A&A 379, 245-256 (2001)
P. Morel - G. Berthomieu - J. Provost - F. Thévenin
Département Cassini, UMR CNRS 6529, Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France
Received 9 July 2001 / Accepted 14 September 2001
We have revisited the calibration of the visual binary system Herculis with the goal to give the seismological properties of the G0IV sub-giant HerA. The sum of masses and the mass fraction are derived from the most recent astrometric data mostly based on the HIPPARCOS ones. We have derived the effective temperatures, the luminosities and the metallicities from available spectroscopic data and TYCHO photometric data and calibrations. For the calculations of evolutionary models we have used updated physics and the most recent physical data. A minimization is performed to approach the most reliable modeling parameters which reproduce the observations within their error bars. For the age of the Her binary system we have obtained Myr, for the masses and , for the initial helium mass fraction , for the initial mass ratio of heavy elements to hydrogen and for the mixing-length parameters and using the Canuto & Mazitelli (1991, 1992) convection theory. Our results do not exclude that HerA is itself a binary sub-system as has been suspected many times in the past century; the mass of the hypothetical unseen companion would be , a value significantly smaller than previous determinations. A calibration made with an overshoot of the convective core of HerA leads to similar results but with a slight increase of +250 Myr for the age. The adiabatic oscillation spectrum of HerA is found to be a complicated superposition of acoustic and gravity modes. Some of these waves have a dual character. This greatly complicates the classification of the non-radial modes. For the modes all have both energy in the core and in the envelope; they are mixed modes. For there is a succession of modes with energy either in the core or in the envelope with a few mixed modes. The echelle diagram used by the observers to extract the frequencies will work for . The large difference is found to be of the order of Hz, in agreement with the Martic et al. (2001) seismic observations.
Key words: stars: binaries: visual - stars: evolution - stars: fundamental parameters - stars: individual: Her
Herculis (40Her; BD+312884; STF2084; ADS10157; IDS16375+3147; WDS 16413+3136; HR6212; HD150680; HIP81693; , (2000)) is a well known bright visual and single lined spectroscopic binary system of naked-eye brightness. According to the CDS Simbad data base the system is composed of a 2.90 Vmagnitude G0IV sub-giant star and by a 5.53 Vmagnitude G7V dwarf star. The binarity was discovered by Herschell as early as 1782 (Aitken 1932) and the system has been carefully observed by visual binary observers for more than six revolutions back to the first reliable measurements by W. Struve in 1826. Several orbital solutions have been published from the early nineties to the present day. The latest are by Heintz (1994) and Söderhjelm (2000). The orbital elements are well determined. They have recently allowed improvements of Sproul Observatory and HIPPARCOS trigonometrical parallaxes. About a century ago, a duplicity of HerA was detected from micrometer and meridian observations (Lewis 1906). A period yr and a semi-major axis of was obtained for the sub-binary system HerAa. Ten years later, Comstock (1917) noted that the small irregularities of the areal velocity in the orbit have the effect of an invisible companion having a period of 18yr and an amplitude less than . Later, a careful reanalysis of the available astrometric and spectroscopic observational material led Berman (1941) to conclude that" the presence of a third body revolving about the brighter component of Herculis is not definitely indicated''. The comparison between the observed position angles and distances and their values derived from its astrometric orbit allowed Baize (1976) to claim that HerA is split in two stars of respectively and with an orbital period yr and a semi-major axis . McCarthy (1983) reported that Her contains at least one unseen companion easily detected at m by infrared speckle interferometry. A note in the Fifth Catalog of Orbits of Visual Binary Stars (Hartkopf et al. 2000) stipulates: "No evidence in the speckle or Hipparcos data for the large-amplitude third-body orbit given by Baize''.
Lebreton et al. (1993) tried to determine both age and chemical composition of the system, modeling the two components simultaneously, but they did not succeeded in modeling the secondary consistently. Based on a precise spectroscopic analysis Chmielewski et al. (1995, hereafter C95) succeeded in modeling both components. They derived for the age Gyr and for the masses of components respectively and . Since 1995 the HIPPARCOS's parallax of Her has been available; it was recently improved by Söderhjelm (2000). The TYCHO & HIPPARCOS magnitudes are also available (Fabricius & Makarov 2000); they have been connected to the B, V and I magnitudes (Bessell 2000). New improved theoretical data are also available, viz. opacities, nuclear reaction rates and equation of state. The seismic observations of HerA recently carried out by Martic et al. (2001) indicate the presence of solar like oscillations.
The aim of the present paper is firstly to revisit the calibration of the Her binary system using updated physical data and theories and improved astrometrical and photometrical observational material, and secondly to give seismological properties of HerA which will be useful to exploit future asteroseismological observations.
Based on the reasonable hypothesis of a common origin for both components,
i.e. same initial chemical composition and age, the calibration of a binary
system consists of determining a consistent evolutionary history for the
double star, given (1) the positions of the two components in the HR diagram,
(2) the stellar masses and, if possible, (3) the present-day surface chemical
compositions. The goal is to compute evolutionary models that reproduce the
The calibration yields estimates for the age, the
initial helium mass fraction and the initial metallicity
which are fundamental quantities for our
understanding of the galactic chemical evolution. Within the error bars
provided by astrometry, one also determines mass values
consistent with the stellar structure modeling.
According to the convection theory applied, one derives
values for the "mixing-length parameter'' or "convection parameter''
ratio of the mixing-length to the pressure scale height.
Once the physics is fixed, the modeling of the two components A and B of a binary
system requires a set
of seven so-called modeling parameters:
The knowledge of individual masses of each companion is one
cornerstone of any calibration of a binary system. As illustrated by
calibrations of Cen, (e.g. Morel et al. 2000),
the resulting age is very sensitive to the values adopted
for the masses of the components. We do not reproduce here
the large bibliography on relevant astrometric data (e.g. C95); nevertheless
we emphasize the discussion concerning the estimate of masses.
For Her we are in the fortunate position of having two
precise and independent determinations of the
trigonometrical parallax and also improved orbital elements; from these
data, two estimates of the sum of masses can be derived independently.
A first determination is provided by a standard photographic
parallax (Heintz 1994) using
an improved relative orbit and 152 mid-nights of several decades
of Sproul Observatory long focus photographic observations. The
second one is the recently improved adjustment of parallax and orbital
elements by Söderhjelm (2000)
based on HIPPARCOS data collected during 3.25yr, old ground based observations and recent
Table 1 lists the relevant orbital elements of these two
recent astrometric orbits; they are so close
that we can safely adopt the means listed in Table 3.
The sum of masses is provided by Kepler's third law:
The outcome of standard long focus photographic parallax
measurements is the
so-called relative parallax
It must be reduced
the absolute parallax,
by adding a correction representing the dependence weighted parallax of
reference stars. The correction, of order 1 to 5 mas, is not accurately known
(van deKamp 1967).
For Her the correction is
that leads to a photographic absolute parallax of
The improved absolute parallax based on HIPPARCOS data and orbital
ground-based measurements amounts to
We adopt respectively for the parallax and the distance modulus of Her:
Her is one of the rare binary systems for which the mass fraction:
The second estimate of the mass fraction is provided by the spectroscopic orbit.
For a single lined spectroscopic binary, with known orbital elements and
parallax, the mass fraction is given by
(e.g. Heintz 1971; Scarfe et al. 1983):
The mass fraction is not derived in the analysis of Söderhjelm (2000) despite an angular distance larger than the grid step ( ) of HIPPARCOS (Martin et al. 1997), and changes of and respectively in position angle and separation along the flight of the satellite.
For further investigations we shall adopt as the mass fraction of
Her the weighted mean:
With the values retained for
the individual masses are:
With classical 1.5m to 2.0m telescopes Her appears as a single star under average seeing conditions because of the large magnitude difference and of the small angular distance. Therefore isolated spectra of each component cannot be obtained. The available spectroscopic data in C95 allows the derivation of effective temperature only for HerA. We start with an estimate of from Magain's (1987) calibration, assuming a metallicity of and color index (B-V)=0.65 (see Table 2). Then we perform a standard LTE detailed analysis using the curve of growth technique with models atmosphere from Gustafsson et al. (1975). Equivalent widths used are from C95 and the oscillator strengths are from Thévenin (1990). The microturbulence is fixed to 1.5kms-1. With the updated solar iron abundance (Holweger 1979; Asplund 2000) we derived the same metallicity dex as C95. An ionization equilibrium is obtained from curves of growth of FeI and FeII which permits us to derive the surface gravity . We correct the [Fe/H] value in Magain's (1987) formula and derived an improved temperature value of K and re-iterate the curve of growth analysis. Finally we deduce and . With these improved input parameters the scattering of the curve of growth decreases and is satisfactory. C95 have derived a slightly smaller value for the gravity invoking non- LTE effects. Such stars with solar abundances are not suspected to suffer from NLTE overionization (Thévenin & Idiart 1999), therefore the surface gravity of HerA can be considered as being well determined. It is remarkable that this new effective temperature value we derived, using both TYCHO's photometry (Fabricius & Makarov 2000) and spectroscopic analysis, is very close to K, the value obtained by C95 using both the photometry available in 1995 and the profile of H corrected by the presence of the light of the secondary. The uncertainty on has been estimated by varying and in their extremes and with the accuracy of the fit resulting from the quality of equivalent widths. The precision obtained for the metallicity is compatible with a value of 0.05dex, which is the more pessimistic estimate in C95. The precision obtained for the effective temperature results from the partial derivative of Magain's (1987) temperature formula combined with the abundance uncertainty and errors of the TYCHO magnitudes listed Table 2. For the effective temperature of HerA, C95 have obtained K, a similar estimate.
The projected rotational velocity of HerA amounts to 3.9kms-1 (Fekel 1997). Assuming a parallel axis for rotation and orbital motion, , HerA is therefore a slow rotator. We can safely infer that it is the same for the less massive B component. So we can neglect the small rotational velocity in modeling the internal structure of both components.
Table 2 lists the bolometric corrections, the bolometric magnitudes and luminosities computed with as the solar bolometric magnitude to be used with the Bessell et al. (1998) calibration. The Bessell (2000) photometric calibration allows us to derive the standard Johnson B and V magnitudes either from TYCHO magnitudes, or from and HIPPARCOS magnitudes. The values obtained from each differ from each other by more than is expected from the accuracy of measurements. Moreover, fixing the color index (B-V), the effective temperatures calculated with different photometric calibrations (Magain 1987; Alonso et al. 1996; Flower 1996) differ by more than 150K. This indicates that the errors on data derived with the whole procedure may not have a standard Gaussian distribution. Therefore the uncertainties of quantities derived from photometric data are added in absolute value ( norm) instead of standard quadrature ( norm). Table 3 allows comparisons between the observational constraints retained in C95 and in this paper. The main differences are for the luminosity values owing to the smaller bolometric corrections and parallax we have used.
|this paper||Chmielewski et al. (1995)|
Models have been computed using the CESAM code (Morel 1997). About 600 mass shells describe each model; this number increases up to 2100 for the model used in seismological analysis. Around 400 and 30 models are needed to describe the evolutions of Her A and B respectively.
Basically the physics employed is the same as in Morel et al. (2000). The ordinary assumptions of stellar modeling are made, i.e. spherical symmetry, no rotation, no magnetic field and no mass loss. The evolutions are initialized with homogeneous zero-age main-sequence models ( ZAMS). In the absence of satisfactory treatment of microscopic diffusion for stars with mass larger than , we do not take into account the diffusion of chemical species. This important assumption is discussed in Sect. 5.
|Calibration without overshoot for HerA|
|Calibration with overshoot for HerA|
We have computed the evolution of models with
initial helium mass fraction, initial ratio of heavy elements to hydrogen,
masses and mixing length parameters respectively in the ranges
For the models of HerA, as soon as
the evolution is halted.
Table 4 lists the set of "best'' modeling parameters
we adopt for the Her A & B models
with and without overshooting of the convective
core of the primary. Figure 1 shows the corresponding
evolutionary tracks in the HR diagram. For the
calibration with core overshooting of HerA,
we have not recomputed specific sets of models
to undertake another
minimization. We fit the
observable constraints within error boxes by adjustments of
the other modeling parameters having the values obtained without
overshooting. The larger value obtained for the age (+250Myr)
results from the larger
amount of nuclear fuel available in the overshoot convection core.
|Figure 1: Evolutionary tracks in the H-R diagram for models of HerA, & B without overshooting (full), with overshooting of for the convective core of HerA (dot-dash-dot) and with a larger metallicity of 0.05 dex for HerB (dotted). The open squares correspond to the locus of HerB assuming this larger metallicity. Dashed rectangles delimit the uncertainty domains. Top panel: full tracks from ZAMS. The "+'' signs denote 1Gyr time intervals along the evolutionary tracks. Bottom left and right panels: enlargements around the observed HerA & B loci.|
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For our HerA model without overshooting, we have computed a
set of adiabatic frequencies of the normal modes
in the frequency range 300 to Hz.
The model corresponds to an evolved star which has burnt all the
hydrogen core and presents a radiative core and a convective
envelope starting at
). At age
the convective core which is present during the main sequence evolution
has disappeared about 440Myr ago. A zone of varying chemical
gradient was formed between the outer edge of the initial convective
core and the center. This -gradient gives a rapid
variation of the sound speed and a large value of the maximum of
frequency N (e.g. Unno et al. 1989):
Looking at the set of computed frequencies, we see that for each degree, the oscillation spectrum is no longer composed of two separated sets of modes with acoustic (p-modes) and gravity (g-modes) behavior as in solar-like stars but it is a complicated superposition of these two sets. Some of these waves have a dual character as they behave like pressure waves in the envelope of the star and gravity waves in the core. This greatly complicates the classification of the non-radial modes.
The distinction between the g- and p-modes can be made by considering their normalized integrated kinetic energy (or inertia) (e.g. Christensen-Dalsgaard & Berthomieu 1991). The energy of the p-modes does not depend on the degree. Figure 3 plots this quantity as a function of the frequency for . The g-modes have a much larger energy than the p-modes. It appears that for the modes all have both energy in the core and in the envelop, so they are mixed modes. Figure 4, upper panel, shows the distribution of kinetic energy density of two of these modes along the radius of the star. For we have a succession of modes with energy either in the core or in the envelope with a few mixed modes as seen in Fig. 4, lower panel.
|Figure 2: Propagation diagram: profiles as functions of the nondimensionned stellar radius, of Brunt-Väisälä frequency N and Lamb frequencies for degrees (dotted), (dot-dash) and (dot-dot-dot-dash). The horizontal lines correspond to the two modes Hz and Hz, the first one with gravity behavior and the second one with acoustic behavior. Figure 4 shows their eigenfunctions. The full line indicates the region of the star where the modes are propagative.|
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|Figure 3: Logarithm of the normalized integrated kinetic energy of the modes as a function of frequency (in Hz) for modes of degree (full), (dashed), (dot-dash), (dotted). Note that all the points corresponding to p-modes are on the full line.|
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|Figure 4: Density of kinetic energy of two modes with consecutive radial orders for (upper panels) and (lower panels) as a function of the normalized radius ( is the density and the displacement). Note that the modes are mixed modes while the modes are alternatively g- and p-modes.|
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These values are to be compared to the observations of
Martic et al. (2001) who derive a value around Hz
from the construction of an echelle diagram.
Figure 5 plots the large differences
with respect to frequency.
Except for modes ,
the points corresponding to high frequency modes
are close to the same flat curve around Hz,
with oscillations due to the rapid variation of the adiabatic exponent
helium ionization zone. As in the Sun, the g-modes have small amplitudes at
the surface and thus they will be hardly observable.
Therefore it is probable that the echelle diagram
used by the observers to extract the frequencies, taking
account of their asymptotic distribution, will work for
not for the mixed modes .
The fit of the small differences:
|Figure 5: Frequency differences between acoustic modes of same degree and consecutive radial order as function of the frequency (in Hz).|
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|Figure 6: Variation of the small differences d02 (full dots) and (open dots), as a function of the frequency (in Hz). The full and dotted lines represent their linear fit.|
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The calibrated model of HerA calculated with overshoot of its convective core, has a larger radius inducing a smaller value of as seen in Table 4.
We are aware that our models suffer from the absence of diffusion of chemicals species. We do not introduce it as we do not have a satisfactory description of the physical process which acts against the too large efficiency of the gravitational settling in the envelope of a main sequence stellar model with mass larger than i.e. without a significant outer convection zone (e.g. Schatzman 1969; Turcotte et al. 1998). The derived metallicity of HerA could not be representative of the initial mixture that formed HerA & B. Because of the large difference in mass between the components, the diffusion has been more efficient in HerA than in HerB and therefore the adopted metallicity for HerB needs to be increased. As the mass of HerB is close to the solar one, we can safely assume that the change in metallicity has occurred at the same rate in the Sun and in HerB. From ZAMS to the age Myr, the metallicity of the Sun is increased by about 0.05dex. Using Magain's (1987) formula, an increase corresponds to an increase of +75K in effective temperature. Figure 1 shows that the locus (open square) of HerB in the HR diagram with larger by 75K is closer than before to the evolutionary tracks. To go further one needs a measurement of the metallicity of HerB.
it was not possible to obtain simultaneous satisfactory adjustments,
within the error boxes, for both components.
Realistic solutions are found with
indicates that the suspected duplicity of HerA is perhaps real.
In such a case the hypothetical unseen
component Hera will be less massive than previously announced,
and will be a brown dwarf or a giant planet.
Though Table 4 exhibits insignificant small
differences between mixing length parameters of HerA & B,
we obtained values of the order of unity,
as expected with the Canuto & Mazitelli (1991, 1992)
The differences between the C95
and our calibrations mainly result from the difference in distance.
In the HR diagram the locus of our HerA model is found
in the Hertzsprung gap soon
after the main sequence, as expected for a sub-giant star.
In this part of the HR diagram, the effective temperature of a star model
varies rapidly with respect to time.
So the age is very sensitive to small changes of physics or of
modeling parameter. The small uncertainty we give for the
age corresponds to the crossing time of the error box in effective temperature.
It is valid only
with the physics we used and with the central values of other
In their study C95 obtained a value for the age:
Gyr close to ours. We emphasized the fact that
the crossing time of the C95 error box
in effective temperature (K) is about twenty times smaller
than their estimated accuracy in age.
Table 5 lists the ages, the radii, the effective temperatures and the large differences of HerA models computed with modeling parameters within their error bars, but for mixing-length parameters. The age of the model Ag- (respt. Ag+) corresponds to the time elapsed from ZAMS to the instant where the effective temperature crosses the left (respt. right) limit of the error box in the HR diagram, namely (respt. ). The ages of other models listed in Table 5 corresponds to the time elapsed from ZAMS to the instant where the effective temperature crosses the central value . The variations of reflect the differences in radius and mass of star models. Table 5 shows that one derives unrealistic precision for the modeling parameters by defining their accuracy in such a way that any combination of them, within their error bars, will provide models of HerA & B within the observational constraints. Owing to the weakness of the observing material we do not attempt to derive the accuracy of the modelling parameters using the method developed by Brown et al. (1994). A more sensitive modeling parameter is the age due to the fast post main-sequence evolution of HerA. The accuracies of modeling parameters listed in Table 4 may be optimistic. They mean that, for a value of any modeling parameter within its interval of accuracy, one can find a set of other modeling parameters, each of them within its own accuracy limit, in such a way that they will provide models of HerA & B within the observational constraints.
The adiabatic oscillation spectrum of HerA is found to be a complicated superposition of acoustic and gravity modes. Some of these waves have a dual character. This greatly complicates the classification of the non-radial modes. For the modes are mixed modes with both energy in the core and in the envelope, they are mixed modes. For there is a succession of modes with energy either in the core or in the envelope with a few mixed modes. This will have an implication for the properties of the echelle diagram used by observers which will have a smooth behavior only for the modes . The large difference is found to be of the order of Hz close to the preliminary value derived from observations by Martic et al. (2001).
Six years ago, in the conclusion of their study, Chmielewski et al. (1995) requested " to go further, (i) a better photometry of component B and (ii) the exact value of the parallax''. Our work shows that the calibration of Her remains a difficult task, even with the disposal of improved observational data. Among the binaries to be calibrated with some confidence, Herculis is one of the most interesting owing to the difference of evolutionary state of components resulting from their mass difference. Though difficult, Her is a reachable target for modern spectroscopic and photometric apparatus. Her deserves interest to improve the accuracy of the modeling constraints hence, of stellar models and seismological analysis. It is all the more desirable because seismic observations of HerA will hopefully give new constraints.
We thank M. Martic for having directed our attention to the opportunity to revisit the calibration of the Her binary system and to undertake the seismological analysis of the brightest component. We would like to express our thanks to the unknown referee, for helpful advises. This research has made use of the Simbad data base, operated at CDS, Strasbourg, France and of the WDS data base operating at USNO, Washington, DC, USA. This work has been performed using the computing facilities provided by the OCA program "Simulations Interactives et Visualisation en Astronomie et Mécanique (SIVAM)''.