R-mode in XTE J1751-305

Implications of an r-mode in XTE J1751305: Mass, radius and spin evolution

N. Andersson, D. I. Jones and W. C. G. Ho
Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton SO17 1BJ, UK
Email: na@maths.soton.ac.uk
Accepted 2014 xx. Received 2014 xx; in original form 2014 xx

Recently Strohmayer and Mahmoodifar presented evidence for a coherent oscillation in the X-ray light curve of the accreting millisecond pulsar XTE J1751305, using data taken by RXTE during the 2002 outburst of this source. They noted that a possible explanation includes the excitation of a non-radial oscillation mode of the neutron star, either in the form of a g-mode or an r-mode. The r-mode interpretation has connections with proposed spin-evolution scenarios for systems such as XTE J1751305. Here we examine in detail this interesting possible interpretation. Using the ratio of the observed oscillation frequency to the star’s spin frequency, we derive an approximate neutron star mass-radius relation which yields reasonable values for the mass over the range of expected stellar radius (as constrained by observations of radius-expansion burst sources). However, we argue that the large mode amplitude suggested by the Strohmayer and Mahmoodifar analysis would inevitably lead to a large spin-down of the star, inconsistent with its observed spin evolution, regardless of whether the r-mode itself is in a stable or unstable regime. We therefore conclude that the r-mode interpretation of the observed oscillation is not consistent with our current understanding of neutron star dynamics and must be considered unlikely. Finally we note that, subject to the availability of a sufficiently accurate timing model, a direct gravitational-wave search may be able to confirm or reject an r-mode interpretation unambiguously, should such an event, with a similar inferred mode amplitude, recur during the Advanced detector era.

pagerange: Implications of an r-mode in XTE J1751305: Mass, radius and spin evolutionReferencespubyear: 2014

1 Context

With improved observational sensitivity in both the electromagnetic and gravitational-wave channels, we are approaching the time when we may be able to probe neutron star interiors through asteroseismology. This era may, in fact, already have begun with observations of quasiperiodic oscillations in the tails of magnetar giant flares (see, e.g. Israel et al. 2005, Strohmayer & Watts 2006) and the suggestion that the observed oscillations can be mapped to shear modes of the star’s crust (e.g. Samuelsson & Andersson 2007). However, the main lesson learned from this exercise is that the asteroseismology programme is complicated, owing in part to the unknown nature of the state of matter inside the star and the form of the internal magnetic field (Glampedakis et al., 2006; Gabler et al., 2013). In order to progress further, we need theoretical models that account for as much of the involved complex physics as possible (or, indeed, palatable because the problem becomes increasingly messy). We also need more observational data. In this context, it is interesting to note the recent suggestion by Strohmayer & Mahmoodifar (2013) that a non-radial oscillation mode may have been present in the X-ray data associated with the 2002 discovery outburst of XTE J1751305, an accreting neutron star in a low-mass X-ray binary (LMXB) with spin frequency  Hz. The observed frequency is tantalizingly close to the frequency of the star’s quadrupole r-mode. In Newtonian gravity, this r-mode would have frequency to leading order of a slow-rotation expansion. Noting this proximity, Strohmayer & Mahmoodifar (2013) discuss whether the observations could be consistent with an r-mode excited in the bulk of the star. This would be an exciting result, given the suggestion that gravitational-wave emission may drive this particular mode unstable at some critical rotation rate, providing a mechanism to prevent further spin-up by accretion (Andersson et al. 1999; see also Ho et al. 2011; Haskell et al. 2012, for a recent assessment of this idea).

At first sight, the r-mode interpretation would seem unlikely, because the observed frequency is close to the mode frequency in a frame co-rotating with the star whereas one might expect a distant observer to see the inertial frame mode frequency, which is to leading order. The first part of any r-mode interpretation has to deal with this issue. The required explanation was provided by Numata & Lee (2010), who demonstrated that a nonradial oscillation mode can indeed lead to modulations of an X-ray hotspot being observed at the rotating frame mode frequency. This conclusion is key to the r-mode interpretation of the XTE J1751305 data. However, this is not sufficient to make the connection credible.

Strohmayer and Mahmoodifar concluded in part that to interpret the observed oscillation frequency as an r-mode, it was necessary for the star’s spin rate to be such that it was close to a resonance between a crustal torsional mode and the r-mode frequency. Furthermore, for this to be the case, they noted that the star’s crust must have a shear modulus approximately twice as large as is generally predicted (cf. the results of Strohmayer et al. 1991). One might not be comfortable with such a sizeable change of a relatively well understood part of the star’s physics, but it is a possible explanation.

However, it turns out that one does not need to tweak crust physics to explain the observed frequency in terms of a global r-mode. A large correction to the r-mode frequency is caused by the combined relativistic effects of the gravitational redshift and rotational frame-dragging. These effects were calculated some time ago by Lockitch et al. (2003). According to their analytic expression for a uniform density barotropic model (equation 36 of Lockitch et al. 2003) the quadrupole () r-mode should have a co-rotating r-mode frequency where


to first post-Newtonian (1PN) order. This shows that, for a typical neutron star compactness of , the relativistic corrections is at the 10% level. This is significantly larger than the rotational and crust corrections (unless one invokes resonances) considered in Strohmayer & Mahmoodifar (2013). Moreover, the relativistic effects tend to lower the (rotating frame) mode frequency, exactly as required to explain the observations, which require . This motivates us to take a closer look at the problem.

In Section 2 we go beyond the rough estimate of equation (1) and argue that we can use the observational results to constrain the relationship between mass and radius for this neutron star. If one additionally assumes that the radius lies in the range inferred from radius-expansion burst sources (Lattimer & Steiner, 2013), estimates for the mass itself are obtained. We show that this procedure yields a sensible result. However, this is only part of the story—it is also necessary to explain the observed variation in spin frequency for this system. In Section 3 we first consider the issue of spin evolution including only standard accretion torques and magnetic fields, without any r-mode excitation. We show that the standard models fail to explain the observed frequency variation of XTE J1751305, as different estimates/constraints on the star’s magnetic field fail to agree; we find the same disagreement for another LMXB, IGR J00291+5934. In Section 4 we consider the effect of adding an r-mode into the model, with the large amplitude indicated by the Strohmayer & Mahmoodifar analysis. We show that, in all plausible scenarios (i.e., stable and unstable and unsaturated and saturated), the inclusion of the r-mode over the duration of the X-ray outburst (10 days or so) in fact makes the observed spin variation much more difficult to explain (Strohmayer & Mahmoodifar noted this problem while considering only the unstable scenario). We therefore conclude that the r-mode interpretation of the data is hard to sustain, but note that future gravitational wave detectors could, if a sufficiently accurate timing model is available, be able to directly test if an r-mode is present at a similar level in any future outburst of XTE J1751305 or a similar system.

2 Inferring the star’s mass

The ratio of the (rotating frame) r-mode frequency to spin frequency is given by only in the highly idealised case of small oscillations of a slowly rotating perfect fluid star, modelled using Newtonian gravity. More realistic models will lead to different -values, allowing us to use the observed value to learn something of the physics of the real system. As discussed above, the effect of General Relativity is significant. The first post-Newtonian (1PN) result given in equation (1), strictly valid only for a uniform density star (i.e. a polytrope), leads to an estimate for the compactness . This can be viewed as a constraint in the mass-radius plane, as represented by the black line in Figure 1.

Figure 1: Various relations between mass and radius obtained from the observed ratio of spin frequency to ‘mode’ frequency, assuming the mode to be the r-mode. The solid black line includes the effects of General Relativity at the first post-Newtonian level, but neglects the effects of rapid rotation. The dashed red line includes the effects of General Relativity but again neglects rapid rotation. The solid red line includes both General Relativity and an (Newtonian-based) estimate of the effects of rapid rotation. These curves simply give our estimates of the mass as a function of radius—no attempt has been made to bracket uncertainties; see text for discussion.

However, perusal of Figure 1 of Lockitch et al. (2003), which gives the mode frequency as a function of compactness for a fully relativistic (numerical) calculation, shows a significant non-linear variation of with compactness, indicating that the 1PN result is not very accurate for stars with realistic compactness values. Using instead the full relativistic result, easily read-off from Figure 1 of Lockitch et al. (2003), we obtain the (presumably) more accurate estimate . This corresponds to the dashed red curve of Figure 1. This constraint is strictly only valid for uniform density stars, but as shown in Figure 2 of Lockitch et al. (2003), varies by only a few percent when the polytropic index is varied (at fixed ) over the interval , so it seems the variation of with is much more significant than its variation with the equation of state.

As described in Lindblom et al. (1999), there will be rotational corrections to too, of order


where the angular frequency is and is the average density of the corresponding non-rotating star of mass and radius . This indicates that for a star spinning as rapidly as XTE J1751305, rotational corrections are not dominant, but not negligible either. To account for the effect of rotation into our analysis, we write the actual value as the sum of its Newtonian part and the relativistic and rotational corrections:


The combination , as a function of , can be read-off from Figure 1 of Lockitch et al. (2003). The rotational corrections are estimated by Lindblom et al. (1999), who wrote their result in the form


In Lindblom et al. (1999) it was found that, for stars with and for a variety of realistic equations of state, , so we take as a representative value. We then have


Using the observed value , the numerical calculation of of Figure 1 of Lockitch et al. (2003), and the value , the value of can then be varied over some sensible range, and equation (5) is solved numerically for . The results of such a calculation are depicted by the solid red line in Figure 1. Note that our procedure is inconsistent, in that we use the results of Lockitch et al. (2003) for , but an average over realistic equations of state for a star from the results of Lindblom et al. (1999) for . However, as noted above, in both cases the variation in with respect to variations in the equation of state is small, at the level of a few percent, so errors introduced by this inconsistency are presumably at a similar level, much smaller than the variation in when is varied over an astrophysically-plausible interval. Given the relatively crude nature of our approximations, a detailed error analysis is clearly not appropriate, but given the several percent variation of of the relativistic correction with polytropic index, and the variation in the rotational parameter over realistic equations of states, the error in this curve due to our lack of knowledge of the real equation of state is plausibly smaller than 10%.

Assuming that the inferred compactness reflects the nature of the neutron star, we can combine the result with the radius constraint from X-ray burst sources. According to the most recent analysis by Lattimer & Steiner (2013), the allowed radius range for all neutron stars between and is 10.9 to 12.7 km111Note that a lower radius range is found by Guillot et al. (2013). However an important systematic effect (as well as others) is the assumption of a hydrogen atmosphere when fitting X-ray data. A helium atmosphere leads to a larger inferred radius (see, e.g. Servillat et al. 2012; Catuneanu et al. 2013). This difference in composition is approximately taken into account in the analysis of Lattimer & Steiner (2013).. In other words, one would expect  km, or, more appropriately given our likely level of accuracy,  km. Combining this result with the inferred compactness, we conclude that the mass of the XTE J1751305 neutron star should lie in the range , with a preferred value of . This is a perfectly reasonable mass for a neutron star that has accreted enough mass to be spun up to the observed fast spin. More detailed calculations, especially concerning the fast rotation correction in General Relativity (perhaps building on Gaertig & Kokkotas 2008) are needed to improve on this.

The close proximity (in frequency) of a crustal toroidal mode could have a significant effect on the r-mode frequency, and change the above estimates. However, as Strohmayer & Mahmoodifar noted, proximity to such an avoided crossing would require an unexpectedly large value for the crustal shear modulus, so this solution does not seem very likely. There are other pieces of physics, beyond the inclusion of full relativity, rapid rotation, and the crust, that could affect the r-mode frequency. These include magnetic fields and superfluidity. The weak magnetic fields of LMXBs indicate that magnetic corrections should be negligible [see, e.g. Lander et al. 2010, for normal (rather than superconducting) interiors at least]. Similarly, studies of Newtonian stars with superfluid cores indicate that the r-mode we are considering here is relatively insensitive to this aspect (Passamonti & Andersson, 2012).

3 Observations and spin evolution

Having established that the observed frequency would be consistent with a global r-mode oscillation in XTE J1751305, let us now turn to the implications for and constraints from the measured spin evolution of this system. We will begin by applying ‘standard’ accretion theory to this system, which involves only accretion and magnetic fields, deferring the inclusion of r-modes to Section 4.

The 2002 X-ray outburst in XTE J1751305 occurred from April 3 to April 30, had a rise time of less than 4 d, peaked around April 4–5, and declined exponentially with a decay timescale of (Markwardt et al., 2002). Pulsations at the spin-frequency of 435 Hz were only detectable until April 14, with the pulse fraction decreasing from 5% to 3% during this early period, and no pulsations were detected afterwards, including during a small outburst that occurred April 28–30 (pulse fraction limit ). A reanalysis by Papitto et al. (2008) of the outburst found that coherent pulsations could be detected for 9 d (after which the flux decreased dramatically) and that a frequency increase (spin-up) occurred during this time. They estimated the spin-up rate to be


with an error at the 90% confidence level. The total change in the spin-frequency was found to be


The analysis in Papitto et al. (2008) was not able to establish whether there was a dependence of on the mass accretion rate . We note that the distance to XTE J1751305 is estimated to be (Markwardt et al., 2002; Papitto et al., 2008), which yields a peak 2-10 keV luminosity of for the 2002 outburst (Riggio et al., 2011), corresponding to .

XTE J1751305 has subsequently exhibited outbursts in 2005, 2007, and 2009. These outbursts were weaker (at 14%, 18%, and 27%, respectively, of the 2002 flux) and shorter (with decay timescales ) (Riggio et al., 2011). Pulsations were only detected in the 2009 outburst for 1 d, with an amplitude of 7–8%, and there was no significant frequency evolution within this outburst. The analysis in Riggio et al. (2011) suggests a long-term decrease in frequency (spin-down) of the pulsations detected in 2002 and those in 2009, with


The picture for XTE J1751305 is thus one where the powerful outburst in 2002 was associated with an episode of rapid spin-up, followed by a slow drop off in the spin-rate. One would expect this behaviour to be naturally explained in terms of a dramatic rise in the accretion torque during the initial outburst and standard magnetic dipole spin-down in between outbursts. Assuming such a scenario will allow us to constrain the stellar magnetic field.

Before estimating the torques required to explain the spin-evolution, we note that the evolution of XTE J1751305 is qualitatively similar to that of another accreting pulsar, IGR J00291+5934. This system underwent outbursts in 2004 and twice in rapid (34 d) succession in 2008 (and possibly in 1998 and 2001, as well), with each 2008 outburst being about half as luminous as the one in 2004 (Patruno, 2010). For the 2004 outburst, the peak bolometric luminosity is , assuming a distance of 5 kpc (Falanga et al., 2005), which corresponds to . The 2004 outburst lasted for 14 d, during which the frequency increased with (Patruno, 2010; Papitto et al., 2011)


The first 2008 outburst had a rise-time of , lasted for 5 d, and decayed with a timescale of , while the second 2008 outburst lasted for 12 d (Hartman et al., 2011; Papitto et al., 2011). The 598.89 Hz pulsations associated with the spin of the neutron star had an amplitude of during both 2008 outbursts but was not detected in between these outbursts (amplitude ) (Hartman et al., 2011). The long-term decrease in frequency of pulsations detected in 2004 and those in 2008 is , yielding a spin-down rate (Patruno, 2010; Hartman et al., 2011; Papitto et al., 2011)


We now consider the implications of the observed spin evolution of XTE J1751305 and IGR J00291+5934, in particular using the measured changes of spin rate (or torque) to constrain the magnetic field of each neutron star. If the neutron star is not accreting between outbursts and the long-term spin-down is due to magnetic dipole radiation loss, then the magnetic field at the pole is given by Gunn & Ostriker (1969) as


or more accurately, by Spitkovsky (2006) as


where is the angle between the stellar rotation and magnetic axes (see also Contopoulos et al. 2014). Figure 2 shows, as a vertical band, the inferred magnetic field from the observed and its uncertainty [see equations (8) and (10)] and using equation (12).

Figure 2: Magnetic field of XTE J1751305 (top) and IGR J00291+5934 (bottom) as determined by the observed spin-down rate, spin-up rate and fastness parameter . The observed spin-down rate (with 90% uncertainty) produces the vertical band. The observed spin-up rate (with 90% uncertainty) is the horizontal band. The dotted and solid lines show the maximum calculated spin-up rates due to accretion torques and , respectively. The magnetic field determined by is where the solid line crosses the bottom axis.

The horizontal band in Figure 2 shows the observed spin-up rate during outburst and its uncertainty [see equations (6) and (9)]. If the spin up of the neutron star (and hence the torque on the star ) is the result of accretion (see below), then the magnetic field is determined by where the calculated spin-up due to the accretion torque intersects this horizontal band. For example, the simple accretion torque given by , where is the Alfvén radius and is the dipole moment (see, e.g. Pringle & Rees 1972), is shown as the dotted line. This torque produces a spin-up


where and . In other words, for a measured and , the neutron star magnetic field is


Here and in Figure 2, we use the maximal distance and peak X-ray flux during outburst to determine . This yields the maximum torque, and the shaded region below the dotted line denotes lower flux. We see that the spin-up rate and spin-down rate constraints on the magnetic field are wholly inconsistent for IGR J00291+5934, while they are consistent, given the observational uncertainties, for XTE J1751305.

However the simple accretion torque does not fully account for a magnetic field that threads the accretion disc and can lead to spin-up (when ) or spin-down (when ), where is the fastness parameter and is the Kepler rotation frequency at the Alfvén radius. A more realistic accretion torque is calculated by Ghosh & Lamb (1979), which is well-approximated by (Ho et al., 2014). The spin-up regime for this torque is shown as the solid line and cross-hatched region in Figure 2. We see that this accretion torque produces a spin-up rate far below the observed for XTE J1751305 and IGR J00291+5934. Other accretion torque estimates, such as torque or , where [] is the corotation radius, and models, such as those from Wang (1996); Kluźniak & Rappaport (2007); Tauris (2012), all fail to produce a strong enough spin-up torque.

Finally we can obtain a third estimate of the magnetic field by noting that accretion occurs only when the fastness parameter . This yields a magnetic field


That is,


indicated in Figure 2 by where the solid line drops to zero [since this line denotes ]. This estimate is close to that obtained from the dipole spin down [see equation (12)], but it does not explain the spin-up rate during outburst [see equation (14)]. The obvious conclusion is that either the accretion torque is severely underestimated or the spin of the star changes due to some other mechanism. For example, a possible contributing factor is timing noise caused by motion of the X-ray hot spot, such that the measured is not actually a change in the neutron star spin frequency (Patruno et al., 2009). However Patruno et al. (2009) and Patruno & Watts (2012) note that timing noise is weak in the 2002 (for XTE J1751305) and 2004 (for IGR J00291+5934) outburst data when was measured (see equations 6 and 9). Therefore it appears unlikely that timing noise could explain the entirety of the significant discrepancy between observed and calculated . Alternatively, it seems reasonable to suggest that the X-ray outburst may be associated with a crust fracture or slight rearrangement of the stellar magnetic field. This could lead to a shift in the stellar moment of inertia and a glitch similar to those seen in young radio pulsars. Noting that the observed , a level typical for smaller glitches, this explanation seems a possibility. Of course, pulse-by-pulse tracking of the signal during outburst could constrain this scenario.

4 Is there a consistent r-mode scenario?

So far we have arrived at two main conclusions. First of all, the value of the frequency of the observed oscillation in XTE J1751305 is consistent with an r-mode, without any particular adjustments to our current understanding of the physics of neutron stars, and does correspond to sensible values for mass and radius. Secondly, we have seen that, when r-modes are not included in the modelling, the observed evolution in the spin frequency (a spin up) of the star during the X-ray outburst is not (completely) explained by the standard accretion torque. We will now reconsider the evolution of the spin frequency, with the r-mode included; there are a variety of possible scenarios depending on the state of the mode: stable versus unstable and unsaturated versus saturated. It is easy to anticipate that inclusion of the r-mode will make the spin evolution even more difficult to explain, not easier. Note that Strohmayer & Mahmoodifar (2013) briefly described the unstable scenario and its difficulty in explaining the oscillation of XTE J1751305.

The amplitude of the mode is parameterised by ; see Owen et al. (1998) for a definition. The results of Strohmayer & Mahmoodifar (2013) suggest . One can immediately say that such a large value for an r-mode amplitude is extremely surprising on the basis of known results. As is discussed in detail in Mahmoodifar & Strohmayer (2013), one can place upper limits on r-mode amplitudes of intermittently accreting neutron stars in two ways, providing one assumes the mode is active all the time (not just during outburst). Firstly, one can use the core temperature, inferred from the star’s quiescent luminosity, to place a limit on the rate at which an r-mode heats the star, and hence limit the mode’s amplitude. Secondly, if one assumes that the long-term time-averaged change of spin frequency, averaged over many cycles of outburst and quiescence, is zero, with a steady r-mode driven spin-down providing the balance to the (intermittent) spin-up from accretion, one again gets an upper limit on . Such estimates typically lead to amplitudes in the range to . Using the first of these procedures, Mahmoodifar & Strohmayer (2013) find for XTE J1751305, much smaller than the value suggested by their measurement from the X-ray lightcurve.

However the oscillation in XTE J1751305 was observed only during the 2002 outburst itself. One can therefore take the conservative approach of setting aside the upper limits quoted above, and consider what effect the presence of an r-mode in this system would have on the spin frequency evolution, restricting attention to the interval of the outburst. Two features of the observations are relevant: (i) The mode amplitude has a typical value of over the outburst duration, and (ii) the star spins up during the outburst, at an average rate given by equation (6) above, with a total accumulated spin up given by equation (7). We will show that, even when attention is restricted to these limited observational facts, we fail to find a self-consistent explanation for the spin evolution of the star.

The basic reason for this lies in the well known result that an r-mode of steady amplitude produces a spin-down torque that would tend to make the spin frequency decrease at the rate (Owen et al., 1998):


where for a polytrope and is the timescale associated with gravitational radiation reaction effects; hour for XTE J1751305 (see equation 2.9 of Owen et al. 1998; the negative value reflects the fact that gravitational-wave emission tends to destabilize r-modes). Inserting numerical values, we obtain a spin-down rate of


where . This spindown rate is more than times greater than the observed over the outburst, and of opposite sign (see equation 6). Clearly, it is going to be difficult to reconcile the r-mode interpretation with the observed spin evolution. Nevertheless, let us examine the different possible scenarios.

To do so we can use the evolution model of Owen et al. (1998). This makes use of the canonical energy of the r-mode:


where for a polytrope, and also the total angular momentum of the system (star + r-mode):


where (again for a polytrope; the constants and are related by ). Together with expressions for the rate at which energy and angular momentum are radiated, these can be used to give evolution equations for the parameters :


The dimensionless amplitude is expected to saturate due to non-linear effects for an unstable mode, at some value . The above pair of evolution equations are valid only for . When saturation is achieved, so that takes the constant value , the evolution in spin frequency is given by


which is simply equation (17) with . Note that the equations above neglect the (weak) spin-up accretion torque.

The current thinking is that this saturation amplitude is small, due to coupling between the r-mode and a sea of other (shorter length scale) inertial modes. The initial estimates for this effect, due to Arras et al. (2003), lead to a saturation amplitude of order for a star spinning at 435 Hz, close to the value inferred from observation. More recent work predicts a much smaller amplitude, of (Bondarescu & Wasserman, 2013). However, the saturation amplitude depends upon the driving mechanism, and, as we will argue below, it is difficult to reconcile an r-mode interpretation of the observations with the standard unstable r-mode scenario, so it may be that these mode saturation studies are not applicable here.

Let us now consider the different possible regimes. The r-mode may be either stable or unstable. Let us first consider the (more familiar) unstable case. Before considering the evolution equations, a first obvious question, given the context provided by the r-mode literature from the last fifteen years or so, has to be if it is conceivable that XTE J1751305 lies inside the r-mode instability window. Since this system is not the fastest or hottest among the accreting LMXBs, one would not expect this to be the case (see, e.g. Figure 2 of Ho et al. 2011). Nevertheless, we will explore the consequences of the assumption of instability.

If the mode is saturated, then equation (23) immediately predicts the rapid spindown described above, inconsistent with the observations. Suppose instead the mode is, at the beginning of the observation at least, not saturated. Then equation (21) shows that, for the observed value of , a sufficiently large value of the timescale will give a small spin-down rate due to gravitational wave emission, opening up the possibility of a small net spin-up, once the accretion torque is included. However, in this case equation (22) indicates that the unstable mode will grow on the short timescale hour, leading to saturation on a timescale much shorter than the observation duration of days. Then the rapid spin-down of equation (23) again applies, in conflict with the observations. Clearly, there is no consistent scenario in which the mode is unstable: the mode is either saturated throughout the observation, or saturates very quickly, in both cases leading to rapid spin-down, in contradiction with the observed spin-up.

Now consider the (less familiar) case of stable r-mode evolution. In fact, the excitation of stable r-modes and their gravitational-wave detectability has already been considered in the context of isolated pulsars (Santiago-Prieto et al., 2012), and an exploration of the corresponding problem in accreting millisecond pulsar systems is currently underway (Santiago-Prieto et al., 2014). In such scenarios, it is necessary to supply some mechanism to excite the mode in the first place, perhaps connected with a glitch in an isolated pulsar or with the process that initiates bursts in accreting systems. We will therefore suppose some process suddenly excites the mode on some short dynamical timescale, up to an amplitude . If this happens at fixed angular momentum, equation (20) implies a corresponding spin-up of the star:


This is larger than the total spin-up measured over the course of the outburst, as given in equation (7). However, once excited, equation (22) shows that the only way the mode can remain at (or close to) this level for the long duration of the observation is if there is a near-cancellation between the and terms, so that . In such a case, equation (21) again gives rapid spin-down and total spin-down over the course of the observation


Comparing equations (24) and (25), we see that spin-down due to the gravitational-wave emission exceeds the initial spin-up due to the kick by a factor , so that the strong gravitational-wave torque clearly results in a net spin-down over the course of the observation, in conflict with the observations.

One might attempt to evade this conclusion by, instead of exciting the mode once with an impulsive kick, having some excitation mechanism acting in a continuous manner. If one adds energy to at a rate , then it is straight-forward to show that the evolution equations become




There is no difficulty in choosing a value of such that, in equation (27), can be made zero (or very small) such that the mode amplitude remains constant (or nearly constant) over the observation period at . However, eliminating from (27) using (26) leads to


By assumption, a nearly constant mode amplitude over the observation period means that (this follows from and ). Then this equation reduces to the familiar strong spin-down of equation (17) above, again in conflict with the observations.

The unavoidable conclusion is that, within the standard phenomenological model for r-modes (be they unstable or stable), it is not possible to reconcile the presence of a mode excited to the suggested amplitude with the observed spin evolution of the system.

5 Discussion

We showed that the observed frequency of the oscillation seen in XTE J1751305 during the 2002 outburst, if interpreted as an r-mode, would correspond to sensible values of mass and radius for the neutron star. However, we also argued that the presence of an r-mode with the large amplitude that the observations suggest, cannot be reconciled with the observed spin-up observed over the duration of the outburst. We conclude that the presence of an r-mode in XTE J1751305 during the 2002 outburst is not consistent with our current understanding of neutron star dynamics and must therefore be considered unlikely.

Nevertheless, one can look to gravitational-wave observations to see if they might be able to rule out (or indeed support) the r-mode scenario for similar events in the future. For the purpose of such a search in the case of XTE J1751305, it is interesting to note that the frequency of the signal would be known. The observed electromagnetic signal would carry the rotating frame frequency, but a gravitational-wave counterpart would be found at the inertial frame frequency. This means that one ought to search at


Moreover, it is easy to make a back-of-the-envelope estimate of the effective gravitational amplitude. The associated strain can be obtained from


where we take a conservative distance to the source of  kpc and (Owen, 2010). Inserting a mode amplitude of , this leads to (for the preferred values of mass and radius from Section 2)


Assuming that the mode remains at this amplitude over the outburst, this would be a periodic signal, the detectability of which increases with the observation time. The effective amplitude is then


The outburst in XTE J1751305 in fact preceded the first LIGO science run (S1) by a matter of a few months, so there is no possibility of carrying out a gravitational-wave search on archival LIGO science data. However, we note that this would have corresponded to a signal-to-noise of about if present in LIGO S5 data, and about in Advanced LIGO data, assuming that one can perform a fully phase coherent search over the observation period. Strohmayer & Mahmoodifar (2013) did indeed find that the observed oscillation was highly coherent, quoting the oscillation frequency (in Hertz) to the sixth decimal place, i.e. at the resolution of the Fourier transform ( days). See also their Figure 4. Clearly, the oscillation itself was intrinsically stable, and also the orbital solution was known sufficiently accurately to allow the effects of Doppler modulation due to the orbit to be largely removed over the observation interval. Should such an accurate timing solution be available for any future outburst, a fully coherent gravitational wave search could be carried out. Alternatively, in the absence of such an accurate timing solution, a search over a parameter space that allows for the orbital uncertainties could be carried out, but this would increase the signal-to-noise required to claim detection, as discussed in detail in Watts et al. (2008). Clearly, the signal analysis procedures for carrying out such a search merit further investigation and could be put to use if this (or a similar accreting system) were to display similar behaviour during a future science run of an advanced gravitational-wave detector.


DIJ acknowledges and thanks Ignacio (‘Nacho’) Santiago Prieto, Ik Siong Heng, and James Clark, with whom he has spoken about issues closely related to some of those discussed here over the last few years. The authors thank Tod Stohmayer and Simin Mahmoodifar for feedback on an early version of the manuscript. NA and DIJ acknowledge support from STFC via grant number ST/H002359/1, and also travel support from CompStar (a COST-funded Research Networking Programme).


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