Bumpy neutron stars in theory and practice

Transcription

1 Bumpy neutron stars in theory and practice Nathan K. Johnson-McDaniel TPI, Uni Jena NS2013, Bonn In collaboration with Benjamin J. Owen and William G. Newton. See PRD 86, (2012), arxiv: , and arxiv:

2 Outline Motivation (GW observations) How to make a bump on a neutron star Calculating the maximum possible bump Uncertainties in the calculation Results for various models GW upper limits bounds on the shape of the star s surface Conclusions

3 Why are deformed neutron stars so interesting for GW observers? They are the only potential source for ground-based GW detectors that resides in our own Milky Way (besides, of course, galactic supernovae, or something completely unexpected). In the case of continuous waves, one has the opportunity to learn much about the source, since one would observe a billion (10 9 ) or more cycles in a year, and is sensitive to changes of a cycle or less. And with known neutron stars (either observed as pulsars or SGR flares), one can significantly reduce the considerable computational burden of these searches using the known sky location and the pulsar s frequency and phase evolution, or the time of the SGR burst. But the observers also carry out all-sky searches, and the possibility of detecting an unknown neutron star with gravitational waves is indeed tantalizing.

4 Why are deformed neutron stars so interesting for GW observers? (cont.) Finally, as Finn and Romano [arxiv: ] have recently shown, one can make a highly precise Rømer delay-style measurement of the speed of propagation of gravitational waves using a continuous wave source in the ground-based detector band.

5 But are large enough deformations on NSs theoretically plausible? The problem of the size of the deformations one expects for neutron stars in our universe is complicated, and involves a variety of (astro)physics. First, one has to consider how to support the deformation. This can be done either by elasticity (e.g., in the crust, or in a solid phase in the core), or with a magnetic field. Then one might want to know the largest deformation it is possible to sustain. In the elastic case this depends primarily upon the star s constituents (e.g., is there an exotic solid phase in the core?). In the magnetic case (which we will not discuss much further), the maximum magnetic field is constrained to be G by the virial theorem. Fields close to the maximum produce very large deformations, but most fields will be much less than this, and will give deformations too small to be observed in current GW searches.

6 But are large enough deformations on NSs theoretically plausible? (cont.) But the really difficult problem in the elastic case is how to create a (relatively) large deformation in the absence of accretion. One might hypothesize that neutron stars might have some residual frozen-in deformation from their violent birth in a supernova, but this has not progressed past the hand-waving stage. Moreover, electromagnetic observations of the spin-down of many pulsars constrain their deformations to be much smaller than anything that could be observed in current GW searches However, in certain cases, the searches are sensitive enough to go below this spin-down limit, most notably for the Crab and Vela pulsars. In particular, for the Crab pulsar, the LIGO/Virgo gravitational wave searches have constrained gravitational waves to contribute only 2% ( 0.5% with more recent, preliminary work [C. D. Gill s thesis]) of the star s overall spin-down power.

7 But are large enough deformations on NSs theoretically plausible? (cont.) Thus, the searches are saying something interesting even when one accounts for the fact that most of the star s spin-down power is almost surely not going into gravitational waves. (One has to power the pulses and the nebula, for one thing...) Crab pulsar and nebula, Chandra

8 Calculating the maximum (quadrupole) deformation for elastic deformations: Ingredients There are three primary inputs to the maximum elastic quadrupole computation. The equation of state and composition of the neutron star (nuclear/particle physics). The shear modulus (how strongly the material resists strains) and breaking strain (how much the material can be strained before it yields) of the solid component (condensed matter/materials physics). Stellar perturbation theory (GR). Dany Page s NS figure.

9 Quadrupole deformation calculation ingredients: Shear moduli Figure 2. A scientifically-accurate rendition of the composition of the stellar crust (courtesy of Sanjay Reddy). Figure from Sanjay Reddy. Possible solid phases in NSs: neutrons and the excess neutrons go into the formation of a dilute likely superfluid neutron vapor; this signals the transition from the outer to the inner crust. At a neutron-drip density of about g/cm 3, 118 Kr is unable to retain any more neutrons. As alluded earlier, at densities approaching nuclear-matter saturation density ( g/cm 3 ) uniformity in the system will be restored. Yet the transition from the highly-ordered crystal to the uniform liquid calculation is both interesting by Baiko MNRAS and complex. 416, This 22 (2011) is because and Contib. distanceplasma scales that Phys. were 52, well 157 separated (2012).] in both the crystalline phase (where the long-range Coulomb interaction dominates) and in the uniform Lattice of (heavy) nuclei and possible pasta phases in the crust but (relatively) small shear modulus. [Detailed shear modulus

10 Quadrupole deformation calculation ingredients: Shear moduli (cont.) Possible solid phases in NSs (cont.): Significantly larger shear moduli from larger charge separations in exotic lattices in the core. Most notably the hadron quark mixed phase (careful calculation by NKJ-M and Owen), but also possibly meson condensates (pions or kaons). Crystalline superconducting strange quark matter, either throughout a strange quark star (very speculative!) or in the core of a hadron quark hybrid star. [Shear modulus estimated by Mannarelli, Rajagopal, and Sharma, PRD 76, (2007).]

11 Large shear moduli, from the lab to the heavens µ eff (erg cm -3 ) Diamond Steel Crystallized 12 C WD NS crust CSC SQM Hybrid core (Hy1) ρ (g cm -3 ) Shear modulus versus density for terrestrial and astronomical materials.

12 Quadrupole deformation calculation ingredients: Breaking strain The breaking strain of the crust (albeit above neutron drip, and thus not for the densest portions of the crust, with the largest shear moduli) has been calculated by Horowitz and Kadau PRL 102, (2009) [and more recently by Hoffman and Heyl MNRAS 426, 2404 (2012)] and found to be quite large: 10 1 (compared with a maximum of 10 2 for terrestrial materials). FIG. 2 (color online). Polycrystalline sample with 12:8! 106 However, the strength is thought to come ions mostly the consisting offrom 8 differently oriented grains with an average grain diameter of 3961 fm at strains of 0.0, 0.05, 0.1, and Upper standard panel: The radial distribution system s high pressure (which prevents most failurefunctions at different shear states exhibit the characteristic peaks of the bcc structure for up to 0.1.higher-density At 0.15 strain some degree of amorphization modes), so it seems likely to be applicablestrains to the might be possible. Lower panel: At 0.0 strain the eight different are shown in different colors. At strains of 0.05, 0.1, and part of the crust (and even higher density,grains more uncertain phases), 0.15 the dark gray (red) color indicates plastic deformation, i.e., deviations of the ions away from an ideally uniformly sheared as we shall take it to be. bcc lattice in the range of 0 to 2 times the nearest neighbor distance. Note that the top and bottom layers are frozen and moved to impose the shear.

13 Quadrupole deformation calculation ingredients: Stellar perturbation theory Finally, the stellar perturbation theory calculation was only done in GR very recently [NKJ-M and Owen] Newtonian calculations (usually with an even further approximation) were standard before this (even though there have been GR calculations of magnetically deformed stars for quite some time). We found that GR significantly 10 suppresses the maximum 39 Newt. Cowling Newt. no Cowling quadrupole, particularly for GR (total) massive, compact stars, mostly due to the boundary conditions (cf. the no-hair property) Q 22 (g cm 2 ) M (solar masses) Illustration of the GR suppression of the maximum quadrupole for the Douchin and Haensel SLy EOS ε fid

14 Most significant uncertainties in the calculation Obviously, the star s EOS and composition are a significant (but well appreciated) uncertainty. But the uncertainties in the elastic properties of even the crust are significant, and are not all well appreciated in the literature: Perhaps most important is that the elastic properties of the lattice are highly anisotropic, even for spherical nuclei (i.e., no pasta phases): There is a factor of 7 difference between the maximum and minimum shear elastic constants. It is standard to take the crust to have a polycrystalline structure, so its elasticity is described by a single shear modulus (though the lattice will tend to align itself with the magnetic field, so this is probably only a good approximation if the field is tangled), but even there one has a factor-of-2 uncertainty due to the method of averaging: The standard (Voigt) average gives an upper bound, while the (Reuss) lower bound does not appear to be widely known in the astrophysics literature.

15 Most significant uncertainties in the calculation This uncertainty also affects the calculations of torsional oscillation frequencies, so one is likely to be unable to infer too much about neutron star properties from fitting calculated frequencies to observations without significantly more detailed modelling of the crust s elastic response.

16 Maximum quadrupoles: The crust The crustal shear modulus depends upon its composition, and thus on nuclear physics parameters (e.g., the density dependence of the symmetry energy, given by L). Particularly important is whether pasta phases are present. µ eff (erg cm -3 ) SLy RMF L = 25 MeV RMF L = 55 MeV RMF L = 95 MeV n (fm -3 ) Here the SLy results are for the Douchin and Haensel unified crust and core EOS A&A 380, 151 (2001). We also show preliminary results for some of the suite of crustal models including pasta phases from Will Newton. These are an improvement on the suite calculated in Newton, Gearheart, and Li ApJS 204, 9 (2013), where one matches to a relativistic mean field model for the core EOS instead of the modified Skyrme-like model used in the paper. In all cases we show both the upper and lower bounds on the effective shear modulus.

17 Maximum quadrupoles: The crust (cont.) SLy LKR1+SLy PNM L = 70 MeV RMF L = 25 MeV, ζ = 0 RMF L = 25 MeV, ζ = 0.02 RMF L = 95 MeV, ζ = Q 22 (g cm 2 ) ε fid M (solar masses) Here we show the maximum quadrupoles for the unified SLy EOS and crust in addition to the larger ones possible for a much stiffer EOS, which gives much less compact stars and a crust that s more than twice as thick. We also show the dependence on composition and the high-density EOS (ζ, which gives the strength of omega meson self-interactions smaller values of ζ give stiffer EOSs; note that one also obtains a stiffer EOS around saturation density for larger values of L, the slope of the symmetry energy) for some of the Newton et al. RMF models. In all of these, we show the uncertainty due to shear modulus averaging. For the L = 25 MeV, ζ = 0 case, we also show how much of the upper bound comes from the part of the lattice with no pasta phases.

18 Maximum elastic quadrupoles: Hadron quark hybrid stars and crystalline strange quark stars Q 22 (g cm 2 ) LKR1 Hy1 Hy1 generic generic SQM1 Vela Crab ε fid M (solar masses) The strange quark star quadrupoles use an EOS from Kurkela et al. PRD 81, (2010), and the Mannarelli et al. shear modulus with a superconducting gap parameter of = 10 MeV. We also show the LIGO/Virgo upper bounds (published and preliminary) for the Crab and Vela pulsars. One can obtain even larger strange quark quadrupoles by increasing the gap parameter. However, both the shear modulus and stellar perturbation theory calculations become less accurate as one does so... Also, all of the queadrupoles use the upper bound on the shear

19 Converting upper bounds on the star s ellipticity to upper bounds on its l = m = 2 surface deformation As we have seen, GW observers like to quote upper bounds on the quadrupole deformations of known neutron stars in terms of a dimensionless ellipticity. In the constant density, Newtonian case, this indeed gives a direct measure of the star s deformation. However, the relation is not clear in the more realistic cases, particularly when including relativity. But it turns out that the star s surface metric can be obtained from its quadrupole moment (and its unperturbed mass and radius), with very reasonable assumptions about the star s constituents at is surface (perfect fluid stress-energy tensor and force-free magnetic field).

20 Ellipticity to surface deformation (cont.) Since the star s polar (s polar ) and equatorial (s polar ) circumferences are given in terms of this metric, one can thus use these to define a completely relativistic measure of the shape of the star s surface in terms of its quadrupole moment, by ɛ surf := (s polar, max s eq )/s eq. One can also define a similar measure for the star s surface deformation due to rotation. While there is no reason to assume any sort of correlation between the two deformations, the rotational deformation gives a convenient scale against which to compare the l = m = 2 deformations.

21 Constraints on the Crab pulsar s shape Crab maximum l = m = 2 surface deformation (cm) Crab rotational surface deformation (cm) M (solar masses) M (solar masses) Dimensionful surface deformations for the Crab pulsar: Upper bound on the l = m = 2 deformation (left) and the expected rotational deformation (right), for a variety of EOSs. The Crab pulsar s dimensionless surface deformation due to rotation is about the same as the Earth s ( ) for the EOSs with larger radii, while the bound on its dimensionless l = m = 2 deformation is larger than the Earth s ( ), except for the strange quark EOS, with its large radius, for which it is comparable for the new bound.

22 Constraints on the Crab pulsar s shape (cont.) Crab surface deformation (cm) 100 APR, 22 APR, 20 BBB2, 22 BBB2, 20 SLy, 22 SLy, 20 WFF1, 22 WFF1, M (solar masses) The LIGO observations have thus constrained the Crab pulsar s l = m = 2 surface deformation to be smaller than its rotational deformation, for all the causal EOSs we have considered, and for all but the very largest masses for an equation of state that becomes acausal. (And the preliminary updated results are well below the rotational deformation even for the acausal EOS.) This could not have been concluded solely from the E&M observations of the spin-down, which give a bound that is 7 times as large. Even the smaller upper bound from Palomba A&A 354, 163 (2000), obtained by folding in the braking index is still 3 times larger.

23 Conclusions Bumpy neutron stars are an exciting (and convenient) potential source for gravitational wave detectors....and there are theoretically well-motivated scenarios that yield maximum quadrupoles which are large enough to have been detected in current searches from known neutron stars, even though the prospects for crustal quadrupoles are less optimistic than they once were, with the GR suppression of the quadrupole. However, there is still work to be done on modelling elasticity, to help put some remaining factor-of-2 (or possibly more) uncertainty to rest. (And this uncertainty also affects calculations of crustal torsional modes.) On the observational side, one can convert the current upper bounds on the quadrupole moment into bounds on the star s shape, and find that the LIGO bound has constrained the Crab pulsar s largest global (l = 2) surface deformation to be due to rotation.