331
CALCULATING THE MASS OF HORIZONTAL-BRANCH STARS WITH HIPPARCOS
Klaas S. de Bo er, Michael Ge ert, Hans-Joachim Tucholke, Jelena H.K. Schmidt
Sternwarte, Univ. Bonn, D-53121 Bonn, Germany
of UV sp ectrophotometric capabilities namely the ABSTRACT
IUE satellite .
Horizontal-branch stars in globular clusters turned
Surface gravity, log g . The gravityat the surface
out to have a mass of '0.4 M , which followed from
of the star in the layer with optical depth =1 is
T and log g as derived from photometry, Balmer
e
notoriously dicult to determine.
line sp ectroscopy plus Balmer pro le mo dels , the lu-
The continuum sp ectral energy distribution de-
minosity of these stars from UV + visual sp ectropho-
p ends somewhat on log g , but not in a decisiveway.
tometry, and the distance to the globular cluster.
Fitting mo del continua gives only crude values for
Such a mass value is in contradiction with values from
log g , and the log g values dep end then strongly on
the theory of evolution of stars.
the quality of the mo del.
The Balmer lines are formed near the surface of
Hipparcos parallaxes to a few eld-HB stars have
the star and the standard metho d is to compare the
b een used to do the same analysis. The input data
observed shap es of the Balmer lines with Balmer pro-
are T and log g from the literature, the luminos-
e
les calculated with mo dels. Normally, a range of T
e
ity from IUE + visual sp ectrophotometry, and the
and log g will t well to a given pro le. Therefore,
Hipparcos distance. Here the mean of the masses is
the sp ectro-photometrically derived T is used to x
e
found to b e M =0:38 0:07 M .
HB
log g .
The masses determined for globular cluster HB stars
and for eld HB stars agree with each other, but de-
Luminosity, L . The apparent luminosity or the
HB
viate from stellar evolution theory. The cause for the
integral over the observed sp ectrum l , can b e ob-
HB
discrepancy is discussed. The most likely cause is
tained from sp ectrophotometry covering at least the
problems with the determination of the gravity from
maximum of the sp ectral energy distribution for HB
the comparison with theoretical Balmer pro les. An
stars the visual and the UV . If the distance is known,
unlikely increase of the distance to the globular clus-
l can b e translated to the value of L .
HB HB
ter by 0.4 mag would bring the mass of its HB-stars
up to 0.6 M . However, the mass of the eld HB
Radius, R . No direct access to radii of HB stars
stars stays xed at 0.38 M , based on the Hipparcos
is p ossible. Radii follow from the equations when
parallaxes.
the other parameters have b een determined, or radii
follow from mo dels, given a sucient set of observa-
tionally determined surface parameters.
Key words: horizontal-branch stars; stellar mass;
globular clusters; Cepheids.
Mass, M .
HB
The mass can b e `measured', only if the HB star is
part of a binary, of which then all further parameters
1. BASIC STELLAR PARAMETERS
must b e known.
The mass can b e calculated once the other relevant
parameters have b een determined. The formalism is
The basic parameters describing a star are mass M ,
given in the equations b elow. It requires, however,
luminosity L, radius R , surface temp erature T , and
that the distance to the star is known since the cal-
e
surface gravity log g . These parameters can be de-
culation is based on the luminosity L .
HB
rived in various ways.
The mass may follow from a comparison with mo d-
els for HB stars. This comparison uses the lo cation
in the T versus log g diagram. Such determinations Temp erature T .
e e
are not unique, since evolution of HB stars follows From a sp ectral index. The temp erature of the
tracks whichmayintersect in the mentioned param- stellar surface can be derived from the slop e of the
eter space. sp ectral energy distribution in a well cho osen wave-
Evolution theory predicts that a HB-like star has a length range.
Helium core of ab out 0.5 M , surrounded by a Hy- From the overall sp ectral distribution based on ac-
drogen envelop e of up to 0.9 M , but whichmay also curate mo del atmosphere calculations. For the hotter
be vanishingly thin. stars this p ossibility exists only since the availability
332
Table 1. Field Horizontal Branch stars: Hipparcos paral laxes and distances.
Name V B V A d d ,d M M
max
V min V V
mas p c
HD 86986 7.99 +0.12 0.09 3.78 0.95 265 355,210 +0.79 0.55
HD 109995 7.62 +0.04 0.00 4.92 0.89 205 250,170 +1.08 0.40
HD 130095 8.13 +0.08 0.31 5.91 1.08 170 205,145 +1.68 0.40
HD 139961 8.85 +0.10 0.31 4.50 1.19 220 300,175 +1.81 0.60
HD 161817 6.96 +0.16 0.06 5.81 0.65 170 195,155 +0.72 0.25
M given is due to the uncertainty in the parallax only.
V
Table 2. HB star parameters.
a b b c
Name log l log L T log g log log log M
HB HB HB
e
4
cgs L K cgs g /T M M M
HB HB
e
HD 86986 7.768 1.57 7900 3.1 0.10 0.32 0.32 0.48
HD 109995 7.666 1.45 8300 3.15 0.20 0.44 0.36 0.37
HD 130095 7.679 1.28 8800 3.4 0.15 0.49 0.31 0.32
HD 139961 7.788 1.39 8750 3.3 0.20 0.48 0.44 0.33
HD 161817 7.402 1.58 7500 2.95 0.05 0.37 0.15 0.42
R
a
l = I d from IUE and scanner data.
HB
b
T and log g are b est values from the literature.
e
c 4
log g /T is total error of the pro duct of g and T .
e e
For details see de Bo er et al. 1997a .
3. ABSOLUTE MAGNITUDES 2. INTERRELATIONS
The parameters describing the surface of a star in
relation to the Sun are given by the following equa-
Parallaxes for eld HB stars from the Hipparcos In-
tions:
put Catalogue Turon et al. 1992 as determined by
Hipparcos ESA 1997 have b een used to calculate
g M R
HB HB HB
the absolute magnitudes for the eld stars de Bo er et
log = log 2 log 1
al. 1997a . A subset of the data is discussed here and
g M R
presented in Table 1. Figure 1 shows that the values
L R T
of M spread around the lo cation of the theoretically
HB HB HB
V
log = 2 log + 4 log 2
determined horizontal branch.
L R T
It is p ossible to eliminate the radius, so that the com-
The observed absolute magnitudes t well to the the-
bined equation has only the mass, the temp erature
oretical lo cation of the Zero Age HB from mo dels see
and gravity, and the luminosityasvariables. Thus:
Figure 1 . Note that the red HB app ears from the
Hipparcos eld stars at M ' +1 mag.
V
M
HB
= log
M
log g 4 log T + log l + 2 log d +15:11 3
HB HB HB
where l is the integral of the extinction cor-
HB
rected sp ectral energy distribution as measured at
R
4. MASS DETERMINATIONS
the Earth, I d, and d is the distance of the star
in p c the parallax =1=d, with in arcsec . The
numerical constant is determined as:
The programme of investigations of the core Helium
18
log 4 + 2 log 3:09 10 log L +
burning sdB and HB stars carried out in Bonn aims
not only at characterising the stars, but also at in-
4 log T log g = 15.11
vestigating the spatial distribution in the galaxy, in-
cluding the determination of the orbits de Bo er et
33 1
al. 1988, Mo ehler et al. 1990, Theissen et al. 1993, with solar values L =3:85 10 erg s , T = 5800
Theissen et al. 1995, Schmidt et al. 1997, Colin et K, and log g =4:44 in cgs units. It is Equation 3
al. 1994, de Bo er et al. 1997b . In the course of the which can be used in connection with the various
pro ject it was realised that a veri cation of the mass observationally determined parameters to calculate
of the HB stars was urgently needed. the mass of the stars.
333
6. MASS OF GLOBULAR CLUSTER STARS
The investigations of HB-like stars in clusters is avail-
able from several studies see Table 3 . The trend is
that the hot stars come out with masses in line with
mo dels for the sdB stars, the co ol stars come out with
masses clearly b elow the exp ected values. However,
pAGB stars de Bo er 1985 have masses of ' 0:5M
de Bo er 1987 .
The value for the mass of the cluster HB stars is com-
pletely in line with the results for the eld horizontal-
branch stars. This must mean two things: a the
cluster stars and the eld stars b ehaveinavery sim-
ilar manner, in spite of p ossible di erences in, e.g.,
age, original mass, or original metallicity; and b
the problem with the masses lies in asp ects of the
analysis of the data.
Figure 1. The absolute magnitude of the eld HB stars
from de Boer et al. 1997a is presentedtogether with the
theoretical HB [Fe/H]= 1.03, Y =0.252 from Dor-
HB
7. IS THE GRAVITY DETERMINATION AT
man 1992 .
FAULT?
Of the data involved in the full parameterisation of
HB stars according to Equation 3, the temp erature
5. MASS OF FIELD STARS USING
and the integrated sp ectral intensity are all quite
HIPPARCOS PARALLAXES
accurate. The uncertainty of the distance is solely
based on the uncertainty in the parallax. The mass
of the stars, which follows if the gravityis xed, is
supp osed to come out in line with evolutionary exp ec-
The mass for the eld HB stars can now b e calculated
tations. HB stars are thoughttohave a mass ranging
using the relations given ab ove. They are plotted in
from 0.5 for hot stars to ab out 0.6 for HBA stars
Figure 2. For that, further parameters for these stars
near T = 7500 K.
e
have b een taken from the literature. The relevant
parameters and the results are given in Table 2.
From b oth the eld stars with temp eratures b etween
7500 and 9000 K and the cluster stars with temp er-
atures b etween 10 000 and 20 000 K the determined
For these eld HB stars, whichhave temp eratures in
masses are M ' 0:4 M . The only wayto solve
HB
the range of 7500 e this discrepancy is to admit that the gravities from erage of the individual values for the masses is 0.38 sp ectroscopic analysis are to o small see also de Bo er 0.07 M . Note that the uncertaintyin each indi- et al. 1995 . vidual value is large. Taking the argument in the other direction, if one assumes the mass of the HB stars is 0.50 to 0.60 M indeed p ost AGB stars in globular clusters have masses between 0.54 and 0.56 M ; de Bo er 1987 , then the data for the eld HB stars and the stars in NGC 6397 can b e brought to consistency if the loga- rithmic gravities of the HB stars are made larger by 0.2. 8. PROBLEMS WITH ATMOSPHERE MODELS? In studies of hot Population I stars one has noted that the masses derived from sp ectroscopic data come out smaller than those exp ected from other lines of ar- gument. Recently Lanz et al. 1996 argue, that the gravities may b e at fault. They show that if one do es Figure2. The mass of the eld HB stars is plotted against not use fully blanketed non-LTE mo dels for the cal- temperature from de Boer et al. 1997a , together with culation of the shap e of the Balmer pro les, the sp ec- the theoretical HB from Dorman 1992 . The numeric troscopically derived gravities come out to o small and average of the individual values is 0.38 M . The crosses in consequence the calculated masses are to o small as represent the mass for some of the HB stars investigated well. With the HB stars wehave tried to make clear in NGC 6397 de Boer et al. 1995 . that to o small masses are found when sp ectroscopic gravities are used. 334 Table 3. Masses derived for globular cluster horizontal branch stars. Cluster d range of T mass ref. e kp c K M M15 10.0 20000 { 30000 0.7: Mo ehler et al. 1995 10000 { 20000 0.3: NGC 6397 2.0 8000 { 12000 0:39 0:05 de Bo er et al. 1995 NGC 6752 4.0 25000 { 30000 0:50 0:04 Mo ehler et al. 1997 12000 { 20000 0:30 0:06 Non-LTE fully blanketed mo dels have to b e applied Mo ehler, S., Odenkirchen, M., Schmidt, J.H.K., to the conditions of HB star atmospheres to investi- Theissen, A., 1994, A&A, 287, 38 gate if they lead to di erent Balmer pro les such that de Bo er, K.S., 1985, A&A, 142, 321 gravities and thus mass values arrived at in sp ectro- de Bo er, K.S., 1987, in Second Conf. on Faint Blue scopic analyses are more in line with masses from Stars, IAU Coll. 95, eds A.G.D. Philip, D.S. evolution theory. Hayes, J.W. Lieb ert; Davis Press, Schenectady, p. 95 de Bo er, K.S., Heb er, U., Richtler, T., 1988, A&A, 9. CONSEQUENCES OF A REVISED 202, 113 GLOBULAR CLUSTER DISTANCE SCALE de Bo er, K.S., Schmidt, J.H.K., Heb er, U., 1995, A&A, 303, 95 Feast & Catchp ole 1997 have suggested that the de Bo er, K.S., Tucholke, H.-J., Schmidt, J.H.K., p erio d{luminosity relation for Cepheids has to be 1997a, A&A, 317, L 23 changed compared to the one of Caldwell & Laney de Bo er, K.S., Aguilar S anchez, Y., Altmann, M., 1991 . The consequence would b e, that the LMC is Ge ert, M., Odenkirchen, M., Schmidt, J.H.K., farther awayby 10 p er cent, that its RR Lyrae are far- Colin, J., 1997b, A&A, submitted ther away, and that the globular clusters are farther away. In consequence, the globular cluster NGC 6397 Dorman, B., 1992, ApJS, 81, 221 would b e farther away, making the luminosities of its ESA, 1997, The Hipparcos and Tycho Catalogues, HB stars larger by a factor 1.2. Inserting this in the ESA SP{1200 mass determination the mass of those stars b ecomes Feast, M.W., Catchp ole, R.M., 1997, MNRAS, 286, 20 per cent larger. With this change, the HB star L1 masses in NGC 6397 now b ecome di erent from those in the eld, the latter b eing based directly on Hip- Lanz, T., de Koter, A., Hub eny, I., Heap, S.R., 1996, parcos parallaxes, all other parameters b eing based ApJ, 465, 359 on identical analysis metho ds. Mo ehler, S., Heb er, U., de Bo er, K.S., 1990, A&A, 239, 265 The change of the distance scale byFeast & Catch- Mo ehler, S., Heb er, U., de Bo er, K.S., 1995, A&A, p ole thus leads to a discrepancy within the research 294, 65 of HB stars. We feel this is an additional argument against the prop osed p erio d{luminosity revision. Mo ehler, S., Heb er, U., Rupprecht, G., 1997, A&A, 319, 109 Schmidt, J.H.K., Mo ehler, S., Theissen, A., de Bo er, 10. CONCLUSIONS K.S., Heb er, U., Greb el, E.K., 1997, A&A, to b e submitted Theissen, A., Mo ehler, S., Heb er, U., de Bo er, K.S., The ongoing in-depth studies of samples of 1993, A&A, 273, 524 horizontal-branch like stars have con rmed existing Theissen, A., Mo ehler, S., Heb er, U., Schmidt, ideas of evolution in some resp ects, but have uncov- J.H.K., de Bo er, K.S., 1995, A&A, 298, 577 ered discrepancies in other resp ects. The low masses 4 found for stars with T between roughly 10 and Turon, C. et al., 1992, The Hipparcos Input Cata- e 4 logue, ESA SP{1136 2 10 K p oint at problems with the gravities derived from sp ectroscopy. However, eld HB stars and clus- ter HB stars b ehave identically. Adopting a revised P/L scale increases the number of problems in the eld of HB star research. REFERENCES Caldwell, J.A.R., Laney, C.D., 1991, IAU Symp. 148, p. 249 Colin, J., de Bo er, K.S., Dauphole, B., Ducourant, C., Dulou, M.R., Ge ert, M., LeCampion, J.-F.,