CONSTRAINTS ON THE COMPACT OBJECT MASS IN THE ECLIPSING HIGH-MASS X-RAY BINARY XMMU J013236.7+303228 IN M 33

Based on photometry of XMMU J013236.7+303228, Pietsch et al. (2009) estimate that the companion is a 10.9 M_☉ object with T_eff = 33, 000 K and log (g) = 4.5, with χ² = 2.4 for their best-fit model. They then assume a distance of 795 kpc to M 33, and calculate that the star has an absolute magnitude M_V ∼ −4.1 and the line-of-sight extinction is A_V = 0.6, so derive a stellar radius of 8.0 R_☉.

We deduce the spectral type by comparing our spectra to Walborn & Fitzpatrick (1990) and the Gray spectral atlas.⁵ The absence of He ii lines (Figure 2) implies a spectral type later than O, while the relative strengths of the Mg ii 4482 Å/He i 4471 Å lines point to a spectral type earlier than B3. The strength of He i lines and a weak feature at 4420 Å indicate a spectral type around B1 for main-sequence stars. A bump blueward of H8 3889 Å is characteristic of spectral type B2. The weakness of C iii 4650 Å and the relative strengths of C iii/O ii near 4650 Å refine the spectral type to between B1 and B2 for both dwarfs and giants. Finally, the weakness of Si iv Hδ 4101 Å gives a spectral type of B1.5. To determine the luminosity class, we note that the O ii 4415–4417 Å and Si iii 4552 Å lines are present, but are weak as compared to He i 4387. We conclude that the donor is a B1.5IV star, with rough uncertainties of 0.5 spectral subclasses and one luminosity class.

Tabulated values of stellar parameters are usually provided for luminosity class III and V stars. For B1.5V stars, T ≈ 23, 000 K, log (g) = 4.14, and M_V = −2.8. For B1.5III stars, T ≈ 22, 000 K, log (g) = 3.63, and M_V = −3.4 (Cox 2000). The B1.5IV target will have values intermediate to these. This inferred temperature is significantly cooler than T ∼ 33,000 K reported by Pietsch et al. (2009). However, the absence of He ii lines at, e.g., 4541 and 4686 Å clearly excludes such a high temperature.

The colors of a star depend on its temperature and surface gravity. These expected colors can be compared to the observed colors to calculate the reddening and extinction. The expected color is (B − V)₀ = −0.224 for a 22,000 K subgiant star, and (B − V)₀ = −0.231 for a 23,000 K main-sequence star (Bessell et al. 1998). From Pietsch et al. (2009), the mean magnitudes are $m_{g^\prime } = 21.03 \pm 0.02$ and $m_{r^\prime } = 21.36 \pm 0.02$. Using the Jester et al. (2005) photometric transformations for blue, U − B < 0 stars,⁶ m_V = 21.21 ± 0.03 and (B − V)_obs = −0.09 ± 0.04. The color excess is E(B − V) = (B − V)_obs − (B − V)₀ = 0.14 ± 0.04. Using the standard ratio of total-to-selective extinction, R_V = 3.1, we get A_V = 0.43 ± 0.12. For comparison, the foreground extinction to M 33 is A_V = 0.22.

We measure various stellar parameters by fitting our combined, flux-calibrated spectrum with model atmospheres (taking into account the instrumental broadening; see Section 3.2). For a Roche-lobe-filling companion (discussed in Section 4), we expect a radius of 6–10 R_☉, surface gravity log (g) ≃ 3.7 (consistent with our luminosity class), and projected rotational velocity v_rotsin i ≃ 250 km s⁻¹. We use these values as starting points to select model atmospheres from a grid calculated by Munari et al. (2005). These templates are calculated in steps of 0.5 dex in log (g), so we use models with log (g) = 3.5, 4.0. For the initial fit, we assume v_rotsin i = 250 km s⁻¹ and solar metallicity. The only free parameters are a normalization and an extinction. We use extinction coefficients from Cox (2000), assuming R_V = 3.1. We find that the best-fit model for log (g) = 3.5 has T = 22, 100 ± 40 K and A_V = 0.401(3), with χ²/dof = 1.13 for 3600 degrees of freedom, while for log (g) = 4.0, we get T = 23, 500 ± 50 K and A_V = 0.425(3) with χ²/dof = 1.22. The temperatures are consistent with those expected for a B1.5 subgiant star, and the extinction is within the range derived from photometric measurements. Pietsch et al. (2009) obtained a higher extinction for the target, which explains why they estimated the source temperature to be higher. Munari et al.'s (2005) templates are calculated in temperature steps of 1000 K in this range. For further analysis, we use the best-fit template: log (g) = 3.5 and T = 22, 000 K. Since log (g) is slightly higher than this value, for completeness we also give results using the best-fit template for log (g) = 4.0, which has T = 23, 000 K. For both these templates, the best-fit extinction is A_V = 0.395(3). We then keep T and log (g) constant and vary v_rotsin i. For both the log (g)and T combinations, we measure v_rotsin i = 260 ± 5 km s⁻¹. Finally, using the same templates but with varying metallicity, we get the best fits for [M/H] = 0. The 0.5 dex steps in [M/H] are too large to formally fit for uncertainties.

Next, we calculate the luminosity of the object to obtain a radius–temperature relation. The distance modulus to M 33 is (m − M)_M33 = 24.54 ± 0.06 (d = 809 kpc; McConnachie et al. 2005; Freedman et al. 2001), which gives, accounting for the reddening of A_V = 0.4, M_V = −3.74 ± 0.07. The bolometric luminosity of a star is related to its temperature and radius by L_bol∝R²T⁴. To obtain the visual luminosity, one must apply a temperature-dependent bolometric correction, BC = M_bol − M_V. Torres (2010) gives formulas for bolometric correction as a power series in log (T). We calculate BC = −2.11 (− 2.21) for T = 22, 000 (23, 000) K (which are consistent with Bessell et al.'s (1998) tables for main-sequence stars). After some basic algebra, we obtain

$$\begin{eqnarray} \hspace{-6pc}5 \log&& \left(\frac{R}{R_\odot }\right) + 10 \log \left(\frac{T}{T_\odot }\right) + {\rm BC}(T)\nonumber\\ \hspace{-6pc}&&= M_{\rm bol,\odot } - m_V + (m-M)_{\rm M33} + A_V \nonumber\\ \hspace{-6pc}&&= 8.48 \pm 0.07. \end{eqnarray} \tag{ 1 }$$

Here, M_{bol, ☉} = 4.75 (Bessell et al. 1998). The resultant radius–temperature relationship is shown in Figure 4. For T = 22, 000 (23, 000) K, we infer R = 9.1 (8.7) ± 0.3 R_☉. The absolute magnitude and radius are both consistent with a B1.5 subgiant.

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We measure the radial velocities of the B star using model stellar spectra by Munari et al. (2005) as follows. For each observed spectrum, we measure the seeing using the width of the spectral trace. We then generate an instrument response function by taking a Gaussian matched to the seeing, truncating it at the slit size, and convolving it with the pixel size. Flux-calibrated synthetic spectra (templates) are convolved with this instrument response and then redshifted to a test velocity. Then we redden the template using the measured value of extinction, A_V = 0.395, with coefficients from Cox (2000). We use IDL⁷ mpfit (Markwardt 2009) to calculate the reddening and normalization to match this spectrum with the observed spectrum and measure the χ². By minimizing the χ² over test velocities, we find the best-fit velocity and the error bars. This velocity is converted to a barycentric radial velocity using the baryvel routine in Astrolib (Landsman 1993). Table 1 lists the radial velocities for all spectra, measured using the best-fit stellar template: T = 22,000 K, log g = 3.5, [M/H] = 0.0, and v_rotsin i = 250 km s⁻¹.

Red-side spectra are not useful for radial-velocity measurement for several reasons. The flux of the B star at redder wavelengths is lower than at blue wavelengths, and there are fewer spectral lines in this range. Also, the background noise is higher, from the large number of cosmic rays detected by the LRIS red side and from intrinsic sky emission. Hence, all further discussion omits red-side spectra.

We calculate an orbital solution for the B star using these radial velocities. Owing to the short 1.73 day period of the system, we assume that the orbit must be circularized. The orbital solution is then given by

$$\begin{equation} v(t) = \gamma _{\rm opt} + K_{\rm opt} \sin \left(2 \pi \frac{t - T_0}{P} \right), \end{equation} \tag{ 2 }$$

where γ_opt is the systemic velocity, K_opt is the projected semi-amplitude of radial velocity, and T₀ is the epoch of mid-eclipse. We adopt P = 1.732479 ± 0.000027 and T₀ = 2453997.476 ± 0.006 from Pietsch et al. (2009). We obtain γ_opt = −80 ± 5 km s⁻¹ and K_opt = 64 ± 12 km s⁻¹. The best fit has χ²/dof = 5.1 for 16 degrees of freedom, which is rather poor. We find that an additional error term Δv = 15 km s⁻¹ needs to be added in quadrature to our error estimates to obtain χ²_red = 1. We attribute this to the movement of the star on the slit. If the target has a systematic offset of 01 from the slit center over the entire 30 minute exposure, it shifts the line centroids by about 0.45 pixels or 18 km s⁻¹. For comparison, van Kerkwijk et al. (2011) find a similar scatter (13 km s⁻¹) in their observations of a reference star when using LRIS with a similar configuration (600/4000 grating, 07 slit). Future observations should orient the spectrograph slit to obtain a reference star spectrum to correct for such an offset.

When we fit Equation (2) to data including the 15 km s⁻¹ error in quadrature, we obtain γ_opt = −80 ± 5 km s⁻¹ and K_opt = 63 ± 12 km s⁻¹ (Figure 5, Table 2). For the epoch of observations, the uncertainty in the phase is 0.019 days. If we allow T₀ to vary, we get T_{0, fit} = 2453997.489 ± 0.019 (Heliocentric Julian Date), γ_opt = −80 ± 5 km s⁻¹, and K_opt = 64 ± 12 km s⁻¹. These values are consistent with those obtained using the Pietsch et al. (2009) ephemeris. Hence, for the rest of this paper, we simply assume their best-fit value for T₀.

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Table 2. System Parameters for XMMU J013236.7+303228

Property	Value
From Pietsch et al. (2009)
Period (P)	1.732479 ± 0.000027
HJD of mid-eclipse (T0)	2453997.476 ± 0.006
Eclipse half-angle (θe)	306 ± 12
This work
Systemic velocitya (γopt)	−80 ± 5 km s−1
Velocity semi-amplitudea (kopt)	63 ± 12 km s−1
HJD of mid-eclipseb (T0)	2453997.489 ± 0.019
Spectroscopically inferred
OB star spectroscopic mass (Mopt)	11 ± 1 M☉
NS mass (MX)	2.0 ± 0.4 M☉
Distance-based calculations
OB star mass (Mopt)	18.1+2.0− 1.9 M☉
NS mass (MX)	2.7+0.7− 0.6 M☉

Notes. ^aCalculated using P = 1.732479 ± 0.000027 and T₀ = 2453997.476 ± 0.006 from Pietsch et al. (2009). ^bWhen we allow T₀ to vary, we obtain γ_opt = −80 ± 5 km s⁻¹ and K_opt = 64 ± 12 km s⁻¹. The analysis in this paper uses the Pietsch et al. (2009) value of T₀.

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To investigate the sensitivity of the result to the choice of the stellar template, we repeat the measurement with a variety of templates. We vary the temperature from O9 (33,000 K) to B3 (18,000 K) spectral classes. As before, we use templates with log (g) = 3.5, 4.0, v_rotsin i = 250 km s⁻¹ and solar metallicity. Repeating the orbit calculations for each of these models, we find that the systemic velocity γ_opt may change between the models being fitted: the extreme values are −77 ± 6 km s⁻¹ and −90 ± 5 km s⁻¹, a 1.7σ difference. We suspect that the reason for this variation is the difference in shape of the continuum, as the extinction was held constant in these fits. For hotter templates with a steeper continuum, the red side of a line has lower flux than the blue side, so lowest χ² will be obtained at a slightly higher redshift, as seen. The magnitude of this effect should be independent of the intrinsic Doppler shift of the spectrum and should not affect the velocity semi-amplitude K_opt. This is indeed the case: K_opt is constant irrespective of templates. The extreme values are 62 ± 12 km s⁻¹ and 63 ± 12 km s⁻¹, differing by less than 0.1σ. Dynamical calculations depend only on K_opt, and hence are robust to the selection of template.

In the following, we will denote the masses of the compact object and donor star by M_X and M_opt, respectively. The mass of the compact object (M_X) is related to the radial velocity of the B star (K_opt) as follows:

$$\begin{equation} M_{\mathrm{X}} = \frac{K_{\mathrm{opt}}^3 P(1 - e^2)^{3/2}}{2\pi G \sin ^3 i} \left(1+\frac{1}{q}\right)^2, \end{equation} \tag{ 3 }$$

where q = M_X/M_opt is the ratio of masses, defined so that higher values of M_X relate to higher values of q. P is the orbital period of the binary, e is the eccentricity, and i is the inclination of the orbit. For eclipsing systems, the inclination is constrained by

$$\begin{equation} \sin i = \frac{\sqrt{1 - \beta ^2\left({R_{\rm L}}/{a} \right)^2}}{\cos \theta _{\rm e}}, \end{equation} \tag{ 4 }$$

where R_L is the volume radius of the Roche lobe, a is the semi-major axis, and β is the Roche lobe filling factor (Joss & Rappaport 1984). For XMMU J013236.7+303228, the eclipse half-angle is θ_e = 306 ± 12 (Pietsch et al. 2009). Owing to the short orbital period, we assume that the orbit is circular and the B star rotation is completely synchronized with its orbit. For corotating stars, Eggleton (1983) expresses R_L/a in terms of q:

$$\begin{equation} \frac{R_{\rm L}}{a} = \frac{0.49 q^{-2/3}}{0.6q^{-2/3} + \ln (1 + q^{-1/3})}. \end{equation} \tag{ 5 }$$

The constant, relatively high X-ray luminosity, sustained over the non-eclipsed parts of the orbit, strongly indicates that accretion is occurring via Roche lobe overflow. In Roche lobe overflow, matter flowing through the Lagrangian point may form a disk around the compact object before being accreted onto it. This disk can occult the compact object, causing periods of low X-ray luminosity. Both these characteristics are seen in the X-ray light curves of XMMU J013236.7+303228 (Pietsch et al. 2009). If mass is being accreted onto an object in a spherically symmetric manner, the accretion rate is limited by the Eddington rate, $\dot{M}_{\rm Edd}$, and the peak luminosity is L_Edd/L_☉ = 3 × 10⁴ M/M_☉. For a 1.5–2.5 M_☉ compact object, L_Edd = 1.8–3 × 10³⁸ erg s⁻¹. At its brightest, the source luminosity in the 0.2–4.5 keV band was 2.0 × 10³⁷ erg s⁻¹—about 0.1 L_Edd. This luminosity was sustained throughout Chandra ObsID 6387, which covered about 0.72 days of the non-eclipsed orbit (Pietsch et al. 2009). Comparable flux was observed in the non-eclipsed parts of the orbit (0.73 days) in Chandra ObsID 6385. Such high luminosity sustained over significant parts of the orbit is not observed in wind-fed systems, which have typical luminosities an order of magnitude smaller. Further, the short 1.73 day orbital period is not consistent with Be X-ray binary or wind-fed systems. We conclude therefore that the B star fills its Roche lobe.

The Roche lobe radius as a function of B star mass is plotted in Figure 6 for various neutron star masses. For a Roche-filling companion, we infer 3.6 ⩽ log (g) ⩽ 3.8. For an NS mass in the range 1.4–2.4 M_☉ and B star mass 8–20 M_☉, the B star radius lies in the range 6–10 R_☉. Further, the assumed synchronous rotation requires the surface rotational velocity to be in the range 200 km s⁻¹ ≲ v_rot ≲ 285 km s⁻¹. These values are consistent with those derived in Section 3.1.

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Figure 7 plots M_X as a function of M_opt, calculated by solving Equations (3)–(5) under known constraints. If we know M_opt, we can calculate M_X. We use the physical properties of the primary (Section 3.1) to estimate the mass of the primary by comparing it with stellar evolutionary models, assuming that binary evolution has not drastically changed the mass–luminosity relation. First, we place the primary on an H-R diagram (Figure 8) using models by Brott et al. (2011). We conservatively allow for a 0.5 mag error in luminosity. We also plot evolutionary tracks on a log (g)–T figure to utilize the stricter constraints on log (g) from Roche lobe arguments. From these plots, we see that the primary is approximately a 15 Myr old, ∼11 M_☉ star. This is consistent with the typical mass of a B1.5IV star (Cox 2000).

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For M_opt = 11 M_☉, we calculate M_X = 2.0 ± 0.4 M_☉. From evolutionary tracks, we estimate that the uncertainty in M_opt is ∼1 M_☉ (Figure 8), corresponding to ΔM_X = 0.12, much smaller than the uncertainty arising from ΔK_opt. Adding this in quadrature with the uncertainty in the M_X–M_opt conversion, we conclude M_X = 2.0 ± 0.4 M_☉.

In Section 3.1, we calculated the radius of the primary from its apparent magnitude, temperature, and the distance to M 33 (Equation (1)). Since the primary is filling its Roche lobe, the stellar radius is equal to the Roche lobe radius (R_L). This additional constraint can be used in Equations (3)–(5) to solve for M_X and M_opt.

We calculate the probability density function (PDF) of component masses as follows. For every pair of assumed masses (M_X, M_opt), we use the period P to calculate the semi-major axis a. Then we calculate R_L/a from Equation (5) and substitute it in Equation (4) to calculate sin i. Using P, a, and sin i we calculate the expected semi-amplitude of the radial velocity:

$$\begin{equation} K_2 = \frac{2\pi a \sin i}{P} \frac{M_{\rm X}}{(M_{\rm X} + M_{\rm opt})}. \end{equation} \tag{ 6 }$$

Next, we calculate the probability for obtaining a certain value of R_L and K₂, given the measured radius R (Section 3.1) and K_opt (Section 3.2):

$$\begin{eqnarray} &&P(R_{\rm L}, K_2) = \exp \left(-\frac{(R_{\rm L} - R)^2}{2\cdot \Delta R^2}\right) \exp \left(-\frac{(K_2 - K_{\rm opt})^2}{2\cdot \Delta K_{\rm opt}^2}\right).\nonumber\\ \end{eqnarray} \tag{ 7 }$$

Here, we are making a simplifying assumption that the Roche volume radius (Equation (5)) is same as the effective radius from photometry (Equation (1)). We convert this PDF to a probability density as a function of M_X, M_opt by multiplying by the Jacobian ∂(R, K_opt)/∂(M_X, M_opt).

The results are shown in Figure 9. Red contours show the 68.3% and 95.4% confidence intervals for masses for the best-fit template (log (g) = 3.5 and T = 22, 000 K). The panel on the left shows the PDF for M_X marginalized over M_opt. Similarly, the lower panel shows the PDF for M_opt marginalized over M_X. In these panels, the solid, dashed, and dotted lines show the peak and 68.3% and 95.4% confidence intervals, respectively. We obtain M_X = 2.7^+0.7_{− 0.6} M_☉ and M_opt = 18.1^+2.0_{− 1.9} M_☉. For completeness, fits for the less likely scenario with log (g) = 4.0 and T = 23, 000 K are shown in blue. In this case, M_X = 2.5 ± 0.6 M_☉ and M_opt = 16.1^+1.8_{− 1.7} M_☉.

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To test the validity of this technique, we apply it to two well-studied targets: LMC X-4 and SMC X-1. We find that the basic application of our method is overestimating the mass (Table 3). Part of this discrepancy is likely related to equating the Roche volume radius to the effective photometric radius (Equation (7)). The Roche volume radius (Equation (5)) is the radius of a sphere with the same volume as the Roche lobe of the star. The effective photometric radius (Equation (1)) is the radius of a sphere with the same surface area as the star. Since an ellipsoid has a larger surface area than a sphere of the same volume, the actual Roche volume radius will be smaller than the effective photometric radius. Using a larger radius for the Roche lobe increases the masses of both the components in the binary.

Table 3. System Parameters for SMC X-1 and LMC X-4

Property	SMC X-1	LMC X-4
Period (P)	3.89 days	1.41 days
Pspin	0.708 s	13.5 s
aXsin i (lt-s)	53.4876 ± 0.0004	26.343 ± 0.016
Eclipse half-angle (θe)	26°−305	27° ± 2°
Mean systemic velocity (γopt)	−191 ± 6 km s−1	306 ± 10 km s−1
Velocity semi-amplitude (kopt)	20.2 ± 1.1 km s−1	35.1 ± 1.5 km s−1
Companion spectral type	B0I	O8III
Companion Teff	29000 K	35000 K
Distance (d)	60.6 ± 1 kpca	49.4 ± 1 kpcb
Visual magnitude (mV)c	13.3 ± 0.1	14.0 ± 0.1
Extinction (AV)d	0.12 ± 0.01	0.25 ± 0.04
NS mass (MX)
van der Meer et al. (2007)	1.06+0.11− 0.10 M☉	1.25+0.11− 0.10 M☉
Our calculation	1.32+0.16− 0.14 M☉	2.05+0.24− 0.23 M☉
OB star mass (Mopt)
van der Meer et al. (2007)	≈15.7 M☉	≈14.5 M☉
Our calculation	21.1+3.2− 2.9 M☉	29.1+4.8− 4.4 M☉

Notes. Data from van der Meer et al. (2007). ^aHilditch et al. (2005). ^bFreedman et al. (2001). ^cConservative 0.1 mag errors assumed. ^dForeground extinction to galaxy only. Error bars are approximate, the dominant uncertainty has been assigned to m_V.

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Putting it another way, if we use the Roche volume radius to calculate the brightness of the star, we will get a number lower than the observed brightness. A similar discrepancy is observed by Massey et al. (2012) in massive binaries in the LMC. They find that the absolute magnitude of LMC 172231 calculated using a spherical approximation is 0.45 mag fainter than observed, while for the triple system [ST92]2-28, the numbers are consistent within errors. Using system parameters derived by van der Meer et al. (2007), we find a similar offset of 0.45 mag for LMC X-4 and 0.2 mag for SMC X-1. If we incorporate this uncertainty by allowing offsets of 0–0.4 mag, we get M_X = 2.2^+0.8_{− 0.6} M_☉ and M_opt = 13 ± 4 M_☉.

Thus while this method has potential, more detailed modeling of the primary is clearly required to accurately infer component masses.