of a non-dissipative interaction between the two components - known as entrainment, firstly introduced by Andreev & Bashkin (1976) in the framework of He-He superfluid mixtures - reduces the angular momentum associated to the superfluid component. Given the large estimates of the entrainment effect in the crust (Chamel, 2012), it is impossible to explain the glitching activity of some pulsars, such as that of the Vela pulsar (Andersson et al., 2012; Chamel, 2013; Delsate et al., 2016).

On the other hand, other models account for the possibility of a superfluid extended to the whole star. In order to implement the early idea of Ruderman & Sutherland (1975), according to which the motion of the superfluid is columnar, Pizzochero (2011) proposed that the core corotates with the superfluid in the crust and thus participates in storing angular momentum: an array of straight vortices is assumed to fill the whole superfluid domain, and the angular momentum reservoir naturally extends into the core111 Note that if the ensemble of vortices is not assumed straight but has negligible collective tension at the macroscopic scale, the absence of a normal matter layer is not sufficient by itself to guarantee significant decoupling of the core superfluid from the normal component. In this case the core is expected to be strongly coupled to the crust (Alpar et al., 1984b), unless an additional pinning mechanism in the core is invoked (Jones, 1991; Ruderman et al., 1998). because theoretical calculations indicate the absence of a normal matter layer dividing the P-wave superfluid in the core and the S-wave one in the crust (Zuo et al., 2004). The likely presence of hyperons (or other exotic phases in the inner core) suppresses the presence of superfluid neutrons in that region but, as discussed in Antonelli & Pizzochero (2017a), this does not change significantly the moment of inertia of the reservoir if the array of vortices can resist bending due to an enhanced collective rigidity.

Clearly, in order to overcome the difficulty posed by strong entrainment, it is also possible that the only type of superfluid involved in the glitch phenomenon is that in the singlet state (Ho et al., 2015), which naturally extends beyond the crust-core boundary. In this case, the superfluid reservoir also depends on the model for the superfluid gap and on the internal temperature of the neutron star, whose estimate is based on the inferred age of the pulsar and on the particular cooling model used.

In any of the cases considered above, the moment of inertia of the pinning region depends on the unknown properties of nuclear matter near and above the saturation density. In fact, different equations of state (EoSs) have different stiffness, which implies different structural proprieties of the star such as maximum mass, crustal thickness and free neutron fractions.

In particular, thanks to the formalism developed in Antonelli & Pizzochero (2017a), it has been possible to set an upper limit on the mass of a glitcher by means of its largest glitch (Pizzochero et al., 2017, hereafter Paper-I). This limit does not depend on the actual extension of the superfluid domain (as long as it completely overlaps the region where pinning is possible) and on the strength of entrainment. Moreover, general relativistic corrections on this mass upper bound are of the order of few percent, allowing for Newtonian models (Antonelli et al., 2018).

In this work we extend the exploratory results of Paper-I that go beyond the calculation of the mass upper bound: we focus on the proposed method to estimate a glitcher’s mass by employing its largest observed glitch and its mean waiting time between large glitches. In Paper-I, the estimate has been made in the particular case of a superfluid reservoir extended to the whole star. The aim of this work is that of relaxing this hypothesis, by evaluating the dependence of this mass estimate on different extensions of the S-wave superfluid domain, thus implicitly considering the possibility of different superfluid gaps for the state, similarly to what has been proposed by Ho et al. (2015).

We consider a sample of glitchers with observational criteria that allow us to estimate the typical timescale between two large glitches, and we study how the distribution of the mass estimates for these objects varies for different reservoirs related to the extension of the -wave gap.

2 General approach

In the following subsections we describe the general approach underlying the method developed in Paper-I. The first step is to choose a particular dynamical model for the evolution of the angular momentum reservoir. After integration of the model, a time-dependent upper bound on the observed glitch amplitude has to be compared with the observed timing properties of a given pulsar. This allows for a test of the model (and the inputs needed to set it) or, conversely, to give mass estimates of glitching pulsars.

2.1 Following the evolution of the maximal glitch amplitude

In order to uniform with previous works, we follow the standard notation for two-component models of superfluid neutron stars, indicating with the subscript the quantities related to the normal component (the crustal lattice and everything tightly coupled to it), which is assumed to be rigid (Easson, 1979); the subscript is used to indicate the superfluid neutrons (Andersson & Comer, 2007).

In absence of precession, the total angular momentum of a slowly rotating neutron star in general relativity (in the sense provided by Hartle, 1967) can always be split as:

(1)

where is the total moment of inertia of the star and is the angular velocity of the rigidly rotating normal component as seen from an inertial observer at spatial infinity (Antonelli et al., 2018). The functional represents the extra angular momentum due to the possible presence of a lag between the two components. In the slow rotation approximation, this functional is linear in the lag . Hence, we have that

(2)

where we can bring the time derivative inside the functional because of its linearity. The positive parameter represents the observed secular spin down of the pulsar.

A glitch occurs on timescales over which the spin down effect is negligible; integrating the above equation between a time before and after a glitch and ignoring the term give

(3)

An upper limit on the observed glitch amplitude can be found by imposing that at a generic time the star is instantaneously in a state of average corotation222 The pulsar is instantaneously in a state of average corotation at a certain time if . This is an hypothetical condition that may never be realised during real glitches, unless an overshoot of the normal component occurs.. Note that it is not required that the lag (which can be non-uniform inside the superfluid domain) is everywhere null, but only that at a given instant after the triggering of the glitch. Therefore, we define the maximal glitch amplitude at a generic time as

(4)

Since a possible overshoot of the normal component would occur within the current black window for timing observations due to the fast processes which regulate the spin up (Antonelli & Pizzochero, 2017a; Graber et al., 2018; Haskell et al., 2018), the quantity in Equation (4) sets an upper limit to the amplitude of a glitch that has been triggered at time , when the lag is .

We now need a prescription to obtain the time dependence of the lag. A way to proceed would be that of employing a set of two-fluid hydrodynamic equations encoding macroscopic mutual friction (Andersson et al., 2006) and the effect of pinning (see e.g. Alpar et al., 1984a; Antonelli & Pizzochero, 2017a; Khomenko & Haskell, 2018, for different examples on how to implement the effect of pinning in macroscopic two-fluid equations). Such equations would depend explicitly on the observed angular velocity of the star and on the secular spin down rate, implying that the hydrodynamical problem should be integrated for every different pair of rotational parameters of a given pulsar. Moreover, the dynamical equations will also depend on some structural properties of the star, like the EoS and the total mass, as well as the presence of entrainment and pinning.

Let us assume to fix a particular pulsar of angular velocity and spin down rate . We impose , and solve the assumed hydrodynamical equations for and . A first problem is how to choose an appropriate initial condition for the lag: it is possible to circumvent it by taking , but every initial lag such that may be used as well: since we are interested to simulate the system on the timescale of years, little difference is introduced by considering different initial conditions for the lag, provided the fact that it is null on the average at (eventual differences should manifest as transients at the beginning of the simulation but the long-term dynamics is driven by the global conservation of the angular momentum). Clearly this problem related to the arbitrariness of the initial condition does not exist if the assumed dynamical equations are based on a body-averaged rigid model (like the ones used in the simulations of Sourie et al., 2017).

This procedure means that the dynamics of the particular pulsar is simulated starting from the instantaneous corotation condition (at a large glitch with overshoot has just occurred). The integration of the dynamical model gives us , so that it is also possible to track the dynamics of the quantity by means of Eq. (4).

Once the theoretical curve has been obtained, we still need to find a way to compare it with some information extracted from the observed timing behaviour of the particular pulsar under study.

2.2 Contrasting the model with pulsar’s timing data

In the previous sections we saw how the quantity sets a theoretical limit for the glitch amplitude at time in a pulsar that emptied its reservoir at . However, we do not know when an observed pulsar actually empties its reservoir of angular momentum (maybe never). In principle there is no systematic argument for saying that the pulsar reaches corotation during its largest observed glitch: we have to assume it, bearing in mind that typically in glitching pulsars, only a small fraction of the accumulated angular momentum is expected to be released at each relaxation event.

A sequence of maximal glitches would result in a strong positive correlation between the glitch amplitudes and the waiting time between them, in contrast with the idea of glitches as random events in a self-organised system. It is indeed expected that the momentum released in each event do not necessarily correlate with the momentum accumulated since the previous glitch: the effect of a finite-size reservoir that can be eventually emptied is expected to generate only weak correlations between the glitch amplitude and the waiting time to the previous glitch (Melatos et al., 2008b; Haskell & Melatos, 2015). So far, these correlations induced by the finite size of the reservoir have not been observed in any pulsar except only for Vela at the low confidence level of (Melatos et al., 2018).

Given the lack of evidence for backward waiting time-size correlation, our assumption that maximal glitches can occur in real pulsars may be satisfied only for very few events in some pulsars. Hence, we tentatively extend it to all pulsars showing large glitches of amplitude rad/s, but only for their largest event in size. Continuous monitoring of glitching pulsars will provide more and more secure identification of an observational bound to the glitch amplitude in each object, and, therefore, of the maximum amount of momentum that can be exchanged between the two components.

We now need to find a value for the typical timescale between two events that may empty significantly the angular momentum reservoir. To do this we need to rely on an intrinsic property of the pulsar under study, the absolute activity . Because of the random and impulsive nature of glitch sequences and of the slowness of the spin-down which drives the system, it is quite difficult to extrapolate good estimates for from glitch databases, except for few pulsars (we will discuss this in greater detail in Sec 4). For a pulsar which have undergone glitches of size during an extended time interval , the absolute activity is estimated as

(5)

Called the largest among the observed glitches, it is possible to introduce a parameter which roughly counts the number of times that average corotation has been realized during the observation,

(6)

The choice to use the largest observed glitch in this formula is coherent with the hypothesis that the largest glitch in a puslar is a maximal glitch; however, we stress that also a sequence of smaller glitches can be effective in emptying the reservoir. Therefore, an estimate of is given by

(7)

Finally, as discussed in Paper-I, the condition provides a refinement of the (more robust and less model dependent) absolute upper bound given by emptying the fully-replenished reservoir, formally obtained when tends to the critical lag for vortex unpinning (see below).

3 Newtonian unified model

Despite the general form of the hydrodynamical equations is known, modelling mutual friction introduces some degree of arbitrariness, which is unavoidable due to the still poorly-understood vortex dynamics in neutron stars. The dynamical equations are therefore always assumed at a certain level, in particular for what concerns problems related to the unpinning and repinning of vortices.

For this reason, we now use the general concepts presented in the previous section employing the particular model already presented in Paper-I, which may be dynamically inaccurate but captures in a simple way the most important feature we are interested in: neglecting the finite-temperature effects on vortex dynamics, pulsars are slowly driven systems which internal clock is set by the parameter .

Following Paper-I, we now restrict to the Newtonian limit of Eqs. (1) and (2); this approximation is still acceptable when calculating the maximum glitch, as seen in Antonelli et al. (2018). To date, the only cylindrical model accounting for realistic stratification, non-uniform entrainment and differential rotation of the superfluid is described in Antonelli & Pizzochero (2017a), in which vortex lines are assumed to be parallel to the axis of rotation333 Cylindrical coordinates are used, with representing the cylindrical radius, the azimuthal angle and the coordinate along the rotation axis of the neutron star. The radial coordinate from the centre of the star is . . In this case, it is useful to introduce an auxiliary variable defined as:

(8)

where is the entrainment parameter (Prix, 2004). In this way, the rescaled lag

will depend on only, even if the entrainment parameter depends on . The dependence of on turns out to be

(9)

where is the star radius, is the curve that describes a straight vortex line placed at a distance from the rotation axis and is the superfluid mass density. As a last step we recall the particular prescription used to obtain the critical lag for the unpinning of vortices, obtained by equating the total Magnus and pinning forces along the line :

(10)

where is the quantum of circulation and is the pinning force per unit length (Antonelli & Pizzochero, 2017a). This value corresponds to the maximum achievable lag between the two components as a function of .

At this point, a possible way to proceed would be that of employing the set of two-fluid hydrodynamic equations described in Antonelli & Pizzochero (2017a). As discussed in Sec. 2.1, those equations should be solved for every pulsar. We circumvent this complication by introducing a common unified timescale for pulsars with very different secular spin down rates. By taking as the moment in which the star is at corotation, we define a nominal lag as . This is a rescaling of time which allows us to treat all pulsars within an approximate but unified model, regardless of their particular rotational parameters (see also the discussion in Antonelli & Pizzochero, 2017b). Within this unified model, the increasing value of determines the actual lag built between the two components since corotation:

(11)

Note that within this prescription the lag will depend - besides on the nominal lag - only on the mass of the star, once the microphysical inputs like EoS, pinning force and entrainment parameters are fixed. Thus, also the maximal glitch444 We make a distinction between maximal and maximum glitch, which is a maximal glitch that occurs when the lag saturates to the critical lag value everywhere, i.e. . Actually, the maximum glitch is truly entrainment independent (Antonelli et al., 2018); since the evolution of is entrainment dependent, the maximal glitch is entrainment dependent too. will depend only on the mass of the star, namely .

According to the general approach outlined in Sec. 2.2, we estimate the typical nominal lag elapsed between two large glitches as

(12)

Finally, if we measure a maximum glitch amplitude for a particular pulsar and we calculate its activity, we can invert the relation in order to obtain an estimate for its mass.

Up to this point, we have not assumed anything about the location and extension of the region in which the neutron superfluid resides. In the case of the maximum glitch amplitude corresponding to the critical lag in Eq. (10), we obtain

(13)

It is easy to show that does not depend on the vortex extension inside the star, provided that they extend at least in the pinning region of the crust of the star and we do not have any pinning in the core:

(14)

On the other hand, the maximal glitch amplitude of Eq. (4) will be different according to the region in which we assume the presence of superfluid, due to the dependence in Eqs. (9) and (10) on . As we will see in Section 5, the assumption of considering the superfluid limited in spherical shells ending at different depths will change the value of , and therefore of .

4 Pulsar sample

# J-name
[rad/s] [ rad/s rad/s] [ rad/s] [ rad/s]
1 J0205+6449 95.61 28.19 6 2.39 3.6332 104.24 48.90 7.05
2 J0534+2200 188.16 237.21 27 2.19 0.8862 3636.98 2232 292
3 J0537-6910 389.72 125.18 45 16.66 2.6774 491.13 30.62 0.17
4 J0631+1036 21.83 0.79 15 1.55 0.7161 3.79 5.37 0.86
5 J0742-2822 37.68 0.38 8 1.14 0.0347 2.99 3.23 1.07
6 J0835-4510 70.34 9.84 20 11.67 2.1805 148.44 13.49 0.14
7 J1048-5832 50.81 3.96 6 2.86 1.5461 18.98 9.53 1.04
8 J1105-6107 99.43 2.49 5 2.30 0.9661 10.48 7.37 1.04
9 J1119-6127 15.40 15.18 4 1.99 0.8917 81.29 50.91 10.74
10 J1341-6220 32.50 4.25 23 5.46 1.0004 26.09 6.56 0.42
11 J1413-6141 22.00 2.57 7 2.38 0.5301 6.65 3.86 0.30
12 J1420-6048 92.16 11.24 5 3.33 1.8606 37.00 13.62 0.37
13 J1709-4429 61.32 5.57 4 3.08 1.7612 28.54 12.68 2.11
14 J1730-3350 45.05 2.74 3 2.65 1.4426 18.78 10.28 0.19
15 J1737-3137 13.95 0.43 4 1.18 0.1873 2.06 1.64 0.28
16 J1740-3015 10.35 0.80 36 3.67 0.2762 7.73 2.27 0.07
17 J1801-2304 15.11 0.41 13 4.22 0.0977 3.77 0.97 0.03
18 J1801-2451 50.30 5.15 5 2.69 1.8890 23.07 10.98 0.73
19 J1803-2137 47.01 4.72 5 3.34 2.2530 30.74 12.65 0.76
20 J1826-1334 61.91 4.59 6 3.33 2.2170 39.86 17.31 2.68
21 J1833-1034 101.53 33.14 4 2.04 0.0076 27.86 21.27 2.54
22 J1841-0524 14.10 0.74 5 1.85 0.1455 1.92 0.95 0.10
23 J1932+2220 43.49 1.73 3 2.13 1.9449 6.39 4.31 0.70
24 J2021+3651 60.57 5.59 4 2.82 1.8461 22.09 11.47 0.48
25 J2229+6114 121.71 18.45 6 2.42 1.4873 52.66 28.48 3.58
Table 1: Sample of pulsar used in this work. The data has been obtained by crossing the information obtained from the ATNF Pulsar Catalogue (www.atnf.csiro.au/research/pulsar/psrcat, see also Manchester et al. 2005) and the Jodrell Bank Glitch Catalogue (www.jb.man.ac.uk/pulsar/glitches.html, see also Espinoza et al. 2011). The data has been retrieved in August 2018.
Figure 1: The glitch activity of the four pulsars with , see Table 1. The Crab pulsar (in the first plot) has been added for comparison because of its peculiar behaviour. The red step-like curve represents the cumulated glitch amplitude, its width representing the observed uncertainty on both the glitch amplitude and the glitch date (this is particularly visible in the case of J1341-6220). The blue line is a linear fit to the data. Since the time on the horizontal axis has been reported in terms of the nominal lag, the slope of the blue line gives the value of the dimensionless parameter .
Figure 2: Every pulsar with the largest glitch exceeding rad/s and with rad/s is shown in the plane. Each pulsar is represented by an arrow, whose tail (marked by a dot) indicates the point . The head ends at the nominal lag corresponding to , so that the length of the arrow is proportional to . There are some pulsars in this sample with a degenerate arrow (since is extremely close to unity): these single glitchers are marked with a blue name and a single dot. The five pulsars of Fig. 1 are marked in red. Lines of constant are also shown, for the values , , and so on until .

In this work we have chosen a sample from all the glitching pulsars stored in the Jodrell Bank Glitch Catalogue (Espinoza et al., 2011). In particular, the sample has been selected by asking specific conditions:

  • The total number of glitches should be at least 3, in order to have the least number to fit the activity of the star.

  • The cumulative size of the glitches should be at least 1.1 times the size of the biggest glitch (this quantity corresponds to the parameter ), in order to eliminate the single glitchers from the sample.

  • Finally, we ask that rad/s, in order to eliminate the pulsars that evolve slowly (and so requiring a lot of time in order to replenish the angular momentum reservoir) or that have not been observed for a long period.

In this way, we obtain a sample of 25 stars, which we report in Table 1. The nominal lag has been calculated starting from the absolute glitching activity . In particular, it is possible to define a dimensionless activity as:

(15)

The dependence of on and can be easily obtained from its definition, and turns out to be:

(16)

We show in Figure 1 the activity fit for five particular pulsars, i.e. those with > 4 and the Crab pulsar. The data are plotted in a nominal lag - cumulative glitch amplitude plane, so that the slope of the line corresponds to the dimensionless parameter .

Clearly, the particular cutoff that we have imposed on the observed value of is quite arbitrary: the low threshold 1.1 has been chosen in order to select, as a first tentative step, a large number of potentially interesting objects with diverse rotational parameters and glitch amplitudes. This is shown in Fig 2, where we plot all the known pulsars with the largest glitch exceeding rad/s and with rad/s, according to the Jodrell Bank catalogue. Pulsars with rad/s but rad/s are all single glitchers, in the sense that is very close to one. It is interesting to notice how most pulsars of our sample have and are therefore roughly distributed on a line when plotted in the plane.

5 Results

We now study the dependence of the mass estimate on the extension of the superfluid reservoir. In order to do so, we perform different spherical cutoffs in the extension of the region involved in glitch. First of all, to make contact with Paper-I, we study the case of the superfluid reservoir extended to the whole star. We stress that, despite this extreme scenario is the one proposed in Pizzochero (2011), we use here a different dynamical model which is only similar to the original “snowplow model” reviewed in Haskell & Melatos (2015). Then, we consider the case of superfluid extended from the neutron drip density to , , and , where is the benchmark for the nuclear saturation density (Chamel & Haensel, 2008). Finally, we also consider the case of reservoir limited to the crust.

The cutoffs are performed by imposing for , where is the baryon number density. The choice of those cutoffs is justified by physical motivations: the region between the crust-core interface and is the region where most of the superfluid gaps of singlet state ends. In particular, corresponds to the value where in a neutron star with temperature the superfluid region ends, considering a SFB superfluid gap (Schwenk et al., 2003).

We consider two unified EoSs, SLy4 (Douchin & Haensel, 2001), BSk20 (Goriely et al., 2010), and a relativistic mean field model, DDME2 (Lalazissis et al., 2005). Unfortunately the last EoS do not have any consistently calculated superfluid neutron fraction in the crust. Therefore, we glued the crust from the SLy4 EoS to the DDME2, keeping the crust-core transition density to be the one of SLy4. The gluing was carried out by ensuring the continuity of the chemical potential, as discussed by Fortin et al. (2016). This procedure, while ensuring thermodynamic consistency, also produces a quite strong first order phase transition at the core-crust interface: the profile of the DDME2+SLy4 EoS turns out to be flat for between fm and fm (namely, and ).

Figure 3: Critical lag as a function of the normalised radius for a star described by the BSk20 EoS and for some of the different cutoffs employed.

As an example, we present in Figure 3 the critical lag for straight vortex lines of Equation (10) as a function of the normalised cylindrical radius and for all the different cutoffs considered here. The calculation has been performed by employing the pinning force of Seveso et al. (2016) and the entrainment parameters obtained in Chamel & Haensel (2006) for the core and Chamel (2012) for the crust. As expected, has higher values in the central region of the star for a smaller superfluid reservoir. In fact, since the superfluid extends in a smaller spherical layer, the superfluid vortices are less subject to Magnus force, while the influence of the pinning force is constant: the Magnus force thus needs a bigger lag to overcome the pinning force. On the other hand, in the outermost cylindrical region, due to the fact that the vortex lines are there completely immersed in the crust, the peak of is unchanged for every cutoff. Since the critical lag is different between cutoffs, it is evident how the lag (hence via Equation (4)) evolves differently. However, we have to stress that does not depend on the cutoff we are considering, as the different form of the critical lag is compensated by the second integral over in Equation (9).

Figure 4: Maximal glitch as a function of the nominal lag for different masses , in the case of the BSk20 EoS. The nominal lag spans four orders of magnitude, from rad/s to rad/s. The different curves represent different masses, starting from (the highest curve) up to (the lowest one). To facilitate the visualisation, the and the curves are plotted with black dashed lines instead of grey dotted. We also plot the largest observed glitch and the nominal lag for the glitchers in the sample described in the text. We display in grey stars with , in red stars with , and in blue stars with .

Let us now study the dependence of the maximal glitch as a function of the nominal lag and for different masses. The results are shown in Figure 4, for the BSk20 EoS. As we can see, the maximal glitch raises faster as a function of for more extended reservoir, in particular for lower masses. We have included in the Figure the sample of glitchers discussed in Section 4. We can notice, in particular, that two of them (J0742-2822 and J1833-1034) do not fit the lowest curve corresponding to the highest mass achievable from BSk20. This is probably due to the fact that these stars may have not reached a glitch big enough to be fitted, but we expect bigger glitches to be observed on these stars. This fact resembles the case of the Crab pulsar (PSR J0534+2200): this star has recently shown his largest glitch, more than twice the preceding largest glitch (Shaw et al., 2018). Due to this fact, we have been able to fit the Crab’s mass, while it has not been possible in Paper-I. The estimate is that of a massive star, with a mass of about for the BSk20 EoS. On the other hand, PSR J1932+2220 seems to be the most problematic, as it is marginally fitted in the crust case by the curve. However, we also have to notice that in the case the star is well within the 1-1.4 region. Thus, there is the need to extend the superfluid reservoir for a small region in the outer core. Finally, it is interesting to notice how the stars of the sample seems to follow the form of the curves, in particular in the case of the smallest superfluid reservoirs (see the crust and the cases): as we told before, this may be a coincidence due to the fact that most pulsars of our sample have .

Figure 5: Mass estimate as a function of the largest observed glitch for all the fitted stars in the sample and for the BSk20 equation of state. In each panel we show in red different cutoffs for the superfluid reservoir: the crust only case, the , the and the . In all panels we also show the case of the whole star in blue, as a comparison. We omitted the case of the cutoff, since it is almost identical to the whole star case.

We plot in Figure 5 the mass estimate as a function of the largest observed glitch for all the pulsars in the sample. We have omitted the cutoff at since the mass estimates are identical to the case of the whole star: it thus seems that the inner core does not play an important role, if we do not consider pinning in the core.

It is possible to notice some general trends in our results.

First of all, if we extend the superfluid reservoir to deeper regions of the star we usually fit less masses than in the case of a smaller reservoir: in Figure 4 some pulsars with small largest glitch and small nominal lag can be fitted fitted only in the cases of more external cutoffs.

Secondly, the stars with small are usually more sensitive to the variation of the cutoff than those with a large one. In fact, some pulsars with small nominal lag show masses around 1.0- in the case of reservoir limited to the crust, while they show much larger masses or they do not even get fitted in subsequently more extended cutoffs. On the other hand, the Crab pulsar and PSR J1119-6127, for instance, which have two of the biggest nominal lag of all the sample and correspond to masses greater than , have its mass almost unaltered between the different cutoffs, as can be noticed also in Figure 5. The reason for this is the following: when is big enough, the lag as a function of time (11) has reached the critical value (10). As a consequence, the maximal glitch reaches a plateau, which is represented the maximum glitch amplitude (14). Thus, for those stars with big, the maximal glitch corresponds to the maximum glitch, and so they are independent on the superfluid reservoir, but strongly dependent on the pinning force considered.

Finally, it is interesting to notice how for the crust limited reservoir the masses of the pulsar are - besides two exceptions, namely Crab and J1119-6127 - all quite low, peaked around and even less than in some cases. This fact indicates that the crustal reservoir alone is not enough to describe pulsar glitches, as already noticed by (Andersson et al., 2012) and chamel2013.

Figure 6: Mass estimate as a function of the cutoff baryon density for the four stars in the sample which have and for the three EoSs considered: SLy4 (yellow), BSk20 (blue) and DDME2 (purple). All lines start with a star, which indicates the cutoff at the crust-core interface. In the case of J1801-2304, the lines end with a dot at the maximum mass allowed by the EoS. The shaded regions indicate the uncertainty on the mass estimate. The plateau in the DDME2 curves is a by-product of the presence of a strong first-order phase transition at the core-crust interface.

Finally, we performed some comparisons between the three different EoSs considered, and we show the results in Figure 6. In this plot we consider the four pulsars of the sample with , namely J0537-6910, J0835-4510 (Vela), J1341-6220 and J1801-2304. For each of these pulsars, we plot the inferred mass as a function of the superfluid region cutoff. As we can see, the general trend of lower masses for smaller superfluid reservoir is preserved. As one may expect, a stiffer EoS like DDME2 show higher masses than the other two EoSs. Moreover, it has to be stressed the presence of a region of constant mass for this EoS: this region corresponds to the first order phase transition discussed above. We can also notice the same trend showed in the previous plots: the star with the highest in this Figure, i.e. J0537-6910, show small variability in mass between the cutoffs, in comparison to the star with the smallest nominal lag, J1801-2304, which reaches the maximum mass achievable for every EoS before .

6 Conclusions

In Paper-I, a mass estimate for glitchers based on the absolute activity and the largest observed glitch amplitude has been introduced, for the particular case of superfluid reservoir which extends into the whole star. In this work, we have relaxed this assumption, by studying the dependence of the mass estimate on the extension of the superfluid reservoir. We have considered the neutron superfluid involved in the glitch mechanism to be limited in spherical shells starting from the neutron drip and ending at different cutoff densities near the crust-core interface. The reason for this choice is that thermal effects may shrink the region where superfluid resides, thus reducing the associated angular momentum reservoir (Ho et al., 2015). For this reason, we have chosen values for the cutoffs similar to those expected for the gap of a superfluid, trying to take into account for the uncertainties in pairing gaps and temperature inside a glitching pulsar. As an extreme case, we have also considered the case of reservoir limited to the crust. Finally, we have compared the results to the case of superfluid extended in the whole star.

As in Paper-I our results are biased by some simplifications that have been explicitly employed, but that can be relaxed in more refined studies, as discussed in Section 2: the model does not account for general relativistic effects, we consider only crustal pinning and we employ a simplified and unified model for the dynamics of the lag between the two components as a function of time. Nonetheless, we can still study the dependence of the mass estimates on the extension of the superfluid reservoir for a large sample of pulsars.

As a result, we have observed a difference in behaviour for different pulsars. Glitchers with small are strongly dependent on the reservoir considered: the observed tendency is that for smaller superfluid reservoirs correspond smaller masses. This is particularly true for the crust-limited case, in which the mass estimates rarely exceeds the canonical value of . This fact is another piece of evidence of the fact that the superfluid in the crust of neutron star may not suffice for explaining glitching activity of pulsars, if entrainment is considered, but a small part of the core may be needed. On the other hand, the masses of stars with big , such as J1119-6127 and Crab (that is, however, exceptional with respect to the rest of the sample, as seen in Fig 2), do not depend on the cutoff considered. This is due to the fact that for high tends to , which has been seen to be independent on the superfluid extension: in this case, the strength of the pinning force should play a more important role than that of the superfluid reservoir. Finally, we also tested the dependence of the model on EoS with different stiffness. The general trend of smaller masses for more restricted reservoir is preserved, and stiffer EoSs show higher mass estimates.

Acknowledgements

We thank the PHAROS COST Action (CA16214) for partial support. Marco Antonelli acknowledges support from the Polish National Science Centre grant SONATA BIS 2015/18/E/ST9/00577, P.I.: B. Haskell. The authors would like to thank Morgane Fortin for providing the non-unified SLy4+DDME2 equation of state.

References

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