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5. Constraints on the fundamental parameters

 

5.1. The minimum radius

 

The true rotation period tex2html_wrap_inline2187 combined with the projected rotational velocity tex2html_wrap_inline2189 =27.0tex2html_wrap2197 gives a lower limit for the radius of the star:
 equation475
This is only valid if the star is assumed to be spherical. Due to its relatively long rotation period it seems unlikely that the star is flattened by rotation; it might, however, approach its Roche-lobe limit and become distorted. The assumption of sphericity is thus equivalent to the assumption that the star does not fill its Roche-lobe. So far, there is no observation indicative of mass-transfer in the system, so this assumption seems to be justified. Note that here and in the following we do not distinguish between the inclination of the axis of rotation and that of the orbital axis. Since the orbit is circularized and the rotation is synchronized with the orbit it is natural to assume that these two axes are aligned. This is supported by the results of Stawikowski & Glebocki (1994).

5.2. The Roche-lobe limit

 

In the following the index 1 (``primary'') indicates the visible component of the binary (the K1III star) and the index 2 the secondary. The orbital parameters determine tex2html_wrap_inline2201. If the primary would fill its Roche-lobe, the ratio of the effective radius of the Roche-lobe tex2html_wrap_inline2203, i.e. the radius of the sphere that has the same volume as the Roche-lobe, and the distance between the two stars, i.e. tex2html_wrap_inline2205 is entirely determined by the mass-ratio tex2html_wrap_inline2207 of the two components:
 equation489
(Eggleton 1983). Since tex2html_wrap_inline2209 and tex2html_wrap_inline2211 increases with tex2html_wrap_inline2213, we obtain a lower limit for the mass ratio q, if we replace tex2html_wrap_inline2217 by tex2html_wrap_inline2219 in Eq. (8 (click here)):
 equation511

5.3. The secondary

 

The mean SOFIN spectrum has been subtracted from each of the individual spectra and the residuals carefully examined. There are variations exceeding the noise, they are, however, inside the spectral features, i.e. their displacement with respect to the primary star is less than 27.0tex2html_wrap2255. They cannot be due to the secondary, because from the limit of the mass-ratio it can be concluded that the displacement should reach sometimes at least tex2html_wrap_inline2223tex2html_wrap2257. The variations are also not systematic over the whole 2 years period; they are thus most probably features caused by surface inhomogeneities.

Thus, even with the high-resolution, high S/N SOFIN spectra, no trace of the secondary (in this wavelength region) can be found. Since the S/N of the SOFIN spectra is generally well above 100, we can conclude, that the luminosity of the secondary does not exceed 1% of the luminosity of the primary. Adopting tex2html_wrap_inline2227 for the K1III primary (Gray 1988, Appendix B), the secondary must be fainter than tex2html_wrap_inline2229tex2html_wrap_inline2231, which would make it a main-sequence star of type G6 or later, or a compact object (white dwarf, neutron star). As a star of type G6V or later, its mass would be less than tex2html_wrap_inline2233, as a white dwarf less than the Chandrasekhar-limit of tex2html_wrap_inline2235. Ayres et al. (1984) analyse the far-UV spectra of tex2html_wrap_inline2237 Gem and from the lack of any trace of a hot continuum conclude that the secondary is probably a late-type dwarf.

If we adopt for the secondary tex2html_wrap_inline2239 and take into account the mass function from the orbital solution as well as the lower limit for the mass ratio from the Roche-limit, we obtain a stringent lower limit for the inclination:
 eqnarray526
Here, min(x) and max(x) give the minimum and maximum of all possible values for x. This limit on i is interesting, because it means that tex2html_wrap_inline2249 Gem has a high inclination of its rotational axis and is therefore suitable for surface imaging. The value of tex2html_wrap_inline2251 adopted for the interpretation of photometry of tex2html_wrap_inline2253 Gem e.g. by Poe & Eaton (1985) is perfectly consistent with this limit.

5.4. The limits on the mass of the primary

 

If we now combine tex2html_wrap_inline2267 with the mass function and the upper limit for the inclination, tex2html_wrap_inline2269, we obtain an upper limit for the mass tex2html_wrap_inline2271:
 eqnarray540
A lower limit on tex2html_wrap_inline2275 results from the lower limit on q and tex2html_wrap_inline2279:
 eqnarray552
Unfortunately, this limit is not very useful: a star of such low mass would probably not have become a giant yet in the 10 Gyr or so that the Galaxy exists. Probably, the mass of the primary is larger than 1 tex2html_wrap_inline2283.

5.5. The primary's gravity

 

The surface gravity tex2html_wrap_inline2297 of the primary can be written as:
 equation568
The last term corrects for the influence of the centrifugal force, which is tex2html_wrap_inline2299, if we adopt as a value for the observed tex2html_wrap_inline2301 that in the center of the stellar disk, which is at a latitude tex2html_wrap_inline2303. The correction for centrifugal forces consists thus only of observables and cannot be varied.

The limits for mass, radius and mass ratio then give also limits for the surface gravity:
 eqnarray577

Note, that the lower limit corresponds to the lower limit for tex2html_wrap_inline2305, which is too low. If we assume tex2html_wrap_inline2307, then
 eqnarray604
Therefore, the true gravity is very probably higher than logtex2html_wrap_inline2309.

5.6. The mass-mass-plot of the allowed stellar parameters

 

 figure621
Figure 4:   The stellar parameters in the mass-mass-plane. See text for detailed explanation

In Fig. 4 (click here), the results of the previous paragraphs are presented in graphical form in the tex2html_wrap_inline2315-plane. The oblique lines to the left are the mass-ratios q=1.0 (for orientation) and tex2html_wrap_inline2319. The curves are lines of constant radius of the primary, and since tex2html_wrap_inline2321sini is fixed, lines of constant inclination as well. Since the mass function is also fixed, each value of tex2html_wrap_inline2325 for given tex2html_wrap_inline2327 thus corresponds uniquely to a value of tex2html_wrap_inline2329. Lines tex2html_wrap_inline2331 and tex2html_wrap_inline2333 are also shown. Note, that if tex2html_wrap_inline2335 the secondary needs to be a white dwarf or low mass neutron star, because a main-sequence star would have been detected in the spectra, and above tex2html_wrap_inline2337 it must be a neutron star. To emphasize the importance of the gravity, the lines of constant logtex2html_wrap_inline2339 are added to the figure. We note, that the line logtex2html_wrap_inline2341 is almost always above the Roche-limit tex2html_wrap_inline2343 and the line logtex2html_wrap_inline2345 entirely in the region with tex2html_wrap_inline2347. Note, that for a normal radius, tex2html_wrap_inline2349 (Dyck et al. 1996), the secondary is most probably too massive to be a white dwarf.


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