A NON-RADIAL OSCILLATION MODE IN AN ACCRETING MILLISECOND PULSAR?

As noted in Section 1 above, when a pulsation mode periodically perturbs an X-ray-emitting hot spot that is fixed in the rotating frame of the star, the co-rotating frame mode frequency is imprinted on the light curve seen by a distant observer. Further, rapid rotation tends to increase the co-rotating frame frequency of the l = m = 2 r-mode from the slow-rotation limit of ω = 2Ω/3, while the influence of a solid crust may decrease it. Based on the discussion above, a reasonable frequency range to search is then 2/3 − k₁ ⩽ ω/Ω ⩽ (2/3 + k₂), where k₁ represents a plausible reduction in the frequency based on the possible crustal effects to the r-mode, and k₂ represents a reasonable maximum increase for $\kappa _2 \bar{\Omega }$ given the observed spin frequency of J1751 and various possible masses, equations of state and interior compositions for the neutron star. Based on the calculations of Yoshida & Lee (2001) and Alford et al. (2012; see their Figure 3), plausible values for k₁ and k₂ are 0.25 and 0.09, respectively. This defines a search range from 0.4166 ⩽ ω/Ω ⩽ 0.75667. A search in that range reveals one candidate peak in slightly more than 77.59 million independent Fourier frequency bins. Figure 5 shows a portion of the full-resolution spectrum in the vicinity of this peak. It appears at a frequency of 0.5727597 × ν_spin = 249.332609 Hz, and has a fractional Fourier amplitude of 7.455 × 10⁻⁴.

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To assess the significance of this peak we first convert its fractional Fourier amplitude to a Leahy-normalized power and then estimate its single-trial probability using the expected noise power distribution, which for a single power spectrum is the χ² distribution with 2 degrees of freedom. The peak Leahy-normalized power is then 49.26, which corresponds to a single-trial probability of 2 × 10⁻¹¹. Accounting for the number of trials by multiplying by the number of independent Fourier frequencies in the search range, 77.6 × 10⁶, gives a significance of 1.6 × 10⁻³, which is a little better than a 3σ detection. We then used a portion of the power spectrum at higher frequencies (from 1.6 to 2.2 times the pulsar spin frequency) to investigate how accurately the distribution of noise powers follows the expected χ² distribution. The result is shown in Figure 6, where the red dashed line denotes the probability to exceed a given Fourier power for the χ² distribution with 2 degrees of freedom, and the Leahy-normalized power spectral data are plotted as a histogram. Over the range of Fourier powers present in the data the power spectral values show a good match to the expected distribution. The Fourier power of the candidate peak is marked by the vertical dash-dotted line, and as indicated above, has a single trial probability of 2 × 10⁻¹¹. Based on this we think our significance estimate is reasonable.

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We next averaged the full resolution power spectrum in order to search for any broader bandwidth signals that might be present. Figure 7 shows two such averaged power spectra over the full frequency range. The black and green histograms have frequency resolutions of 1/2048, and 1/128 Hz, respectively. The pulsar signal is still easily detected in each case, but we do not find any other significant features at these or other frequency resolutions. The horizontal dashed black line marks the amplitude of the candidate signal at 249.33 Hz discussed above. We can place upper limits on any signal power in our defined search range at these frequency resolutions of 1.64 × 10⁻⁴ and 1.42 × 10⁻⁴, respectively. The horizontal red dashed line in Figure 7 marks an amplitude given by 1/(N_tot)^1/2 = 1.50 × 10⁻⁴, which gives a reasonably close approximation to the quoted upper limits for broader band signals.

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As discussed in Section 1, if an oscillation mode modulates emission over the entire neutron star surface rather than simply perturbing a hot spot fixed in the co-rotating frame, then one would expect a pulsation signal at the mode's inertial frame frequency, ω_i = 2Ω − ω, where ω is the co-rotating frame frequency. Thus, to search the range of inertial frame frequencies corresponding to the range of co-rotating frame frequencies just discussed in Section 2.1 we need to search the frequency range 2 − (2/3 + 0.09) < σ/Ω < 2 − (2/3 − 0.25), which reduces to 1.243 < σ/Ω < 1.583. A search reveals no significant peaks in this range. The highest peak appears at a frequency of 1.565327ν_spin, with a fractional Fourier amplitude limit of 6.6 × 10⁻⁴.

J1814 was discovered by RXTE in 2003 June using data obtained with the Galactic bulge monitoring program then being conducted with the PCA on board RXTE. The pulsar has a 314.36 Hz spin frequency and an orbital period of 4.275 hr (Markwardt et al. 2003; Papitto et al. 2007). The discovery outburst lasted for ≈50 days. This object was the first neutron star to exhibit burst oscillations with a significant first harmonic (Strohmayer et al. 2003), and indeed, the persistent pulse profile also shows substantial harmonic content. This source is also known to exhibit significant timing noise, that is, systematic timing residuals remain after modeling the binary Doppler delays (Papitto et al. 2007; Watts et al. 2008b). This noise is still not completely understood, but may represent movement of the accretion hot spot relative to the stellar spin axis as the accretion rate changes during an outburst (Patruno 2010). Here we use data from the first 12 days for which such variations were less significant (Watts et al. 2008b). We used data beginning on 2003 June 5 at 02:34:20 UTC and constructed two light curves, each sampled at 2048 Hz and with 2³⁰ time bins. There are a total of 31, 361, 962 X-ray events in the two light curves. We first modeled the orbital variations in a similar manner as described above for J1751. Our orbit parameters are consistent with those of Papitto et al. (2007). Figures 8 and 9 show the resulting orbit-corrected phase residuals for the two data segments used to construct our light curves. One can see that the second interval (Figure 9) shows more systematic timing noise than the first interval (Figure 8). We then computed power spectra for each interval in the same manner as described for J1751. We searched the power spectra in the same frequency ranges as described above for J1751 and for each data interval separately as well as the average power spectrum computed from both intervals. We did not find any significant features in the power spectra. Figure 10 shows two average power spectra computed from both intervals, the black and green histograms have been averaged to frequency resolutions of 1/2048 and 1/128 Hz, respectively. The pulsar fundamental and first harmonic are clearly evident (at 1 and 2 in these units). The horizontal dashed line (black) marks the upper limit of 7.8 × 10⁻⁴ on any signal power at the full frequency resolution of the power spectrum. The horizontal dashed (red) line marks an amplitude given by 1/(N_tot/2)^1/2 ≈ 2.5 × 10⁻⁴, which again gives a reasonably close approximation to the upper limits for broader band signals.

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Pulsations at 205.89 Hz were detected with RXTE from the globular cluster source NGC 6440 X−2 on 2009 August 30 (Altamirano et al. 2009). On this date the source was observed for a single RXTE orbit, yielding ≈3000 s of exposure, revealing a 57 minute orbital period (Altamirano et al. 2010). A subsequent outburst with detectable pulsations was observed with RXTE on 2010 March 21, for an additional three RXTE orbits and 6600 s of exposure. We used all these data in our search. As for J1751, we first barycentered the data using the best determined position from Heinke et al. (2010). Because the available data for this source are too sparse to enable calculation of a single, coherent Fourier power spectrum, we separately modeled the orbital variations in each of the four data segments. We then generated light curves using the orbit-corrected arrival times, computed a Fourier power spectrum for each, and then averaged them. The light curves were sampled at 8192 Hz, yielding a Nyquist frequency of 4096 Hz. The resulting averaged power spectrum is shown in Figure 11. Since there are many fewer Fourier frequencies compared to either J1751 or J1814, we show the spectrum at the full frequency resolution. Two spectra are shown in Figure 11, the black curve is plotted at the full frequency resolution (3.125 × 10⁻⁴ Hz), and the green curve has been averaged by a factor of 32 to a resolution of 0.01 Hz. The pulsar fundamental is clearly evident (at 1 in these units), but there are no other candidate detections. At these resolutions we can place upper limits on any signal power in our search ranges of 5.6 × 10⁻³ (at 3.125 × 10⁻⁴ Hz resolution), and 2.8 × 10⁻³ (at 0.02 Hz). Because much less data is available for X−2 than for either J1751 or J1814, the limits are not as constraining as for those sources.

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As briefly mentioned above, the g-modes are low frequency non-radial oscillation modes of neutron stars with buoyancy as their restoring force. In a three-component neutron star model composed of a fluid core, a solid crust and a fluid ocean, g-modes might be excited in the core and/or in the ocean, but the finite shear modulus excludes them from the crust (Bildsten & Cutler 1995). As a result core g-modes are unlikely to have observable effects on the radiation observed from the surface of the star. Therefore, here we focus on the surface g-modes that are confined to a thin layer at the surface of the star. There have been many studies on g-modes in neutron stars (McDermott & Taam 1987; Strohmayer & Lee 1996; Bildsten & Cutler 1995; Bildsten et al. 1996; Bildsten & Cumming 1998). We are particularly interested in the surface g-modes in AMXPs. The conditions at the surface of these objects evolve slowly due to the accretion and their g-mode spectrum is different from that of the isolated and non-accreting neutron stars. The g-modes at the surface of an accreting neutron star can be divided into several different categories according to their different sources of buoyancy, such as an entropy gradient or density discontinuity. Bildsten & Cutler (1995) studied surface g-modes in accreting systems with thermal buoyancy as the restoring force. They obtained an analytic result for the mode frequency in the non-rotating limit (Ω → 0)

$$\begin{eqnarray} f_{th} &=& 6.26\ {\rm Hz} \left(T_8\frac{16}{A}\right)^{\frac{1}{2}} \left(1+\left(\frac{3n\pi }{2\ln (\rho _b/\rho _t)}\right) ^2\right)^{-1/2}\nonumber \\ &&\times \left(\frac{10\ {\rm km}}{R}\right) \left(\frac{l(l+1)}{2}\right)^{\frac{1}{2}}, \end{eqnarray} \tag{ 1 }$$

where T₈ = (T/10⁸ K), A is the mass number, l is the spherical harmonic index, n is the number of radial nodes in the displacement eigenfunction, R is the stellar radius, and ρ_b and ρ_t are densities at the bottom and top of the ocean, respectively. Strohmayer & Lee (1996) studied thermal g-modes in a steady state accreting and nuclear burning atmosphere, and found some modes can be excited by the mechanism (perturbations in the nuclear burning). For example, see their Table 3 for results on the oscillation periods of g₁ and g₂ modes. Bildsten & Cumming (1998) studied the effect of hydrogen electron captures on g-modes in the ocean of accreting neutron stars. They found that the sudden increase in the density at the hydrogen electron capture layer supports a density discontinuity mode with a non-rotating limit frequency

$$\begin{equation} f_d \approx 35\ {\rm Hz} \left(\frac{X_r}{0.1}\right)^{\frac{1}{2}} \left(1-\frac{\Delta Z}{\Delta A}\right)^{\frac{1}{2}}\left(\frac{10\ {\rm km}}{R}\right) \left(\frac{l(l+1)}{2}\right)^{\frac{1}{2}}, \end{equation} \tag{ 2 }$$

where X_r is the residual mass fraction of hydrogen and ΔZ and ΔA are the changes in the charge and mass of the nuclei from one side of the discontinuity to the other.

Piro & Bildsten (2004) studied non-radial oscillations at the surface of a helium-burning neutron star. They found one unstable mode that resides in the helium atmosphere and is supported by the buoyancy of the helium/carbon interface. The frequency of this mode in the non-rotating limit is given by

$$\begin{equation} f_{th-He}\approx (20\hbox{--}30\ {\rm Hz}) \sqrt{\frac{l(l+1)}{2}}, \end{equation} \tag{ 3 }$$

which depends on the accretion rate. Similar to the results of Strohmayer & Lee (1996), they find that this mode can also be driven unstable by the -mechanism, and they also compute results for higher accretion rates. Now, the orbital period of J1751 is very short (∼42 minutes; Markwardt et al. 2002), which means that it is a very compact system and thus the donor star is likely a helium white dwarf. This suggests that the accreted material is helium-rich and it therefore seems plausible that the system might show the unstable shallow surface waves that are obtained for a helium atmosphere.

It is important to note that the g-mode frequencies just discussed are obtained in the non-rotating limit (Ω → 0) and, as alluded to above, they will be modified by rapid rotation of the star. Bildsten et al. (1996) studied the effect of high spin frequencies on the g-mode spectrum in the so-called "traditional approximation," in which mode propagation is confined to a thin shell, the radial component of the coriolis force is neglected, and the radial displacements produced by the modes are assumed to be much less than the horizontal displacements. These approximations are reasonable for surface g-modes as long as the coriolis force remains less than the buoyant force (see Section 2 of Bildsten et al. 1996). This condition can be expressed as N² ≫ (ΩωR)/h, where N, R and h are the Brunt Väisällä frequency (which sets the strength of buoyancy), stellar radius, and characteristic scale height in the surface envelope, respectively. Assuming the candidate frequency in J1751 represents the mode frequency in the co-rotating frame (and using the 435 Hz spin frequency of J1751), then ω = 0.573Ω, and we would require N² ≫ 4.3 × 10⁶(R/h) Hz².

From this analysis Bildsten et al. (1996) found that stellar rotation "squeezes" the eigenfunctions toward the equator within an angle cos θ < (1/q), where θ is measured from the pole, q = (2Ω/ω), and the oscillation frequency of the mode (in the co-rotating frame) in a non-rotating star, ω_{l, 0}, is related to the mode frequency at arbitrary spin frequencies by the following equation:

$$\begin{equation} \omega ^2=2\Omega \omega _{l,0}\left[\frac{(2l_{\mu }-1)^2}{l(l+1)}\right]^{1/2} \end{equation} \tag{ 4 }$$

where l_μ is the number of zero crossings in the angular displacement between cos θ = −(1/q) and +(1/q). Thus, the surface displacements of the rotationally modified g-modes are strongly confined to the equatorial region at high spin frequencies and the modes are exponentially damped for cos θ ⩾ 1/q (Bildsten et al. 1996). For the spin frequency of J1751, q ≃ 3.5 (assuming that the 249.33 Hz candidate frequency is associated with a co-rotating frame mode frequency). This leaves open the question of what mode amplitudes would be needed to effectively perturb a hot spot located near the rotational pole.

Moreover, the relevant scale height, h, will depend on details of the surface envelopes in question; however, a typical value for the He-rich envelopes of Piro & Bildsten (2004) is h ≈ 200 cm, thus, for a 10 km radius neutron star we would require N² ≫ 2.2 × 10¹⁰ Hz² in order to satisfy the assumptions associated with the traditional approximation. We note that this condition appears to be technically violated for these envelopes as N is everywhere less than about 1 × 10⁵ Hz (see their Figures 2 and 3). This suggests the need for more theoretical work in order to more accurately determine the surface g-mode properties for rotation rates appropriate to the faster spinning AMXPs (such as J1751). In addition, more work similar to that done in the context of r-modes by Numata & Lee (2010) should be done for the rotationally modified g-modes to determine how efficiently these modes can perturb a hot spot located near the spin axis of the star and the resulting light curves.

Keeping in mind these caveats, we can nevertheless rearrange Equation (4) to express the frequency of the mode in a non-rotating star, f_{l, 0}, in terms of the observed mode co-rotating frame frequency, f_obs, stellar spin frequency, ν_spin and the mode indices l, m and l_μ. If the observed frequency is directly related to the modes co-rotating frame frequency (the "co-rotating frame scenario"), we find

$$\begin{equation} f_{l,0}=\left(f_{{\rm obs}}^2/(2\nu _{{\rm spin}})\right)*\sqrt{(l(l+1)/(2l_{\mu }-1)^2}. \end{equation} \tag{ 5 }$$

However, if the observed oscillation frequency is the mode's inertial frame frequency (the "inertial frame scenario"), then we must first relate this to the co-rotating frame via f_obs = mν_spin − f_{obs, i}, where f_{obs, i} is the (observed) inertial frame mode frequency, yielding

$$\begin{equation} f_{l,0}=((m\nu _{{\rm spin}} - f_{{\rm obs}, i})^2/(2\nu _{{\rm spin}}))*\sqrt{(l(l+1)/(2l_{\mu }-1)^2} . \end{equation} \tag{ 6 }$$

We can then find plausible non-rotating g-modes that can be consistent with the candidate frequency. Possible identifications are summarized in Table 1 and discussed in more detail below.

From the discussion above we can see that the candidate peak at 0.5727 × ν_spin = 249.33 Hz in J1751 may be identified as an l = 2, m = 1(l_μ = 3) g-mode that resides in a helium atmosphere and has a non-rotating frequency of ∼35 Hz as observed in the co-rotating frame. This is based on the assumption that this surface mode perturbs the hot spot periodically and the candidate frequency is related to the frequency of the mode in the co-rotating frame. This mode is consistent with the g₂ mode given in Table 3 of Strohmayer & Lee (1996) with a period of 29.04 ms and $\dot{M}/\dot{M}_{{\rm Edd}}=0.7$ in a pure helium shell. The thermal g-mode computations discussed above have been done under the assumption of steady-state nuclear burning in a thermally stable envelope. Now, stable burning of the accreted material in the envelope requires a high, near-Eddington accretion rate (Piro & Bildsten 2004); however, the average accretion rate of J1751 was about 2.1 × 10⁻¹¹ M_☉ yr⁻¹ (Markwardt et al. 2002), which is low relative to the Eddington rate, and therefore the assumption of steady-state nuclear burning may not be applicable in this case. However, we note that the relevant accretion rate for the thermal stability calculation is the local value (per unit area) which might be higher depending on the accretion geometry, for example, if accretion is restricted to a portion of the neutron star's surface.

If we assume that the amplitude of this mode is high enough that it can modify the temperature distribution at the surface of the star and produce observable X-ray variations then the inertial frame scenario is relevant (see Equation (6) above). In this case the candidate oscillation in J1751 may be consistent with an l = m = 1 shallow surface wave in the helium layer with a non-rotating frequency of 18.7 Hz (this is slightly less than the lower limit of 20 Hz given in Piro & Bildsten (2004)).

Another possibility for the candidate at 249.33 Hz would be an l = l_μ = 2 (with m = 0 or m = 2) density discontinuity g-mode due to hydrogen electron captures in the ocean of the star with a non-rotating limit frequency of f_d ≃ 58.34 (see Equation (2)) as measured in the co-rotating frame. However, whether or not sufficient hydrogen is present to support a density discontinuity mode in such a compact and presumably helium-rich system as J1751 remains an open question.

Carroll et al. (1986) showed that in the presence of strong magnetic fields the frequencies and displacements of modes that reside in the ocean, in particular g-modes, will be modified. For magnetic fields B₀ > 10⁵ G, these modified g-modes (magneto-gravity modes) change with increasing B from predominantly g-modes with constant periods to predominantly magnetic modes with periods proportional to $B_0^{-1}$ (see their Equation (42) and Figure 4). Piro & Bildsten (2004) estimated the maximum magnetic field before the shallow surface mode would be dynamically affected to be B_dyn ≈ 5 × 10⁷ G((ω/2π)/21.4 Hz) which is about 6 × 10⁸ G for a rotationally modified shallow surface wave with a frequency of 249.33 Hz. This is close to the estimated value of the magnetic field of J1751 obtained from spin-down measurements due to magneto-dipole radiation which is about 4 × 10⁸ G (Riggio et al. 2011). However, Heng & Spitkovsky (2009) also explored the effect of a vertical magnetic field on shallow surface waves, and their results suggest that for the spin rate and likely magnetic field strength appropriate to J175, the field does not strongly modify the mode frequencies (see the "magneto-Poincare modes" in their Figure 2). The above results support the conclusion that the magnetic field likely does not exert a dramatic influence on these g-mode frequencies.

As we discussed earlier, another class of non-radial oscillation modes that may have frequencies consistent with the candidate signal in J1751 are the r-modes. A three-component neutron star model may have unstable r-modes in the ocean and/or in the core. The frequency of the r-modes in the slow-rotation limit ($\bar{\Omega } \equiv \Omega /(GM/R^3)^{1/2} \rightarrow 0$) is given by ω₀ = 2mΩ/[l(l + 1)]. As the rotation frequency of the star increases, the co-rotating frame frequency of the r-modes in the surface layer of the star deviates appreciably from this asymptotic form and becomes almost insensitive to Ω (see Figure 4 in Lee 2004). According to Table 2 in Lee (2004) the frequencies of the surface r-modes are always less than 200 Hz for the spin frequencies and mass accretion rates that are relevant to LMXBs. For example, for the l = |m| = 2 fundamental r-modes of radiative envelopes Lee (2004) found that the co-rotating frequency is in the range of 101 to 173 Hz for the stellar spin frequencies of 300 to 600 Hz and $\dot{M}=0.02\ \dot{M}_{{\rm Edd}}$ to $0.1\ \dot{M}_{{\rm Edd}}$. Lee (2010) also studied the low frequency oscillations of rotating and magnetized neutron stars and found no r-modes confined in the ocean in the presence of a magnetic field even as low as B₀ ∼ 10⁷ G in a three-component neutron star model. Thus, the candidate frequency at 249.33 Hz in J1751 doesn't appear to be consistent with that of a surface r-mode.

Although the amplitudes of the ocean g- and r-modes tend to be confined to the equatorial regions, this is not the case for l = |m|r-modes in the fluid core. In fact Lee (2010) showed that the displacement vector of these core r-modes have large amplitudes around the rotation axis at the stellar surface even in the presence of a surface magnetic field B₀ ∼ 10¹⁰ G.

As we briefly mentioned in the previous sections, the co-rotating frame frequency of l = m = 2 core r-modes (which are the most unstable ones) in the $\bar{\Omega } \rightarrow 0$ limit is equal to ω₀ = (2/3)Ω, which is larger than the frequency of the candidate peak at ω = 0.5727Ω and adding the corrections due to high spin rates only slightly increases the slow-rotation limit value. Yoshida & Lee (2001) studied the effect of a solid crust on the r-mode oscillations of a three-component neutron star model. At sufficiently small values of Ω the coupling between r-modes and crustal toroidal modes is negligibly weak, but at higher spin frequencies they found that the core r-modes are strongly affected by the mode coupling with crustal toroidal modes, and because of the avoided crossings with the crustal toroidal modes, the core r-modes will lose their simple form of eigenfrequency and eigenfunction. The r-mode frequency increases as the spin frequency of the star increases, and at some point it meets the frequency of the crustal modes which results in avoided crossings. Depending on the thickness of the crust and therefore the number of modes in the crust with relevant frequencies, there might be several avoided crossings between core r-modes and crustal toroidal modes (Levin & Ushomirsky 2001; Glampedakis & Andersson 2006). The spin frequencies at which the avoided crossings occur are given by Ω_cross ≈ (l(l + 1)/m)ω_t(0) where ω_t(0) is the oscillation frequency of the toroidal mode at Ω = 0 and it is a function of the shear modulus of the crust. As shown in Figure 3 of Yoshida & Lee (2001), in the presence of a solid crust and at high rotation frequencies, the r-mode frequency in the co-rotating frame deviates from its simple form in the $\bar{\Omega } \rightarrow 0$ limit. For fundamental r-modes with l = m = 2 they showed that κ can decrease from its slow-rotation limit and span a range of values from (2/3) to less than 0.4 depending on the spin frequency of the star and the properties of the solid crust, such as its shear modulus. We note that the value of μ/ρ is almost constant in the crust of a neutron star, μ/ρ ≃ 1–6 × 10¹⁶ cm² s⁻² (see, for example, Figure 1 in Glampedakis & Andersson 2006). The results of Yoshida & Lee (2001) given in their Figure 3 and Table 1 suggest that for κ ∼ 0.57 at $\bar{\Omega } \simeq 0.2$ (relevant for J1751) one needs the shear modulus of the crust to be a few times higher than the standard values given by Strohmayer et al. (1991) for a bcc crystal at the higher densities in the crust. This suggests that observations of r-mode-induced oscillations in the X-ray flux of neutron stars could be useful in probing the structure and properties of the crust.

In addition to the r-modes, the Coriolis force also supports the more general class of inertial modes which have both significant toroidal and spheroidal angular displacements, whereas the r-modes are principally toroidal. A number of authors have studied the properties of inertial modes, and in particular their relationship to other low-frequency modes such as the g-modes (Yoshida & Lee 2000a, 2000b; Passamonti et al. 2009; Lee 2010). For example, Passamonti et al. (2009) have computed time evolutions of the linear perturbation equations in order to explore the oscillations of rapidly rotating, stratified (non-isentropic) neutron stars. They find that the g-modes in stratified stars become strongly modified by rapid rotation, with each g-mode frequency approaching that of a particular inertial mode associated with the corresponding isentropic (i.e., no bouyancy) stellar model. Earlier work by Yoshida & Lee (2000b) reached a similar conclusion, but the more recent results of Passamonti et al. (2009) have explored the connection to much higher rotation rates. These studies, as well as the recent calculations of Lee (2010), all find some inertial modes with co-rotating frame frequencies that appear at least qualitatively consistent with the candidate oscillation in J1751. For example, the ³i₁ and ⁴i₂ modes of Passamonti et al. (2009) have frequencies near ω = 0.573Ω (see their Table 2 and Figures 3 and 11). Note that for their stellar models Ω/(Gρ_c)^1/2 ≈ 0.5 is appropriate for the 435 Hz spin frequency of J1751. Similarly, the l₀ − |m| = 2, m = 2 prograde, isentropic inertial mode of Yoshida & Lee (2000a), and the non-isentropic modes labeled g₋₁(2) < − − > i₋₁(2) in Figure 9(a) of Yoshida & Lee (2000b) have frequencies near our candidate oscillation. It should be noted, however, that all these calculations have significant simplifications that likely make detailed quantitative comparisons with our observed frequency problematic. For example, they all employ rather simplistic stellar models, such as the use of polytropic equations of state, and the models do not have a solid crust. Additionally, the calculations of Yoshida & Lee (2000a, 2000b) were for relatively modest spin rates, and extrapolation to the higher spin rate appropriate for J1751 is perhaps risky.

In addition, Lee (2010) has presented oscillation mode calculations for rotating, and magnetized neutron stars using three-component (ocean, crust, core) models. He also finds prograde inertial modes with frequencies approximately consistent with our candidate oscillation (see, for example, the |m| = 2 modes for B₀ = 10¹⁰ G near the lower right corner of Figure 5). These calculations were for a low mass, 0.5 M_☉, neutron star and are also only strictly valid for modest rotation rates, so, again, caution should be exercised when making quantitative comparisons with observed frequencies. We emphasize that all of the above calculations were for global stellar modes, and not restricted to only surface displacements. Similar to the global r-modes these inertial modes will likely have appreciable surface amplitudes closer to the rotational poles than the surface-based, rotationally modified g-modes investigated by Bildsten et al. (1996) and Piro & Bildsten (2004). Based on the above discussion it seems possible that inertial modes could be relevant to our candidate oscillation in J1751, but clearly new theoretical work is needed to explore such modes in more realistic, rapidly rotating neutron star models before any firm conclusion should be drawn. Further, new calculations to determine how effectively inertial modes can perturb an X-ray emitting hot spot, and the resulting light curves, are certainly warranted.

The candidate power spectral peak in J1751 is narrow, which means that the oscillation frequency has to be steady over most of the time span used to compute the power spectrum, which is about six days. Thus, if the candidate peak is due to some non-radial oscillation of the star, its frequency has to be almost constant during that time span. Between surface g-modes and core r-modes which might be consistent with the observed candidate peak as discussed above, r-modes are expected to have steady frequencies over such a short time span because they reside in the core and conditions there are not expected to change over such timescales. Among surface g-modes that are consistent with the candidate oscillation in J1751, thermal g-modes of a helium burning neutron star reside in the shallow layers close to the surface of the star, but the density discontinuity g-modes due to hydrogen electron capture reside in deeper layers close to the ocean-crust interface. If the temperature and elemental composition of the ocean doesn't change during the time span used to compute the light curve, the frequency would be steady which is expected to be the case if the accretion rate varies little. In fact, it has been shown by Piro & Bildsten (2004) that the g-mode frequency scales approximately as $\dot{m}^{1/8}$ where $\dot{m}$ is the local accretion rate, and therefore a small change in the accretion rate will not have a large effect on the g-mode frequencies.

The light curve of J1751 (see Figure 1) shows variation in the count rate at the level of 30–40 counts s⁻¹, which likely suggests some variation in the accretion rate. Although we note that X-ray flux (or count rate) is known to not always correlate linearly with the accretion rate. While this suggests the mode frequency may change, a second effect likely limits the rate at which it can vary, and that is set by the time, t_acc, required to change conditions in the surface layers at a column depth where the mode frequency is set. This can be roughly approximated as $t_{{\rm acc}} \approx y / \dot{m}$, where y is the relevant column depth in g cm⁻², and $\dot{m}$ is the accretion rate per unit surface area. For an accretion rate of 2 × 10⁻¹¹ M_☉ yr⁻¹, and a characteristic column depth of 10⁸ g cm⁻², t_acc is about 11.6 days. So, while accretion rate variations can, in principle, change the g-mode frequencies, for timescales much less than t_acc the frequency is likely reasonably stable.

As can be seen in several of our power spectra (see, for example, Figure 7), an upper limit on the modulation amplitude is approximately given by 1/(N_tot)^1/2, where N_tot is simply the total number of X-ray events in the light curve from which the power spectrum is computed. The approximation is better as one averages more frequency bins, meaning it is a more precise limit in the context of broader band-width signals. For the full resolution spectra presented here the derived upper limits are reasonably approximated as ≈4/(N_tot)^1/2.

This is not too surprising, as the fractional Poisson error on the average count rate within a time interval is just 1/(N_tot)^1/2. Thus, this limit is simply a statement that one cannot measure a fractional modulation amplitude of the X-ray count rate that is smaller than the precision with which that rate can be determined. Assuming that other necessary capabilities are present in future observatories—such as adequate high frequency time resolution—then a simple way to estimate the amplitude sensitivity for future detectors is just to scale up the expected count rates appropriately. The above considerations are valid in the case that the source count rate dominates any background rate.

The largest effective area for fast X-ray timing presently being planned is ESA's LOFT (Feroci et al. 2012). The Large Area Detector on LOFT would consist of ≈12 m² of silicon detectors and due to the larger collecting area and better (flatter) response above 6–7 keV would provide an increase in source count rate compared to the PCA on RXTE of about a factor of 30 (though the exact scaling would depend on the X-ray spectrum of the source being considered). The other way to increase the total counts that can be included in a light curve is to more densely sample an outburst, and to Fourier analyze longer continuous time intervals. For the sake of argument, if we scale based on the most sensitive observation reported here, that is, the single ≈6 day interval for J1751, and assume that a LOFT observation has twice the duty cycle and extends for twice as long, then we might expect to reach an amplitude limit of a_amp ≈ 1/(2*2*30*44 × 10⁶)^1/2 = 1.4 × 10⁻⁵.

While this represents a limit on the Fourier amplitude of X-ray flux modulations that could be detected, the corresponding amplitude of an oscillation mode would depend on the details of how the oscillation mode perturbs the X-ray emission. Numata & Lee (2010) show from their light curve modeling that the observed Fourier amplitude is proportional to the normalized amplitude of the stellar oscillation (see their Figure 6). The details of the scaling depends on the particular oscillation mode and other details, but a rough estimate indicates that the Fourier amplitude a_amp ≈ 1–2 × A, where A represents the maximum horizontal displacement produced by a mode divided by the stellar radius (A = max(|ξ_θ|/R, |ξ_ϕ|/R)). Based on this simple scaling one can expect that future sensitivities with LOFT would be such that A ≈ 1 × 10⁻⁵ could be probed. We note that this corresponds to a 10 cm maximum surface displacement for a 10 km neutron star.

In the case of r-mode oscillations, A is approximately equal to α/2, where α is the dimensionless amplitude of the mode, defined in Equation (1) of Lindblom et al. (1998). We note that for the candidate oscillation in J1751, A ≈ 7 × 10⁻⁴, and α ∼ 10⁻³. This is much larger than the upper limits on α given in Mahmoodifar & Strohmayer (2013), which is less than 10⁻⁷ for J1751 (see also Haskell et al. 2012). A global r-mode with an amplitude of the order of 10⁻³ would cause a rapid spin-down of the star. Using the corresponding equation for spin-down due to gravitational wave emission from unstable r-modes (Owen et al. 1998), dΩ/dt ≃ −(2Ω/τ_V)Qα², where τ_V(T, Ω, α) is the viscous damping timescale of the mode, and $Q \equiv {3\tilde{J}}/{2\tilde{I}}$ (Mahmoodifar & Strohmayer 2013) gives a spin-down rate of ∼−1.3 × 10⁻⁹ for J1751 assuming a core temperature of ∼3 × 10⁷ K (Mahmoodifar & Strohmayer 2013). We note that even with a higher core temperature of ∼3 × 10⁸ K the spin-down rate would be ∼−2.8 × 10⁻¹¹, which would still dominate the accretion spin-up rate and therefore is inconsistent with the observations (Patruno & Watts 2012). Further, if the amplitude of the mode is saturated at α_s ∼ 10⁻³, τ_V in the spin evolution equation should be replaced by τ_G, the gravitational radiation timescale. This would cause an even larger spin-down rate of ≈ − 1.5 × 10⁻⁷. Such a large amplitude for a global r-mode, even assuming that it is large only during the outburst and would be damped in quiescence, would cause a large change in the frequency of J1751 which would be easily detectable in the data. In addition, the maximum saturation amplitude due to nonlinear mode coupling, computed by Arras et al. (2003), α_s ≈ 8 × 10⁻³(ν_s/1 kHz)^5/2, that in the case of J1751 is ∼6 × 10⁻⁵ (see also Watts et al. 2008a and Bondarescu et al. 2007), and the upper limits on α (∼10⁻⁴) from gravitational wave searches with LIGO (Owen 2010; Aasi et al. 2013) further support the notion that the candidate oscillation is unlikely to be a global r-mode. This argues that a g-mode or inertial mode interpretation is more likely. While we think the present evidence is strongly suggestive, future, more sensitive observations will likely be needed to confirm the presence of non-radial oscillation modes in J1751 and/or other AMXPs.