4U1700-37

3 Mass determination for the system components

Since no X-ray pulsations have been convincingly measured for 4U1700-37 - and hence no determination of a_xsini is possible - the orbital solution cannot be uniquely determined. However, following Heap & Corcoran (1992) and Rubin et al. (1996) we may estimate the mass of the companion using a Monte Carlo method. The mass of the companion can be calculated using a series of equations relating eclipse and orbital parameters. The Roche lobe filling factor $\Omega$ is defined by

$$R_{\ast} = \Omega R_{L}$$ (1)

where $R_{\ast}$ is the radius of the O-star and R_L is the Roche lobe radius which is related to the semi-major axis of the system a and the mass ratio q (= $M_{\rm ast}$/$M_{\rm x}$) by

$$\frac{R_L}{a} = A + B \log{q} + C (\log{q})^2$$ (2)

and the coefficients A, B and C are

$$A = 0.398 - 0.026 \Gamma^2 + 0.004 \Gamma^3$$

$$B=-0.264 + 0.052 \Gamma^2 - 0.015 \Gamma^3$$

$$C=-0.023 - 0.005 \Gamma^2$$

where $\Gamma$ is the ratio of the rotational angular frequency of the companion to its orbital angular frequency (Rappaport & Joss 1983). The radius of the O-star is related to the semimajor axis by the inclination i and eclipse semiangle $\theta_{\rm E}$ by

$$\frac{R_{\ast}}{a} = \sqrt{ {\rm cos}^2 i + {\rm sin}^2 i {\rm cos}^2 \theta_{\rm E}}$$ (3)

while the companion mass function f is given by

$${M}_x^3 {\rm sin}^3 i = f ({M}_{\ast} + {M}_x)^2$$ (4)

and f is related to orbital parameters by

$$f = 1.038 \times 10^{-7} K_{\ast} P (1-{\rm e}^2)^{3/2}$$ (5)

where $K_{\ast}$ is the radial velocity semi-amplitude in km s^-1, P the period in days and e the ellipticity. Finally by Kepler's third law

 

a³ = 75.19 (1+q) M_x P². (6)

We combine these equations in a similar way to Rubin et al. (1996) to obtain

$$R_{\ast}^2 = \frac{R_L^2}{R_{La}^2} - 17.81 P^{4/3} f^{2/3} (1+q)^2 {\rm cos}^2 \theta_{\rm E}$$ (7)

where R_La = R_L/a. This equation can then be solved numerically for q. Values of various system parameters (listed in Table 2) are selected randomly either from a Gaussian distribution (if observed) or uniformly if constrained between certain values. Equation (7) is solved for q which gives a from Eq. (2) which then gives M_x from Eq. (6). Consistency can be checked by requiring sin i < 1 and i > 55 degrees (e.g. Rubin et al. 1996). This procedure is followed 10⁶ times to gain a distribution of M_x and $M_{\ast }$ for a considerable number of possible parameter combinations (noting that as expected there is a very strong positive correlation between the two masses). We find that $M_x =2.44\pm 0.27~M_{\odot}$. As shown by the histogram of M_x in Fig. 4 the distribution is very asymmetric beyond the $1 \sigma$ limits (determined by the 16th and 84th percentile of the cumulative distribution function), with only 3.5 per cent of the sample having a mass of less than 2 $M_{\odot }$, and none less than $1.65~M_{\odot}$ (which is significantly higher than the upper limit to the range found for binary pulsars by Thorsett & Chakrabarty 1999). We note that none of the 10⁶ trials were rejected from inclination constraints suggesting that the range of stellar radii adopted for the modeling are unlikely to be significantly in error (which, for the fixed eclipse length, would lead to unphysical solutions for the orbital inclination).

The errors on M_x are significantly smaller than previous work (e.g. Rubin et al. 1996) due to the far more stringent limits on $R_{\ast}$, which constrain the orbital and eclipse parameters far more strongly. This is not surprising as the eclipse parameters are used to work out the orbital parameters and the eclipse constraints rely strongly on the O-star radius.

 

Table 1: Stellar parameters for HD 153919 derived from the NLTE modeling described in Sect. 2.2.

Parameter	Value
E(B-V)	$0.54\pm0.02$
$T_{\rm eff}$	$35~000\pm1000$ K
log($L_{\ast}$/$L_{\odot}$)	$5.82\pm0.07$
$R_{\ast}$	21.9 +1.3-0.5 $R_{\odot }$
$\dot{M}$	$9.5\times10^{-6}~M_{\odot}$ yr-1
$v_{\infty}$	1750 km s-1
$\log g$	3.45-3.55

Figure 5 shows the O-star mass distributions around $M_{\ast} = 58 \pm 11~M_{\odot}$. Again the distribution is anti-symmetric with 32 per cent of trials between 50-60 $M_{\odot }$, 26 per cent between 40-50 $M_{\odot }$ and only 2 per cent less than 40 $M_{\odot }$. Therefore, the mass implied for HD 153919 appears to be consistent with both that expected from its spectral classification and relevant evolutionary tracks (see Fig. 6), and that suggested by its high terminal wind velocity (Sect. 2). Additionally the $\log g$ determined from the He  I and Balmer line wings (Sect. 2.2) indicates a minimum mass of 50 $M_{\odot }$ (and maximum of $\sim$60 $M_{\odot }$), again fully consistent with the results of the Monte Carlo simulation. Therefore, given the consistency between mass estimates based on spectral type, evolutionary tracks (when compared to the stellar temperature and luminosity derived from modeling), surface gravity and the Monte Carlo simulations, we have confidence that the mass of HD 153919 lies in the range 50-60 $M_{\odot }$. This resolves the problem that the star is undermassive by a factor of $\sim$2.

However, the mass of the compact companion is more problematic given that it is significantly in excess of the observed mass range for NS, but apparently considerably lower than those found for BH candidates (e.g. Fig. 7). If a minimum mass of 50 $M_{\odot }$ is adopted for HD 153919 the minimum value of M_x that may be obtained is 1.83 $M_{\odot }$, while for values of M_o between 50-60 $M_{\odot }$ only 0.17 per cent of trials result in $M_{x} <2~M_{\odot}$. This will be returned to in Sect. 5.

Recent reanalysis of spectroscopic data by Hammerschlag-Hensberge et al. (in prep.) suggests that the eccentricity of the orbit is somewhat uncertain, and that the orbital velocity curve is equally well fit by an orbit of eccentricity $e \sim 0.22\pm0.04$ as it is by a circular orbit. In order to address this uncertainty we modified the above equations for the more general case of an elliptical orbit and repeated the simulations with $e=0.22\pm0.04$. This resulted in significantly higher masses for both components, with $M_{\ast}=70\pm7~M_{\odot}$ and $M_x=2.53\pm0.2~M_{\odot}$. Therefore, the mass of the O star in the case of an elliptical orbit is significantly higher than expected for an O6.5 Iaf⁺ star (only 0.002 per cent of the trials result in a mass $\leq$50 $M_{\odot }$, and 5 per cent give a mass between 50-60 $M_{\odot }$). Such high values for $M_{\ast }$are inconsistent with the measured $\log g$ and we note that 95 per cent of trials are rejected due to the inclination constraints, suggesting that a low eccentricity solution is more likely.

If such extreme values for $M_{\ast }$ are adopted, the mass of the compact object is still less than that observed for the lowest mass black hole candidate known ($\sim$4.4 $M_{\odot }$; Sect. 5) and remains significantly greater than any known neutron star. Indeed, the lowest mass estimates for both components were derived in the case of a circular orbit; therefore the value of $M_x =2.44\pm 0.27~M_{\odot}$ represents a lower limit for the mass of the compact object, and we suggest that these results favour a low eccentricity solution for the orbit (we note that the orbital eccentricity of Vela X-1 is overestimated from optical observations when compared to the value derived from timing analysis, cf. Barziv et al. 2001).

Figure 4: Histogram of the results of the Monte Carlo simulations for the mass of the compact object in 4U1700-37. The results indicate a mass in the range of $2.44\pm 0.27$ $R_{\odot }$ with only 3.5 per cent of simulations indicating masses of less than 2 $M_{\odot }$, and none <1.65 $M_{\odot }$.

4U1700-37