Calculation of the Masses of the Binary Star HD 93205 by Application of the Theory of the Apsidal Motion

We present a method to calculate masses for components of both eclipsing and non-eclipsing binary systems as long as their apsidal motion rates are available. The method is based on the fact that the equation that gives the rate of apsidal motion is a supplementary equation that allows the computation of the masses of the components, if the radii and the internal structure constants of them can be obtained from theoretical models. For this reason the use of this equation makes the method presented here model dependent. We apply this method to calculate the mass of the components of the non-eclipsing massive binary system HD 93205 (O3V+O8V), which is suspected to be a very young system. To this end, we computed a grid of evolutionary models covering the mass range of interest, and taking the mass of the primary (M_1) as the only independent variable, we solve the equation of apsidal motion for M_1 as a function of the age of the system. The mass of the primary we find ranges from M_1= 60+-19 msun for ZAMS models, which sets an upper limit for M_1, down to M_1= 40+-9 msun for an age of 2 Myr. Accordingly, the upper limit derived for the mass of the secondary (M_2= Q M_1) is M_2= 25 msun is in very good agreement with the masses derived for other O8V stars occurring in eclipsing binaries.


INTRODUCTION
The motion of the apse of a binary is mainly a direct consequence of the finite size of its components. If both stars were spherical objects and General Relativity corrections were negligible, they would move on a Keplerian, fixed orbit. However, the presence of the companion object, and also its rotation, makes the structure of each star depart from spheres. In such a situation, there appears a finite quadrupolar (and higher) momentum to the gravitational field of each object that forces the orbit to modify the position of the apse. This is an effect well known from long time ago (Cowling 1938;Sterne 1939). The rate of motion of the apse is dependent on the internal structure of each component; thus, if we are able to determine the main characteristics of a binary, it provides an observational test of the theory of stellar structure and evolution (Schwarzschild 1958;Kopal 1959).
In spite of the age of the idea, apsidal motion of binary systems has been systematically studied only recently in a series of papers by Claret & Giménez (e.g. Claret & Giménez 1993) and Claret (e.g. Claret 1995, 1997, 1999. Perhaps, one of the main reasons for such a situation is that the rate of motion of the apse is very dependent on the stellar structure. Thus, the apsidal motion test has been useful only recently because of the availability of accurate stellar evolutionary models such as those of Claret (1995). These authors have performed detailed stellar models computing the coefficients that determine the rate of motion of the apse and applied them to compare with observational data from eclipsing binaries.
The detection of apsidal motion in non-eclipsing binary systems is an elusive subject. It has to be determined through the time variation of the shape of the radial velocity curve due to change in the longitude of periastron.
Generally, the observed radial velocities in binary systems have large uncertainties that mask this effect in many cases. Moreover, the fact that times of light minima are usually determined with high precision for eclipsing binaries, acts as a selection effect in favour of detecting the motion of the line of apsides in such systems: a hundred cycles are usually enough to notice the change in relative position of the secondary minimum with respect to the primary minimum. Observations over much longer periods of time are needed to find evidences of apsidal motion in systems where eclipses are not seen and the only observable effect is the change in shape of the radial velocity orbit.
Empirical determinations of masses are scarce for early O-type stars, (e.g. Burkholder et al. 1997;Schönberner & Harmanec 1995). Few early O-type stars are known to be members of double-lined binaries, and from them, those showing eclipses or some kind of light variations that enable the estimate of the orbital inclination (and then, absolute masses), are rare. The so-called "Mass Discrepancy", first described by Herrero et al. (1992) and recently reviewed (Herrero, Puls, & Villamariz 2000), relates to the difference between the masses derived via numerical evolutionary models and those obtained from spectral analysis (plus model atmospheres) or binary star studies. This discrepancy, amounting to 50% in 1992, has been partially solved with the use of new evolutionary models that consider the effect of stellar rotation (Meynet & Maeder 2000) and new model atmospheres. But large differences between 'predicted' and 'observed' masses are still present for the hottest and youngest (non evolved) stars. A recent study of the massive double-lined O-type binary system HD 93205 (Morrell et al. 2001) yields minimum masses of 31.5±1.1 M⊙ and 13.3 ± 1.1 M⊙ for the O3 V primary and O8 V secondary components, respectively. This leads to a probable mass of ∼52-60M⊙ for the O3 V star, if a mass value according to those derived for other O8 V stars in eclipsing binaries is assumed for the secondary O8 V. This is much less than the 80 to 100 M⊙ predicted from the position of this star on a theoretical HRD compared to stellar evolutionary tracks (see Fig. 7 in Morrell et al. 2001).
Notably, HD 93205 is the first early O-type noneclipsing § massive binary for which the rate of motion of the apse has been determined with some accuracy. Morrell et al. (2001) derived an apsidal motion period of 185 ± 16 years, that considering the orbital period of 6.0803 ± 0.0004 days, yields an apsidal motion rate̟ = 0. • 0324 ± 0. • 0031 per orbital cycle. This is very interesting because, from a mathematical point of view, the rate of motion of the apse provides another equation to be applied to the system apart from the standard ones. HD 93205 is an early-type massive short period binary in a highly eccentric orbit (e = 0.370 ± 0.005; Morrell et al. 2001), lying in the Carina Nebula, a galactic massive star forming region. Thus, we can assume its components are on, or very close to, the Zero Age Main-Sequence (ZAMS). Consequently, if we consider the evolutionary stage of HD 93205 as known, the equation of apsidal motion can § Phase dependent light variations with full amplitude of ∼ 0.02 mag in visual light were reported by Antokhina et al. (2000). These authors stated that the observed light variations are probably related to tidal distortions rather than eclipses. be written in a way that we can solve it for the mass of the primary star. The aim of the present paper is to detail this method and to apply it to HD 93205.
The paper is organized as follows: in Section 2 we present the method we use to obtain the masses for components of non-eclipsing binary systems and we describe our evolutionary code and the calculations we have carried out. In Section 3 we present a test of our method by applying it to some eclipsing binary systems. Section 4 is devoted to showing the results obtained for the massive binary system HD 93205 and finally, in Section 5 we give some concluding remarks about the implications of our results.

COMPUTATIONAL DETAILS
Here we present an original method, as far as the authors are aware, to calculate the masses of the components of binary systems provided the knowledge of the rate of motion of the apse and the evolutionary status of the stars. In addition, we describe the main characteristics of our evolutionary code and the calculation of the internal structure constants (ISC) on which the rate of the apsidal motion depends.

Equations of apsidal motion and description
of the method Sterne (1939) has shown that if the classical gravitational potential of each component of a binary system is expanded in a series of spherical harmonics, and terms up to the quadrupolar contribution are kept, then the rate of motion of the apse is given by ¶ ̟2 where̟2 is the rate of the secular motion of the apse calculated considering only the quadrupolar contribution (the lowest order) of the gravitational potential; ki,j are the ISCs that depend on the internal mass distribution of the stars (see below, Subsection 2.2 for more details); i denotes the i-th multipolar momentum considered (i.e. i = 2 throughout this paper); whereas j denotes the component of the binary system. G is the gravitational constant, Ω is the mean orbital angular velocity, A is the semiaxis of the relative orbit, a1 and a2 are the mean radii of the stars, ω1 and ω2 are their angular velocities of rotation, M1 and M2 are the stellar masses; and finally f2(e) and g2(e) are functions of the orbital eccentricity e given by f2(e) = 1 + 3 2 e 2 + 1 8 e 4 1 − e 2 −5 , g2(e) = 1 − e 2 −2 .
In the following we shall reformulate Eq. (1) in order to write it to become an implicit equation for the mass of the ¶ It is assumed that rotation of both components is perpendicular to the orbital plane.
primary star M1. The semiaxis A is not directly known from observations, however the projected semiaxis D given by where sin i is the sine of the inclination of the orbit, can be observationally assessed. Let us define the mass ratio Q and the angular velocities ratio qω as In order to eliminate sin i we can use the mass function, defined as which can be determined from observations (Batten 1973).
In the same trend, projected tangential velocities v1 = V1 sin i and v2 = V2 sin i are also observable quantities and their ratio q = v2/v1 can be used to eliminate qω.
One important point is that rotation modifies the internal structure of the stars. In a recent paper, Claret (1999) has shown that, within the quasi-spherical approximation, rotation can be taken into account in the apsidal motion analysis simply by reducing the ISC k2,i by Here, [k2,i] sph denotes the ISC obtained from spherical models and the parameter λi is defined by where i denotes the component of the binary system and Vi, ai and gi are, respectively, the tangential velocity, the radius and the surface gravity of the component. Up to this point, we have only considered the contributions to the motion of the apse due to Newtonian gravity. However, it is well known that General Relativity predicts a secular motion of the apse which is independent of the classical contributions. The angular velocity of the apse due to General Relativistic effects̟GR is given by (Levi-Civita 1937) ̟GR Defining F2 as and incorporating both, rotation and relativistic effects we get after some algebraic manipulation where k2,i are the ISCs corrected by the effects of rotation. This is the fundamental equation for our purposes. As mentioned above, in Eq. (11) some quantities are determined observationally (Ω,̟2, e, D, Q, v1, f, q). On the other hand, k2,1 and k2,2 must be computed from evolutionary models and, if we are dealing with non-eclipsing systems, as is the case for HD 93205, then the radii a1 and a2 must be obtained from theoretical models as well. Now, if we assume that both components of the binary system have the same age, and we use the observational constraint Q = M2/M1, then k2,1, k2,2, a1 and a2 can be derived from evolutionary calculations as a function of M1 and the age of the system.
The method presented here to calculate M1 is as follows: we compute a grid of evolutionary models covering the range of masses of interest with a small mass step. Using this grid, we construct isochrones starting at the ZAMS with a given time step and for each isochrone we seek the solution of Eq. (11). In this procedure, the only independent quantity is M1 so when the solution of Eq. (11) is found, the corresponding value of M1 is the mass of the primary star that corresponds to the age of the isochrone.
Thus, for a given age of the system we have one solution for the mass of the primary star M1, and using Q = M2/M1 we can derive the mass of the secondary. However, the age of the system must be constrained by other means. In addition, the masses of the components that can be found with the present method are model dependent. Notice especially the sensitivity of the tidal and rotational terms in Eq. (11) to the value of the stellar radius. It is clear that we need accurate stellar models in order to get a physically reliable value of M1.
Finally, it is worth mentioning that the solution of Eq. (11) is subject to some constraints. Two of them were already mentioned: the age of both components in a binary system must be the same, and M2 = Q M1. In addition, the value of the mass function imposes a minimum value for M1 and also, the condition for the system to be detached must be fulfilled.

Evolutionary models and calculation of the ISCs
As stated above, in order to solve Eq. (11), the ISCs and radii of both components must be obtained from evolutionary calculations. This leads us to the necessity of having a set of evolutionary tracks of objects covering the range of masses expected for the components of the system. In the case of HD 93205, the primary O3 V star is candidate to be one of the most massive stars known. Thus, we have carried out calculations up to quite large stellar masses such as 106 M⊙ because, as far as we are aware, there is no computation of the ISCs for such massive stars available in the literature.
The calculations have been carried out with the stellar evolution code developed at La Plata Observatory. It is essentially the same code employed for studying white dwarf (see, e.g., Benvenuto & Althaus 1998) and intermediate mass stars (Brunini & Benvenuto 1997) and has been adapted for properly handling the case of massive stars.
Let us briefly describe the main ingredients of the code. The equation of state employed is that of OPAL (Rogers, Swenson & Iglesias 1996). Radiative opacities are the latest version of OPAL  while for low temperatures they are complemented with the Alexander & Ferguson (1994) molecular opacities. Conductive opacities and neutrino emission rates are the same as in Benvenuto & Althaus (1998). Nuclear reaction rates are taken from Caughlan & Fowler (1988) and weak electron screening is taken from Graboske et al. (1973).
As we are dealing with massive stars, it is important to mention that we have accounted for the occurrence of overshooting by employing the formalism described in Maeder & Meynet (1989). We have adopted the distance of overshooting dov to be a fraction of the pressure scale height HP at the canonical border of the convective zone: dov = αovHp. Also, we allowed for mass loss following De Jager et al. (1986).
Using the evolutionary code just described, we have calculated a set of evolutionary sequences covering the mass range from 4M⊙ to 106M⊙ with a mass step of ≈ 5%. We followed the evolution starting at the ZAMS till the depletion of hydrogen at the centre of the star. The initial helium content of our models is Y = 0.275 and the adopted value for metallicity is Z = 0.02 while two values for overshooting, αov = 0.25 and 0.40, were considered.
After convergence of each model was reached, we solved the Clairaut-Radau differential equation (Sterne 1939) that accounts for the apsidal motion to the lowest (second) order subject to the boundary condition η2 = 0 at a = 0. In this expression a is the mean radius of a given equipotential, ρ(a) is the density at a andρ(a) is the mean density interior to a, ηi is given by and the radius r of the distorted configuration and a are related by, Yi(a, θ) where Yi(a, θ) describe the amplitude of the distortions. In order to integrate Eq. (14) we recall that ρ(a) andρ(a) are provided by the structure of the evolving model. Close to the centre ρ(a), and consequentlyρ(a) and η2(a), are expanded following an analogous treatment to that presented in Brooker & Olle (1955). The integration of Eq. (14) is started at the mesh point adjacent to the centre. Numerical integration is carried out with a standard Runge-Kutta routine (Press et al. 1986) up to the surface of the stellar model in order to get η2(ai). Then, the ISC k2,i is finally given by

A TEST OF THE METHOD EMPLOYING ECLIPSING BINARY SYSTEMS
The method we are presenting here is, to our knowledge, original. In view of this fact, we have applied this method to some previously studied massive eclipsing binary systems with the aim of testing the method before applying it to HD 93205. We have focused our attention on detached systems in which both components are massive stars in the main sequence (MS) and have a rather well measured apsidal motion. We have finally selected the following systems: EM Car, QX Car, GL Car, Y Cyg and V478 Cyg. Observational parameters for these systems are summarized in Table 1. In order to test the method, we compare the masses it yields with the observed ones, i.e. those obtained from the simultaneous analysis of light and radial velocity curves of the systems.
Components of a binary system must have the same age, so the first test we apply to the evolutionary code is that for each system considered there must be a single isochrone fitting the mass and radius of both components on the M −R plane. The results of our calculations for the choice αov = 0.25 can be appreciated in Fig. 1, in which we show the mass and radius of the components of the selected binary systems. Note that for each system there is one isochrone that fits well both components, so the constraint that the ages of the components impose on evolutionary calculations is clearly satisfied by our models. In Fig. 2 we show the effective temperatures derived from our models for each star as a function of the observed effective temperature. Again, a good agreement between our evolutionary calculations and observations is found. As we have already stated, we have also considered a larger amount of overshooting, by fixing αov to 0.40. We find no significant differences with the case of a smaller amount of overshooting, so we adopt the lower value (αov = 0.25) as the standard one in our calculations.
where c2,i is given by Even when the quotient̟2/Ω can be assessed from observation, it is not possible to separate the contribution of each component to the rate of apsidal motion. Instead, we can determine k 2,obs and it is this value that is currently used to contrast evolutionary models with observation. In Fig. 3 we show the theoretical values for k2 derived from our models against the observed ones. We find that our models predict mean ISCs that are in reasonable good agreement with the observed ones for the less concentrated models (those with a higher value of k2). The most discrepant case we find is that of EM Car, which is the most evolved system considered by us, as we find that its primary star has spent ≈ 60% of its life on the MS. In this case, our models result less concentrated than what we should expect from the observed value k 2,obs . However, as stated by Andersen & Clausen (1989), the apsidal motion rate for this system is based on observations covering only about 1/6 of the apsidal motion period, thus the accuracy of apsidal motion parameters is still limited. In addition, information on its chemical composition is also missing. The theoretical values k 2,theo we find for the other systems are in good agreement with the observed values and, for QX Car and Y Cyg, also with those derived theoretically by Claret (1997).
We present below some of the results of applying our method to the solution of Eq. (11). First of all, let us emphasize that for an assumed age of the system, the solution of Eq. (11) is very well determined, i.e. only one solution is found as F2 is a very well-behaved, monotonously decreasing function of the independent quantity M1. We illustrate this general behaviour with one example: in Fig. 4 we show F2 as a function of M1 for the case of V478 Cyg assuming an age of 6 Myr for both components. In view of these results, we find the method to be very reliable, from a mathematical point of view, in yielding a well determined value of M1. In Fig. 5 it is shown the mass M1 of the primary component of EM Car obtained as a function of the age of the system. The figure corresponds to the choice αov = 0.25 for overshooting. We find a good agreement between our theoretical prediction for M1 and its observed value for the whole range of ages considered, within a ±1σ error. The upper limit for the age considered is given by the fact that, for larger ages, M1 falls below the minimum mass derived for this system from its mass function f . Results are very similar if an overshooting amount of αov = 0.40 is considered, the only main difference being that solution curve is slightly shifted to larger ages (≈ 10%). Fig. 6 shows the results obtained for V478 Cyg again for αov = 0.25. For this binary system, an excellent agreement is achieved between our method and the observed mass. The same trend as before is found, i.e. the larger overshooting shifts the solution toward ages about 10% larger. Note also that the larger the age considered, the smaller the mass of the primary (and also the mass of the secondary) that can account for the observed rate of apsidal motion.

CALCULATION OF THE MASSES OF HD 93205 AND RELATED PARAMETERS
From the results of the previous sections, we judge our method to be good enough to be employed in the mass estimation of the components of HD 93205. As stated before in the Introduction, HD 93205 is a highly eccentric system, which strongly suggests that it must be very young. However, the age estimates for such early O-type stars are very uncertain, either one tries to derive them considering the region in which the star is located, or comparing the star's position on the theoretical H-R diagram with isochrones calculated from evolutionary stellar models. HD 93205 belongs to the open cluster Trumpler 16, the most massive stars of which have an age spread between 1 Myr and 2 Myr (DeGioia-Eastwood et al. 2001). Besides, there is evidence of ongoing star formation in the molecular cloud complex associated with the Carina Nebula (Megeath et al. 1996). Consequently, a lower limit to the age of the members of Tr 16 cannot be established. On the other hand, de Koter, Heap, & Hubeny (1998) showed that if we increase by about 10% the effective temperature of O3-type stars, the age would decrease from 2 Myr to 1 Myr. Regarding the interpretation of theoretical isochrones for the most massive stars, these authors stated: "The derived T eff values are so similar because the isochrone for ∼ 2 Myr runs almost vertical and because the distance in temperature between the isochrones of 1 and 3 Myr is very small". Taking into account the problem in the age determination described in the previous paragraph we choose to solve Eq. (11) for a whole set of isochrones ranging from the ZAMS up to 2 Myr. We consider as zero age isochrone the one corresponding to the time when the stellar radius reaches its minimum value. In our models, this happens for ages of a few ten thousand years.
In Fig. 7, we present the mass M1 of the primary component of HD 93205 as a function of the age of the system. Two curves are shown, each of them corresponding to a particular choice of the overshooting parameter (αov = 0.25, 0.40). Let us emphasize that the amount of overshooting that actually occurs is a rather uncertain quantity so we consider it as a free parameter and study its influence on the solution of Eq. (11). As can be seen from Fig. 7, for a given age, M1 is almost insensitive to our different choices of αov. Both curves are almost overlapped over the whole range of ages, though differences tend to increase with age. This is not surprising because for a given mass the initial model (a ZAMS model) is the same in both cases so no initial discrepancy exists between them. As models evolve both sequences depart from one another and different internal mass concentrations slowly arise. In view of this insensitivity, we shall concentrate ourselves on the case αov = 0.25 but there is no particular reason to prefer this value instead of the higher one.
Let us consider again Fig. 7. The mass of the primary is a decreasing function of the age of the system. We find that its maximum value, corresponding to ZAMS models, is M1 = 60 ± 19M⊙. This is the upper limit for the mass M1 of the O3 V component of HD 93205. At increasing ages, it rapidly decreases and reaches 53 M⊙ at just 0.3 Myr and 46.5M⊙ at 1 Myr approximately and finally M1 = 40±9 M⊙ at 2 Myr. Within observable quantities, the main source of uncertainty in determining M1 is the apsidal motion rate (known up to a 9% accuracy) and to a smaller extent the projected semi-axis and the projected rotational velocities, so better determinations of these quantities (especially the apsidal motion rate) are needed in order to decrease the error in the determination of M1. We recall here that the apsidal motion rate is a critical parameter because the necessity of very long time baseline (decades) of high-quality observations. Having the mass M1 determined, it is straightforward to calculate the mass M2 of the secondary if we recall (Table  2, see Morrell et al. 2001 for further details) that the mass ratio Q = M2/M1 for HD 93205 is 0.423 ± 0.009. We find that M2 ranges from 25.3 ± 8 M⊙ at the ZAMS down to 17 ± 4M⊙ if a rather large value of 2 Myr is adopted for the age of the system. These mass values are in good agreement with those expected for an O8 V star like this one (consider, particularly, the well known short-period eclipsing binary EM Car, whose primary component is an O8 V and its mass is 22.89 ± 0.32M⊙, Andersen & Clausen 1989). Once M1 is determined it is easy to obtain the inclination i of the orbit  Andersen & Clausen (1989), (2) , (3) Giménez, Clausen & Jensen (1986), (4) Andersen et al. (1983), (5) Simon, Sturm & Fiedler (1994), (6) Hill & Holmgren (1995), (7) Petrova & Orlov (1999) and references therein.
from Eq. (6). In Fig. 8 it is shown the resulting inclination from the set of calculations corresponding to αov = 0.25. It can be seen that the inclination of the system increases as age does. This is a direct consequence of the behaviour of M1 (which decreases as age increases) but it is worth noting that within the whole range of ages considered the resulting inclination does not allow eclipses to occur. Indeed, if we assume that HD 93205 is not older than 2 Myr we find that 54 • ≤ i ≤ 68 • , in coincidence with Antokhina et al. (2000) who found a most probably value of i = 60 • .
However, a problem arises when we try to compare the luminosity derived from the corresponding models to the observed value for HD 93205. Let us explain this with an example: if we consider the 60 M⊙ model, it predicts, for zero age, a radius R1 = 10.7 R⊙, and a log T eff = 4.68, resulting in a luminosity, log L = 5.72 L⊙. This corresponds to a bolometric magnitude, M bol = −9.55, which is almost one magnitude fainter than M bol = −10.41 derived by Morrell et al. (2001) from the visual magnitude of the O3 V component of HD 93205, the distance modulus of 12.55 obtained by Massey & Johnson (1993) for Tr 16, and the bolometric correction (BC) for an O3 V star taken from the calibration by Vacca, Garmany & Shull (1996). This large disagreement between the expected and observed bolometric magnitudes, points to a large error in some (or any) of the involved assumptions. If the distance modulus is right, then we can suspect that the BC must be wrong by about one magnitude. On the other hand, the distance modulus of the Carina Nebula is still a matter of discussion. Distance modulus of the order of that derived by Massey & Johnson (1993) arise from the consideration of color-magnitude diagrams for the stellar component of the clusters. Some other independent determinations, like the recently obtained by Davidson et al. (2001) from kinematic study of the Homunculus nebula surrounding Eta Car, give distance modulus as low as 11.76, which would significantly decrease the referred discrepancy. But, if we suppose this last distance modulus to be correct, then all of the stars in Tr 16 will have MV about 0.8 magnitudes fainter than the values accepted to date. Here we arrive at a point whose importance is obvious for many astrophysical issues, and deserves to be carefully studied. The referred discrepancies might also arise in a combination of different sources of error (BCs, distances, and adopted absolute magnitude scale for ZAMS stars). A detailed discussion of these issues will be presented in a forthcoming paper.
Finally, let us comment briefly that tidal contribution to the apse motion of HD 93205 is the most important, ranging from about 60% at the ZAMS to 70% at 2 Myr. The rotational contribution ranges from 30% to 20% and the relativistic one is almost constant and approximately 10% of the apsidal motion rate is due to this effect. In this sense, HD 93205 could be classified as a relativistic binary system (Claret 1997).
We present a method to calculate masses for components of non-eclipsing binary systems if their apsidal motion rate is provided. The method consists in solving Eq. (11) if the radius and the internal structure constant of each component can be obtained from a grid of stellar evolution calculations. In order to test this method, we have selected some eclipsing binary systems and have derived the masses of their components. A very good agreement was achieved between masses obtained with our method and those derived from the analysis of their radial velocity and light curves.
The main goal of this article, besides presenting the method, is to calculate the masses of the components of HD 93205. This is an O3 V + O8 V system. Its O3 V component has the earliest known spectral type of a normal star found in a double-lined close binary system, thus potentially being a very massive star. Although HD 93205 is not an eclipsing binary, Morrell et al. (2001) have measured its apsidal motion rate and found it to be̟ = 0. • 0324 ± 0. • 0031 per orbital cycle so we have been able to apply the method presented here to this system. The resulting mass of the primary star (M1) is obtained as a function of the assumed age of the system. HD 93205 is a highly eccentric system (e = 0.370 ± 0.005) which suggests a very low age. However, we do not adopt a particular value for the age as its determination is quite uncertain, and prefer to consider a range of ages starting at the ZAMS. We find that for zero age models the resulting mass is M1 = 60 ± 19 M⊙ and that it monotonously decreases as age is increased (Fig. 7), reaching M1 = 40 ± 9 M⊙ at 2 Myr. Now, if we take into account the mass ratio Q = 0.423 for HD 93025, the mass of the secondary lies in the range M2 = 25.3 − 17 M⊙ for this range of ages. It is worth mentioning again that these M2 values are in good agreement with the masses derived for other O8 V stars in eclipsing binaries such as the well studied system EM Car (Andersen & Clausen 1989). The mass value derived for M1 is also in the range (52 -60 M⊙) obtained from the observed Q assuming a "normal" mass for the O8 V secondary component (i.e. 22 -25 M⊙). In addition, we have estimated the inclination of the system through Eq. (6) and the results obtained (Fig. 8) are consistent with the non-eclipsing condition of HD 93205.
Our results corresponding to zero age give an upper limit to the mass of the O3 V component of HD 93205, a result that places a strong constraint to the masses of theoretical stellar models for the most massive stars. Also, the luminosity derived from the stellar models for the O3 V component rises a problem when compared with the observed value, being the theoretical M bol almost one magnitude fainter than the value derived from the observations. This discrepancy raises the need of reviewing both the distance and BC scales for the earliest type ZAMS stars, a subject that will be addressed in the near future.