Gravitational waves from `mountains' on neutron stars

Ian Jones, University of Southampton D.I.Jones-at-soton.ac.uk

This article should perhaps begin with a disclaimer: when we talk about `mountains' on neutron stars we speak not of precipitous projections from the surface, but rather of large scale mass asymmetries. Over the last decade a series of papers has appeared in the literature attempting to calculate just how large such mountains might be. There is a strong motivation behind this: In a spinning star such asymmetries directly source gravitational wave emission, and with the LIGO and GEO600 detectors up and running, and VIRGO close behind, accurate modeling of this class of potential source is vital. As emphasized recently by Owen (2005), having estimates of possible mountain sizes enables us to assess the significance not just of gravitational wave detections but (currently more usefully!) of upper bounds, telling us when the physics at least would permit a detection.

More accurately, if a neutron star spins about some axis $Oz$, we are concerned with the difference between the $I_{xx}$ and $I_{yy}$ pieces of its moment of inertia tensor, as a difference between these produce a characteristic gravitational wave strain of (Abbott et al. 2007):

$$h = \frac{4G}{c^4} \frac{(I_{yy}-I_{xx}) \Omega^2}{r} ,$$ (5)

where $\Omega$ denotes the star's angular spin frequency and $r$ its distance from Earth. The asymmetry $(I_{yy}-I_{xx})$ is often parameterized in dimensionless form as a so-called ellipticity parameter

$$\epsilon = \frac{I_{yy}-I_{xx}}{I_{zz}} ,$$ (6)

but it is worthwhile remembering that only the combination $\epsilon I_{zz} \equiv I_{yy}-I_{xx}$ appears in the formula for $h$. As will be described below, typical calculations for canonical neutron stars place $\epsilon$ somewhere around $10^{-6}$ or less, corresponding to an equatorial radius difference of no more than $1$ cm, so this is rather a far cry from Mount Everest. Nevertheless, an ant unlucky enough to find himself placed on such a star would have to expend $10^{11}$ ergs to climb up the slope!

If neutron stars consisted of nothing but a ball of self-gravitating perfect fluid, no such asymmetry could exist. However, strains in the solid crust or Lorentz forces connected with the internal magnetic field can support deformations, and it is to these that theorists have turned to try and compute possible $\epsilon$ values. Crustal strains have attracted more attention, possibly because of the large field strengths needed to make the magnetic contribution important.

Historically, strains in the solid crust were of interest to early modelers, who thought that crust fracture might explain the phenomenon of pulsar glitches. Calculations soon showed that, for the Vela pulsar at least, such an explanation wasn't viable--there simply wasn't enough elastic energy available to account for the frequent large glitches.

However, interest in crustal deformations was rekindled in 1998, the inspiration coming from X-ray physics. New satellites succeeded in measuring oscillations in the Low mass X-ray binary (LMXB) systems, and it seemed the derived spin frequencies were unexpectedly tightly clustered in the interval $300$-$600$ Hz. As Bildsten (1998) pointed out, this was suggestive of the spinning-up stars hitting a `gravitational wave wall', where the spin-up accretion torque was balanced by a gravitational wave spin-down torque, the steep spin frequency dependence of the latter serving to cluster the equilibrium frequencies. Such a hypothesis had been advanced previously, first in outline by Papaloizou & Pringle (1978), and then in more detail by Wagoner (1984). However, Bildsten suggested that mountains (rather than fluid oscillation modes) might be responsible for the emission, and even suggested an ingenious way of manufacturing the necessary $\epsilon \approx 10^{-7} \rightarrow 10^{-8}$ deformations: if temperature asymmetries were (for whatever reason) intrinsic to the accretion process, the temperature-dependent nuclear reactions undergone by the accreting crustal matter would be shifted, with the resulting density perturbations providing the necessary mass quadrupole to produce the gravitational wave torque.

This idea was taken up in detail by Ushomirsky, Cutler & Bildsten (2000; hereafter UCB), who found that temperature asymmetries at the $5\%$ level in the inner crust would provide the necessary mountain. They also calculated a bound on $\epsilon$ that assumed only that strains in the elastic crust were responsible for the deformation, making no further assumptions about its cause (e.g. temperature asymmetries). This essentially involved balancing the gravitational and pressure forces (which favor a spherical configuration) against the shear stress forces in the crust (which cause asymmetry). Very roughly, their result was $\epsilon \le 10^{-7} (\sigma_{\rm max}/10^{-2})$, where $\sigma_{\rm max}$ is the breaking strain of neutron star crustal matter. Unfortunately, this number is very uncertain; $10^{-5} < \sigma_{\rm max} < 10^{-2}$ for terrestrial materials, so parameterizing in terms of $\sigma_{\rm max}/10^{-2}$ is probably rather optimistic. Nevertheless, for the remainder of this article we will assume a breaking strain of $10^{-2}$ in all quoted ellipticities; the results can be scaled to any other breaking strain in a linear manner.

This maximization problem was developed further by Haskell, Jones & Andersson (2006), who extended the treatment in a number of ways. They considered a variety of stellar models, including models where the core structure was computed relativistically, stars with both accreted and non-accreted crusts, and models where the perturbations in the gravitational potential were retained (these were neglected in the UCB analysis). The main conclusion was that there was little difference between the maximum $\epsilon$ values for the accreted and non-accreted crust, but that the more accurate treatment of the gravitational potential, together with an improved treatment of boundary conditions, resulted in $\epsilon$ values about one order of magnitude larger than in UCB: $\epsilon \le 10^{-6}$.

However, it is important to remember that our high energy physics colleagues are by no means sure what form the high density equation of state should take. It may well be that neutron stars contain exotic solid cores, or that some or all compact objects are not neutron stars at all but are so-called strange stars, consisting of a mixture of up, down and strange quarks, not arranged into nucleons.

The problem of calculating a maximum mountain size from an exotic star was examined by Owen (2005), who considered two possibilities. First he examined the case of solid strange stars, using a shear modulus proposed by Xu (2003) to explain LMXB quasi-periodic oscillations as torsional oscillations of a fully solid star. The consequent mountain was found to be limited in size to $\epsilon \le 2 \times 10^{-4}$, orders of magnitude larger than for neutron stars. Owen then considered the case of a partly baryonic, partly strange star (Glendenning 1992), with a gradual phase transition between the two. The maximum mountain in this case was somewhat smaller, but still larger than the neutron star estimates: $\epsilon \le 5 \times 10^{-6}$.

A different class of exotic compact object was then studied by Haskell et al. (2007a), who made use of recently developed crystalline color superconducting quark core models (Mannarelli et al 2007). The shear modulus of such matter is sensitive to somewhat uncertain QCD parameters, including the density at which the transition form such an exotic state to normal baryonic matter occurs. Taking optimistic (from the point of view of gravitational wave emission) values can lead to $\epsilon \le 10^{-3}$, about an order of magnitude higher than for the solid quark stars considered by Owen. So, current uncertainties in the high density equation of state do allow for some very large asymmetries indeed.

Before concluding, a few brief remarks on the status of magnetically-supported mountains. Here there is a significant uncertainty that must be borne in mind: It is not at all clear how the strength of the internal magnetic field (which is mainly responsible for the quadrupole generation) is related to the external magnetic field (which is the potentially measurable one, e.g. from pulsar spin-down). In particular, it is not clear how the internal field is arranged, as this depends upon the superfluid/superconducting nature of the core, and also on the star's equation of state (Haskell et al. 2007b).

In the absence of superconductivity, the internal magnetic field is expected to be uniformly distributed in the core. The ellipticity produced by a field of strength $B$ would then produce an ellipticity that can be estimated by taking the ratio of magnetic to gravitational binding energies, giving $\epsilon \approx 10^{-12} (B/10^{12}{\, \rm G})^2$. Superconductivity complicates this picture: If the interior contains a type II proton superconductor, the field is not distributed uniformly, but is confined to a large number of $10^{15} G$ flux tubes (see Cutler 2002 and references therein). This increases the resulting deformation by a factor of $(10^{15} G)/B$. More exotically, if the interior protons form a type I superconductor, the magnetic field is excluded from the core completely, being forced into a thin shell at the base of the crust. As described by Bonazzola & Gourgoulhon (1996), the resulting ellipticity diverges as the thickness of this shell goes to zero, making placing an upper limit problematic. Finally, if a star is accreting from a companion, funneling of the accreted material at the poles might build up a magnetic mountain, a possibility studied recently by Payne and Melatos (2006). Certainly, there are many possibilities when it comes to producing magnetic deformations, and it is not yet clear if these mechanisms are competitive with more conventional stressed elastic crust scenarios.

To sum up, if compact objects really are neutron stars, then crustal strains allow $\epsilon \le 10^{-6}$. This is comfortably large enough to allow for the LMXBs to be spin-limited by gravitational weave emission, and for the spin-down of the millisecond pulsars to have a significant gravitational wave component. This is also just large enough to be of interest for current gravitational wave observations: The S3/S4 results recently posed by the LIGO Scientific Collaboration (LSC) give an upper bound of $\epsilon_{\rm max} = 7.1 \times 10^{-7}$ for PSR J2124-3358, so this non-detection has already told us something non-trivial: this neutron star at least is not maximally strained.

The larger possible mountains that exotic compact objects can provide allow us to make more use of the LSC upper limits. The S3/S4 results contain many upper limits of less than $10^{-4}$, telling us that these objects are not maximally strained strange stars. These results also rule our some of the more extreme magnetic field configurations of Bonazzola & Gourgoulhon (1996).

The sensitivity of these gravitational wave searches scales with the noise level in the detector and as the square root of the duration of the (coherent) observation. This means that as the detectors are improved and longer stretches of data analyzed more and more stars will become potentially detectable, even within the canonical neutron star scenario. Looking ahead, a year's worth of data from Advanced LIGO would provide an upper limit of $\epsilon_{\rm max} \approx 10^{-8}$ for PSR J2124-3358, a level where one no longer feels one has to be optimistic to make a detection.

We will end this summary by posing two questions, both of importance for astrophysical interpretation. Firstly, suppose a positive gravitational wave detection was made. What would we learn? Could we distinguish between, say, a canonical neutron star carrying a large strain, or a less highly strained strange star? Secondly, if even Advanced LIGO detects nothing, what can we conclude? Would this rule out crystalline color superconducting quark cores, or is Nature capable of producing such phases with ellipticities substantially below their theoretical maxima? Clearly, despite the progress of recent years, there are still plenty of important theoretical issues to be examined if we are to extract the maximum information from the search for gravitational waves from neutron stars.

References

B. Abbott et al, Upper limits on gravitational wave emission from 78 radio pulsars, preprint gr-qc/0702039 to appear in Phys. Rev. D (2007)

L. Bildsten, Ap. J. 501 L89 (1998)

S. Bonazzola, E. Gourgoulhon, Astron. & Astrophys. 312 675 (1996)

C. Cutler, Phys. Rev. D 66 084025 (2002)

N. K. Glendenning, Phys. Rev. D 46 1274 (1992)

B. Haskell, N. Andersson, D. I. Jones, L. Samuelsson, Submitted to Phys. Rev. Lett. (2007a)

B. Haskell, Samuelsson, K. Glampedakis, N. Andersson, Submitted to MNRAS (2007b)

B. Haskell, D.I. Jones, N. Andersson, MNRAS 373 1423 (2006)

M. Mannarelli, K. Rajagopal, R. Sharma, The rigidity of crystalline color superconducting quark matter, preprint hep-ph/0702021

B.J. Owen, Phys. Rev. Lett. 95 211101 (2005)

J. Papaloizou & J. E. Pringle, MNRAS 184 501 (1978)

D. J. B. Payne, A. Melatos, Ap. J. 641 471 (2006)

G. Ushomirsky, C. Cutler, L. Bildsten, MNRAS 319 902 (2000)

R. V. Wagoner, Ap. J. 278 345 (1984)

R. X. Xu, Ap. J. 596 L59 (2003)