Radio Pulse Search and X-Ray Monitoring of SAX J1808.4−3658: What Causes Its Orbital Evolution?

In BWs and redbacks, the companion star is being ablated by the pulsar wind and high-energy radiation, thus producing potential mass loss. Observational evidence of this phenomenon comes from the fact that radio pulsations are very often eclipsed by intra-binary material that induces free–free absorption of the pulsed signal. The orbital parameters of SAX J1808.4−3658 are compatible with those of a BW, and its 0.05–0.08 M_⊙ companion is also a semi-degenerate star (Bildsten & Chakrabarty 2001; Deloye et al. 2008; Wang et al. 2013). The only difference between BWs and SAX J1808.4−3658 is that, in the latter system, the companion is in Roche lobe overflow, whereas BWs are thought (at least in some cases) to be detached systems (Breton et al. 2013).

In BWs, as well as in redback pulsars, short-term effects on the orbital evolution occur on timescales that are generally orders of magnitude shorter than the predicted (secular) ones from angular momentum loss due to gravitational waves and/or magnetic braking. The seven BWs and redbacks that have a measured ${\dot{P}}_{b}$ show orbital evolution timescales from 100 Myr to less than 1 Myr (see Table 3). Four of them have a negative orbital period derivative (the orbit is shrinking), whereas only one (PSR J1959+2048) has a positive value (the orbit is expanding). In at least two cases (PSR J2051–0827 and PSR J2339–0533), the sign seems to be changing cyclically over timescales of the order of a few years (Pletsch & Clark 2015; Shaifullah et al. 2016).

Table 3.  BWs and Redbacks with Measured ${\dot{P}}_{b}$

Name (PSR)	Type	Pb (hr)	${\dot{P}}_{b}$	Error
J1023+0038	RB	4.8	−7.3 × 10−11	0.06 × 10−11
J1227−4853	RB	6.9	−8.7 × 10−10	0.1 × 10−10
J1723−2837	RB	14.8	−3.5 × 10−9	0.12 × 10−9
J2339−0533a	RB	4.6	[−5.8, 2.7] × 10−10	0.01 × 10−10
J1731−1847	BW	7.5	−1.08 × 1010	0.07 × 10−10
J2051−0827a	BW	2.4	[−2.03, 1.41] × 10−11	0.08 × 10−11
J1959+2048	BW	9.2	1.47 × 10−11	0.08 × 10−11

Note.

^aThe orbital period of these pulsars shows a roughly cyclic behavior; see Shaifullah et al. (2016) and Pletsch & Clark (2015).

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Similarly, in the case of SAX J1808.4−3658, ${\tau }_{\mathrm{ev}}=\tfrac{{P}_{b}}{{\dot{P}}_{b}}\,\approx 70\,\mathrm{Myr}$, whereas from Equation (7) one would have expected a timescale of a few Gyr (varying slightly with the exact neutron star and companion mass chosen), meaning that the ${\dot{J}}_{\mathrm{gw}}$ term is not the dominant one. It is important to stress that these apparent orbital period derivatives are epoch-dependent in all BWs and redbacks and change on years-long timescales. However, there is at least one situation where, besides the short-term fluctuations discussed above, there is also a decade-long (secular?) variation of the orbit that corresponds to an overall shrinkage of the binary orbital separation (see, e.g., PSR J1023+0038; Jaodand et al. 2016).

In the few compact LMXBs where an orbital period derivative can be measured, we see a roughly equally distributed sign (six positive and four negative signs and three upper limits observed so far; see Table 4). The magnitude of the orbital period derivative is always much larger than expected from conservative binary evolution ($\dot{J}=0$) and/or from angular momentum loss via gravitational waves and/or magnetic braking. In the literature, a number of different mechanisms have been suggested to explain the large ${\dot{P}}_{b}$. These include mass loss from the companion (Burderi et al. 2010; Ponti et al. 2017), enhanced magnetic braking (González Hernández et al. 2014), the presence of a third body (Iaria et al. 2015), spin–orbit coupling (Wolff et al. 2009; Patruno et al. 2012), and, in some cases, even modified theories of gravity (Yagi 2012). The group of LMXBs appears to be the most heterogeneous among the different binaries that we are considering here. Indeed, this group comprises transient LMXBs with black hole accretors (XTE J1118+480 and A0620–00), transient LMXBs with neutron star accretors (EXO 0748–676, MXB 1658–298, SAX J1748.9–2021, AX J1745.6–2901, and SAX J1808.4−3658 ), and persistent neutron star LMXBs (Her X–1, 2A 1822–37, and XB 1916–053). Furthermore, some of these LMXBs are accreting pulsars and have orbital periods determined via timing of their pulsations (e.g., Her X–1, 2A 1822–37, SAX J1808.4−3658, and SAX J1748.9–2021), whereas others are eclipsing systems and their periods are determined via X-ray and/or optical photometry. No other orbital period derivative has been measured so far for any other LMXB. In Table 4, we list the LMXBs in the sample with their orbital period derivatives and the main proposed explanation given in the literature.

Table 4.  LMXBs with Orbital Period Derivatives

Name	Pb (hr)	${\dot{P}}_{b}$	Transient	Companion Type	Sign	Proposed Model	References
Neutron Star LMXBs
EXO 0748–676a	3.8	1.9 × 10−11	Yes	MS	+	SOC	Wolff et al. (2002, 2009)
2A 1822–37	5.5	1.51(8) × 10−10	No	MS	+	Mass Loss	Burderi et al. (2010), Iaria et al. (2011)
SAX J1808.4–3658	2.0	3.5(2) × 10−12	Yes	SD	+	SOC/Mass Loss	di Salvo et al. (2008),Patruno et al. (2012)
MXB 1658–298	7.1	8.4(9) × 10−12	Yes	MS	+	Unknown	Paul & Jain (2010)
XB 1916–053	0.8	1.5(3) × 10−11	No	SD	+	Third Body	Iaria et al. (2015)
SAX J1748.9–2021	8.8	1.1(3) × 10−10	Yes	MS/Sub-G	+	Mass Loss	Sanna et al. (2016)
AX J1745.6–2901	8.4	−4.03(32) × 10−11	Yes	MS/Sub-G	−	Mass Loss	Ponti et al. (2017)
Hercules X–1	40.8	−4.85(13) × 10−11	No	MS	−	Several	Staubert et al. (2009)
XTE J1710–281b	3.2	[−1.6, 0.2] × 10−12	Yes	MS	UL	...	Jain & Paul (2011)
IGR J00291+5934c	2.5	[−5, 6] × 10−12	Yes	SD	UL	...	Patruno (2016)
4U 1323–619d	2.9	[−5, 21] × 10−12	No	MS	UL	...	Gambino et al. (2016)
Black Hole LMXBs
XTE J1118+480	4.0	−6(1.8) × 10−11	Yes	MS	−	Enhanced MB	González Hernández et al. (2012, 2014)
A0620–00	7.8	−1.9(3) × 10−11	Yes	MS	−	Enhanced MB	González Hernández et al. (2014)
Nova Muscae 1991	10.4	−6(4) × 10−10	Yes	MS	−	Enhanced MB	González Hernández et al. (2017)

Notes. UL—upper limits; MS—main sequence; Sub-G—sub-giant; SD—semi-degenerate; MB—magnetic braking; SOC—spin–orbit coupling.

^aThis source shows segments of data where a constant P_b is required. The error on ${\dot{P}}_{b}$ is not given, and confidence intervals are determined via the maximum likelihood method (Wolff et al. 2009). ^b1σ upper limit. ^c90% confidence level upper limit. ^d1σ upper limit.

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The binary EXO 0748–676 is an eclipsing binary with a 0.4 M_⊙ donor that shows sudden variations in ${\dot{P}}_{b}$, which were proposed to be due to spin–orbit coupling (Wolff et al. 2002, 2009). Wolff et al. (2002) analyzed two segments of X-ray data (1985–1990 and 1996–2000) and showed that the period had increased by about 8 ms. However, the period increase shows jitters and cannot be fit with a constant ${\dot{P}}_{b}$. Further work by Wolff et al. (2009) extended the analysis until 2008 and observed a similar behavior.

The eclipsing binary 2A 1822–37 is an accreting pulsar with an ∼0.3–0.4 M_⊙ companion, and it has a relatively steady increase of the orbital period measured over a baseline of 30 yr (see, e.g., Burderi et al. 2010; Iaria et al. 2011; Chou et al. 2016). This system is an accretion-disk corona system showing extended partial eclipses of the central X-ray source. Burderi et al. (2010) and Iaria et al. (2011) suggested that the binary contains an Eddington-limited accreting neutron star whose irradiation of the donor is inducing severe mass loss that can explain the large orbital period derivative observed. The positive sign of ${\dot{P}}_{b}$ is ascribed to the response of the radius of the donor to mass loss, with ${R}_{{\rm{c}}}\propto \,{M}_{{\rm{c}}}^{\zeta }$ and n < 1/3. Burderi et al. (2010) suggested that the donor in 2A 1822–37 has a deep convective envelope with ζ = −1/3 (see, e.g., Rappaport et al. 1982), thus justifying the positive ${\dot{P}}_{b}$.

The source AX J1745.6–2901 is an eclipsing binary with an accreting neutron star and a negative ${\dot{P}}_{b}$ (Ponti et al. 2017). The donor mass is constrained to be M_c ≲ 0.8 M_⊙. A strong mass loss is also suggested in this case; but, since the system is shrinking, the mass–radius index needs to be n > 1/3. The data on the T_asc collected over a baseline of about 30 yr show significant scatter of up to several tens of seconds.

The source XB 1916–053 is an ultra-compact (P_b ≈ 50 minutes) persistent dipping source monitored for over 37 yr by X-ray satellites. Hu et al. (2008) studied the first 24 yr of data and found that a quadratic function (i.e., a constant ${\dot{P}}_{b}$) was able to describe the data correctly. However, Iaria et al. (2015) found that, when considering the entire 37 yr of observations, a quadratic function was unable to fit the orbital evolution, and a model with a sinusoidal variation in addition to the quadratic component was required. A third body with a mass of ∼0.06 M_⊙ was invoked to explain the observations with an orbital period of ≈26 yr. It is instructive to note that a deviation from a quadratic function was not apparent in the first 24 yr of data, which suggests that a very long timescale periodicity (or quasi-periodicity) might still be present even in binaries where a constant ${\dot{P}}_{b}$ is observed over baselines of a few decades.

The globular cluster source SAX J1748.9–2021 is an intermittent AMXP (Gavriil et al. 2007; Altamirano et al. 2008; Patruno et al. 2009a; Sanna et al. 2016), and it has been observed in outburst five times. Its companion star is likely to be an ∼0.8 M_⊙ star close to the turnoff mass of the globular cluster NGC 6440, although much smaller masses down to 0.1 M_⊙ cannot be excluded (Altamirano et al. 2008). The orbital evolution has been studied by looking at the orbital ephemeris calculated with coherent timing in a way similar to what has been done in this work. Sanna et al. (2016) described the orbital evolution with a quadratic function, although the fit shows large deviations of the order of 100 s from the best-fit function (which translates into a poor χ² of 78.4 for 1 dof). These authors interpreted the large orbital expansion with a highly nonconservative mass-loss scenario in which the binary is losing more than 97% of the mass flowing through the inner Lagrangian point L₁.

Her X–1 shows a steady decrease of the orbit whose value is compatible with both a conservative and a nonconservative mass transfer scenario (Staubert et al. 2009).

Finally, the orbital evolution of the transient black hole LMXBs XTE J1118+480 and A0620–00 (González Hernández et al. 2012, 2014) was measured with radial velocity curves determined via optical spectroscopy for a period of ∼10 and 20 yr, respectively. These observations were carried out during the extended periods of quiescence of the binaries. The orbit shows a steady shrinkage interpreted as being due to enhanced magnetic braking. The two binaries have a companion mass of 0.2 M_⊙ (XTE J1118+480; González Hernández et al. 2014) and 0.4 M_⊙ (A0620–00; Cantrell et al. 2010; González Hernández et al. 2014).

It is well known that some cataclysmic variables show an anomalous orbital period derivative as well, with some of them proposed to be transferring mass at a higher rate than expected due to irradiation of the companion (Knigge et al. 2000; Patterson et al. 2015, 2016). Even some Algol-type binaries (i.e., a semi-detached system composed of a detached early-type main-sequence star and a less massive subgiant/giant star in Roche lobe overflow) have been reported to evolve on a very short timescale (Erdem & Öztürk 2014). In this case, a nonconservative mass transfer scenario is expected to take place, since the red giant will emit a significant wind. However, for a few of these systems (i.e., all the converging ones), the required mass loss is larger than the highest theoretical value for wind mass loss in giant stars. These observations might suggest that short-term effects have some influence on the orbital evolution of accreting and nonaccreting neutron stars, persistent and transient systems, and white dwarf/black hole/main-sequence stellar accretors. It is worth noting that no neutron star + white dwarf binary (both accreting and nonaccreting) has been observed (as yet) to evolve on anomalous timescales. The only exception is the ultra-compact LMXB 4U 1822–30, which, however, is located in a globular cluster, and therefore its large ${\dot{P}}_{b}$ might simply be due to the contamination induced by the gravitational potential well of the cluster (Jain et al. 2010; Perera et al. 2017). This suggests that, if there is a common reason behind this behavior for all types of binaries (which is, of course, not necessarily true), it must be related to the type of companion (main-sequence or semi-degenerate star) rather than the type of accretor.

A few cautionary words are necessary at this point on the cataclysmic variables (CVs), Algol-type binaries, and some LMXBs. For these types of systems, where the ${\dot{P}}_{b}$ is detected by looking at the eclipse times in optical data, some selection effects might be present. This means that systems with a ${\dot{P}}_{b}$ in line with the theoretical predictions might be more difficult to measure/detect, and therefore the reported values are invariably skewed toward large/anomalous ${\dot{P}}_{b}$ values. Something similar also applies to most LMXBs, with the exception of systems where the ${\dot{P}}_{b}$ is measured via pulsar timing, in which case the sensitivity of the timing technique potentially allows the detection of values orders of magnitude smaller than those in the CVs and Algol binaries. For the CVs, there is ample literature on the topic, and several different systems with a large ${\dot{P}}_{b}$ are reported. In this paper, we include only the T Pyxidis and IM Normae systems, which are the two best-studied cases and have the highest orbital period variation (Patterson et al. 2015, 2016). We also include NN Serpentis (Brinkworth et al. 2006), which is an eclipsing post–common envelope binary where no mass transfer is currently ongoing. We summarize the information on the orbital period evolution of all the binaries discussed in this work in Figure 4. From the figure, it is clear that all sources with short orbital periods that should be losing angular momentum via gravitational-wave emission are evolving on timescales that are at least an order of magnitude shorter than expected. The binaries with wider orbits, where magnetic braking should dominate, also show shorter evolutionary timescales than predicted, although a larger scattering is observed, and some sources are close to the theoretical predictions.

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If the companion star experiences severe mass loss—for example, because of ablation due to the irradiation from a pulsar wind or from the X-rays originating close to the compact object—then the orbital period of the binary changes dramatically (see Section 6.1).

In this case, the orbital period derivative will depend on the amount of mass lost from the companion,

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{P}}_{b}}{{P}_{b}}=-2\displaystyle \frac{{\dot{M}}_{{\rm{c}}}}{{M}_{{\rm{c}}}},\end{eqnarray} \tag{ 8 }$$

where M_c is the companion mass (see, e.g., Frank et al. 2002; Postnov & Yungelson 2014). Applying this model to SAX J1808.4−3658, it is possible to explain the observed ${\dot{P}}_{b}$ if the donor is losing mass at a rate of about 10⁻⁹ M_⊙ yr⁻¹ (di Salvo et al. 2008; but see also Chen 2017).

For the donor to lose a substantial amount of mass, there must be a way to efficiently inject energy into the donor star. Whatever mechanism is chosen, the amount of energy necessary to create such a strong mass loss must be consistent with the total energy budget available to the binary. In SAX J1808.4−3658, it has been proposed that the mass loss is driven by a pulsar wind and high-energy radiation impinging on the donor surface (di Salvo et al. 2008; Burderi et al. 2009). For a circular binary orbit, the total angular momentum is (Frank et al. 2002)

$$\begin{eqnarray}&&J\propto {M}_{\mathrm{NS}}{M}_{{\rm{c}}}{M}^{-1/3}\,{P}_{b}^{1/3},\end{eqnarray} \tag{ 9 }$$

where P_b is the orbital period, M = M_NS + M_c is the total binary mass, and M_c < M_NS. The orbital energy E_orb is

$$\begin{eqnarray}&&-{E}_{\mathrm{orb}}\propto J/{P}_{b}.\end{eqnarray} \tag{ 10 }$$

From Equation (9), an increase in P_b requires M_c to decrease because, for an isolated system, J and M cannot increase. And, from Equation (10), we see that −E_orb must decrease, making the orbit less tightly bound. In other words, an orbital period increase requires energy injection from somewhere. Marsh & Pringle (1990) showed that energy injection by the secondary star is too slow for observed P_b changes, as this is governed by the star's thermal timescale.¹³

The only energy source left is the spin energy of the neutron star. This is

$$\begin{eqnarray}&&{E}_{\mathrm{spin}}\sim {k}^{2}\,{M}_{\mathrm{NS}}\,{v}_{K}^{2}\sim {k}^{2}\,{{GM}}_{\mathrm{NS}}^{2}/{R}_{\mathrm{NS}},\end{eqnarray} \tag{ 11 }$$

where k is the radius of gyration, R_NS is the physical radius, and ${v}_{K}={({{GM}}_{\mathrm{NS}}/{R}_{\mathrm{NS}})}^{1/2}$ is the breakup spin velocity. This is a huge reservoir, since

$$\begin{eqnarray}&&{E}_{\mathrm{spin}}/(-E)\sim {k}^{2}\displaystyle \frac{{M}_{\mathrm{NS}}}{{M}_{{\rm{c}}}}\displaystyle \frac{a}{R}\sim 400\mbox{--}1000\end{eqnarray} \tag{ 12 }$$

with k² ∼ 0.4. But of course, not all of E_spin can be used to drive mass loss from the system and so increase P_b. We can estimate the required minimum efficiency η for spin energy conversion into orbit energy for SAX J1808.4−3658.

The total orbital binding energy of the binary is

$$\begin{eqnarray}&&{E}_{\mathrm{orb}}=-\displaystyle \frac{{{GM}}_{\mathrm{NS}}{M}_{{\rm{c}}}}{2a}.\end{eqnarray} \tag{ 13 }$$

If we use the third Kepler law, then we find that the orbital period is very well determined (P_b = 2.01 hr, from the X-ray pulse timing). The total mass of the binary is ill constrained, mostly because of the unknown neutron star mass. If we assume a range of total binary mass from 1.4 to 3 M_⊙, then the variation in a is of the order of 20% ($6\mbox{--}8\times {10}^{10}\,\mathrm{cm}$). Here, we will assume a = 6.4 × 10¹⁰ cm (M_NS = 1.4 M_⊙, M_c = 0.08 M_⊙). The orbital energy is therefore

$$\begin{eqnarray}&&{E}_{\mathrm{orb}}\sim 3\times {10}^{47}\,\mathrm{erg}.\end{eqnarray} \tag{ 14 }$$

The semimajor axis of the binary changes according to the third Kepler law:

$$\begin{eqnarray}&&\dot{a}=\displaystyle \frac{2a}{3{P}_{b}}{\dot{P}}_{b}.\end{eqnarray} \tag{ 15 }$$

The value of ${\dot{P}}_{b}$ is measured from observations as ∼3.5 × 10⁻¹². Therefore, $\dot{a}=2\times {10}^{-5}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$. The orbital energy variation is (we assume that $\dot{M}$ terms are negligible)

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{orb}}=\displaystyle \frac{{{GM}}_{\mathrm{NS}}{M}_{{\rm{c}}}\dot{a}}{2\,{a}^{2}}\sim 9\times {10}^{31}\,\mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 16 }$$

The total spin-down power is

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{sd}}=I\omega \dot{\omega },\end{eqnarray} \tag{ 17 }$$

where ω = 2π ν, $\dot{\omega }=2\pi \dot{\nu }$, and ν and $\dot{\nu }$ are the spin frequency (401 Hz) and spin-down (1.65(20) × 10⁻¹⁵ Hz s⁻¹) observed in SAX J1808.4−3658 (Patruno et al. 2012). Numerically, ${\dot{E}}_{\mathrm{sd}}\approx 2.6\times {10}^{34}$ erg s⁻¹.

We need an efficiency η of at least the ratio between the two powers:

$$\begin{eqnarray}&&\eta =\displaystyle \frac{{\dot{E}}_{\mathrm{orb}}}{{\dot{E}}_{\mathrm{sd}}}\sim 0.003.\end{eqnarray} \tag{ 18 }$$

For the parameters assumed above, the Roche lobe radius is R_L = 0.16 R_⊙ (see, e.g., Eggleton 1983), and the fraction of intercepted power is $f={({R}_{L}/2a)}^{2}$, which is approximately 0.8%. This value suggests that the donor star must be extremely efficient in converting the incident power into mass loss, since $\epsilon$ ∼ η/0.8 ≈ 40%. Campana et al. (2004) estimated that the irradiating power¹⁴ required to explain the bright optical counterpart of SAX J1808.4−3658 (observed with Very Large Telescope (VLT) data in 2002) amounted to ${L}_{\mathrm{irr}}\,={8}_{-1}^{+3}\times {10}^{33}$ erg s⁻¹ (see also Homer et al. 2001 for a similar estimate made with observations taken in 1999). Therefore, the optical data require a conversion of a fraction $\xi \,={L}_{\mathrm{irr}}/{\dot{E}}_{\mathrm{sd}}\approx 0.2\mbox{--}0.6$ of incident power into thermal radiation by the donor star (the range provided takes into account the 1σ error bars on the irradiation luminosity and the possibility that the distance is 2.5 kpc rather than 3.5 kpc). Since from the observational constraint we have that $\xi +\epsilon =0.5\mbox{--}1.0$ and $\xi +\epsilon$ is bound to be equal to 1 by the conservation of energy, this scenario is energetically plausible for a range of parameters compatible with the observations.

If we assume that the mass loss from the donor of SAX J1808.4−3658 is constant, then we still need to explain the T_asc of the 2011 outburst. This data point deviated by approximately 7 s from the predicted value that can be obtained in Equation (2) by using a constant ${\dot{P}}_{b}=3.5\times {10}^{-12}\,{\rm{s}}\,{{\rm{s}}}^{-1}$. The 7 s deviation can be explained if the mass-loss rate has increased by about ∼70% during the 2008–2011 period, which would require a proportionally larger spin-down power than assumed above. Indeed, the relation between the mass loss and the spin-down power is linear (see, e.g., a discussion in Hartman et al. 2009):

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{c}}}={\dot{E}}_{\mathrm{sd}}{\left(\displaystyle \frac{1}{2a}\right)}^{2}\,G\,{M}_{{\rm{c}}}\,{R}_{{\rm{c}}}.\end{eqnarray} \tag{ 19 }$$

Since we have seen that the donor needs to convert the incident spin-down power into mass loss with extraordinary efficiency, close to 40% ($\epsilon \sim 0.4$), the spin-down power is larger than initially estimated, either because of a larger moment of inertia of the neutron star or because the spin-down $\dot{\omega }$ is slightly larger than observed. In the first case, one would need I ≳ 1.7 × 10⁴⁵ g cm², and this can be used in principle to constrain the equation of state of ultra-dense matter (under the assumption that mass loss is the main mechanism responsible for the binary evolution). In Figure 5, we plot an illustrative example of the type of constraints that can be obtained when using this method for a selection of equations of state (Fortin et al. 2016).

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In conclusion, this mechanism is energetically feasible if

• 1.  
  SAX J1808.4−3658 has a neutron star with a large moment of inertia and
• 2.  
  the incident pulsar power can be converted into mass loss with an ≈40% efficiency.

When looking at the whole sample of pulsar binaries, the behavior of other BW pulsars cannot be explained by this model, since at least two of them are shrinking their orbits. Therefore, if the mass-loss model is correct, we need two different mechanisms to explain the population of binary pulsars. Furthermore, in a recent work (Patruno 2016), we studied the orbital evolution of another AMXP, namely, IGR J0029+5934. This system can be considered a "twin" system of SAX J1808.4−3658, since its orbital and physical parameters are extremely similar (see, e.g., Patruno & Watts 2017). However, the orbit of the binary is varying at a very slow pace (τ > 0.5 Gyr), compatible with a conservative scenario in which the binary evolution is driven by gravitational-wave emission. We constrained the efficiency of the pulsar spin-down to mass-loss conversion to be ≲5%. It is not clear, therefore, why IGR J0029+5934 is unable to convert spin-down power into mass loss, whereas SAX J1808.4−3658 is so efficient. We stress that the observations of both systems suggest a donor irradiated by a pulsar wind/high-energy radiation, and the donor mass is almost identical.

An exchange of angular momentum between the stellar spin and the orbit can generate variations of the orbital period. These variations of the orbital angular momentum are encoded in the term ${\dot{J}}_{\mathrm{soc}}$ in Equation (6).

The Applegate model was developed by Applegate (1992) and Applegate & Shaham (1994) to explain the orbital variability observed in a sample of eclipsing variables and later applied to the case of the BW pulsar PSR B1957+20 (Applegate & Shaham 1994). The model has been further extended to Roche lobe–filling systems such as cataclysmic variables (Richman et al. 1994). The model can be briefly summarized as follows. If the donor star has internal deformations, then the gravitational potential outside of the active star is (terms higher than quadrupolar are ignored here)

$$\begin{eqnarray}&&\phi (x)=-\displaystyle \frac{{GM}}{r}-\displaystyle \frac{3}{2}{{GQ}}_{{ik}}\displaystyle \frac{{x}_{i}{x}_{k}}{{r}^{5}},\end{eqnarray} \tag{ 20 }$$

where x_i and x_k are Cartesian coordinates measured from the center of mass of the star and Q_ik is the quadrupole tensor (related to the inertia tensor). For simplicity, one can assume a circular orbit (which is legitimate to do in a binary such as SAX J1808.4−3658), an alignment between the donor spin axis and orbital angular momentum, and that the stellar spin and orbit are synchronized. If the Cartesian system is chosen so that the z-axis is the angular momentum axis and the x-axis points from the center of mass to the companion star, then Q_ik reduces to Q_xx = Q. In a circular orbit, the relative velocity can be written as

$$\begin{eqnarray}&&{v}^{2}=r\displaystyle \frac{d\phi }{{dt}}.\end{eqnarray} \tag{ 21 }$$

Therefore, from Equation (20), one can see that the relative velocity v is related to the time-varying quadrupole Q. Since v is also related to the orbital period of the binary, it is clear that a time-varying mass quadrupole term induces a variation of the orbital period. Applegate (1992) suggested that the cause of the time-varying quadrupole Q might be related to magnetic cycles of period P_mod (in a way analogous to the familiar 11 yr magnetic solar cycles). During these cycles, the magnetic field induces a redistribution of the angular momentum in different layers of the star and allows a transition between different equilibrium configurations. The strength of the surface magnetic field B required to explain a variation ΔP of the orbital period can be written as

$$\begin{eqnarray}&&{B}^{2}\sim 10\displaystyle \frac{{{GM}}^{2}}{{R}^{4}}{\left(\displaystyle \frac{A}{R}\right)}^{2}\displaystyle \frac{{\rm{\Delta }}P}{{P}_{\mathrm{mod}}}.\end{eqnarray} \tag{ 22 }$$

The transport of angular momentum inside the star of course needs some energy, which Applegate (1992) suggested might come either from the donor internal nuclear burning reservoir or from tidal heating (Applegate & Shaham 1994). This is, in essence, the Applegate model, which has been proposed as a viable way to explain the behavior of many binary systems. It seems therefore natural to extend it to the case of LMXBs. For the case of SAX J1808.4−3658, one needs to assume a value for P_mod, since the T_asc variations observed so far do not show a complete cycle. By assuming P_mod = 50 yr, R = 0.1 R_⊙, M = 0.07 M_⊙, and A = 7 × 10¹⁰ cm, one finds B ∼ 1 kG.

There are, however, two main problems with the Applegate model applied to SAX J1808.4−3658 and to many other compact binaries considered here. The first is that the required B field is of the order of 1 kG, which is much larger than the typical values thought to be present in fully convective stars. However, some isolated low-mass stars and brown dwarfs have been observed with relatively strong surface B fields (Morin et al. 2010) and, in some cases, with fields larger than 1 kG (Reiners 2012). Furthermore, recent studies have provided observational evidence for the presence of magnetic activity in at least four fully convective stars, suggesting that the dynamo mechanism that produces stellar magnetic fields also operates through convection despite the absence of the tachocline, which is the boundary layer between the radiative and convective envelope where the magnetic fields are generated (Wright & Drake 2016).

The second problem, which seems more difficult to circumvent, was discussed in a critical review by Brinkworth et al. (2006), who found that, for very low-mass stars such as NN Serpentis (which is a nonaccreting, post–common envelope binary with a 0.15 M_⊙ companion and an orbital period of 3 hr), the internal energy budget of the donor star might be insufficient to generate the required donor distortion. Even if one invokes the tidal-heating mechanism proposed in Applegate & Shaham (1994), there will still be insufficient energy available to generate the required stellar distortions (see, e.g., Burderi 2015). As was the case for the mass-loss model, the only source of energy left is the spin-down energy of the pulsar. In this case, there needs to be a viable mechanism to transport energy deeper in the donor star that is able to generate a varying-mass quadrupole. As noted by Applegate (1992), if the donor star becomes more oblate, then the mass quadrupole ΔQ > 0 and the orbital period decreases. The opposite happens if ΔQ < 0: the orbital period increases. The observed behavior of most binaries considered in this work might then be explained if, for some reason, some of them (like SAX J1808.4−3658 and other diverging binaries) have ΔQ < 0, whereas all other converging binaries have ΔQ > 0. This idea remains highly speculative, since the problem of what happens in the deep layers of irradiated stars has not been investigated yet.

This model could explain the sign and strength of ${\dot{P}}_{b}$ in SAX J1808.4−3658 only if the donor magnetic field is sufficiently strong. Indeed, the mass lost by the donor cannot be larger than that estimated in Section 6.3.1, since the energy budget does not allow it. As an example, we follow the recipe provided by Justham et al. (2006), in which the angular momentum lost by the binary via magnetic braking is

$$\begin{eqnarray}&&{\dot{J}}_{\mathrm{mb}}=-{{\rm{\Omega }}}_{d}\,{B}_{s}\,{R}_{c}^{13/4}{\dot{M}}_{w}^{1/2}{(G{M}_{c})}^{-1/4},\end{eqnarray} \tag{ 23 }$$

where Ω_d is the angular rotational frequency of the donor star, B_s is its dipolar magnetic field at the surface, and ${\dot{M}}_{w}$ is the amount of wind loss. If we assume that SAX J1808.4−3658 is changing orbital parameters mainly because of angular momentum loss, then, by rearranging Equation (5), we obtain

$$\begin{eqnarray}&&\dot{J}=\displaystyle \frac{\dot{a}}{{a}^{1/2}}{M}_{\mathrm{NS}}{M}_{{\rm{c}}}{\left(\displaystyle \frac{G}{M}\right)}^{1/2}\approx 2\times {10}^{35}\,{\rm{g}}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-2}.\end{eqnarray} \tag{ 24 }$$

Assuming that the donor is tidally locked, Equation (23) gives the strength of the minimum B_s field required, which is of the order of 10³–10⁴ G for a maximum wind-loss rate of 10⁻⁹ M_⊙ yr⁻¹, even stronger than the value calculated for the Applegate model. This shows that the magnetic-braking model is unlikely to be the correct one. Furthermore, such an explanation cannot work in several other binaries because, at least in some neutron star LMXBs, the sign of the observed orbital period derivative is opposite to that expected when magnetic braking is the main driver of binary evolution.