MULTIBAND STUDIES OF THE OPTICAL PERIODIC MODULATION IN THE X-RAY BINARY SAX J1808.4−3658 DURING ITS QUIESCENCE AND 2008 OUTBURST

Two sets of light curve data obtained from previous Gemini imaging observations were included in this work. The observations for the first and second set were carried out in 2007 March and 2008 May, and were reported in Deloye et al. (2008) and Wang et al. (2009), respectively. The first set mainly contains two g' light curves, one 2.5 hr long and the other 2.8 hr long, while two additional i' brightness data points were obtained at the end of each g' light curve. The second set contains three 3 hr r' light curves obtained over 5 days. Detailed descriptions of the data sets can be found in Deloye et al. (2008) and Wang et al. (2009).

Time-resolved imaging of J1808.4 was carried out with Gemini South Telescope on 2008 August 4 and 7. To obtain nearly simultaneous multi-band light curves, Sloan g' and i' filters were alternately used for imaging in each night. The detector was the Gemini Multi-Object Spectrograph (GMOS; Hook et al. 2004), which consists of three 2048 × 4608 EEV CCDs. In our observations, only a section of 300 × 300 pixel² in the middle CCD (CCD 02) was used. The pixel scale is 0073 pixel⁻¹.

The total observation time in the first night was 3 hr, with 23 and 24 useful g' and i' images respectively made. The exposure times were 200.5 s in g' and 150.5 s in i'. The observing conditions were variable, with the seeing (full width at half-maximum (FWHM) of the point-spread function (PSF) of the images) increasing from 06 to 09 over the course of the observation. On August 7, the total observation time was 5.5 hr, during which 47 and 48 useful g' and i' images, respectively, were made. The exposure time at each band was the same as that in the first night. The observing conditions were relatively stable, with the average seeing being ≃07.

The starting of the 2008 outburst of J1808.4 was reported by Markwardt & Swank (2008) on 2008 September 22. We subsequently requested CFHT target of opportunity observations of the source. The observations were carried out in the queued service observing mode on 2008 September 29, 30, October 3, and 4. The detector was the wide-field imager MegaCam, which consists of 36 2048 × 4612 pixel² CCDs. The field of our target was imaged on CCD 22. The pixel scale is 0187 pixel⁻¹. A Sloan r' filter was used for imaging.

During the four nights, we obtained 60, 63, 84, and 72 images, respectively. The exposure time for each image was 20 s. The total observation times were 1.1, 1.2, 1.6, and 1.4 hr, respectively. The observing conditions in the first night were good, with an average seeing of 08. In the second night, the seeing condition was not good, having a large range of 08–18 and an average of 11. Therefore six images with large seeing values were excluded from the data. The third night had good observing conditions, with an average seeing of 07. The observing conditions in the fourth night were relatively variable, with the seeing in a range of 06–12 and an average of 08.

We used the IRAF packages for data reduction. The images were bias subtracted and flat fielded. We performed PSF-fitting photometry to measure the brightnesses of the source and other in-field stars. A photometry program DOPHOT (Schechter et al. 1993) was used.

Before photometry, we positionally calibrated our Gemini images (made during the quiescent state) to reduce possible contamination from the two nearby stars, since the two stars had similar brightnesses and were 06 and 10 away from the target (e.g., Wang et al. 2009). We first made a reference image by combining three best-quality i' images on August 7. All Gemini images were then calibrated to this reference image. We determined the positions of our target and the two nearby stars in the reference image and fixed them at these positions for photometry of the Gemini images. Differential photometry was performed to eliminate systematic flux variations in the images. Six isolated, non-variable bright stars in the field were used. The brightnesses of our targets and other stars in each frame were calculated relative to the total counts of the six stars. Standard stars 95–100 and 95–96 (Landolt 1992) were observed on August 4 and used for g' and i' flux calibration, respectively. To convert Landolt (1992) Vega magnitudes of the standard stars to Sloan filter magnitudes, transformation equations given by Fukugita et al. (1996) were used.

For CFHT data, because J1808.4 was in outburst, approximately 30 times brighter than in quiescence, and the exposure time was short, no effort was made to separate the two nearby stars from our target. Differential photometry was also performed, with 10 in-field bright stars used for calibrating out systematic flux variations among the images. No standard stars were requested in our CFHT program. Using eight in-field bright stars, we flux calibrated our r' images to those in Wang et al. (2009).

The obtained Gemini and CFHT light curves are shown in Figures 1 and 2, respectively. We note that during our Gemini observations, the source was brightening. Over 2 days, the source's g' and i' brightnesses changed by 1.3 and 0.9 mag, respectively. In addition, the amplitudes of modulation decreased from ≃0.4 mag to ≃0.2 mag.

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At the time of the outburst, J1808.4 was visible to CFHT for only approximately 1 hr. Therefore each CFHT light curve covered half of the binary orbit. It can be seen that the first two light curves are sinusoidal-like with an amplitude of 0.2 mag, similar to those seen in quiescence. The latter two light curves changed to a smaller amplitude of 0.1 mag, and the average source brightness at the time decreased by approximately 0.3 mag.

We fitted the data using the Icarus⁶ light curve model for irradiated companions in binaries (Breton et al. 2012). The model was adapted from the code developed by van Kerkwijk (e.g., Stappers et al. 2001), and has been recently used for two other systems (van Kerkwijk et al. 2010, 2011; Breton et al. 2012). The model is similar to the ELC program (Orosz & Hauschildt 2000) that was employed by Deloye et al. (2008) for their earlier work on J1808.4.

In this model, the surface of the companion is constructed by solving the hydrostatic equilibrium equation for a rotating body, which in our case was tidally locked to the neutron star. We implemented a new stellar grid parameterization that uses a triangle tessellation obtained from the subdivision of the primitives of an icosahedron. This parameterization yields uniform coverage on the projected sphere surrounding the star and allows for better performance of the code. We scaled the star so that its size matches that of the Roche lobe, which is justified from the fact that J1808.4 regularly undergoes outbursts involving mass transfer from the companion. A base temperature for the companion is chosen and effects of gravity darkening accounted using Lucy's law (Lucy 1967), with a power-law index of 0.08. To this, we apply the irradiative flux from the neutron star in the way prescribed by Orosz & Hauschildt (2000). Finally, we integrated over the visible surface the emerging flux in a selected photometric filter, which was interpolated from BTSettl atmosphere models (Allard et al. 2003, 2007, 2011), in order to obtain the total flux at a given orbital phase and for specific orbital parameters.

We approximated the disk contribution by adding an orbital-independent, constant value to the companion's flux. The disk flux values inferred at different photometric bands at one epoch constitute a spectral energy distribution, which were checked for consistency with arising from an accretion disk (Section 4.2). We also made the implicit presumption that the system is seen sufficiently face-on so that there is no mutual eclipse/shadow arising between the disk and the companion. This presumption is motivated by the fact that light curves of J1808.4 in quiescence are symmetrical and do not show obvious signs of flux changes that could result from the interaction between the two components (Wang et al. 2009). Moreover, the estimated orbital inclination from previous work (e.g., Deloye et al. 2008) as well as that inferred from this work (see Section 3.5—the maximum value we found is only 56°) indicates that the companion is unlikely to be obscured by the disk for any reasonable disk scale height.

We used a Bayesian framework to make statistical inferences on our light curve modeling parameters. The Bayesian inference on a set of parameters θ is obtained from their posterior probability, which is conditional to the prior knowledge, I, that we have of these parameters and the experimental data D. The posterior probability can be written as the product of the priors times the likelihood of the data: p(θ|I, D) = p(θ|I)  p(D|θ, I)/p(D|I), where the denominator is a normalization factor to ensure that the posterior probability integrated over the parameter space is unity (see Gregory 2005 for an introduction to Bayesian analysis).

3.3.1. Inclination and Radial Velocity Semi-amplitude

3.3.2. Base and Day-side Temperatures

3.3.3. Distance Modulus and Reddening

3.3.4. Disk Flux

3.3.5. Noise Parameter

The light curve fitting is a multi-dimensional nonlinear optimization problem that we tackled using an MCMC method (Gilks et al. 1995). This is particularly efficient to make Bayesian inference since the full posterior distribution is evaluated. In the limit that the observed magnitude errors are normally distributed, the likelihood function can be written as p(D|θI) ∝ exp(−χ²/2), where χ² is the conventional chi-square.

For the MCMC, we chose the stretch move algorithm that is part of a family of MCMC methods called ensemble samplers with affine invariance described by Goodman & Weare (2009). In a nutshell, the algorithm is executed by simultaneously running several chains, all initialized at random locations of the parameter phase space. Every step of the MCMC, a move is proposed for each chain by choosing a complementary chain at random and drawing point along a line passing through the last position recorded in the two chains. How big a step away from the previous position is determined using a distribution that is affine invariant (i.e., g(z⁻¹) = zg(z)). The proposed move is accepted with a probability that is slightly different than that of the usual Metropolis algorithm in order to satisfy the detailed balance requirement (see Goodman & Weare 2009 for the details). An advantage of the stretch move algorithm is that the acceptance rate, which controls the efficiency of the chain, can be tuned using a single parameter in the proposal distribution, as opposed to one per dimension for a random walk Metropolis algorithm. It should also perform better for highly skewed and badly scaled distributions.

We ran the stretch move algorithm using 30 simultaneous chains for a total of 100, 000 steps each. The proposal distribution's tuning parameter was determined from a previous trial run and the obtained acceptance rates of the chains were found to be in the range 20%–30%, as prescribed by Roberts & Rosenthal (2001). For subsequent analysis, we discarded the first 30, 000 points (e.g., the burn-in period) and thinned the remaining by keeping every other 50 points in order to reduce the auto-correlation. In order to ensure that our MCMC has converged we visually inspected the trace of the parameters and performed the Geweke test (Geweke 1992) by comparing the mean of 10% subsections of the first half of the chain to the remaining second half.

We originally aimed to compare the modulations in quiescence and outburst by fitting the light curves separately. However the combination of the short light curves during the outburst and their relatively large uncertainties does not provide tight constraints. We also tested fitting all data together, but for the same reason the results were hardly changed compared to those including the quiescent data only. We therefore fit all the quiescent data obtained from 2007 March to 2008 August with our model. In Table 1, we provide a summary of the fit results as well as several additional quantities inferred from the marginalized posterior probabilities of our irradiated companion modeling. The one- and two-dimensional posteriors are displayed in Figure 3.

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We show all folded light curve data points, model light curves of the median parameter values, and residuals in Figure 4. As can be seen, our model generally reproduces the observations quite well with the data points evenly distributed along the model light curves. However it can be noted that a few light curves have relatively large variations around our model fits, which are also reflected by the noise parameter values. These variations, noted in previous studies (Deloye et al. 2008; Wang et al. 2009), were likely intrinsic and stochastic and cannot be described by our current model.

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Our inferred orbital inclination, $i = 50^{+6}_{-5}$ deg, is consistent with previous estimates from Deloye et al. (2008) and Cackett et al. (2009; the latter found $i=55^{+8}_{-4}$ deg by relativistic Fe line fitting; see also Papitto et al. 2009; Patruno et al. 2009) despite the day-to-day disk and irradiated-side temperature variability, which probably impairs our ability to combine multi-epoch data obtained in different bands. Consequently we constrained the neutron star and companion to have masses of $0.97\pm ^{+0.31}_{-0.22}\, M_{\odot }$ and $0.04^{+0.02}_{-0.01}\, M_{\odot }$, respectively. Our fitting has narrowed the possible mass ranges for both the neutron star and companion and points to a low-mass neutron star in J1808.4, while we note with caution that the 3σ range for the mass of the neutron star is still wide, 0.47–2.52 M_☉ (Table 1). In addition, we note that our measurement of M_ns is in agreement with the independent estimate of M_ns < 1.5 M_☉ from X-ray pulse shape modeling (Morsink & Leahy 2011).

The results of the distance modulus and extinction, DM = 13.08 ± 0.11 and A_J = 0.26 ± 0.06, are slightly larger than the previous estimates. The posteriors are very close to being normally distributed, and have standard deviations nearly the same as that of the priors. Our inferred companion's projected radial velocity semi-amplitude, $360^{+17}_{-16}$ km s⁻¹, is in agreement with the value, 370 ± 40 km s⁻¹, from Elebert et al. (2009). We observed that the uncertainty on K_comp is only 50% as large as that of the priors, which implies that the data contribute to constraining the companion's velocity. The most likely reason is that given the distance and absorption are relatively well determined, and the fact that the companion fills its Roche lobe, the observed luminosity constrains the actual radius of the companion, which is tied up its orbital velocity.

The day-side temperatures of the companion are consistent with being a constant before the 2008 outburst (Figure 5), and we found that a temperature of 7972 ± 243 K best fits them (χ² = 1.1 for six degrees of freedom). We note that the data points may suggest a trend of temperature decreasing over the time, but no conclusion can be drawn due to the large standard deviations. Our result is relatively surprising, as one might have naively expected increased activity as the system was going toward an outburst. The behavior of the disk flux contribution (Figure 6) is also consistent with being constant in each band over time, except for the last observation (on 2008 August 7) in which the flux increased by a factor ≳ 2 within 3 days. Detailed analysis of the disk components is given below in Section 4.2.

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Previously, by fitting the first two sets of data (obtained on 2007 March 8 and 10; Table 1), Deloye et al. (2008) found M_ns > 1.8 M_☉, which favored a high-mass neutron star in J1808.4. Our results are drastically different, now pointing to a light, M_ns < 1.3 M_☉ neutron star. In addition to the constraint on the companion's radial velocity from the K_Bowen measurements (see also the discussion in Section 5.1), the causes of the difference can be understood from the following. Nearly simultaneous multi-band light curves are crucial in order to break parameter degeneracies and, for instance, tell the stellar temperature apart from the orbital inclination (Breton et al. 2012), but Deloye et al. (2008) had very limited multi-band information of the binary (see Section 2.1). The other important aspect is that the disk contribution acts as a DC component on an observed light curve and is almost completely degenerate with the orbital inclination. However, we were able to disentangle the orbital inclination from the disk in our data since the contribution from the latter varies over time whereas the orbital inclination remains constant. Finally our MCMC fitting allowed to efficiently explore the entire parameter space whereas Deloye et al. (2008) restricted their fitting to well-defined regions with a coarse grid.

3.5.1. Outburst Results

It has been shown that the orbital periodicity of J1808.4 can be determined from time-resolved photometry with a phase-coherent timing technique (Wang et al. 2009). Using three observations made in 5 days, Wang et al. (2009) have found an uncertainty of 2.8 s on the orbital period P. With this uncertainty, it can be inferred that such timing observations within a maximum time span of 3.6 month are needed in order to avoid losing track of the optical periodicity phase. Because our Gemini observations in 2008 August satisfy this requirement, which further improves the accuracy of the period measurement, we attempted to phase connect all of our light curves obtained during the quiescent state. The orbital period of J1808.4 has been found to increase at a rate of 1.2 × 10⁻⁴ s yr⁻¹ from pulsar timing (Hartman et al. 2008, 2009; Di Salvo et al. 2008; Burderi et al. 2009; Patruno et al. 2012), and this large value, which is not well understood because it is an order of magnitude larger than that predicted by standard theoretical calculations for such a binary (however see Patruno et al. 2012), might be confirmed from optical observations.

Since the modulation was variable over different bands and times of the observations (Figure 4), we fit each set/band of the light curves individually with a sinusoidal function m = m_c + m_hsin [2π(t/P + ϕ₀)] (where t is time, ϕ₀ the starting phase, and P is the orbital period fixed at 7249.157 s), subtracted the obtained constant magnitude m_c from them, and normalized them with the obtained semi-amplitudes m_h. The uncertainties resulting from this normalization were added in quadrature to the original uncertainties of the magnitudes. The normalized light curves were then fit with a single sinusoid again, and we found that the best fit has an orbital period P = 7249.151 ± 0.003 s, while χ² = 3170 for 331 degrees of freedom. The large χ² value reflects systematic uncertainties resulting from photometry, intrinsic scattering of the data points from a single sinusoid (Wang et al. 2009), and probably errors from normalizing short light curves. In Figure 7, the folded light curve, best-fit sinusoid, and residuals to the sinusoid are shown. A few data points deviating more than 3σ from the sinusoid are marked by squares. We estimated the uncertainties by scaling them by (χ²/DoF)^1/2 and obtained P = 7249.150 ± 0.008 s and ϕ₀ = 0.675 ± 0.003 at MJD 54599.0 (TDB), where phase ϕ = 0.0 corresponds to the ascending node of the pulsar orbit. These results are consistent with that obtained from pulsar timing (Hartman et al. 2009) within the uncertainties. Since the measurement accuracy of a period is generally proportional to the length of the time span, it is unlikely to be able to measure the orbit change rate via long-term optical photometry of the source.

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The accretion disk components resulting from our MCMC fitting are shown as a function of wavelengths in Figure 8. The data points before the onset of the brightening on 2008 August 7 can be reproduced by a simple accretion disk model, whose structure in quiescence is known to be different from that of the standard, steady-state disk model (e.g., Dubus et al. 2001; Baptista & Bortoletto 2004). In both states the temperature profile can be described by an exponential function, T(r)∝r^ξ, where r is disk radius, but the former is usually flatter than the latter (ξ = −0.75 is the standard value for the latter case). We fit the quiescent data points with the temperature profile. Free parameters were disk temperature at r = 10¹⁰ cm, T₁₀, and ξ. Source distance, reddening, and inclination angle were fixed at the median values found from our MCMC fitting, and the inner and outer edges of the disk were assumed to be at the light cylinder of the pulsar (≃ 1.2 × 10⁷ cm) and 90% of the Roche-lobe radius of the pulsar (≃ 3.2 × 10¹⁰ cm; Frank et al. 2002), respectively. We found that $T_{10}=6200^{+100}_{-200}$ K and ξ = −0.5 ± 0.1 provide the best fit (the minimum χ² = 0.3 for seven degrees of freedom). The small χ² value reflects the large standard deviations of the data points. The spectrum of the best-fit disk model is shown as the solid curve in Figure 8.

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The data points on August 7 are 2–4 times brighter than those obtained 3 days earlier, indicating the onset of the disk brightening. In addition, there is possibly a spectral change from rising to falling although we have only two data points and their standard deviations are too large for such a conclusion to be drawn. For example, if ξ = −0.5 is fixed (the same as that obtained above in quiescence), the disk model with T₁₀ = 8200 ± 700 K provides a best fit (the minimum χ² = 1.4 for 1 degree of freedom), which is plotted as the dotted curve in Figure 8. For a comparison, a steeper profile such as ξ = −0.8 (T₁₀ = 8300 K) is also plotted (the dashed curve) in the figure. We note that the brightness at the time (average i' ≃ 19.9) was approximately seven times lower than that at the outburst peak (average i' ≃ 17.8; Elebert et al. 2009). It is of great interest to study how the disk evolved during the one and a half month period from August 7 to September 22 directly preceding the reported X-ray outburst detection. Since J1808.4 is known to have an outburst every 3 years (Galloway 2008), a close monitoring of the source before an outburst might help us learn the disk evolution over such a period.

Since the orbital period P and the projected semi-major axis of the J1808.4 pulsar have been accurately measured from X-ray timing (Chakrabarty & Morgan 1998; Hartman et al. 2009), the semi-amplitude of the pulsar's projected velocity K₁ is known. Combining with it, the K_comp measurement from optical spectroscopy provides a strong constraint on properties of the binary, particularly the neutron star's mass M_ns. It can be shown that

$$M_{\rm ns}=\frac{1}{2\pi G}\frac{K_{\rm comp}^3P}{(1+K_1/K_{\rm comp})^2} \frac{1}{\sin ^3i},$$

where G is the gravitational constant. If K_comp is known, M_ns is determined only by the orbital inclination i (similar discussions were also given by Cornelisse et al. 2009 and Elebert et al. 2009). This M_ns–i relation is shown in Figure 9, with 68% and 99.7% confidence-level regions resulting from our fitting (low left panel in Figure 3) overplotted. As can be seen, our fitting results are consistent with the analytical expectations.

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If K_comp = 299  ± 23 km s⁻¹ instead, as reported by Cornelisse et al. (2009), the M_ns–i relation is expected to shift toward lower values. We explored this effect by fitting our model using K_Bowen = 248  ±  20 km s⁻¹ (Cornelisse et al. 2009) as a prior. As expected, we found similar results but noticeably i = 45 ± 5° and $M_{\rm ns}=0.58^{+0.20}_{-0.14}$ M_☉ (both uncertainties are 1σ), which are both smaller than the values obtained using the Elebert et al. (2009) measurement as the prior. The neutron star now has an unrealistically low mass. In Figure 9, we show the obtained M_ns–i regions from our fitting in detail. As a result of the change of K_Bowen to the smaller value, the regions are shifted accordingly, but only the upper corner of the 3σ region allows ⩾1 M_☉ neutron star mass. This suggests that their measurement is in general too low to provide a realistic mass range for the neutron star.

On the basis of the current K_comp measurements, K_comp ⩽ 410 km s⁻¹, since the K-correction factor is derived assuming that K_Bowen indicated the orbital motion of the L1 point. Therefore from our fitting, M_ns  ⩽  1.3 M_☉ at 1σ level, which confirms the point raised analytically by Cornelisse et al. (2009) and Elebert et al. (2009) that a light neutron star in J1808.4 is favored.

While our short coverage of the outburst modulation prevents us from conclusive analysis, we have likely detected modulation changes during the outburst. The second light curve appears to systematically deviate from the simple sinusoidal-like shape, as seen in Figure 2. This deviation could have been caused by disk flux variations during the outburst, with a timescale of 20 minutes. On the other hand, a modulation with a narrower flux peak can also explain this. For example using our model, we qualitatively tested this and found that narrower light curves can indeed be produced with smaller irradiated areas, although much higher temperatures are required to account for similar maximum luminosity. It is plausible that during the outburst, a hot spot could have formed on the surface of the companion due to strong irradiation by X-rays from the central pulsar.

The third and fourth light curves, 3 days after the second one, changed to have a modulation with 0.1 mag amplitude. Elebert et al. (2009) have reported their detection of a slow-rise and a fast-decline modulation on 2008 October 1, which would strongly suggest the appearance of superhump modulation since or at least on October 1 (Retter et al. 1997; Wang & Chakrabarty 2010). As observed in cataclysmic variables (CVs), superhumps do often appear a few days after the start of an outburst (or superoutburst as in CVs). Before an accretion disk dumps nearly all its stored mass to the companion, the disk in the beginning of the outburst expands beyond the 3:1 resonance radius, which induces the tidal instability. As a result, the disk develops into an eccentric form, producing superhump modulation (e.g., see Osaki 1996 for details). However in our data, not only are the light curves significantly different from that obtained on October 1 by Elebert et al. (2009; for example, its modulation amplitude was 0.2 mag), but also no clear phase shift of the modulation is seen. In the last two panels of Figure 2, we marked superior conjunction of the companion (i.e., when the pulsar is at 270° from the ascending node) with two vertical dash-dotted lines, and as can be seen, the maximums of the modulation were in phase with superior conjunction of the companion. For superhump modulation in J1808.4 and given its mass ratio q = 0.04/1.0, the expected phase shift would be 0.1 day⁻¹, estimated from the empirical relation between superhump and orbital periods and q among known superhump binaries (Patterson et al. 2005; Wang et al. 2009).

Our last two outburst light curves are similar to that obtained on October 7/8 by Elebert et al. (2009). We note that there is marginal evidence that an additional component appears in the third light curve and that a weak trend exists in the residuals of the fourth light curve from our model fit. Although the 0.1 mag amplitude modulation was in phase with that of the companion, we speculate that it could consist of modulation components such as from the disk but that modulation from the companion still dominated at the time. In order to understand this modulation, better observational coverage of the source's outbursts and detailed analysis of each component's possible modulation should be needed. In any case, given that the modulation showed significant variations over a few days, further observational studies of the source's modulation over its future outbursts are warranted.

The day-side optical emission from the companion is due to reprocessing of energy flux from the central pulsar. The required irradiation luminosities in quiescence in our model were in a range of (8–12) × 10³³ erg s⁻¹. The known energy output from the pulsar in quiescence consists of its rotational energy loss rate L_sd and X-ray emission, where the latter is two orders of magnitude lower than the former. Therefore the energy output absorbed by the companion is approximately (1 − η_*)f_bL_sd ≃ 9 × 10³³(1 − η_*)f_b erg s⁻¹, where η_* is the albedo of the companion in quiescence and f_b is defined as the beaming factor for the pulsar's emission toward the companion. Comparing this value to the irradiation luminosity range, η_* ≃ 0 is required if f_b ≃ 1 (isotropic emission case). However, from similar studies of black widow pulsar binaries (Stappers et al. 2001; Reynolds et al. 2007), η_* was found to be around 0.6. If this value is taken for our case, f_b ≈ 2.5 beaming for the pulsar's energy emission would be required.

Recently Takata et al. (2012) have proposed a model that irradiation could be due to γ-ray emission from pulsars for APMP systems in quiescence. According to their calculations for J1808.4, a high-mass (∼2 M_☉) neutron star is preferred considering that the irradiation luminosity is L_γ ∼ 10³⁴ erg s⁻¹. Their model is therefore not fully supported by our derived limits on the neutron star's mass, but is compatible within 3σ.

The required irradiation luminosity during the outburst on 2008 September 29 was found to be 6.2 × 10³⁵ erg s⁻¹, while the data on the other three nights are not considered because of the deviation of their modulations from a simple sinusoid-like shape. Nearly at the same time (MJD 54738.28), the 2.5–25 keV X-ray luminosity of the pulsar was 3.2 × 10³⁶ erg s⁻¹ (Hartman et al. 2009), where 4.1 kpc source distance (Table 1) is used. Comparing the two values, the X-ray albedo of the companion η_X ≃ 0.8 for the f_b ≃ 1 isotropic emission case. Such a large albedo value has been found in studies of accretion disks' X-ray reprocessing (de Jong et al. 1996). On the other hand, if η_X ≃ 0.6 still holds, f_b ≃ 1/2 would be required. The difference in the albedo or beaming factor during quiescence and the outburst probably provides additional evidence for having a different heating source in the quiescent state.