Download Duplicity and masses

J. Phys. IV France 1 (2001)
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c EDP Sciences, Les Ulis
Duplicity and masses
F. Arenou 1 , J.-L. Halbwachs 2 , M. Mayor and S. Udry 3
1
2
3
Observatoire de Paris, UMR 8633 CNRS
Observatoire Astronomique de Strasbourg, UMR 7550 CNRS
Observatoire de Genève, 51 chemin des Maillettes, CH–1290 Sauverny, Suisse
Abstract
1
INTRODUCTION
Duplicity is a key question in astrophysics, and there is no doubt that the Milky Way would
look entirely different if stars were single. Binaries provide direct measurements of stellar or
galactic quantities and several astronomical phenomena occur only in binary systems. The
binary rate, the distribution of mass ratio between components inform on stellar formation
in relation with the place of formation. Stellar physics needs masses, luminosities and radii
obtained through the studies of binaries. Galactic physics benefits also from this studies, e.g.
the galactic potential can be tested using wide binaries, and the chemical evolution depends
on binaries through the supernovae Ia process.
Remembering the large number of doubles detected or suspected by Hipparcos, so large
that the need of an adequate classification emerged – five different categories in the Double
and Multiple Stars Annex – one could simply consider that GAIA will rescale the number of
double stars to a Catalogue of one billion stars. But there is something more important, and
what will be learned from GAIA may be easily summarized: GAIA will be a complete on-orbit
observatory since it will provide homogeneous astrometric, photometric, and spectroscopic
measurements; GAIA will at last provide large and unbiased samples of all kind of binaries,
for an extremely large range of periods and mass ratios, at all evolution stages.
Some of the questions which will be solved by GAIA may be mentioned. First, the parallaxes
and proper motions will allow to distinguish between optical doubles and physical binaries. The
binary rate will be known down to low mass companions and in particular may allow to study
the low mass companion rate at large separation. The mass ratio distribution will be shown
either random or as a formation product. The formation of close versus wide binaries will be
better known. For large separations, the mass ratio distribution and binary rate will allow to
know the evolution process of such systems. For multiple system, the hierarchy and relative
inclination will be accessible. For contact binaries, the accurate knowledge of the masses will
allow to address the question of mass transfer, evolution and space distribution. . .
The GAIA study report [1] describes largely the science case concerning binaries. After a
brief remainder from this report concerning the general GAIA capabilities, we will mention a
few complementary applications.
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JOURNAL DE PHYSIQUE IV
GAIA CAPABILITIES
GAIA will discover about 60 million binaries, that is 3/4 of the nearby binaries brighter than
20m , and about 35% of them at 1 kpc. The final mission product could be more than 107
astrometric binaries, 106 eclipsing binaries, 106 spectroscopic binaries, 107 resolved binaries
within 250 pc and 10, 000 stars with masses within 1%.
Due to the huge number of detected binaries, even short-lived evolution stages will be
reachable. And the duplicity will be studied in the field, in clusters or star-forming regions,
where a majority of young stars are in binary systems [2]. Even at large separations, the
high precision of the obtained three-dimensional velocity will permit to locate easily ‘common
proper motion’ pairs.
Orbits of nearby visual binaries can be recovered up to 40 yr periods, and precise masses
will be obtained for short enough periods, with the astrometry alone or in combination with
the spectroscopic data. This will allow to put severe constraints on stellar evolution models,
where a 2% precision on masses is needed [4].
The impressive number of expected astrometric binaries is due to the extreme GAIA sensitivity to non-linear proper motions. As an example, a pair of one solar-mass stars with a 1 000
yr period would have an acceleration of 10µas.yr−2 (i.e. significant for the brightest stars) at
300 pc [3].
3
ON-BOARD DOUBLE STAR DETECTION
The small angular size (37 mas) of the pixels in the GAIA astrometric focal plane, with a PSF
of FWHM=2.2 pixels, will allow to easily detect close (0.1”) visual binaries. In the Astrometric
Sky Mapper (ASM) where the on-board detection takes place, the sample size is 2 × 2 pixels,
and the readout noise higher than in the astrometric field [5], but the detection of double stars
in ASM is of interest for at least two reasons: a) the detection defines the patches which are
observed in the astrometric field, and an undetected duplicity would render the astrometric
and photometric data reduction much harder if the patches are too small; b) a subsample of
well-behaved (single) stars will be needed for the data reduction.
Fig. 1. Detection limit of double stars in the Astrometric Sky Mapper, from [6]
The Figure 1 shows the minimum separation between components for which a duplicity de-
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tection could be done, and it also shows where larger patches would be needed. This theoretical
minimum separation will most probably not be reached by the on-board detection algorithm.
Moreover spurious double star detection may occur since the PSF is larger for a red star, and
this may be attributed to duplicity as nothing is known about the star’s colour when it enters
the sky mapper.
4
PHOTOCENTRIC BINARIES
Apart from the 16 CCD dedicated to astrometric measurements, each GAIA astrometric instrument also includes 4 broad bands photometers (BBP) primarily used for chromaticity
corrections. However, these BBPs could also be useful for the duplicity detection: the photocentre of an (unresolved) double pair will not be identically positioned in each of these BBP,
provided that the two companions do not have the same color. GAIA could then introduce a
new category of photocentric or ‘color-induced’ binaries in the terminology of [7, 8].
Using only the BBPs of the Astro 1 instrument, where the along-scan resolution√is the same
as in the astrometric fields, the expected precision in one BBP will then be about 2 × 16 less
precise than the final astrometric precision. Assuming this precision for a G-type primary, and
projecting randomly the separation between components for each of the 67 transits expected on
the average during the mission, main-sequence secondaries of various periods and magnitude
differences have been simulated in [9] for different distances. Then the separation ρ, position
angle, magnitude and color of each components have been recovered by least-square.
As may be seen in Figure 2, where are indicated the solutions with ρ > 3σρ , GAIA may be
able to resolve the brighter binaries down to the milliarcsec level, provided that the magnitude
difference is in the range 1 to 3 mag.
V magnitude difference = 1 mag
V magnitude difference = 6 mag
100
Separation between components (mas)
Separation between components (mas)
100
10
0.1 yr
0.5
1
5
10
50
100
500
1000 yr
1
0
12
14
16
Primary V apparent magnitude
18
10
0.1 yr
0.5
1
5
10
50
100
500
1000 yr
1
0
12
14
16
Primary V apparent magnitude
18
Fig. 2. Simulation of 3σ detection of main-sequence binaries in the BBP, for a 1m magnitude difference
between components (left) and 6m (right). The oblique line is a empirical fit, and gives approximately
the minimum detectable separation between components as a function of the apparent magnitude of the
primary.
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MASS RATIO DISTRIBUTION WITH GAIA SPECTROSCOPIC BINARIES
Thanks to its spectroscopic measurements, GAIA will obtain radial velocity curves up to about
magnitude 17. In what follows, we will assume that GAIA will achieve a 2.3 km/s precision
per field transit at G = 14. This is slightly lower than what is predicted by [10] for a 0.5Å/pix
dispersion spectrum for late-F to early-M stars.
From our experience in orbit computation, we assume that an orbit will be obtained for
the single–lined spectroscopic binaries (SB1) with
p
K1 1 − e2 sin i > 5 × 2.3 = 11.5 km/s
if this orbit is well covered by the observations (where K1 is the semi-amplitude of the radial
velocity, e the eccentricity and i the orbital inclination). In practice, if the primary is G0V,
the reflex orbit due to a M0V secondary of period smaller than 1.5 year will be computable;
at the transition between brown and red dwarfs (M2 = 0.08), the orbit will be obtained if the
period is smaller than 3 days. For double-lined binaries (SB2) with components with nearly
equal brightness, the condition above is no more valid since the spectral lines are blended; in
practice, the limit in K1 is six times larger than for SB1, giving the condition K1 > 69 km/s.
In what follows we will study how GAIA will contribute to the knowledge of the mass-ratio
distribution. Many binary studies suffer from biases, since the definition of an unbiased sample
is far from obvious. The situation will radically change with GAIA. If we consider solar type
stars brighter than 14, the volume from which a complete sample can be extracted is limited by
the apparent magnitude, not by the astrometric precision on distance. This limiting distance
is about 500 pc for G stars, while it will be 150 pc for K stars, neglecting the interstellar
extinction.
In order to compute how many stars are included in this volume, one may extrapolate
what is known from the solar neighborhood. The Gliese Catalogue contains 10 spectroscopic
binaries of period smaller than 4 years, closer than 21.7 pc with declination above −15◦ . Half
of these are F7-G, the other being K stars. Thus, for each of these two categories, the density
is about 1.85 10−4 SB/pc3 whether we assume a constant volumic density or 0.0054 BS/ pc2
for a constant surfacic density. This would give between 4 240 and 96 900 G-type SB of period
less than 4 years within 500 pc, and between 380 and 2 620 K-type SB within 150 pc.
For a given mass-ratio q, a small inclination of the orbit implies a smaller K1 . Assuming
isotropic orientations of the orbit, half of the inclinations are above 60◦ , so half of our SB
sample will have
p
11.5
K1 1 − e2 >
= 13.3 km/s
sin(60◦ )
and the limiting period corresponding to a 50 % detection rate will be
3
212.9
q3
P (0.5) =
M1
days
13.3
(1 + q)2
For each limiting period P (0.5), the proportion of binaries with P < P (0.5) in our sample
is computed. A constant distribution of log P is assumed, from 3 days until 4 years. We
consider a 0.8 M primary, at the transition between G and K, and we assume that q will be
directly known for the SB2, when q > 0.7, provided that the radial velocity curves of both
components can be obtained. For q < 0.7, the mass-ratio will be obtained in combination
with the astrometric orbit, although it could be deduced from the mass function using the
iterative method from [11]. The astrometric orbit will have a minimum reflex semi-major axis
a1 (D) = P (0.5)days ×13.3×103 /(212.9×Dpc ×365.252/3 ) mas at the limiting distance D = 150
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or 500 pc. Getting an astrometric orbit will easily be obtained with GAIA if a1 > 10σ$ ≈ 0.1
mas.
The results are indicated in Table 1. The largest achievable periods, for q near the SB1–
SB2 transition, is larger than 1 year. For q > 0.2, there will be enough stars to get a good
determination of the mass ratio distribution, provided that the selection effects are corrected
(as may be seen from the variations of P (0.5) with q). For mass ratio of the same order of
the flux ratio, the astrometry will prove less useful, since the motion of the photocentre will
be small. Then, above q > 0.7, the determination of the mass-ratio will effectively depend on
the GAIA capability of deconvolving the primary and secondary spectra.
Table 1. Half-sample limiting period, proportion of stars and limiting semi-major axis as a function
of mass ratio for GAIA spectroscopic binaries.
q
P (0.5) days
prop(P ) %
a1 (150pc) mas
a1 (500pc) mas
0.1
2.7
0.02
0.006
0.2
18
29
0.15
0.045
0.3
52
46
0.43
0.13
0.4
107
62
0.87
0.26
0.5
182
66
1.49
0.45
0.6
277
73
2.26
0.68
0.7
389
79
3.18
0.95
1.0
3.8
Assuming this will be possible, the mass ratio distribution for q > 0.2 will be obtained using
the GAIA spectroscopic capabilities, which will allow to study its variation as a function of
spectral type, population, formation place, etc. Now, below q = 0.2, the mass-ratio will hardly
be known through astrometry of resolved components, since a magnitude difference between
components larger than 10 mag (corresponding to q < 0.1) will hardly be observed. Smaller
mass ratios will be obtained with the astrometric binaries, though here an assumption on the
mass of the primary will be needed.
6
ECLIPSING BINARIES AND LIGHT-TIME TRAVEL EFFECTS
One unexpected result of Hipparcos has been the number of new eclipsing binaries, even bright
(7m ) ones, showing that the completeness of eclipsing and contact binaries was not assured.
Concerning GAIA, several million eclipsing binary light curves will probably be obtained.
The multi-epoch photometry (between 60 to 300 measurements) and the individual precision
will be of the order of 0.001 mag at G = 14 to 0.05 mag at G = 20. Combining the G
band measurements to the 4 broad bands and the 11 medium bands and to the spectroscopic
measurements will allow to obtain masses and radii to better than one per cent for thousands
of binaries [10]. The colour and fundamental parameters (luminosity, temperature) will be
obtained for total eclipses.
Short periods eclipsing binaries are especially of interest for contact binaries, and more
generally for interacting binary systems. As the GAIA sample will be unbiased down to faint
magnitudes, W UMa-type or Algol-type contact binaries will be common, and their mass
distribution, structure, evolution, kinematics and distribution in space will be better known,
combining the radial velocity measurements with the astrometric measurements.
Another kind of duplicity detection may be done using eclipsing binaries (or other variable
stars with stable short-period variations) which host a long period companion. In this case,
the reflex motion of this ‘clock in orbit’ can be detected through a light-time travel effect. An
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1
1000
0.1
1 Msun
0.01
10
0.1 Msun
0.001
10-5
0.01
1
10 Mjup
0.0001
Radial velocity (km/s)
Light-time (day=173 A.U.)
100
0.1
0.1
1
10
100
Period (yr)
Fig. 3. Maximum amplitude for light-time travel (plain, left) and radial velocity (dash, right) as a
function of period, for a one solar mass primary, and a secondary of 1, 0.1 and 0.01 solar mass.
example of the combination of the Hipparcos astrometric data with the Hipparcos photometric
data and ground-based times of minima may be found with R CMa [12].
Compared to the radial velocity method, this method is sensitive to long periods (Figure 3)
and returns the same orbital elements (a1 sin i, P, T, e, ω), which would give the full orbital
elements when combined with the astrometric data, and the mass of the companion if the
primary mass is assumed. The detection would be obtained if an acceleration term is present
both in the astrometry and in the light curve, possibly complemented by ground-based light
curves. In the GAIA Catalogue, a light-time travel effect could then be tested for a 10 Jupitermass companion of period larger than 11 yr on more than 105 stars [13].
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