UPPER BOUNDS ON r-MODE AMPLITUDES FROM OBSERVATIONS OF LOW-MASS X-RAY BINARY NEUTRON STARS

Here we compare two different methods for constraining the r-mode amplitude, α, from observations of LMXB NS transients. The first one, which gives larger values for α, is based on the spin equilibrium assumption (Brown & Ushomirsky 2000; Ho et al. 2011; Haskell & Patruno 2011; Watts et al. 2008) where we assume that in an outburst-quiescence cycle all the spin-up torque due to accretion during the outburst is balanced by the r-mode spin-down torque due to gravitational radiation in the whole cycle. This is similar to the prescription considered by previous authors (Brown & Ushomirsky 2000; Ho et al. 2011), but rather than using a "fiducial" torque estimated from the long-term average $\dot{M}$, we can now use the observed spin-up rates and outburst properties to directly constrain the torque. Therefore we have

$$\begin{equation} 2\pi I \dot{\nu } \Delta = \frac{2 J_c}{\tau _G} \end{equation} \tag{ 17 }$$

where $\dot{\nu }$ is the spin-up rate during outburst and Δ = (t_o/t_r) is the ratio of the outburst duration, t_o, to the recurrence time, t_r. We can estimate the left-hand side of Equation (17) from X-ray observations of LMXBs and since the right-hand side is a function of α through J_c (see Equation (2)), we can determine α. In Table 2 we give the results for α for three sources for which there are now estimates of the spin-up rate due to accretion. The values of α are given for the three different NS models considered in this work, and they are all in the range of ≈1–3 × 10⁻⁷.

Table 2. Constraints from Spin Equilibrium

Source	$\Delta =\frac{t_o}{t_r}$	$\dot{\nu }_{{\rm su}}$	αsp.eq	αsp.eq	αsp.eq
		(Hz s−1)	(1.4 M☉)	(2.0 M☉)	(2.21 M☉)
IGR J00291	$\frac{13}{1363}$	5 × 10−13	1.46 × 10−7	1.22 × 10−7	1.41 × 10−7
SAX J1808 − 3658	$\frac{40}{2\times 365}$	2.5 × 10−14	3.20 × 10−7	2.66 × 10−7	3.08 × 10−7
XTE J1814 − 338	$\frac{50}{19 \times 365}$	1.5 × 10−14	2.12 × 10−7	1.76 × 10−7	2.04 × 10−7

Notes. Duty cycle Δ, spin-up rate $\dot{\nu }_{{\rm su}}$ (Patruno & Watts 2012), and upper limits on r-mode amplitudes from the "spin-equilibrium" condition α_sp.eq, for three neutron star transients with assumed masses of 1.4 M_☉, 2.0 M_☉, and 2.21 M_☉.

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At these amplitudes, the inferred r-mode spin-down rate would be competitive with the magnetic dipole spin-down rate which almost certainly exists in these LMXBs and which is quite likely the dominant spin-down mechanism. Moreover, the amplitudes are also comparable to those deduced assuming r-mode spin equilibrium with the "fiducial" accretion torques estimated by Brown & Ushomirsky (2000). Those authors also demonstrated that at such amplitudes some of these objects should have significantly higher quiescent luminosities due to r-mode reheating than observed. These results suggest that the r-mode torque does not balance the accretion-driven spin-up torque and that r-mode amplitude estimates based on the "spin equilibrium" assumption will overestimate the true amplitude.

Here, we use the same thermal equilibrium argument outlined by Brown & Ushomirsky, but rather than estimating the quiescent luminosity using the r-mode amplitude deduced from "spin equilibrium," we use observations of the quiescent luminosity of LMXBs to directly constrain the amplitude of the r-mode. This works because in a steady-state, gravitational radiation pumps energy into the r-mode at a rate given by $W_d = (1/3) \Omega \dot{J}_c = -2E_c/ \tau _G$. This expression has the familiar relationship for a power dissipated by an applied torque, and in this case it is simply the r-mode torque due to gravitational radiation. In a thermal steady-state, all of this energy must be dissipated in the star. Some fraction of this heat will be lost from the star due to neutrino emission and the rest will be radiated at the surface. It should be mentioned that the thermal steady-state is not an assumption but a rigorous result when the mode is saturated, and in particular it is independent of the cooling mechanism (Alford & Schwenzer 2012). We further assume that all of the energy emitted from the star during quiescence is due to the r-mode dissipation inside the star. This is equivalent to setting H = 0 in Equation (12c). The resulting r-mode amplitude limits are upper bounds in the sense that the observed luminosity reflects the contribution from r-mode heating as well as any additional sources of heat that are present, such as, for example, due to accretion and the nuclear processing of accreted material, so-called deep crustal heating (Haensel & Zdunik 2003; Gupta et al. 2007). If any such sources of heat are present, then the actual r-mode amplitude will be less than the upper bounds given here. For the sources that we study in this work, since we know the values of surface temperatures and quiescent luminosities from observations, we can estimate the core temperature and therefore determine the neutrino luminosities to estimate the total amount of heat deposited in the core of these systems by gravitational radiation.

To compute the core temperatures, we use Equation (A8) in Potekhin et al. (1997), which relates the effective surface temperature of the star T_eff to the internal temperature T_b, which is the temperature at a fiducial boundary at ρ_b = 10¹⁰ g cm⁻³ for a fully accreted envelope and is valid at not-too-high temperatures (T_b ⩽ 10⁸ K),

$$\begin{equation} \left(\frac{T_{{\rm eff}}}{10^6\ {\rm K}}\right)^4=\left(\frac{g}{10^{14}\ {\rm cm}\ {\rm s} ^{-2}}\right)\left(18.1\frac{T_b}{10^9\ {\rm K}}\right)^{2.42} \end{equation} \tag{ 18 }$$

where $g={GM}/(R^2 \sqrt{1-r_g/R})$ is the surface gravity and r_g = 2GM/c². Here we assume that the NS's core is isothermal and since the thermal conductivity of the crust is high (Brown & Cumming 2009), we have T_core = T_b to good approximation. If we instead use Equation (A9) in Potekhin et al. (1997), which gives T_b for a partially accreted envelope, and a column depth of (P/g) = 10⁹ g cm⁻² (Haskell et al. 2012), we get core temperatures slightly higher than those derived from Equation (18). The right-hand side of the error bars on the temperatures in Figure 1 shows this difference. It is really only relevant for a single source, 4U 1608−522, but even in this case it is less than a 50% increase, and the difference is always small enough that it does not qualitatively change our results in the remainder of the paper. To compute T_b we have used the effective surface temperatures, T_eff, given by Heinke et al. (2007, 2009) and Tomsick et al. (2004).³ Note that those temperatures are computed for a 1.4 M_☉ NS with a 10 km radius and we have used the appropriate redshifts to compute T_core for our NS models.

Having surface and core temperatures for these sources, we can use Equations (13) and (14) to evaluate their neutrino and thermal luminosities for different stellar models. The values of the core temperatures, as well as the photon and neutrino luminosities for different sources for the 1.4 M_☉ and 2.21 M_☉ NS models, are computed and given in Table 3. Note that in the case of 1.4 M_☉ NSs the neutrino luminosity is only due to the modified Urca reactions, but for the 2.21 M_☉ NS it is due to both direct and modified Urca reactions. By comparing the neutrino and thermal luminosities in Table 3 one can see that at temperatures relevant for the LMXBs the standard neutrino cooling from modified Urca reactions is negligible compared to the photon emission from the surface of the star. However, if the star is massive enough to enable direct Urca reactions in the core (such as for our 2.21 M_☉ NS model) then the neutrino emission will dominate the cooling process for surface temperatures higher than about 34 eV (Brown et al. 1998).

The thermal equilibrium condition can be written as W_d = L_ν + L_γ, where reheating due to r-mode dissipation is given by W_d = (− 2E_c/τ_GR) and is a function of r-mode amplitude, α. Therefore, α can be written in terms of luminosities as

$$\begin{equation} \alpha =\frac{5\times 3^4}{2^8 \tilde{J} M R^3 \Omega ^4}\left(\frac{L_{\gamma }+L_{\nu }}{2 \pi G}\right)^{1/2} \end{equation} \tag{ 19 }$$

where $L_{\gamma } = 4\pi R^2 \sigma T_{{\rm eff}}^4$ is the thermal photon luminosity at the surface of the star. Here, R and T_eff are the stellar radius and surface temperature, respectively, and the neutrino luminosity is given by

$$\begin{equation} L_{\nu }=\frac{4 \pi R^3\Lambda _{{\rm QCD}}^3 \tilde{L}_{{\rm DU}}}{\Lambda _{{\rm EW}}^4}T^6+\frac{4 \pi R^3 \Lambda _{{\rm QCD}} \tilde{L}_{{\rm MU}}}{\Lambda _{{\rm EW}}^4}T^8 \end{equation} \tag{ 20 }$$

where T is the core temperature, R_DU is the radius of the core where direct Urca neutrino emission is allowed, and $\tilde{L}$ is a dimensionless parameter given in Table 1. The thermal equilibrium condition for an NS with standard neutrino cooling (1.4 M_☉ and 2.0 M_☉ NSs in this study) can be approximated as W_d ≃ L_γ, since the neutrino cooling in this case is negligible compared to the surface photon luminosity.

As can be seen in Figure 1, out of the 11 sources considered in this paper all but 2 of them are likely to have unstable r-modes, meaning that they are above the r-mode instability curve. The two most slowly rotating sources, XTE J1807 and XTE J0929, are outside the instability region for our NS models, which means they likely can no longer spin-down due to gravitational radiation from an r-mode.⁴ Therefore, we only evaluate the upper bounds on the r-mode amplitude for those nine sources within the instability window. Using Equation (19), we have evaluated α for all of those sources using the three different NS models considered in this work. The values of α are given in Table 4. As can be seen for the 1.4 M_☉ and 2.0 M_☉ NSs, where there is no enhanced neutrino emission, the values of α range from 1.07 × 10⁻⁸ to 1.54 × 10⁻⁶, where NGC 6440—which has the lowest spin frequency—has the highest r-mode amplitude, as expected. In the case of 2.21 M_☉ NSs, the upper bounds on α are larger, since due to the direct Urca neutrino emission W_d can be larger, and α in this case ranges from 3.30 × 10⁻⁸ to 8.03 × 10⁻⁵. 4U 1608 − 522 has the highest temperature among the sources considered, which implies a very large neutrino luminosity for the 2.21 M_☉ NS model and as a result a large r-mode amplitude. The large value of the r-mode amplitude in this source may be ruled out by measurement of its spin-down rate, as is explained further in the next section. The high temperature of 4U 1608 − 522 could be explained if it has a lower mass, but it should also be noted that this system has been accreting for a long time and both its high quiescent luminosity and surface temperature may be due to the long-term accretion and our assumption in ascribing all the quiescent luminosity to the heat that comes from inside of the star may not be a good estimate for this system, but here we are only interested in obtaining upper limits on r-mode amplitude.

Table 4. Upper Bounds on r-mode Amplitudes and NS Spin-down Rates

Source	αth.eq	αth.eq	αth.eq	$\dot{\nu }$ (Hz s−1)	$\dot{\nu }$ (Hz s−1)	$\dot{\nu }$ (Hz s−1)	$\dot{\nu }_{{\rm sd}}$ (Hz s−1)
	(1.4 M☉)	(2.0 M☉)	(2.21 M☉)	(1.4 M☉)	(2.0 M☉)	(2.21 M☉)	Observation
4U 1608 − 522	7.15 × 10−8	6.60 × 10−8	2.61 × 10−5	−1.44 × 10−15	−1.78 × 10−15	−2.08 × 10−10
IGR J00291+5934	1.41 × 10−8	1.32 × 10−8	3.99 × 10−7	−4.42 × 10−17	−5.59 × 10−17	−3.82 × 10−14	−3 × 10−15
MXB 1659 − 29	1.16 × 10−8	1.07 × 10−8	1.49 × 10−7	−1.78 × 10−17	−2.18 × 10−17	−3.16 × 10−15
Aql X-1	3.49 × 10−8	3.27 × 10−8	2.26 × 10−6	−1.49 × 10−16	−1.89 × 10−16	−6.74 × 10−13
KS 1731 − 260	2.35 × 10−8	2.20 × 10−8	6.44 × 10−7	−4.81 × 10−17	−6.09 × 10−17	−3.90 × 10−14
XTE J1751 − 305	5.09 × 10−8	4.76 × 10−8	1.44 × 10−6	−6.13 × 10−17	−7.74 × 10−17	−5.29 × 10−14	−5.5 × 10−15
SAX J1808 − 3658	1.28 × 10−8	1.19 × 10−8	3.30 × 10−8	−2.19 × 10−18	−2.74 × 10−18	−1.57 × 10−17	−5.5 × 10−16
XTE J1814 − 338	1.76 × 10−7	1.67 × 10−7	4.49 × 10−6	−7.49 × 10−17	−9.73 × 10−17	−5.26 × 10−14
NGC 6440	1.54 × 10−6	1.45 × 10−6	8.03 × 10−5	−2.90 × 10−16	−3.71 × 10−16	−8.50 × 10−13

Notes. Upper bounds on the r-mode amplitude from the "thermal equilibrium" condition that are consistent with quiescent luminosity data are given for different neutron star models. The gravitational radiation induced spin-down rates due to unstable r-modes as well as the observed spin-down rate for some of the sources are also given (Patruno 2010; Patruno & Watts 2012; Patruno et al. 2012a).

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To see whether or not these results are plausible and what fraction of the quiescent spin-down of these sources can be due to gravitational radiation from r-mode oscillations, we use our results for α from the "thermal equilibrium" condition and insert them into the right-hand side of Equation (12a) to determine $\dot{\nu }$ in the absence of accretion (N_acc = 0). The derived spin-down values for different NS models are given in Table 4, and are shown graphically in Figure 2 (vertical green lines). Comparing these results with the observed spin-down rates, which exist for IGR J00291, XTE 1751 − 305, and SAX J1808 (red diamonds in Figure 2), we find that in the case of 1.4 M_☉ (× symbols) and 2.0 M_☉ (square symbols) NSs, where there is no fast neutrino cooling present in the star, the r-mode spin-down can only provide about 1% of the observed spin-down rate, which means that other spin-down mechanisms such as magnetic-dipole radiation are responsible for spinning down a 1.4 M_☉ or 2.0 M_☉ hadronic star with no fast cooling process. For the two AMXPs with relatively "slow" spin frequencies, NGC 6440 and XTE J1814−338, we find r-mode spin-down limits that are more competitive with observed values. Indeed, our limit for NGC 6440 is comparable to the measured, quiescent spin-down rate for SAX J1808, and thus spin-down measurements for this source would be particularly interesting.

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On the other hand, the r-mode amplitudes we obtain for the 2.21 M_☉ (circle symbols) hadronic model with direct Urca neutrino emission are only consistent with observations for SAX J1808 (and perhaps MXB 1659), as the inferred spin-down rates are either less than the observed rate for the source—in the case of SAX J1808—or similar to the other observed rates, as for MXB 1659. For the remaining sources considered here, the 2.21 M_☉ limits are likely not consistent with the observations, since such large amplitudes imply very large r-mode spin-down rates, and in the case of IGR J00291 and XTE J1814 they are in fact larger than the observed values. If the neutrino luminosity from these sources was indeed as large as estimated with our 2.21 M_☉ model, then in thermal equilibrium there must be a heat source that can supply it. Since the spin-down measurements for these sources indicate that r-mode heating (for this model) would be insufficient, several possibilities remain. First, there could be some additional source of heat other than r-mode dissipation that supplies the needed energy. However, we note that it would need to supply a substantial luminosity, as the direct Urca neutrino emission for this model outshines the photon luminosity by more than an order of magnitude, and we are not aware of any simple mechanisms that could provide the required luminosity. Second, the actual mass of these systems could be less than that of the model in question (2.21 M_☉). Indeed, if it could be demonstrated that r-mode dissipation were the only mechanism that could produce such a large luminosity, then an upper limit on the mass would follow, and the limit would be the mass for which the neutrino plus photon luminosity matched the r-mode heating produced when the amplitude is large enough to produce a spin-down rate equal to the observed quiescent rate or our theoretical value for a high-mass NS model, whichever is smaller. This would be a conservative limit in the sense that it is likely that the r-mode torque does not account for all of the observed spin-down. Finally, our model assumptions, for example, the EOS and core composition, could be incorrect, with one possibility being the existence of exotic matter, such as kaon or pion condensates, or quark matter in the core, which have smaller neutrino emissivities than nucleon direct Urca processes, or if the pairing gaps for ³P₂ neutrons and ¹S₀ protons were larger than current theoretical values (this will be explained in more detail in the next paragraph). Interestingly, if the masses of these systems were known then one of the possibilities outlined above is precluded and then the observationally derived r-mode limits become sensitive to properties of the core, either the presence of exotic matter or perhaps additional heating physics. In this sense, further spin-down measurements and, where possible, mass constraints could provide interesting new insights into the physics of dense NS matter.

Our theoretical treatment of neutrino emission processes in this study, namely modified and direct Urca, spans the plausible range between "slow" and "fast" neutrino cooling processes. Here we have considered NS models made of non-superfluid hadronic matter with the APR EOS, but more realistically it is likely that neutrons and protons will be in a superfluid phase inside NSs. Therefore, a natural question would be whether or not our conclusions will still hold in the presence of superfluidity. Considering the presence of superfluidity in these sources, assuming that they are still inside the unstable region for r-modes, could have two possible effects on their cooling. The first one is neutrino emission due to Cooper pair breaking and formation (PBF) just below the superfluid critical temperature (Page et al. 2004) and the second one is the suppression of direct and modified Urca neutrino cooling at temperatures below the critical temperature. Here we explain why our qualitative results, and our argument about setting upper bounds on the masses of these sources, are not changed by considering the effect of superfluidity on the neutrino cooling of these sources. In our low-mass NS models (M < 2 M_☉) where there is no fast cooling mechanism in the core, the thermal luminosity is much larger than the modified Urca neutrino luminosity, in fact by more than five orders of magnitude in all of the sources considered here except 4U 1608. Therefore, even if the temperature of these sources were just below the critical temperature of superfluidity, neutrino emission due to PBF, which is only about 10 times stronger than modified Urca neutrino emission, would still be much smaller than the thermal emission. Therefore, suppression of neutrino emission or neutrino emission due to PBF is not important for our low-mass NS models. What about high-mass NSs where fast cooling can happen in the core? Since direct Urca neutrino emission is much stronger than neutrino emission due to PBF, if direct Urca is not suppressed by superfluidity, it will be the dominant cooling mechanism. Now the question is whether or not neutron and proton pairing can suppress direct Urca neutrino emission in the core. Current theoretical results for the pairing gaps in ³P₂ neutron superfluid (Schwenk & Friman 2004; Dong et al. 2013) and ¹S₀ proton superconducting phases (see for example Page et al. 2004; Yakovlev & Pethick 2004), which are relevant in the core of NSs, suggest that both of these pairing gaps are vanishingly small in the inner core of NSs where the direct Urca process can operate. Therefore, superfluidity is unlikely to suppress fast cooling processes in the core of NSs, and thus neglecting the effect of superfluidity does not change our qualitative results, assuming that r-modes are still unstable in the presence of superfluidity in these sources. We also note that the density profile (density versus radius) for an NS made of hadronic matter with the APR EOS is almost flat at the center of the star, which means that as the mass of the star increases above 2 M_☉ (above which direct Urca processes can operate in the core of an NS made of hadronic matter with the APR EOS), there will be a sizable region in the core where direct Urca processes may happen, which can make it easier to obtain an upper limit on the mass of these sources using spin-down measurements.

With typical values of a few × 10⁻⁸, our derived amplitude upper limits suggest that for many LMXB NSs the r-modes are likely not excited to sufficient amplitudes to substantially affect their spin evolution. This begs the question of whether or not unstable, steady-state r-modes actually exist in these NSs. One possibility is that additional damping mechanisms, such as those perhaps associated with crust effects, such as the viscous friction at the crust-core boundary due to the coupling between core r-modes and crustal torsional modes (Levin & Ushomirsky 2001), superfluid mutual friction (Ho et al. 2011) or the existence of exotic matter in the core of NSs (Alford et al. 2012a; Schwenzer 2012), are at work and modify the instability window so as to render these NSs stable to r-mode excitation. An interesting related question is whether the existence of r-modes at the amplitudes estimated here can be inferred directly from observations. Figure 2 shows both r-mode (dashed parallel lines) and magnetic-dipole (solid parallel lines) spin-down laws. The r-mode spin-down braking index, n = 7, is steeper compared to that for magnetic spin-down (n = 3), thus at high-enough spin frequencies (well above a kHz) one might expect that the r-mode torque would eventually become competitive with or dominate the magnetic dipole torque. However, as of yet there are no known NSs spinning fast enough for this effect to become dominant, and depending on the EOS the mass-shedding limit might be reached before the r-mode torque becomes competitive with the magnetic torque.

As discussed above, quiescent spin-down measurements have been typically attributed to the magnetic torque. For a number of sources considered here our r-mode amplitude limits support this presumption. Any spin-down contribution from an r-mode torque would be more easily identifiable if the magnetic field strengths of these NSs were constrained independently from the magnetic spin-down estimate. Moreover, identifying an r-mode spin-down would, in principle, be simpler for those NSs with the lowest magnetic torques, and thus field strengths. At present, the lowest inferred dipolar magnetic field strengths are ≈6 × 10⁷ G. At this level the magnetic spin-down is of the order of $\dot{\nu } \approx 5 \times 10^{-17}$ Hz s⁻¹, which is comparable to our derived r-mode spin-down limits for a number of sources considered here, assuming that their NSs are <2 M_☉. In this regard, quiescent spin-down measurements for more of the sources considered here, in particular the AMXPs XTE J1814 and NGC 6440, would be extremely valuable.

r-modes in NSs are one of the possible mechanisms for GW emission and they can be observationally interesting in newborn NSs and perhaps accreting NSs in LMXBs (Owen 2010). Continuous GW emission from r-modes is dominated by l = m = 2 current quadrupole emission (Lindblom et al. 1998). The GW amplitude h₀ (strain tensor amplitude) is related to the r-mode amplitude α by the following equation (Owen 2010, 2009)

$$\begin{equation} h_0=\sqrt{\frac{8\pi }{5}}\frac{G}{c^5}\frac{1}{r}\alpha \omega _r^3MR^3\tilde{J}, \end{equation} \tag{ 21 }$$

where r is the distance to the source, M and R are the mass and radius of the NS, $\tilde{J}$ is the dimensionless parameter defined by Equation (3), and ω_r is the frequency of the r-mode, which is related to the angular spin frequency of the star Ω (for the m = l = 2 r-mode) by the following equation

$$\begin{equation} \omega _r\approx \frac{4}{3}\Omega. \end{equation} \tag{ 22 }$$

Using the upper limits on the r-mode amplitude of NSs in LMXBs derived above, we can obtain upper limits on the amplitude of the GWs emitted from these sources due to unstable r-modes. For the sources considered in this work, upper limits on the GW strain amplitude h₀ for the 1.4 M_☉ and 2.0 M_☉ NS models are in the range of 1.8 × 10⁻²⁹ to 4.9 × 10⁻²⁸, which is below the anticipated detectability threshold of Advanced LIGO (Abbott et al. 2010; Watts et al. 2008). In the case of the 2.21 M_☉ NS model, since the r-mode amplitudes are larger than those for the low-mass NSs, we get larger values of h₀, but even in this case for most of the sources h₀ is still below the detectability threshold of Advanced LIGO. The highest values of h₀ for a massive star are obtained for NGC 6440 and Aql X-1 with an amplitude of the order of 8.5 × 10⁻²⁷ and 4U 1608 − 522 with an amplitude of 1.59 × 10⁻²⁵. However, it should be noted that the large r-mode amplitudes in these sources, which would cause very large spin-down rates, may eventually be ruled out by future spin-down measurements. In this context it is important to restore the X-ray timing capability that was lost when RXTE was decommissioned in 2012 January. Missions planned or currently in development which could provide such a capability include India's ASTROSAT (Paul 2012), ESA's Large Observatory for X-ray Timing (Feroci et al. 2012), and NASA's Neutron Star Interior Composition Explorer (Gendreau et al. 2012).

The upper limits on GW amplitudes discussed here are related to the GW emission due to unstable r-modes, but our results do not exclude the possibility of having larger GW amplitudes in LMXBs from other GW emission mechanisms such as NS mountains (Ushomirsky et al. 2000; Haskell et al. 2006). It is worth mentioning that indirect upper limits on GW amplitude can be obtained for sources with observed spin-down rates, $\dot{\nu }$, by assuming that all of the observed spin-down is due to GW emission (Owen 2010),

$$\begin{equation} h_0^{sd}=\frac{1}{r}\sqrt{\frac{45G I \dot{P}}{8c^3P}} \end{equation} \tag{ 23 }$$

where P = (2π/Ω) is the observed pulse period and $|{\dot{P}}/{P}|=|{\dot{\nu }}/{\nu }|$. Using this equation for the three sources with measured $\dot{\nu }$ and different NS models (i.e., different masses and radii) we obtain $h_0^{sd}$ values that range from 4.14 × 10⁻²⁸ to 6.53 × 10⁻²⁸.