SPIN EQUILIBRIUM WITH OR WITHOUT GRAVITATIONAL WAVE EMISSION: THE CASE OF XTE J1814-338 AND SAX J1808.4-3658

XTE J1814 is the AMXP that shows the strongest linear correlation between pulse phase and X-ray flux (Patruno et al. 2009). This correlation has been interpreted as possibly being generated by a moving hot spot on the surface of the NS (Patruno et al. 2009, 2010). Such an interpretation is supported by recent results obtained by means of three-dimensional MHD simulations of AMXPs (Romanova et al. 2003; Bachetti et al. 2010). Special care must be taken as flux induced pulse phase variations may resemble spin frequency variations. In Figure 1, the effect of the phase–flux correlation is made evident by overplotting flipped pulse phase residuals (with respect to a constant spin frequency model) of the fundamental and the X-ray light curve.

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It is obvious how variations of the pulse phase closely follow variations in the X-ray flux. In standard coherent pulsar timing the spin frequency derivatives are measured by fitting a quadratic function to the pulse phases; by doing so for XTE J1814 the measured quadratic component will simply be a measure of the effect that the X-ray flux has on the pulse phases.

To measure the spin frequency derivative we have first removed the effect of flux variations by using the linear phase–flux correlation (for both harmonics). After "cleaning" the pulse phases, the residual rms variability is of the order of 0.02 cycles. We then fit a quadratic function to the cleaned pulse phases and calculate the errors on the fitted spin parameters by means of ∼10⁴ Monte Carlo simulations to account for the residual unmodeled variability, as described in Hartman et al. (2008).

The results indicate upper limits on the spin frequency derivative of the order of $|\dot{\nu }|\lesssim \,1.5\times \,10^{-14}\rm \,Hz\,s^{-1}$ at the 95% confidence level. The upper limits are of the same order of magnitude as those reported by Hartman et al. (2008) for SAX J1808. These results strongly contrast with the measurements obtained without cleaning the pulse phases (see, e.g., Papitto et al. (2007) who report $\dot{\nu }\approx -7\times \,10^{-14}\rm \,Hz\,s^{-1}$).

The crust of a NS is a thin layer (≈ 1 km thick) of matter in a crystalline phase, that can support shearing and the presence of a small asymmetry, or "mountain." The crust consists of several layers of different nuclear composition and as accreted matter gets pushed further into the star it undergoes a series of nuclear reactions, including electron captures, neutron emission, and pycnonuclear reactions (Sato 1979; Haensel & Zdunik 1990). The heat deposited in the whole crust due to these reactions is Q_H ≈ 1.5 MeV per accreted baryon, but most of the heat is released by reactions occurring close to neutron drip (Haensel & Zdunik 1990). These reactions will heat the region by an amount (Ushomirsky & Rutledge 2001)

$$\begin{equation} \delta T\approx 10^3 {C}_{k}^{-1}{p}_{d}^{-1}{Q_n}{\Delta M}_{22}\;\mbox{K}, \end{equation} \tag{ 2 }$$

where C_k is the heat capacity in units of k_B (the Boltzmann constant) per baryon, p_d is the pressure at which the reactions occur (in units of 10³⁰ erg cm⁻³), and Q_n is the deposited heat per accreted baryon in MeV. ΔM is the accreted mass in units of 10²² g, which for the systems we consider is approximately the amount of matter accreted after a few days of outburst. If the energy deposition is (partly) asymmetric this would perturb the equilibrium stellar structure and give rise to a quadrupole (Ushomirsky et al. 2000):

$$\begin{equation} Q_{22}\approx 1.3 \times 10^{35} R_6^4\left(\frac{\delta T_q}{10^5\,\mbox{K}}\right)\left(\frac{Q}{30\,\mbox{MeV}}\right)^3 \mbox{g cm$^2$}, \end{equation} \tag{ 3 }$$

where δT_q is the asymmetric (quadrupolar) part of the temperature increase and Q is the threshold energy for the reaction. The quadrupole required for spin equilibrium during an outburst is Q ≈ 10³⁷ g cm² for both systems (Ushomirsky et al. 2000). It is clear from Equations (2) and (3) that even under the most optimistic assumptions it is unlikely to build a "mountain" large enough to balance spin-up during accretion.

It is well known that a magnetic star will not be spherical and, if the rotation and magnetic axes are not aligned, one could have a "magnetic mountain" leading to GW emission (Cutler 2002). However, such deformations are unlikely to be large enough to balance the accretion torques in weakly magnetized systems such as the LMXBs (Haskell et al. 2008) and would persist in quiescence, leading to a rapid spin-down, of the order of the spin-up in Equation (1), which is not observed in SAX J1808 (Hartman et al. 2009).

Another possibility is that the magnetic field lines are stretched by the accreted material as it spreads on the star, giving rise to a magnetically confined mountain. The results of Melatos & Payne (2005) suggest that the quadrupole built this way could balance the accretion torque only if the surface field is significantly stronger than the external dipole component. Furthermore, such a mountain would persist on an ohmic dissipation timescale $\tau _{_{{\rm Ohm}}}\approx 10^2$ yr (Melatos & Payne 2005) and thus should also give rise to a strong spin-down in quiescence.

A more promising scenario is that of an oscillation mode of the NS being driven unstable by GW emission, the main candidate for this mechanism being the l = m = 2 r-mode (Andersson 1998). An r-mode is a toroidal mode of oscillation for which the restoring force is the Coriolis force. It can be driven unstable by GW emission, as long as viscosity does not damp it on a faster timescale. This will only happen in a narrow window in frequency and temperature which depends on the microphysical details of the damping mechanisms (for a review, see Andersson & Kokkotas 2001).

Of particular interest in this context is the situation in which the frequency at which the mode is driven unstable (the "critical" frequency) increases with temperature in the range we consider (T ≈ 10⁷–10⁸ K). This could lead to the system being in equilibrium at the critical frequency and emitting GWs at a level that will balance the accretion torque (Andersson et al. 2002). This would be the scenario if the core contains hyperons (Nayyar & Owen 2006; Haskell & Andersson 2010) or deconfined quarks (Andersson et al. 2002). If this is the case the dimensionless amplitude of the mode α (as defined by Owen et al. 1998) will be approximately constant in time, i.e., $\dot{\alpha }\approx 0$, and if we assume that GW emission is balancing the accretion torque one has

$$\begin{equation} \alpha \approx 1.9\times 10^{-6} \dot{M}_{-10}^{3/7}\nu _{400}^{-7/2}\xi ^{1/4} B_8^{1/7}, \end{equation} \tag{ 4 }$$

where ν₄₀₀ = ν/400 Hz and we have assumed a 1.4 M_☉ mass and 10 km radius, which we now take as standard values.

After the outburst, when accretion ceases, the system will cool and spin-down following the critical frequency curve. The mode amplitude would thus be approximately constant resulting in a spin-down rate $\dot{\nu }\approx -7\times 10^{-14}$ Hz s⁻¹ for both systems. This is considerably higher than the observed quiescent spin-down rate for SAX J1808 of $\dot{\nu }\approx -5.5\times 10^{-16}$ Hz s⁻¹.

Furthermore, the shear from the mode will heat the star at a rate (Andersson & Kokkotas 2001)

$$\begin{equation} \dot{E}\approx 1.8\times 10^{44}\alpha ^2 T_8^{-5/3}\nu _{400}^2\,\mbox{erg s$^{-1}$}, \end{equation} \tag{ 5 }$$

where T₈ = (T/10⁸ K). Let us first of all assume that the core cools due to modified URCA neutrino emission processes, which leads to an energy loss $\dot{E}\approx 8.4\times 10^{31}T_8^8$ erg s⁻¹ (Page et al. 2006). Balancing this energy loss with the viscous r-mode heating in Equation (5) leads to an equilibrium temperature of T ≈ 1.4 × 10⁸ K for SAX J1808 and T ≈ 1.5 × 10⁸ K for XTE J1814. Let us now estimate the temperature in the presence of the strongest possible neutrino emission processes. We therefore assume that nucleon direct URCA processes are possible, which may be the case in SAX J1808 (Yakovlev et al. 2003), and that the core is not superfluid, with $\dot{E}_\nu \approx 4\times 10^{39} T_8^6$ erg s⁻¹ (Page et al. 2006; similar processes would be at work if hyperons or deconfined quarks are present). Note that in the presence of superfluidity these processes would be strongly quenched (for a recent analysis of r-mode heating see Ho et al. 2011). The temperatures we obtain for the stars are T ≈ 1.5 × 10⁷ K for SAX J1808 and T ≈ 1.6 × 10⁷ K for XTE J1814.

How do these temperatures compare with observational constraints? Both systems have upper limits on their temperature from quiescent observations (Heinke et al. 2009). One can map the surface temperature to the temperature at the top of the crust via the relation given by Potekhin et al. (1997) for an accreted crust:

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

where g is the gravitational acceleration at the surface. By assuming that the star is isothermal, which is roughly correct if the thermal conductivity is high, as observations of cooling X-ray transients suggest (Brown & Cumming 2009), one can obtain a core temperature of T < 1 × 10⁷ K for SAX J1808 and T < 4.1 × 10⁷ for XTE J1814.

It would thus appear that the presence of an unstable r-mode is not consistent with the observed quiescent spin-down of SAX J1808, while it could be marginally consistent with the temperature of both sources only in the presence of direct URCA processes. We have not considered the role of the magnetic field, but this is likely to stabilize the mode (Rezzolla et al. 2000), consistently with observations.