Pinning down the superfluid and nuclear equation of state and measuring neutron star mass using pulsar glitches
Wynn C. G. Ho, Cristobal M. Espinoza, Danai Antonopoulou, Nils Andersson
aa r X i v : . [ a s t r o - ph . H E ] M a r Pinning Down the Superfluid and Nuclear Equation of Stateand Measuring Neutron Star Mass Using Pulsar Glitches
Wynn C. G. H o , Crist´obal M. E spinoza , Danai A ntonopoulou , and Nils A ndersson Mathematical Sciences, Physics & Astronomy, and STAG Research Centre, University ofSouthampton, Southampton, SO17 1BJ, United Kingdom Departamento de F´ısica, Universidad de Santiago de Chile, Avenida Ecuador 3493, Estaci´onCentral, Santiago, ChileE-mail: [email protected] (Received July 18, 2016)Pulsars are rotating neutron stars that are renowned for their timing precision, although glitches caninterrupt the regular timing behavior when these stars are young. Glitches are thought to be causedby interactions between normal and superfluid matter in the star. We update our recent work on anew technique using pulsar glitch data to constrain superfluid and nuclear equation of state models,demonstrating how current and future astronomy telescopes can probe fundamental physics such assuperfluidity near nuclear saturation and matter at supranuclear densities. Unlike traditional methodsof measuring a star’s mass by its gravitational e ff ect on another object, our technique relies on nuclearphysics knowledge and therefore allows measurement of the mass of pulsars which are in isolation. KEYWORDS: equations of state of neutron-star matter, neutron stars, pulsars, quantum fluids
1. Pulsar Spin-down and Superfluid Model of Glitch Spin-up
Rotating neutron stars, or pulsars, are born in the collapse and supernova explosion at the end ofa massive star’s life. More massive than the Sun but only ∼
20 km in diameter, pulsars are primarilycomposed of neutron-rich matter near and above nuclear densities. Pulsars emit beamed electromag-netic radiation, and this loss of energy comes at the expense of the pulsar’s rotational energy, causingthe star to spin more slowly over time. While most isolated pulsars are observed to rotate very stablywith a spin period between ∼ T c (below which neutrons becomesuperfluid) is well above the typical crust temperature of pulsars. Unlike normal matter, superfluidmatter in the inner crust rotates by forming vortices whose areal density determines the spin rate of thesuperfluid. To decrease its spin rate, superfluid vortices must move so that the areal density decreases.In the inner crust of a neutron star, these vortices are usually pinned to the nuclei of normal matter [1].While the rest of the star slows down owing to electromagnetic energy loss, the neutron superfluiddoes not. As a result, this superfluid can act as a reservoir of angular momentum. Over many pulsarrotations, an increasing lag develops between the stellar spin rate and that of the neutron superfluidin the inner crust. When this lag exceeds a critical value, superfluid vortices unpin and transfer theirangular momentum to the rest of the star, causing the stellar rotation rate to increase and producingwhat we observe as a glitch [1, 2]. Thus glitches provide a valuable measure of the amount of super-fluid angular momentum, or equivalently moment of inertia, in neutron stars. To probe superfluid andnuclear equation of state (EOS) properties with pulsar glitches, three questions need answering. . Observational and Theoretical Constraints on Pulsar Moment of Inertia For glitching pulsars, G ≡ τ c h A i is the measured parameter that is to be compared to theoreticalmodels. Here τ c = Ω / Ω , Ω and ˙ Ω are the pulsar spin rate and its time derivative, respectively, and h A i = (1 / t obs ) P ∆Ω / Ω is the average activity parameter and the summation is over each glitch withspin rate change ∆Ω during an observation time span t obs . G = . ± .
03% for the Vela pulsar, and G = . ± . − G in order for there to be su ffi cient angular momentum to be transferred during glitches.In our recent work [3], we examine a large new dataset of glitches [4, 5] and explore an ideaproposed in [6] and illustrated in Fig. 1. The neutron star crust (core) is denoted by shaded (unshaded)regions. Vertical dotted lines indicate the density at which the moment of inertia exceeds G = . M Sun . Furthermore,all glitching pulsars with G ≈
1% would have to be of low mass ( < M Sun ) as well. For typicalneutron star masses of 1 . M Sun [7], some small fraction of the core must contribute to themoment of inertia required by glitches seen in, for example, the Vela pulsar.
Fig. 1.
Temperature as a function of baryon number density, using the BSk20 EOS. Curved lines are thesuperfluid critical temperature for nine models from [15] (left panel) and seven of eleven models discussed inSec. 3 (right panel). The vertical solid line indicates the separation between the crust (shaded region) and thecore. Vertical dotted lines denote the density at which the partial moment of inertia is Vela’s G = .
6% of thetotal stellar moment of inertia for neutron stars of di ff erent mass (labeled in units of solar mass). The (nearlyhorizontal) dashed line is the temperature of a 1 . M Sun neutron star at the Vela pulsar’s age of 11000 years.
In the previous analyses of [6, 8–13], pulsar glitches are assumed to tap the angular momentumreservoir associated with superfluid neutrons in the inner crust of the star (and largely ignoring thetemperature dependence of superfluidity). Therefore, by calculating the entire moment of inertia ofthe crust, it is possible to determine the maximum reservoir available for producing glitches, and thisis found to be smaller than that needed to explain observed glitch activity [6, 9, 12, 13] due to thee ff ect of entrainment [14], unless the crust is unusually thick [10, 11]. Here and in [3], we consider asuperfluid reservoir that extends into the stellar core and account for the temperature dependence ofsuperfluidity. Importantly, we use the observed temperature of pulsars to constrain the latter. Figure 1shows models of critical temperature of neutron (singlet-state) superfluidity as a function of baryonnumber density n b : models plotted in the left panel are from [15], while those in the right panel are iscussed in Sec. 3. For some superfluid models, the critical temperature, and hence allowed regionfor neutrons to become superfluid, is confined to the inner crust, i.e., in the shaded region to the left ofthe vertical solid line. Therefore, pulsar glitches can only involve the moment of inertia of the innercrust if one of these superfluid models is the correct one. However, there are superfluid models thatextend into the core, e.g., solid curve labeled SFB or HFB-16nm. For superfluid models such as theSFB model, if the pulsar temperature is low enough so that neutrons in the inner crust and outer coreare superfluid, then pulsar glitches could involve additional moment of inertia from the core. Finally, are enough neutrons in the crust and core actually superfluid, i.e., to the left of one ofthe vertical dotted lines and below the superfluid critical temperature T c in Fig. 1? To answer thisquestion, we need to determine the interior temperature T ( n b ) of a neutron star and evaluate at whatdensities n b the inequality T < T c is satisfied. This will vary for each pulsar, depending on its ageand / or measured temperature. Neutron stars are born in supernovae at very high temperatures butcool rapidly because of e ffi cient neutrino emission. We perform neutron star cooling simulationsusing standard neutrino emission processes to find the interior temperature at various ages [16]. Thedashed lines in Fig. 1 show the resulting temperature profile (at the age of the Vela pulsar) for a1 . M Sun neutron star. The temperature profile is di ff erent for di ff erent mass but not drastically so,unless neutrino emission by direct Urca processes occurs. We find that, among nine superfluid modelswhich span a wide range in parameter space (see left panel of Fig. 1), only the SFB model providesa superfluid reservoir of the required level (see below for more recent results). For superfluid modelsthat are confined to the crust, the reservoir is too small, whereas the reservoir is too large for modelsthat extend much deeper into the core. The latter would be unable to explain the regularity of similar-sized glitches, which requires the reservoir to be completely exhausted in each event.
3. Measuring Pulsar Masses and New Results Since Ho et al. 2015 [3]
The intersection of the three lines in Fig. 1 [vertical dotted line at 1 . M Sun for glitch requirement G = . T c , and horizontal dashed linefor neutron star temperature T ( n b ) at age = The Vela pulsarhas a mass near the characteristic value of . M Sun , and the size and frequency of Vela’s observedglitches are a natural consequence of the superfluid moment of inertia available to it at its currentage . Our results using the BSK20 EOS and SFB superfluid models are summarized in the left panelof Fig. 2, which shows interior temperature T of a pulsar as a function of glitch parameter G (see[3] for results using APR and BSk21 EOS models). Note that G and T are directly determined fromobservational data. The former comes from radio or X-ray glitch measurements. The latter is obtainedeither from the age of the pulsar or by measuring the surface temperature of the pulsar through X-rayobservations. The age gives the interior temperature via neutron star cooling simulations, whereassurface temperature is related to interior temperature via well-known relationships [16]. Thus fora given pulsar that has measured G and T , Fig. 2 allows one to determine the pulsar’s mass. Forexample, using the BSk20 EOS and SFB superfluid models, we find that Vela is a 1 . ± . M Sun neutron star and PSR J0537 − . ± . M Sun neutron star. Results for seven other glitchingpulsars are also plotted in Figure 2.Since [3], we tested an additional two EOS models and eleven superfluid models (see right panelof Fig. 1): two EOS models from [17], along with superfluid models calculated using the same inter-actions [18]; same HFB-16 symmetric and neutron matter superfluid models that are used to constructBSk EOS models [19]; and six superfluid models from [20]. In brief, the models of [17,18] imply lowmasses, while superfluid models HFB-16nm [19] and N3LOsrc + lrc [20] yield reasonable masses.The ability to measure the mass of isolated pulsars has not been previously demonstrated. The ig. 2. Left panel: Neutron star mass from pulsar observables G and interior temperature T . Data points arefor pulsars with measured G from glitches and T from an age or surface temperature observation [3]. Lines(labeled by neutron star mass, in units of solar mass) are the theoretical prediction for G and T using the BSk20EOS and SFB superfluid models. Right panel: Neutron star mass and 1 σ uncertainty using the BSk20 EOS andSFB superfluid models; see Table 2 in [3] for masses derived using APR and BSk21 EOS models. most precise neutron star mass measurements to date are by radio timing of pulsars that are in a binarystar system [7]. Our method of using glitches to measure mass can greatly increase the number ofknown masses, thereby providing constraints on fundamental physics properties such as the nuclearEOS and superfluidity. Although there are currently relatively large systematic uncertainties, thesewill improve as the understanding of dense matter improves. The novelty of our approach is thecombination of pulsar glitch data and the temperature dependence of superfluidity. The method isespecially promising with upcoming astronomical observatories such as the Square Kilometer Array(SKA) in radio and Athena + in X-rays. SKA could discover all observable pulsars in the Galaxy, anda program to monitor glitching pulsars could transform the fields of neutron star and nuclear physics.WCGH thanks M. Baldo, F. Burgio, N. Chamel, K. Glampedakis, A. Rios, and H.-J. Schulze forproviding models and for discussion. WCGH acknowledges support from UK STFC. References [1] P. W. Anderson, N. Itoh, Nature , 25 (1975).[2] G. 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