Szabolcs Barcza
Konkoly Observatory of the Hungarian Academy of Sciences
Except for supergiants and cool stars with effective temperature
Te<4000K a large grid of model atmospheres is available in the literature (Kurucz, 1997 for the latest versions). The conversion of
theoretical fluxes to observational magnitude and colour indices
has been solved (e.g. for Johnson and Kron-Cousins photometries,
Castelli, 1999) opening the way to a determination of the atmospheric
parameters effective temperature,
Te,
and surface gravity
if broad band photometry is available for the target star.
The stellar angular diameter can be derived in
the following manner from atmospheric models. Monochromatic flux
F{lambda}(0)
of the models at optical depth
{tau}=0
is given in tabular form for a large grid of
Te,
g,
metallicity
[M],
etc., furthermore, by integrating
F{lambda}(0)
for a number of photometric systems the fluxes
FX(0)
were computed in physical units
erg s-1 cm-2
as well, and the corrections from interstellar
absorption, reddening were included (Kurucz, 1997).
X
represents the photometric band, e.g.
V
in the Johnson system. On the other hand, observed physical fluxes,
IX,
of stars in band
X
can be derived by using the absolute flux calibration of Vega
(Tüg et al., 1977). If the interstellar extinction is
AX
magnitude, the half angular diameter of a spherical (i.e. non-distorted)
star can be derived from the simple formula
For giant, main sequence, or white dwarf stars the stellar atmosphere
is a thin layer in comparison with
R.
The gravitational acceleration
Problems, dependence of
g,
{theta},
M
on
E(B-V),[M]
were discussed in numerous papers, e.g. Kinman & Castelli (2002), Decin et al. (2003). Accuracy of the data from broad band photometry
is inferior to that from fine analysis of spectra or interferometry,
respectively, and
M
can be determined much better from binary orbits. However, if the more
accurate methods are not available, the photometric determination of
g,
{theta}, and
M
must be appreciated.
The main parameters of atmospheric models are
Te,
g,
[M],
and in addition to them the interstellar reddening, extinction (e.g.
E(B-V),
AX)
must be known. From observational point of view colour indices,
Ci, (i=1,2,...)
are the input parameters and the inverse problem of ordering a model to
them must be solved.
In optimal cases the typical errors of interpolation are
0.1
in
log |g|
while a few hundredth in
FX,
respectively, however, the successful solution
depends on the shape of the functions
Te(C1,...) and
log|g(C1,...)|.
The final step is that theoretical
FX(C1,...)
belonging to the selected model must be compared with
IX
from observed magnitudes according to (eq1).
From inspecting the Kurucz tables some general rules can be inferred, we
mention a few of them. For main sequence or giant stars
UBV photometry must be preferred if
7000 < Te < 104
K while for cooler stars
BVIC
gives reliable results as well. Below
Te=9000 K
the dependence of
log|g|
on
U-B
becomes more pronounced,
U-B
itself is sensitive to
[M]
because of the appearance of metallic lines in ultraviolet. In favourable
cases
Te
and
g
can be determined from a two colour diagram if
[M],
E(B-V)
were already known while the procedure is more uncertain if
[M]
and
E(B-V)
must be estimated from colour indices as well, in this case more
than two colour indices are necessary.
To demonstrate reliability of the method, some date are given in Table 1:
{theta} was derived from observed
V,
B-V,
U-B
(Hoffleit, 1982) by using
FV(U-B,B-V)
of the Kurucz models while for comparison the value
{theta}HB
is given from intensity interferometry corrected for
limb darkening (Hanbury Brown et al., 1974). Using the
Hipparcos parallaxes (ESA, 1997)
R=3.086×1013{theta}/{pi}
km was calculated and the mass was obtained by (eq2) from
log|g(U-B,B-V)|.
It can be noted that the weakest sampling of the spectrum is provided by
(U-B,B-V)
since the wavelength interval
~360-540 nm
is covered in this case. Of course an inclusion of other colour indices
could improve the values of
{theta},log|g|.
P.O. Box 67, H-1525 Budapest, Hungary 
where
d
is the distance to the star,
R
is the radius of
{tau}=0
(Barcza, 2003).
is practically constant in the atmosphere, i.e. at
r~R,
where
{rho}(r), p(r), M, G
are density, pressure, mass, and the Newtonian gravitational constant,
respectively. The surface gravity,
g(R),
is an important parameter of the static atmospheric models given
conventionally as
log|g|
and it can be used to weigh a star if the stellar radius,
R,
is known.
Relation (eq2) was applied for a number of non-variable stars
enhancing our knowledge on stellar masses: e.g. for white
dwarfs purely from
log|g(R)|
of model atmospheres
and
R
Mwd=0.480±0.014
average mass was derived (McMahan, 1989) for a sample of 53 stars
while other methods gave somewhat larger mass,
Mwd=0.58±0.10
for 64 stars (Koester et al., 1979).
The fundamental stellar parameters: distance and mass
(Barcza, 2003) follow from (
where
{sigma}
is the Stefan-Boltzmann constant,
Teq
differs slightly from the average effective temperature
<Te({phi})>.
Leq,Teq allow to locate a variable star
in theoretical Hertzsprung-Russell diagram: of course on the basis of
[colours-Te, FX]
calibration of the used atmospheric models.
must be solved for
d
where
pp
is the projection factor of converting radial to pulsation velocity and
v{gamma}
is the barycentric velocity of the star.
Physical input comes from the time-dependent projection factor. A much more
serious uncertainty of kinematic nature is imported in this procedure by
the error
{Delta}v{gamma}.
If
{Delta}v{gamma} << {overline}|v{gamma}-vradial(t)|
a negligible error is propagated into
{delta}R,
however, it is problematic to achieve this desired accuracy
because the observed radial velocities are an integral of the radial
component of true, non-uniform motions in the atmosphere contaminated by
apparent velocity changes from varying opacity during phases of different
compression. The difficulties from the uncertain value of
v{gamma}
could be circumvented by differentiating
pp(t)vradial(t)
and substituting it for
{theta}d
in (
with
P0=0.3247796±0.0000032
and the values
{delta}k
in Table 2. The summed string length was
{nu}l0(P0,{delta}=0)=0.330
with standard deviation
SD=0.069
magnitude, the folded light curve belonging to it is plotted in Fig. 1a.
Applying the values
{delta}k
of Table 2 reduced
{nu}l(P0,{delta})
to 0.062 with
SD(P0,{delta})=0.012
mag, dots of Fig. 1b are a plot of the folded light curve, its
SD=0.012
does not exceed the expected random error of a
V
point, it is lower than the claimed amplitude 0.028, 0.015 mag for the
second and third periods of T Sex (Hobart et al., 1991). Thus, the conclusion must
be drawn that between
HJD 2434311-2447488
the light curve of T Sex can perfectly be reproduced if
the light curve segments in Fig. 1a are shifted to the
HJDmax
given by (
