THE PECULIAR EVOLUTIONARY HISTORY OF IGR J17480-2446 IN TERZAN 5

A previous coherent timing analysis of IGR J17480-2446 is briefly described in Cavecchi et al. (2011), who reported a pulse frequency derivative of ≈1.4 × 10⁻¹² Hz s⁻¹ with errors calculated by means of Monte Carlo (MC) simulations that take into account the red noise in the timing residuals (Hartman et al. 2008). The steps followed are those used in standard coherent timing of radio and X-ray pulsars: the photon arrival times are referred to the solar system barycenter using the optical position from Pooley et al. (2010), fine clock corrections are applied to the RXTE data, and Earth occultations and intervals of unstable pointing are filtered out.

To measure the TOAs of the persistent pulsations, all bursts are removed from the light curve, then ≈10 to ≈500 s long intervals of data are folded in profiles of N = 32 bins with the preliminary ephemeris reported in Papitto et al. (2011). The length of the folded data segments is chosen to have a nearly uniform statistical error (∼1 ms) for each TOA. The ≈10 s profiles are those folded from the first RXTE data set (ObsId 95437-01-01-00) when the pulse profiles have the highest fractional amplitude and the highest signal-to-noise ratio. All other observations are folded with ≈500 s segments with the exact length slightly varying for each observation to assure that all good photons are retained in the data analysis. We assume that pulsations are detected in a profile if the ratio between the amplitude of a harmonic and its statistical error is larger than 3.5. This guarantees that the expected number of false pulse detections is less than one in the entire data collection (the total number of profiles is ≈650 for each harmonic). A fundamental and two overtones are detected with strong significance and in enough data segments to allow a determination of a precise global timing solution.

The TOAs of each harmonic are fitted using the TEMPO2 pulse timing program (Hobbs et al. 2006), assuming a circular Keplerian orbit plus a pulse frequency and its first time derivative. The entire folding and fitting procedure is then repeated until the final timing solution converges. The orbital parameters are reported in Table 1 along with 1σ error bars obtained by multiplying the statistical error as given by TEMPO2 by $\sqrt{\vphantom{A^A}\smash{\hbox{${\chi ^2}$}}}$. The errors on the pulse frequency and its derivative are instead treated separately to include the strong effect of timing noise which operates on the same timescales over which the two parameters are measured. Details of the procedure have been extensively discussed in the literature, and we refer to Cavecchi et al. (2011) for the results obtained on IGR J17480-2446 and to Hartman et al. (2008) and Patruno et al. (2010) for further details on the procedure.

Here, we will verify more carefully whether the pulse frequency derivative (i.e., the measured $\dot{\nu }$) reported in Cavecchi et al. (2011) is indeed the spin frequency derivative (i.e., a true rotational variation of the NS spin) and not the effect of timing noise. Although Cavecchi et al. (2011) used MC simulations to calculate the long-term average $\dot{\nu }$ of IGR J17480-2446 , no interpretation to the pulse frequency derivative is given and no discussion of the effect of timing noise is provided. Another long-term $\dot{\nu }$ has been reported by Papitto et al. (2011). It is of the same order as, but significantly different from, that given in Cavecchi et al. (2011). The timing analysis of Papitto et al. (2011) did not consider in detail the strong effect of timing noise and the authors reported discordant $\dot{\nu }$ values in their analysis of the fundamental and the first overtone, which deviate by more than 32σ from each other and deviate by more than 3σ from the results of Cavecchi et al. (2011). Identical considerations also apply to the pulse frequency. These results show why extra care has to be taken to verify that the pulse frequency derivative is the spin frequency derivative of IGR J17480-2446. Determining whether the NS is in a spin-up phase is crucial for understanding the evolutionary history of the binary as we will discuss in Section 4.

We split the data into five segments of length between ≈4 and 9 days (see Table 2). The exact values are chosen empirically to have enough data to measure a pulse frequency derivative of the order of 10⁻¹² Hz s⁻¹. The data used and the start and end times of each data segment are reported in Table 2.

We fit the pulse frequency and its first derivative to each data segment. The fundamental frequency and the first overtone are analyzed separately, whereas the second overtone is not detected in enough observations to allow data splitting and therefore will not be considered any further. The harmonic phase residuals relative to our best constant frequency timing model exhibit variability on time scales as short as a few tens of minutes (i.e., close to the shortest timescale we are sensitive to) with an amplitude well in excess of the Poisson noise expected from counting statistics. This variability requires a special treatment of the statistical errors of the pulsar spin parameters since standard χ² minimization techniques (i.e., those used by TEMPO2 or any other standard fitting routine) rely on the assumption that only Poissonian noise is present in the data, which is not a good approximation in IGR J17480-2446. We apply the MC method developed by Hartman et al. (2008) to estimate the effect of timing noise on the pulse parameters. We run 10⁴ MC simulations on time series which have the same power content of the original TOA residuals. The parameters ν and $\dot{\nu }$ are measured for each simulated time series and a distribution of parameters is constructed over the 10⁴ simulations. The 68% confidence interval on the pulse parameters is then determined as the standard deviation of each parameter distribution.

A second test is performed by repeating the entire procedure described above with the difference that we fit only a constant pulse frequency with the derivative set to zero. In this way, we perform a "partial coherent" analysis: coherent timing is used in each of the five data segments to extract a constant pulse frequency (constant in that segment), whereas the long-term pulse frequency derivative is inferred later by fitting the five pulse frequencies with standard χ² minimization techniques. In this way, we can check whether the five constant pulse frequencies increase linearly in time as expected if a constant spin frequency derivative is present in the data. If instead the five frequencies fluctuate with a large scatter around a linear trend, the effect of timing noise is predominant and a presence of a constant spin frequency derivative is questionable.

To establish whether the pulse frequency derivative is consistent with a spin frequency derivative we require that

• 1.  
  the pulse frequency derivative inferred for the two harmonics is consistent with being the same and it also matches with the long-term pulse frequency derivatives measured with coherent timing (i.e., those published in Table 1 by Cavecchi et al. 2011);
• 2.  
  the pulse frequencies have to be consistent with each other for the two harmonics in each of the five segments; and
• 3.  
  there is no large deviation (>3σ) of any of the pulse frequencies from the linear trend. This demonstrates that timing noise is not dominating the fitted pulse frequency parameter in any of the five segments.

The results presented above strongly suggest that the long-term pulse frequency derivative reported by Cavecchi et al. (2011) is indeed the long-term spin frequency derivative of IGR J17480-2446. There are, however, two points that need further discussion before drawing a firm conclusion from our data analysis: the meaning of the outlier in the short-term pulse frequency derivatives and the lack of correlation between $\dot{\nu }$ and f_X.

Although the outlier indicates a significantly larger pulse frequency derivative than the long-term average, the first overtone gives in the same data segment a $\dot{\nu }$ incompatible with the outlier and compatible with the long-term average $\dot{\nu }$. The interpretation we give to this is that the outlier does not represent a true increase in the spin frequency derivative of the NS. This can be still ascribed to timing noise, even if we are carefully calculating the statistical errors with MC simulations. MC simulations verify whether a measured $\dot{\nu }$ is significant with respect to the red noise component in the TOA residuals. This is achieved by testing whether random fluctuations in the residuals are able to produce fake $\dot{\nu }$ as strong as the measured one. Since we are looking at short time series (∼4–9 days), the red noise is not stationary, as evident from Figure 3, where the data segment to which the outlier belongs is shaded. In that data segment, the highest red noise frequencies observable are comparable with the length of the data segment and the data show a simple parabolic trend which is absorbed by $\dot{\nu }$ during the fitting procedure. If this is the case, MC simulations cannot discriminate between a true spin frequency derivative and the effect of timing noise. Therefore, the error bars thus calculated do not differ much from those one obtains when the errors are dominated by uncorrelated Gaussian noise (i.e., white noise). The final values of the MC errors are not, is such a case, a good representation of the true statistical uncertainties. This is a limitation of the MC method. However, a similar behavior is not observed in the same data segment for the first overtone and so we ascribe this behavior to timing noise.

Zoom In Zoom Out Reset image size

It is of course possible that timing noise is the result of true changes of the spin frequency and/or spin frequency derivative of the NS. Fluctuating torques have been proposed in the past as a mechanism to explain fluctuations in the NS rotation (Lamb et al. 1978). However, in this case we would expect to see identical variations of the spin parameters in both harmonics, which is not observed for the outlier in IGR J17480-2446. Furthermore, it has been shown that the timing noise observed in at least two accreting millisecond pulsars (Watts et al. 2008; Patruno et al. 2010) cannot be ascribed to true variations of the NS spin parameters since these require unphysical large torque variations.

The fact that $\dot{\nu }$ is not proportional to f_X can be a consequence of different phenomena. The X-ray flux might be not a good tracer of $\dot{M}$ as has been already seen in several other X-ray binaries (van der Klis 2001). Our data suggest that $\dot{\nu }\approx \,{\rm constant}$ over an X-ray flux variation of a factor ≈2. An approximately constant bolometric flux might therefore easily explain the observations.

Another possibility is that $\dot{\nu }\propto \,f_{X}^{6/7}$ is valid only if the accretion disk is a standard thin disk under the rather unrealistic assumption that radiation pressure is not important in determining the inner disk structure (Psaltis & Chakrabarty 1999). Andersson et al. (2005) suggest that this assumption is not correct already at accretion rates of a few percent Eddington. Since IGR J17480-2446 is accreting at least at $\dot{M}>5\%$ Eddington with peaks exceeding 37%, it is not unreasonable to assume that radiation pressure plays a role in determining the inner disk structure. The disk–magnetosphere interaction region is also more complex than usually assumed (see, e.g., Rappaport et al. 2004; D'Angelo & Spruit 2010; Kajava et al. 2011; Patruno et al. 2012), and $\dot{\nu }$ might have a weaker dependence on the bolometric flux than usually assumed.

We conclude therefore that the interpretation of the long-term $\dot{\nu }$ as being the spin frequency derivative of IGR J17480-2446 is supported by the observations, with the outlier most likely explained by unmodeled timing noise and the lack of correlation between the short-term $\dot{\nu }$ and f_X, which is not surprising, since the expectation of such correlations is based on assumptions about the accretion process which may not be valid.

In the present RLOF spin-up epoch, the 2–16 keV X-ray luminosity of IGR J17480-2446 spans values between 9 × 10³⁶ erg s⁻¹ and 6.5 × 10³⁷ erg s⁻¹ for an assumed distance of 5.5 kpc (Ortolani et al. 2007). This luminosity has to be considered a lower limit given that the bolometric luminosity is expected to be higher than the X-ray luminosity. Linares et al. (2012) estimated a bolometric luminosity at the peak of the outburst of about 50% of the Eddington limit for d = 6.3 kpc (L_Edd = 2.5 × 10³⁸ erg s⁻¹). The maximum mass accretion rate is therefore

$$\begin{equation} \dot{M}= \frac{{\it LR}_{{\rm NS}}}{{\it GM}_{{\rm NS}}} \simeq 5 \times 10^{17} \left(\frac{d}{5.5{\rm \,kpc}}\right)^2 \frac{R_{6}}{M_{1.4}}{\rm \,g\, s}^{-1}, \end{equation} \tag{ 1 }$$

where G is the gravitational constant, M_NS and R_NS are the mass and radius of the NS, and M_1.4 and R₆ are the mass of the NS in units of 1.4 M_☉ and the radius in units of 10⁶ cm, respectively. The average mass accretion rate during the outburst has a value approximately half the peak value, and we assume it is $\langle \dot{M}\rangle \,= 2\times 10^{17}{\rm \,g\,s}^{-1}$, in agreement with the value reported by Degenaar & Wijnands (2011).

The inner radius r_A of the accretion disk is determined by the balance between the magnetic stresses in the NS magnetosphere and the material stresses of the accretion disk (Ghosh & Lamb 1979):

$$\begin{equation} r_{A}= \left(\frac{{\mu }^2}{\dot{M}}\right)^{2/7}\left({\it GM}_{{\rm NS}}\right)^{-1/7}, \end{equation} \tag{ 2 }$$

where μ is the dipole magnetic moment of the star. The Keplerian rotation rate at the inner radius of the disk

$$\begin{equation} \Omega _K(r_A) = \left(\frac{GM_{{\rm NS}}}{{r_A}^3}\right)^{1/2} = \left(\frac{{\mu }^2}{\dot{M}}\right)^{-3/7}\left(GM_{{\rm NS}}\right)^{5/7} \end{equation} \tag{ 3 }$$

determines the torques on the NS applied by the accretion flow and the induced distortions of the magnetic field. These torques operate as long as $\Omega _{\*}$ is not equal to Ω_K(r_A), so that the star spins up or down toward an equilibrium rotation rate Ω_eq = Ω_K(r_A). If any of the QPO frequencies (see Altamirano et al. 2010) is interpreted as the eventual spin equilibrium frequency of the NS, the present rotation rate will give an estimate of the dipole moment of the NS. This identification, however, is not straightforward. QPO frequencies reflect modes of oscillation in the accretion flow that are excited in the interacting disk–NS system to some amplitude to effect the luminosity signal. A particular QPO frequency band might be defined by some oscillation mode of the disk or by a beat or resonance between the NS and a disk mode. Disk modes are fundamentally effected by rotation and imprinted by the frequency scale of the disk rotation, Keplerian in most parts of the disk, but necessarily and significantly deviating from the Keplerian rate at the inner boundary where the effect of the NS magnetosphere is important. This results in frequencies of epicyclic (radial) disk oscillations to be somewhat higher than the non-Keplerian rotation frequency at the inner disk boundary or transition region (Alpar & Psaltis 2008). The presence of a sequence of QPO frequencies may be due to the presence of several multipole components of the NS magnetic field, so that the boundary between the disk and the NS magnetosphere is convoluted with a sequence of decreasing "stopping radii," smaller than the conventional Alfvén radius given in Equation (2) and increasing Kepler frequencies corresponding to higher multipoles of the NS magnetic field (Alpar 2011). We assume that one of the observed range of QPO frequencies ν_i is close to the Keplerian rotation rate Ω_K(r_A) near the Alfvén radius,

$$\begin{equation} \nu _i \cong \frac{\Omega _K(r_A)}{2\pi }, \nonumber \end{equation} \tag{ 4 }$$

yielding the estimates

$$\begin{eqnarray} \mu & \cong & (2\pi \nu _i)^{-7/6} \dot{M}^{1/2} (GM_{{\rm NS}})^{5/6}\nonumber\\ \mu & = & 1.3 \times 10^{26}{\rm \,G\, {cm}}^3\left(\frac{\nu _i}{{\rm kHz}}\right)^{-7/6}\langle \dot{M}\rangle ^{1/2} M_{1.4}^{5/6} \end{eqnarray} \tag{ 5 }$$

for the dipole moment. In these estimates we use $\langle \dot{M}\rangle$ to mean the mass accretion rate averaged over the outburst length and it is normalized to 2 × 10¹⁷ g  s⁻¹.

The values of the magnetic dipole moment are reported in Table 4. If we take the NS to be already in equilibrium at the present spin frequency of 11 Hz, then μ≅2 × 10²⁸ G cm³ and a B field at the poles of 4 × 10¹⁰ G where we have used:

$$\begin{equation} \mu _{30} = 0.6 f^{7/6}\left(\frac{d}{5.5{\rm \,kpc}}\right) {R_{6}}^{1/2} M_{1.4}^{1/3}\nu ^{-7/6}. \end{equation} \tag{ 6 }$$

Here, f takes account of the possibility that the observed frequency is not the Kepler frequency but some related frequency at the disk boundary: e.g., f ∼ 1.5–2 for the epicyclic frequency (e.g., Alpar & Psaltis 2008). This uncertainty makes the field estimates somewhat higher. The torque applied by the accretion disk on the NS is usually assumed to be

$$\begin{equation} N = \dot{M}(G M_{{\rm NS}} r_A)^{1/2} F(\omega) \end{equation} \tag{ 7 }$$

with the dimensionless torque F(ω) describing the dependence on the fastness parameter $\omega \equiv \Omega _{\*}/\Omega _K(r_A)$, is likely of the form

$$\begin{equation} F(\omega) = 1 - \omega ^p \nonumber \end{equation} \tag{ 8 }$$

with the exponent p ∼ 2 (Ertan et al. 2009). If IGR J17480-2446 is as yet at the beginning of its spin-up epoch, then ω ≪ 1, F(ω)≅1, giving constant $\dot{\nu }$ and expected spin frequency derivatives

$$\begin{equation} \dot{\nu } \cong 6 \times 10^{-13}{\rm \,Hz\; s^{-1}} \frac{\langle \dot{M}\rangle M_{1.4}^{2/3}}{ I_{45}\left(\frac{\nu _i}{\rm \,kHz}\right)^{1/3}}, \end{equation} \tag{ 9 }$$

where I₄₅ is the NS moment of inertia in units of 10⁴⁵ g cm². These estimates of $\dot{\nu }$ are also given in Table 4. With any of the QPO ν_i this gives $\dot{\nu } \sim 10^{-12}{\rm \,Hz\, s}^{-1}$. If the system is already in equilibrium $\dot{\nu }$ would be fluctuating in sign with rms values much smaller than 10⁻¹² Hz s⁻¹, which is, however, not observed (see Section 3 and Figures 1 and 2).

Table 4. QPO Frequency Model for IGR J17480-2446

QPO Frequency νi	μ	B Field	$\dot{\nu }$	tsu, QPO
(Hz)	(1027 G cm3)	(109 G)	(10−12)	(107 yr)
48	5	10	1.6	1.8
173	1	2	1.1	2.8
814	0.2	0.4	0.6	4.6

Note. The magnetic field B refers to the value at the NS magnetic poles. This is twice the value at the equator.

Download table as:  ASCIITypeset image

The spin-up age up to the present is then

$$\begin{equation} t_{{\rm su},{\rm QPO}} \cong 5\times 10^5 \;{\rm yr} \frac{ I_{45} {\left(\frac{\nu _i}{{\rm kHz}}\right)}^{1/3}}{\langle \dot{M}\rangle M_{1.4}^{2/3}\Delta }, \end{equation} \tag{ 10 }$$

where Δ is the (unknown) duty cycle of the binary. If we assume Δ ∼ 0.01, the maximum estimated time in the spin-up epoch up to now is ≈5 × 10⁷ yr corresponding to ν_i = 814 Hz (see Table 4). If any of the QPO frequencies is interpreted as the eventual equilibrium frequency, then the time spent in the current spin-up epoch is a small fraction of the age of the system. Even if the system were already in equilibrium at ν = 11 Hz then it must have reached equilibrium in the short time

$$\begin{equation} t_{{\rm su}} \cong \frac{\nu }{\dot{\nu }\,\Delta }\simeq 2.5\times 10^7 {\rm \;yr}. \end{equation} \tag{ 11 }$$

Newborn NSs have a distribution of rotation rates Ω₀ ∼ 1–300 rad s⁻¹ and dipole magnetic moments μ₀ ∼ 10²⁹–10³⁰ G cm³ (Faucher-Giguère & Kaspi 2006). The NS is active as a pulsar until the magnetospheric voltage falls below a critical threshold (the "pulsar death line"). For conventional radio pulsars the epoch of pulsar activity lasts for 10⁶–10⁷ yr. No evidence of magnetic field decay is seen in the pulsar population during this active epoch. The lower limit for an exponential decay time is 10⁷ yr.

If the magnetic dipole moment decayed in proportion to the rotation rate, due to flux-line vortex line coupling (Srinivasan et al. 1990),

$$\begin{equation} \mu (t) = \mu _0 \frac{\Omega (t)}{\Omega _0}, \end{equation} \tag{ 12 }$$

the spin-down would follow a power law with braking index ≅5. More detailed considerations of flux-line motion indicates that the simple scaling in Equation (12) is a likely to be a good description for the evolution (Jahan-Miri 2000).

The vortex–flux coupling leads to the decay of magnetic flux in the NS superfluid neutron-superconducting proton core. For the total dipole moment to decay, the flux expelled from the core must diffuse through the highly conducting crust. The decay time through the crust is estimated to be 10⁷ yr. This is supported by the limits on exponential decay of the dipole moment from population analysis of the radio pulsars. Ongoing dipole spin-down on time scales much longer than 10⁷ yr, that is, beyond the typical duration of the epoch of radio pulsar emission, will be effected by the power-law decay of the dipole moment due to flux–vortex coupling.

Whether pulsar activity has ceased or not, vacuum dipole spin-down will continue until the wind captured from the companion penetrates the light cylinder. The captured wind loses angular momentum in a shock and will accrete toward the NS down to a stopping radius ∼r_A. The wind will be able to affect the torque on the NS when

$$\begin{equation} r_A (\mu _0, \dot{M}_{w{\rm NS}}) \lesssim r_{{\rm LC}} \equiv \frac{c}{\Omega _1}, \end{equation} \tag{ 13 }$$

with $\dot{M}_{w{\rm NS}}$ denoting the wind captured by the NS. After this point, spin-down will proceed under the effect of the inflowing wind. From Equation (13) the rotation rate Ω₁ at the beginning of the wind spin-down epoch is

$$\begin{equation} \Omega _1 \cong 24 {\rm \,rad\, s^{-1}} \mu _{1,29}^{- 4/7} \dot{M}_{w{\rm NS},13}^{2/7} M_{1.4}^{1/7}, \end{equation} \tag{ 14 }$$

where $\dot{M}_{w{\rm NS},13}$ is the mass inflow rate arriving at r_A in units of 10¹³ g  s⁻¹. If we first assume that flux–vortex coupling has no effect, the dipole spin-down proceeds with a constant dipole moment and braking index 3:

$$\begin{equation} \dot{\Omega } = -K \Omega ^{3} = -\frac{4 \mu ^2 \pi }{3 c^3 I} \Omega ^3, \end{equation} \tag{ 15 }$$

where c is the speed of light and I is the NS moment of inertia. The duration of the dipole spin-down epoch, until the rotation rate Ω₁ is reached is given by

$$\begin{eqnarray} t_{n=3} & \equiv & \frac{P_1}{2\dot{P}}=-\frac{\Omega _1}{2\dot{\Omega }} \nonumber \\ & \cong & 10^{10} {\rm \,yr}\; \mu _{1,29}^{-2}\Omega _1^{-2} \nonumber \\ & \cong & 2.5\times 10^6 {\rm \,yr}\; \mu _{0,30}^{- 6/7} \dot{M}_{w{\rm NS},13}^{-4/7} M_{1.4}^{-2/7}, \end{eqnarray} \tag{ 16 }$$

where we have used Equations (14) and (15) and I = 10⁴⁵ g cm². In the last line of Equation (16) we have assumed that the magnetic field does not decay during the dipole spin-down epoch, so that the dipole moment μ₁ at the end of this epoch is still the dipole moment μ₀ of the NS at birth. Since the lifetime thus obtained is not much smaller than the crust diffusion timescale of 10⁷ yr, it might be necessary to take the vortex–flux coupling and the induced decay of the dipole magnetic moment into account. The correct estimate of dipole spin-down epoch, incorporating flux decay according to Equation (12) proceeds with braking index 5. We obtain

$$\begin{eqnarray} t_{n=5} & \equiv & \frac{P_1}{4\dot{P}} = {-}\frac{\Omega _1}{4\dot{\Omega }}\nonumber \\ & \cong & 5 \times 10^{9}\, {\rm yr}\; \mu _{1,29}^{-2}\Omega _1^{-4} \nonumber \\ & \cong & 1.5 \times 10^4\, {\rm yr}\; \mu _{1,29}^{2/7} \dot{M}_{w{\rm NS},13}^{-8/7} M_{1.4}^{-4/7}. \nonumber \\ \end{eqnarray} \tag{ 17 }$$

Relating the magnetic moment μ₁ and rotation rate Ω₁ at the end of dipole spin-down to the initial values μ₀ and Ω₀ with Equation (12)

$$\begin{equation} \mu _{1,29} = 10 \mu _{0, 30} \frac{\Omega _1}{\Omega _0}, \end{equation} \tag{ 18 }$$

and using Equation (14) we obtain

$$\begin{equation} t_{n=5} \cong 2 \times 10^4 y\;\mu _{0,30}^{2/11}\left[\frac{\Omega _0}{10{\rm \,rad\,s^{-1}}}\right]^{-2/11} \dot{M}_{w{\rm NS},13}^{-12/11} M_{1.4}^{-6/11}. \end{equation} \tag{ 19 }$$

This value is shorter than the typical field diffusion time scale through the NS crust (≳ 10⁷ yr) and therefore Equation (19) cannot be used self-consistently for μ₀ ∼ 10²⁹–10³⁰ G cm³. If one ignores the field decay and assumes a typical μ₀ = 10^29–30 G cm³ in Equation (16) then the dipole spin-down age is ∼10⁷ yr.

Wind accretion (Edgar 2004; Theuns & Jorissen 1993; Theuns et al. 1996; Pfahl et al. 2002) proceeds by Bondi–Hoyle-like capture of material from the companion's wind, which is then channeled toward the NS by outward transport of angular momentum in traversing a shock front. The wind mass-loss rate from a ∼1 M_☉ main-sequence companion is $\dot{M}_{{\rm wind}} \sim 10^{12}{\rm \,g\,s}^{-1}$, which is low when compared to the present-day mass accretion rate, driven by RLOF. Even considering a subgiant phase of a ∼1 M_☉ donor star, the wind loss rate will increase only slightly, reaching $\dot{M}_{{\rm wind}} \sim 10^{13}{\rm \,g\,s}^{-1}$ for subgiant stars (see, for example, Reimers 1975, 1978). Only a fraction of this wind will be captured to flow toward the NS, with $\dot{M}/\dot{M}_{{\rm wind}}$ as low as ∼10⁻² (Nagae et al. 2004). However, since the orbit of IGR J17480-2446 shows stringent upper limits on the eccentricity (e < 10⁻³; Papitto et al. 2011) tidal circularization has probably already taken place and it is likely that the donor star is rotating synchronously with the orbit, since the circularization timescale is typically larger than the synchronization timescale (see, for example, Hurley et al. 2002). The large rotation of the donor (v_rot ≃ 70 km s⁻¹) can boost the wind loss rate by a large factor of the order of 10^2–3 when compared with typical isolated stars of similar mass at the same evolutionary stage. The captured wind might therefore be comparably higher, with values of $\dot{M}_{w{\rm NS}}\sim 10^{13\hbox{--}14}{\rm \,g\,s}^{-1}$.

The NS is spinning rapidly within this wind of low specific angular momentum. For our purposes of estimating the duration of the spin-down epoch it will be enough to take a constant torque, giving the spin-down rate:

$$\begin{eqnarray} \dot{\Omega } & = & - \frac{\dot{M}_{w{\rm NS}}}{I} \left(G M_{{\rm NS}} r_A\right)^{1/2} \nonumber \\ & = & - (\dot{M}_{w{\rm NS}})^{6/7}\frac{\mu ^{2/7}}{I}\left(G M_{{\rm NS}}\right)^{3/7} \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} & = & -4.9\times 10^{-15}{\rm \,rad\,s^{-1}}\dot{M}_{w{\rm NS},13}^{6/7}\mu _{1,29}^{2/7}M_{1.4}^{3/7}I_{45}^{-1}. \end{eqnarray} \tag{ 21 }$$

Using Equations (12) and (18) one can find the spin-down in wind and field decay solutions

$$\begin{eqnarray} {\mu (t)}^{5/7} & = & {\mu _1}^{5/7} - 5/7 C^{\prime } t \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} {\Omega (t)}^{5/7} & = & {\Omega _1}^{5/7} - 5/7 C t \nonumber\\ C & \equiv & \left(\frac{\mu _1}{\Omega _1}\right)^{2/7}\frac{{\dot{M}_{w{\rm NS}}}^{6/7}(GM_{{\rm NS}})^{3/7}}{I} \nonumber \\ C^{\prime } & \equiv & \frac{\mu _1}{\Omega _1}\frac{{\dot{M}_{w{\rm NS}}}^{6/7}(GM_{{\rm NS}})^{3/7}}{I}. \end{eqnarray} \tag{ 23 }$$

Since the magnetic moment does not change after the wind spin-down epoch, μ₂ at the end of the wind spin-down epoch is just the magnetic moment in the present RLOF spin-up epoch as estimated above. By assuming that Ω(t_sd) = Ω₂ ≪ Ω₁ and μ₂ ≪ μ₁, we obtain

$$\begin{eqnarray} t_{{\rm sd}} & = & \frac{7}{5} \frac{\Omega _1}{{\mu _1}^{2/7}} \frac{I}{{\dot{M}_{w{\rm NS}}}^{6/7}(GM_{{\rm NS}})^{3/7}} \nonumber \\ & \cong & 3\times 10^{7} \; {\rm yr} \;I_{45}\;\mu _{0,30}^{-6/7}\dot{M}_{w{\rm NS},13}^{- 4/7} M_{1.4}^{-2/7}, \nonumber \\ \end{eqnarray} \tag{ 24 }$$

where we have assumed that μ₁ = μ₀ as found at the end of the previous section.

From the calculations presented in Equation (24) we see that for a captured wind of $\dot{M}_{{\rm wind},{\rm NS}}\sim \,10^{-2}\cdot \,\dot{M}_{{\rm wind}}\sim 10^{13}\hbox{--}10^{14}{\rm \,g\,s}^{-1}$, the spin-down epoch lasts for a time 10^7–8 yr, much shorter than the age of the cluster (∼10¹⁰ yr) for an assumed μ₁ = μ₀ = 10^29–30 G cm³. If more wind is accreted, as described, for example, in the focused accretion model of Podsiadlowski & Mohamed (2007), the wind spin-down epoch would be further shortened if the wind still carries low angular momentum. If the wind has instead large angular momentum because it is focused through the Lagrangian point L₁, the RLOF epoch would be further shortened.

In the previous section, we have seen that IGR J17480-2446 is not close to equilibrium since it exhibits a strong spin-up during the outburst. Several other considerations on the evolutionary history of IGR J17480-2446 also suggest that the NS is currently spinning up. According to the spin evolution history, the present 11 Hz spin frequency does not represent a special frequency. It remains to be explained whether the conclusion that the binary has currently completed only the first few million years since the onset of the RLOF, a phase which lasts for ∼10⁹ yr, requires any special pleading.

If we assume that the observed spin-up and outburst duration represent the typical values of an outburst of IGR J17480-2446, then the pulsar spins up by $\dot{\nu }\;\Delta \,t_{o}\simeq 7\times 10^{-6}{\rm \,Hz}$ at each outburst, where Δ t_o is the outburst length (55 days). If the NS is dominated by magneto-dipole spin-down during quiescence, as proposed for some accreting millisecond pulsars (Hartman et al. 2008, 2009, 2011; Patruno et al. 2010; Papitto et al. 2011; Riggio et al. 2011), then a balance of the spin-up requires a magnetic field B ≃ 10¹¹ G and a recurrence time of ∼15 yr to provide a spin-down with the necessary magnitude calculated via the expression (Spitkovsky 2006):

$$\begin{equation} \dot{\nu }_{{\rm sd}}=-\frac{\mu ^2\left(\frac{2\pi \nu }{c}\right)^3\left(1+{\rm sin}^2\alpha \right)}{2\pi \,I}, \end{equation} \tag{ 25 }$$

where α is the offset angle between the rotational and magnetic field axes. The spin-down would then be of the order of −10⁻¹⁴ Hz s⁻¹.

However, as discussed in Section 1, different estimates of the B field of IGR J17480-2446 suggest B ∼ 10⁹–10¹⁰ G. Such a B value requires a recurrence time of ∼10²–10³ yr, which is not compatible with a binary evolution scenario with the donor in an RLOF phase. The mass transfer rate (equal to the outburst mass accretion rate averaged over the source duty cycle) would then average to $\dot{M}\sim 10^{13}\hbox{--}10^{14}{\rm \,g\,s}^{-1}$, a value too small for an evolved donor or even a main-sequence star with M > 0.8 M_☉ (typical values being ∼10¹⁵–10¹⁶ g s⁻¹).