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Astron. Astrophys. 317, 815-822 (1997)

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2. The mass and evolutionary history of the sdO6 star HD 49798

2.1. The mass of the subdwarf

From spectroscopic and photometric observations, Kudritzki & Simon (1978; hereafter KS78), derived the atmosphere parameters, radius, distance and luminosity of the star, as listed in Table 1.


Table 1. The parameters of HD 49798 as derived from a fine-analysis of its spectrum.

The upper limit to the radius and luminosity are derived from the upper limit to the distance, which has been estimated from the strength of interstellar lines. The lower limit to its radius follows from the assumption of corotation of the star with its orbit, which subsequently sets the lower limit to the distance and luminosity.

The mass function, [FORMULA], has lately been redetermined by Stickland & Lloyd (1994), after including IUE observations of the subdwarf. Once the mass of the compact star [FORMULA] and the orbital inclination i of the system are known, one can derive the mass of the subdwarf [FORMULA]. The relation between the masses of the two stars for several inclinations is shown in Fig. 1. The mass function yields:


The absence of eclipses, together with [FORMULA] (see Table 1) and an orbital radius of between 6 and [FORMULA] (for the masses involved) puts an upper limit on the inclination: [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA], which in Fig. 1 is indicated with the solid line.

[FIGURE] Fig. 1. Companion mass versus subdwarf mass, as found from the massfunction [FORMULA] of HD 49798. The solid line indicates the maximum mass of the subdwarf for a given companion mass corresponding to the maximum inclination for which eclipses are absent ([FORMULA]). Dashed lines relate the masses for several inclinations ([FORMULA]). The dot-dashed line denotes the minimum inclination ([FORMULA]) and minimum subdwarf mass ([FORMULA]) under the assumption of corotation (see text). The heavy dotted line gives the minimum subdwarf mass which follows from its lower luminosity limit. Fine dotted lines give some possible companion star masses.

From the assumption of corotation we can further limit the possible mass range. As then both the rotational period of the star and its maximum radius are known we also know its minimum rotational velocity. If we assume the axis of stellar rotation to coincide with that of the orbit, combination with the upper limit of [FORMULA] yields a lower limit to the inclination of [FORMULA]. In combination with the lower limit to the radius, together with [FORMULA], it results in a lower limit to the subdwarf mass of [FORMULA]. In the figure these limits are indicated by the dot-dashed line. We shall see below that the subdwarf is probably slightly more massive, at least [FORMULA], as it must be able to produce the observed luminosity. This in turn gives a lower limit to the companion's mass of [FORMULA].

To produce the observed rapid regular X-ray oscillations, an accreting magnetized compact object which is rotating rapidly must be involved. For a neutron star companion of [FORMULA] the mass of the subdwarf is in the range [FORMULA] and for a white dwarf companion of mass [FORMULA] the mass of the subdwarf is probably between 0.7 and [FORMULA].

2.2. Evolutionary status of the subdwarf

Table 1 shows that the star, even though it is 'sub-luminous', has a very high bolometric luminosity: [FORMULA]. It also has, for a subdwarf, an unusually low surface gravity, [FORMULA].

It has been suggested (KS78, see also Iben & Tutukov 1993, hereafter IT93) that HD 49798 is the helium-burning core of a star that has lost its envelope in a case B mass transfer event, i.e. before helium was ignited at the tip of the first giant branch (FGB). However, these authors have actually assumed the subdwarf to have a rather high mass of ([FORMULA]) for which there is now no good reason, as Fig. 1 shows. As we shall see in Sect. 2.3, with the implied smaller mass, a case B progenitor is highly unlikely to have produced the present orbit of the binary system. A comparison with stellar models - e.g. by De Greve & De Loore (1976), and unpublished models computed with a recent version of the Eggleton (1971) stellar evolution code - further reveals that it is unlikely that the subdwarf is in a core helium-burning phase. Models of core He-burning stars with approximately the observed surface hydrogen abundance of [FORMULA] and [FORMULA] are much more compact ([FORMULA] and [FORMULA]) and less luminous ([FORMULA]) than is observed.

Therefore, two possibilities remain to explain the evolutionary status of the subdwarf. First, the star may be in a shell helium-burning phase, which would explain both its low surface gravity and its high luminosity. Such stars follow an approximate core-mass luminosity relation (see IT93), and the luminosity of [FORMULA] is consistent with the presence of a CO core of [FORMULA]. The high luminosity is generated by a He-burning shell around a degenerate CO core, surrounded by a thick ([FORMULA]) He envelope.

A shell He-burning subdwarf with a mass in the range between 0.7 [FORMULA] and 1.8 [FORMULA] fits very well with the core of a star which originally was on the early asymptotic giant branch (EAGB), where it had a mass between [FORMULA] and 7 [FORMULA]. After being stripped of its hydrogen-rich envelope in a common-envelope (CE) event, the remaining cores have luminosities, masses and radii in the range inferred above for HD 49798, as can be seen from the evolutionary tracks of such stripped EAGB star cores calculated by IT93. The relevant results of IT93 are summarized in Table 2. At present HD 49798 has a Roche radius of [FORMULA], in between the values considered in the models of IT93. After stripping, these stars still have some hydrogen in their envelope, of order [FORMULA], which in the calculations is subsequently lost by Roche-lobe overflow (RLOF) in a time between [FORMULA] and [FORMULA]  yr. However, mass loss by stellar wind was not included in the calculations, and as IT93 point out, a radiation-driven wind could easily drive a somewhat larger mass-loss rate. This would prevent the remnant from filling its Roche lobe, as is the case with HD 49798. We shall return to the mass loss rate in Sect. 3.3, where we consider wind accretion onto the compact companion. After all hydrogen has been removed, these cores will continue to lose helium at a much higher rate by RLOF, until less than about 0.04 [FORMULA] of helium is left. Then they will shrink and cool into white dwarfs.


Table 2. Summary of the results of the calculations by Iben & Tutukov (1993) for stripped EAGB cores. Columns give: initial mass on the EAGB; assumed final Roche-lobe radius after spiral-in; remnant mass after spiral-in; time during which a H-rich envelope is present; mass-loss rate during the H-rich phase; and growth of the C-O core mass during the H-rich phase.

Alternatively, the subdwarf might be the core of a star which lost its H-rich envelope while on the thermally pulsing (TP) AGB. In this case the star should presently be shrinking and become a white dwarf, having only very recently shed its envelope in a CE phase. The luminosity is then provided (mainly) by shell-hydrogen burning, below a very thin hydrogen envelope. Its present luminosity and mass should be nearly equal to the luminosity and core mass of its TP-AGB progenitor, which are related by the Paczyski (1970) core-mass luminosity relation. From the observed effective temperature and gravity, one can also determine the present luminosity-to-mass ratio: [FORMULA], in solar units. Combination of these two relations would limit the subdwarf mass to very low values, [FORMULA].

There are two reasons why we favor the first possibility, i.e. a shell He-burning star. Firstly, the evolutionary timescale is up to 2 orders of magnitude longer (up to [FORMULA] yr, as opposed to [FORMULA] yr in the case of a shell H-burning proto-white dwarf). Secondly, because of the short time since CE ejection, one would expect the ejected envelope to be ionized by the hot subdwarf and to show up as a planetary nebula. Although some extended emission has been observed around HD 49798, this is believed to be unassociated with the star (Mendez et al. 1988).

We show in the following section that the present orbital period of the system implies that the progenitor must indeed have been on the asymptotic giant branch, most likely on the EAGB.

2.3. The progenitor system of HD 49798

With a present mass of [FORMULA] the progenitor must have been either more massive than [FORMULA], or have had a deep convective envelope, or both. Since the companion is a compact object of [FORMULA], mass transfer would inevitably have led to the formation of a common envelope.

In order to trace back the evolution we therefore use the 'Webbink-formalism' for calculating the decrease in orbital radius during CE-evolution. Denoting the initial and final orbital radii as [FORMULA] and [FORMULA], respectively, one has in this formalism (Webbink 1984, 1992):


where [FORMULA] is the mass of the compact companion, [FORMULA] the mass of the core of the progenitor prior to spiral in, [FORMULA] the mass of its envelope, [FORMULA] the 'efficiency parameter' for CE evolution, [FORMULA] a parameter which depends on the density distribution in the envelope of the progenitor, and [FORMULA] the fractional Roche lobe radius of this progenitor prior to spiral in.

As discussed by Van den Heuvel (1994), [FORMULA] has a value that from an analysis of observed post-CE systems, in combination with results of numerical computations (Taam 1996), is expected to be in the range between 1 and 4 (implying an [FORMULA] parameter in the 'Iben-Tutukov formalism', e.g. see Iben & Livio (1993), of between 0.3 and 1.0). In fact, [FORMULA] -values [FORMULA] 1 imply that apart from the orbital binding energy, other sources of energy were available in the envelope of the giant for expelling this envelope, e.g. thermal energy, recombination energy, and pulsational energy. The thermal energy of the envelope is formally neglected in the 'Webbink-formalism', and its inclusion as an energy source reduces the envelope binding energy by a factor of about 0.5 (if atomic and molecular recombination energy is not included in the thermal energy). Hence, in the absence of other energy sources, the maximum value of [FORMULA] is 2. On the FGB and on the EAGB, [FORMULA] can therefore not be much larger than 2, but on the TP-AGB it may well be [FORMULA]. For the sake of argument, to trace back the original system of HD 49798, we will consider the 'extremes' [FORMULA] and [FORMULA].

We can now use Eq. (2) to compute the initial Roche-lobe radius [FORMULA] that the progenitor of HD 49798 should have had to produce the present orbital separation [FORMULA] (which is in the range [FORMULA], see Sect. 2.1), as a function of progenitor mass [FORMULA], and for certain values of [FORMULA] and [FORMULA]. This requires knowledge of the helium core mass [FORMULA] as a function of mass and stellar radius. We take the radii and core masses from a set of evolutionary calculations computed with the Eggleton (1971) code for single stars with [FORMULA], as described by Pols et al. (1995). The calculations by IT93 indicate that, after a CE event on the EAGB, the remnant mass is only slightly larger than the helium core mass. The remnant masses which we would infer from the core masses of the stellar models indeed correspond within a few per cent with those computed by IT93. Inspection of the stellar models also shows that the typical value of [FORMULA] for a massive FGB star is [FORMULA], while for an AGB star it is [FORMULA], the larger value applying to more evolved stars. We use a value of [FORMULA] throughout in computing the initial Roche radii. We compute [FORMULA] with the formula of Eggleton (1983).

The results are shown in Fig. 2, where we have plotted, as a function of progenitor mass, the progenitor's Roche radius [FORMULA] as computed above, for values of [FORMULA] between 1 and 4 and for [FORMULA] and 1.4 [FORMULA]. These can be compared with the stellar radii at core helium ignition, [FORMULA], and at the start of the TP-AGB, [FORMULA]. Mass transfer starts on the EAGB if [FORMULA] is between these values, on the FGB if [FORMULA], and on the TP-AGB if [FORMULA]. Also shown are lines of equal core mass, indicating the range of possible progenitor masses. For instance, an EAGB progenitor could have had a mass between 3.8 and 7.5 [FORMULA] in the case of a 1.4 [FORMULA] neutron-star companion, but only between 3.8 and 5.0 [FORMULA] in the case of a 1.0 [FORMULA] white dwarf.

[FIGURE] Fig. 2. Possible progenitor masses and radii of the subdwarf component of HD 49798. The dashed and dotted lines indicate, as a function of initial mass, the initial Roche-lobe radius that the progenitor should have had in order to obtain the present orbit, for the indicated values of the CE efficiency parameter [FORMULA], 2 and 4. Dashed lines are for a companion mass [FORMULA] ; dotted lines are for [FORMULA]. We have calculated these curves using Eq. (2) and assuming the remnant mass after CE evolution to be equal to the helium (H-exhausted) core mass. Thick solid curves are the stellar radii at helium ignition ([FORMULA]), at the start of the TP-AGB ([FORMULA]), and at the start of the second dredge-up ([FORMULA]), as a function of initial mass. Thin solid curves are lines of constant He core mass (defined as the mass shell interior to which [FORMULA]), for core masses between 0.6 and 1.8 [FORMULA], as indicated. These values for core masses and radii are taken from a set of evolution tracks computed by Pols et al. (1995).

It is clear from Fig. 2 that, for any possible combination of progenitor mass and companion mass, and for any reasonable value of [FORMULA], the progenitor of HD 49798 should have been on the AGB. In particular, we see that an FGB (i.e. case B) progenitor could almost certainly not have produced the present orbital radius of the system, since a value [FORMULA] would have been required. These FGB stars have relatively smaller core masses, so that a curve of constant [FORMULA] lies at larger radius than for an EAGB progenitor. Even for the largest possible stellar radius on the FGB ([FORMULA] for a [FORMULA] progenitor, which would give a 1.8 [FORMULA] He-burning remnant), a value [FORMULA] would be required. An even larger [FORMULA] is necessary for a smaller progenitor mass (see, however, below).

Fig. 2 also shows that if the progenitor of HD 49798 was on the TP-AGB, the progenitor mass should have been [FORMULA] since, as we saw in Sect. 2.2, in that case the subdwarf mass is [FORMULA]. Furthermore, the CE efficiency should have been small, [FORMULA], in order to produce the present orbit. However, especially a TP-AGB star might be expected to have [FORMULA], i.e. to have extra energy sources available for envelope ejection, since such a star is eventually capable of doing so without the help of a companion.

Also plotted in Fig. 2 is the radius [FORMULA] at the start of the second dredge-up on the EAGB, when the convective envelope starts to eat into the helium core for stellar masses [FORMULA]. If [FORMULA], the remnant of CE evolution may have only a short-lived phase during which a H-rich envelope is present, since the helium core of such a progenitor is already rapidly expanding (which is the very reason for the penetration of the convective envelope). Therefore one may expect the CE remnant to enter quickly into a stage of Roche-lobe overflow at rather high mass-loss rate, during which helium rather than hydrogen is transferred, such as occurs in the later phases of the calculations of IT93. Although no calculations have been done for this case, it seems likely that the progenitor of HD 49798 had a Roche radius [FORMULA] (if its mass was [FORMULA]), since HD 49798 has a H-rich atmosphere and is not filling its Roche lobe.

Finally, it appears from Fig. 2 that the progenitor of HD 49798 must already have evolved some way up the EAGB before the CE phase, since even the [FORMULA] curve is well above the [FORMULA] line. This is consistent with its present luminosity and inferred CO-core mass (see Sect. 2.2), which are somewhat higher than those of the IT93 models (see Table 2). These models were calculated for progenitors that had just reached the base of the EAGB.

Two potentially important effects are not included in the calculations and in the discussion above: (1) convective-core overshooting during the main sequence, and (2) stellar-wind mass loss prior to Roche-lobe filling. Both effects (if present) will reduce the envelope mass relative to the core mass, and therefore facilitate CE ejection. In other words, a smaller value of [FORMULA] would be required to produce the present orbit. Convective overshooting may increase the core mass by as much as 25% (Maeder & Meynet 1989), and also somewhat increases the stellar radii at He-ignition and at the start of the TP-AGB. On the EAGB, and especially on the TP-AGB, stellar winds during preceding phases (e.g. core helium burning on the giant branch) may have further reduced the envelope mass. However, stellar winds cannot reduce the envelope mass substantially before He ignition, since the FGB phase is rather short-lived in these intermediate-mass stars. Consequently, an FGB progenitor of even the largest mass would still require [FORMULA], even if one makes reasonable allowance for both effects (overshooting and mass loss). As we have argued above, [FORMULA] is probably [FORMULA] on the FGB, so that we remain confident that a case B progenitor is extremely unlikely to have produced the present orbit of HD 49798.

We conclude that, regardless of whether the compact star is a neutron star or a white dwarf, and for any reasonable value of [FORMULA], the progenitor of the subdwarf component of HD 49798 was on the AGB, in all likelihood the EAGB. This implies that at present this star has a degenerate CO-core and is in the phase of shell helium burning, which is consistent with the high luminosity of HD 49798.

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Online publication: July 8, 1998