The true rotation period combined
with the
projected rotational velocity
=27.0
gives a lower limit for the
radius of the star:
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).
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 . If the primary would fill its
Roche-lobe, the ratio of the effective
radius of the Roche-lobe
,
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.
is entirely determined by the mass-ratio
of the two components:
(Eggleton 1983).
Since and
increases with
, we
obtain a lower limit for the mass ratio q, if we replace
by
in Eq. (8 (click here)):
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 27.0. 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
.
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
for the K1III primary (Gray 1988, Appendix B),
the secondary
must be fainter than
, 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
, as a white dwarf less than the Chandrasekhar-limit of
. Ayres et al. (1984) analyse the far-UV spectra of
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 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:
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 Gem has a high
inclination of its rotational axis and is therefore suitable for surface
imaging. The value of
adopted for the interpretation of photometry
of
Gem e.g. by Poe & Eaton (1985)
is perfectly consistent with this limit.
If we now combine with the mass function and the upper
limit for the inclination,
, we obtain an upper limit for the
mass
:
A lower limit on results from the lower limit on q and
:
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 .
The surface gravity of the primary can be written as:
The last term corrects for the influence of the centrifugal force, which is
, if we adopt as a value for the
observed
that in the center of the stellar disk, which is at a
latitude
. 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:
Note, that the lower limit corresponds to the lower limit for , which is
too low. If we assume
, then
Therefore,
the true gravity is very probably higher than log.
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 -plane.
The oblique lines to the left are the mass-ratios q=1.0 (for orientation)
and
. The curves are lines of constant radius of the primary,
and since
sini is fixed, lines of constant inclination as well.
Since the mass function is also fixed, each value of
for given
thus corresponds uniquely to a value of
. Lines
and
are also shown. Note, that if
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
it must be a neutron star. To emphasize
the importance
of the gravity, the lines of constant log
are added to the
figure. We
note, that the line log
is almost always above the Roche-limit
and the line log
entirely in the region with
. Note, that for a normal radius,
(Dyck et al. 1996), the secondary is most probably too massive to
be a white dwarf.