A&A 431, 1129-1137 (2005)
DOI: 10.1051/0004-6361:20041219
J. L. Halbwachs1 - M. Mayor2 - S. Udry2
1 - Observatoire Astronomique de Strasbourg (UMR 7550),
11 rue de l'Université, 67000 Strasbourg, France
2 - Geneva Observatory,
51 chemin des Maillettes, 1290 Sauverny, Switzerland
Received 3 May 2004 / Accepted 19 October 2004
Abstract
A sample of spectroscopic binaries and a sample of single planetary
systems, both having main-sequence solar-type primary components,
are selected in order
to compare their eccentricities. The positions of the objects in the
(
P.(1-e2)3/2, e) plane is used to determine parts in the
period-eccentricity diagram
that are not affected by tidal circularization. The
original eccentricities of binaries and planets are derived and compared.
They seem to be weakly or not at all correlated with period in both
samples, but two major differences are found:
(1) The tidal circularization of planetary orbits is almost complete for periods shorter than 5 days, but it is not visible when P.(1-e2)3/2is longer than this limit. This suggests that the circularization occurs rapidly after the end of the migration process and is probably simultaneous with the end of the formation of the planet. By contrast, we confirm that the circularization of the binary orbits is a process still progressing a long time after the formation of the systems.
(2) Beyond the circularization limit, the eccentricities of the orbits of the planets are significantly smaller than those of binary orbits, and this discrepancy cannot be due to a selection effect. Moreover, the eccentricities of binaries with small mass ratios are quite similar to those of all binaries with q<0.8. This suggests that the low eccentricities of exoplanet orbits are not a consequence of low-mass secondaries in a universal process.
These remarks are in favor of the idea that binaries and exoplanets are two different classes of object from the point of view of their formation.
Key words: stars: binaries: general - stars: binaries: spectroscopic - stars: planetary systems - stars: planetary systems: formation
It is well known that the orbits of the exoplanets with periods larger than 5 or 6 days have eccentricities significantly larger than those of the giant planets of the solar system. Several mechanisms were proposed to explain this feature, but, up to now, none is fully convincing. It was proposed that eccentric orbits could be a consequence of the dynamic evolution of systems initially involving several planets (Rasio & Ford 1996; Lin & Ida 1997; Ford et al. 2001; Papaloizou & Terquem 2001; Rice et al. 2003), but these models fail to produce the frequency of giant planets with semi-major axes smaller than about 1 AU. The giant planets close to their harboring stars are often assumed to be produced by migration within a disk (Ward 1997; Masset & Papaloizou 2003, and references therein), but this process hardly produces eccentric orbits (Papaloizou et al. 2001; Thommes & Lissauer 2003), although Goldreich & Sari (2003) and Woolfson (2003) leave some room for hope. Therefore, it is tempting to consider that the exoplanets are generated by the same process as binary stars (Stepinski & Black 2000). This implies that giant exoplanets were not formed by gas accretion onto a heavy rocky core, as usually assumed, but by an alternative process. They could come from disk instabilities (Mayer et al. 2002; Boss 2002, 2003), but inward migration in a disk is then invoked again to explain the short-period orbits; alternatively, planets could be generated by fragmentation of a collapsing protostellar cloud, via filament condensation and capture (Oxley & Woolfson 2004), or even exactly as stellar components in binary systems (see discussion in Bodenheimer et al. 2000, and references therein). However, these models may be efficient in forming massive planets or brown dwarfs, but not planets around 1 Jupiter mass or less.
Note that the binary formation models are not very satisfactory either (see the review by Tohline 2002). The large eccentricities of binaries are explained by fragmentation of collapsing cores and subsequent interactions between the forming stars (Bate et al. 2002; Goodwin et al. 2004), but, as for exoplanets, the simulations do not provide the high frequency of close systems. Moreover, statistical investigations on main sequence binaries (Halbwachs et al. 2003, Paper I hereafter) have shown that the close binaries (i.e. with semi-major axes less than a few AU) consist in two populations: one with large eccentricities and mass ratios less than 0.8 ("non-twins'' hereafter), and one with moderate eccentricities and nearly identical components ("twins''). Additionally, the twins are more frequent among short-period binaries than among the others. At first, these properties were derived from binaries with F7-K primary components, but they are also valid for M-type dwarfs (Marchal et al. 2003).
In the present paper, the period-eccentricity diagram is used to compare the exoplanets with the binary stars: our main purpose is to investigate if the properties of exoplanets may be considered as an extrapolation of the properties of binaries in the range of very low mass ratios. This would indicate whether the formation processes of these objects are similar. In the course of the paper, a few other points are also treated: (1) the correlation between the eccentricity and the period or the angular momentum; (2) the relation between the eccentricity and the metallicity of planets; (3) the original distributions of eccentricities for planets and for binaries, considering the twins separately. Comparisons between planets and binaries in the period-eccentricity diagram were already presented by Mayor et al. (2001), Mazeh & Zucker (2001), and Udry (2001), who concluded that planets and binaries are very similar when periods longer than 50 days are considered. However, their samples contained around 30 or 40 planets, and a many others have been discovered since. The question needs to be re-considered.
The interpretation of the period-eccentricity diagram is rather complex, and our investigations are based on the method presented in Sect. 2. Section 3 is dedicated to the binaries; we investigate if, additionally to the twins, other classes of mass ratio have specific distributions of eccentricities. A similar treatment is applied to exoplanets in Sect. 4. Binaries and exoplanets are compared in Sect. 5, in which we pay attention to the difference in the selection effects of both samples. The consequences of our results are discussed in Sect. 6.
We must pay attention to the fact that the periods (P) and the
eccentricities (e) of the objects are modified by tidal interactions,
especially when P is short.
As a consequence, the (P - e) diagram may schematically be divided into two
parts: the short periods, where the orbits are circular or have low
eccentricities, and the periods
longer than the circularization limit, hereafter called
(Mayor & Mermilliod 1984; Duquennoy & Mayor 1991).
Several theoretical models were proposed to derived
,
and the
treatment is not the same for
binaries (Zahn 1992; Hut 1981, 1982;
Keppens 1997, and references therein)
and for planets (Goldreich & Soter 1966; Trilling 2000).
Moreover, several physical processes are invoked, each of them depending differently
on the mass ratios of the systems.
Despite the complexity of the process, a few simple guidelines may be
drawn. First of all, the
efficiency of tides in modifying the orbits is very sensitive to the
distance between the components.
For a given system, the tidal torque depends on the orientation of the tidal
bulge and on the separation between the components, r. It varies as
1/r6 (Lecar et al. 1976).
Therefore, for systems differing only by
period, the transition from circularized orbits to orbits
practically unaffected by tides is a narrow strip in the (P - e)
diagram (see the simulations by Witte & Savonije 2002).
However, this does not mean that the systems with
may have
any eccentricity. For a given period the systems with eccentric orbits
have components much closer than the semi-major axis during a part of
the period, and below a
certain limit the orbit rapidly becomes circular. As a consequence,
the upper part of the (P - e) diagram is cleared even for
.
Note that the orbits that were originally eccentric
do not keep the same period when they are evolving towards e=0:
when the primary star is a slow rotator, the orbit is circularized keeping
the orbital angular momentum unchanged (Witte & Savonije 2002; Hurley
et al. 2002). Therefore, the semilatus rectum,
,
is conserved and it becomes the radius of the final
circular orbit. As a consequence, the final period is:
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Figure 1:
The mean value of the 1/r6 term of the tidal torque,
compared to
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In practice, however, a real sample is selected up to a maximum period
,
and not all the eccentricities are permitted in the
(
- e)
diagram. For a given
,
the eccentricities range from 0 to the
limit:
The comparison between different samples of orbiting
systems is based on the median eccentricities. Two approaches are used:
the first is a visual examination, and the second is a statistical test.
In the first approach,
the median eccentricity of each sample is derived as a function of
the period. In practice, for any period
,
the median eccentricity is derived from 6 systems taken among the periods
closest to Pi. Except near the limit of
the period range, 3 of these 6 systems have P<Pi and the 3 others have
P>Pi. The second approach
is the statistical test used in Paper I: the two samples are merged, and the
range of periods longer than
is divided into bins, each containing 12 systems, except
for the last one which may contain up to 23 systems. The common median
eccentricity is derived in each bin, and the systems below the median are
counted for one of the two samples. If this sample actually belongs to the same
statistical population as the other, the probability of getting any
count, P(k), obeys the hypergeometric distribution. The rejection
threshold of the null hypothesis, H0: "all the systems are equivalent from
the point of view of the eccentricities'' is then derived. When the count k is
less than half the population of the considered sample, the rejection
threshold of H0 in a two-sided test is
;
on the contrary, it is
when k is larger than the expected
number.
Note that setting the content of the bins to 12 systems is a bit arbitrary. This number is neither too small to derive a reliable median nor too large to have a nearly constant period distribution within each bin. Using other numbers close to 12 would give other results, but it was verified that the differences are not important.
The so-called extended sample of F7-K dwarf binaries selected in Paper I is used again. It consists of 89 spectroscopic binaries (SB) found in the solar neighborhood or in open clusters, with periods of up to 10 years. We already know that the twins have, on average, eccentricities smaller than the other binaries. However, before comparing binaries to exoplanets, it is worthwhile to see if the eccentricities of non-twin binaries depend on the mass ratios.
The mass ratios
of the SB in the sample have been fixed for 58 binaries, thanks to
the combination of the SB orbital elements with Hipparcos astrometric
observations, or with photometric sequences in the open clusters (Paper I).
For the other 31 SB, we derive intervals containing the actual mass
ratios. The minimum limits are computed from the mass functions;
the maxima are obtained differently for the nearby SB and for the
cluster SB: the former all have q < 0.65, since otherwise they would
be double-lined SB with known q, and the latter have limits
coming from their positions in the photometric sequence of the cluster.
In order to make visible a possible relation between the mass ratios
and the eccentricities, the SB are distributed in several groups:
(16 SB),
(20 SB),
and twins (27 SB); we still add a
group containing all the SB with
(62 SB, including the
36 already in the first two groups).
The SB are
plotted in the (
- e) diagram (Fig. 2), to
delimit the part of the diagram affected by tidal circularization.
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Figure 2:
Distribution of the SB in the (
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Several relevant features appear in Fig. 2. First of all, the range
of eccentricities jumps from 0 to almost 1 between 4 and 9 days. All SB with
shorter than 4 days are at present on circular orbits, and
no evidence of circularization is visible when
exceeds 10 days.
The SB with
between 4 and around 10 days have medium
eccentricities, and some of them are even circularized.
At least one of the short period SB may have been generated with a very large
eccentricity, and may be too young for to have been circularized:
KW 181 has the most eccentric orbit among the periods shorter
than 10 days. This system belongs to the Praesepe cluster, and its age is
therefore only
years. Its circularization will be complete
within only
years (Duquennoy et al. 1992).
The other two SB with e>0.1 and P>10 days
may have eccentricities due to
perturbations by a third component (Kozai mechanism or secular perturbations,
see Mazeh & Shaham 1979):
GJ 719 has a CPM companion with a minimum separation of 280 AU
(Zuckerman et al. 1997), and
GJ 233 is a visual binary with a possible period of 200 years
(Heintz 1988). When these 3 SB are discarded, we find
or 10 days, as in Duquennoy & Mayor (1991).
Nevertheless, it is striking that all the SB with eccentric orbits and P<10 days have large mass ratios. In order to see if this feature is significant we look at the period-eccentricity diagram of the 205 SB found by Latham et al. (2002) among stars with large proper motions. They also found a range of period where circular orbits and eccentric orbits both exist, but between around 10 and 20 days (see their Fig. 9). It is visible in their plot that the double-lined SB (SB2), which have the largest q, are not abnormally frequent among the systems with eccentric orbits and short periods. Therefore, we admit that the frequency of twins with large eccentricity and P<10 days is just due to chance.
The sample of Latham et al. contains a few SB with circular orbits and periods
between 10 and 20 days. However, they are stars with large proper motions, and
they generally belong to the old galactic disk or even to the halo. Therefore,
the long periods of some circularized orbits may be an effect of the ages of the
systems, in agreement with theoretical predictions (see e.g. Duquennoy et al.
1992).
Moreover, these old stars are not representative of the stars
observed for planet detection.
For the parent population of stars
harboring planets, the limit of the area affected by
tidal effects in the (
- e) diagram is
days.
The (P - e) diagram of the SB is plotted in Fig. 3. The values of the median of the four classes of q are drawn in this figure for visual comparison.
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Figure 3:
Distribution of the SB in
the period-eccentricity diagram. The symbols are the same as in
Fig. 2. The lines refer to the median eccentricities of the
different classes of mass ratios :
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As shown in Paper I, the twins often have below-average eccentricities
when periods longer than 5 days are considered;
the probability to get so large a discrepancy by chance, as derived by
the two-sided test, is only 2.7%.
By contrast, the
two groups with
have nearly the same
distribution of eccentricities : the significance of the two-sided
test is 100%. In order to check that even the SB with the smallest
mass ratios have the same eccentricity distribution as the others, the test
is done again by comparing the SB with
(10 SB with P > 10 days)
to those with
(32 SB with P > 10 days). Again, exactly
half of the
low-mass ratio SB are below the median, providing a 100% significance.
Therefore, all the non-twin SB
may be considered together in the comparison to the exoplanets.
We now come back to the distribution of e of all SB.
It is clearly visible in Fig. 3 that the median eccentricity
increases with the period. However, we suspect that this may be
entirely explained by tidal circularization, because the orbits having
initially
and
are now circular with
.
In other words, we want to see if
the distribution of eccentricities depends on the periods,
apart from the cut-off at
.
A Spearman test is performed to check this hypothesis.
In order to discard the area affected by tidal circularization,
the test is restricted to the rectangular box (
).
The non-twins and the twins are
considered separately. The Spearman correlation coefficient
is 0.26 for the former, and 0.29 for the latter;
taking into account the numbers of objects,
the probabilities to get by chance values so far from zero are
10% and 20% respectively.
Although these levels of significance are a bit low, they are still too large to
reject the hypothesis that the eccentricities are not correlated with the
periods as soon as the semilatus rectum of the orbit is larger than the
radius of a circular orbit with period
.
Since
is related to the angular momentum of the orbit, it seems
relevant to see if the eccentricities are correlated with this parameter.
This question looks similar to the (P - e) correlation investigated
just above, but it is different in reality, since changing the (P - e)
plane into the (
- e) plane modifies the density in relation with
the distribution of the periods. Therefore, the absence of correlation between
P and e does not necessarily imply an absence of correlation between
and e, and vice versa.
A Spearman test is used again, but in a box in the (
- e)
diagram. Since the selection of the sample was limited by the condition
P<10 years, we must take into account the limit
derived from
Eq. (2) (the thin dotted line in Fig. 2). Therefore,
the limits of the box considered in the Spearman test are
and e<0.8, in order
to avoid the area with
.
Again, the rejection threshold
of the hypothesis that e and
are correlated is between 10 and
20%, and the eccentricities may not depend on the angular momenta of the SB.
Moreover, the planets that were not followed by RV observations during a complete period are also discarded. Since this last condition results in removing the majority of the planets with periods longer than 2200 days, this value is adopted as a selection limit of the sample.
As for the SB, the planets are plotted on the (
- e) diagram
(Fig. 4), in order to investigate the effects of tidal
circularization. The sample is
split into two nearly equal groups, one containing the planets with minimum
mass less than 2 Jupiter, and one with the planets heavier than this limit.
In contrast to the SB, for which circular orbits and moderate eccentricities
are mixed in a small range of
,
the separation between the circularized orbits and the others
is remarkably well determined, at
days.
The most eccentric planetary orbit, HD 80606b (Naef et al.
2001), is actually found for this
period, but Wu & Murray (2003) demonstrated that
it may be excited by a distant companion through the
Kozai mechanism.
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Figure 4:
Same as Fig. 2 for the planets.
The circles are the planets with minimum mass <2 Jupiter, and the
open squares refer to the others.
The thin dotted line is the
limit
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It appears in Fig. 4 that the clustering of planets with
P<10 days (Udry et al. 2003) is even more marked when
is used in place of P:
the planets are concentrated in orbits with the
semilatus recta corresponding to
between 2.5 and 10 days, since we
count 18 planets in this range, but only 1 between 10 and 20 days.
It would be relevant to check if the eccentricities are correlated with
,
but our sample does not permit this: the detection of the
planets is far from complete, and the incompleteness increases with the
period. Therefore, since the eccentric orbits correspond to a longer
period for a fixed
,
the (
- e) diagram of the
planets is biased against large eccentricities. For that reason, the
(
- e) diagram of the planets can be used only for investigating
the circularization limit. As a consequence, it cannot be used to compare
the SB to the exoplanets in Sect. 5.
The (P - e) diagram of the exoplanets is given in Fig. 5. Only 1 planet above 2 Jupiter masses has P between 5 and 70 days. This paucity of heavy-mass planets with short periods has already been pointed out (Zucker & Mazeh 2002; Udry et al. 2003), and it makes the median eccentricity of these planets unreliable in this range of period. A two-sided test based on the common median for P in the range 5 to 2200 days shows that the probability to get differences at least as large as that obtained is 60%. It is then quite possible that the eccentricities of planetary orbits are not related to the masses of the planets. This question is considered again in Sect. 5.2.
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Figure 5:
Same as Fig. 3 for the exoplanets.
The symbols are the same as in Fig. 4. The full line
is the median eccentricity of the planets with
minimum mass <2 Jupiter, and the dashes refer to the masses heavier
than 2 Jupiter. The thin dotted line is the
eccentricity
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The median eccentricities of the exoplanets in Fig. 5 seem approximately constant. A test confirms this impression: the Spearman coefficient of the 52 planets with P > 20 days and e<0.78 is 0.21, providing a threshold between 10 and 20%. It is thus not possible to rule out the hypothesis that, apart from circularization due to tidal effects, the distribution of the eccentricities of the planets is the same for any period between 20 and 2200 days. Therefore, if the eccentricities are modified by migration, they are changed almost independently of the periods. However, this applies essentially to periods longer than 200 days, since we have very few planets between 20 and 200 days.
Santos et al. (2004) have shown that the percentage of stars harboring planets jumps from less than 5% to more than 20% when stars with [Fe/H] larger than 0.2 are considered. In order to see if this limit of 0.2 dex also corresponds to other orbital properties, we use the [Fe/H] of Santos et al. to distinguish the planets orbiting "metallic'' stars and the others. In the (P - e) diagram, we count 27 "metallic'' planets, of which 14 have eccentricities smaller than the median. We conclude with a significance of 79% that metallicity is not related to the eccentricity, confirming the result obtained by Santos et al. (2003) with another test.
Non-twin binaries and planets are plotted in the (P - e)
diagram in Fig. 6. The median eccentricities
are derived from the systems with
.
In order to make the comparison
free of differences in the tidal circularization,
is derived from
Eq. (3) assuming
days for both samples. It appears
clearly
that, although all objects are distributed in the same area of the (P - e)
diagram, the planets have eccentricities that are on average smaller than those
of the SB. The test of the distribution around the common median confirms this
discrepancy: Among 53 planets included in a sample of 102 objects, we count
33 planets below the median eccentricity. The null hypothesis is
rejected at the 1.7% level of significance.
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Figure 6:
The exoplanets compared to the
non-twin binaries in the(P - e)
diagram. The circles refer to the planets, and the open squares to
the SB with
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It appears from Fig. 5 that several planets with long periods and minimum masses below 2 Jupiter have very small eccentricities. Therefore, although we have seen that the (P - e) relation may be the same for all planets, it is relevant to compare the SB only to the planets with masses larger than 2 Jupiter. When the planets above 2 Jupiter masses are compared with all the non-twin SB, the null hypothesis is rejected again at the 4.8% level of significance, confirming that the heavy-mass planets have less eccentric orbits than the binaries.
A direct comparison of the distributions of the eccentricities of
planets and of SB is not feasible, since the possible range of
eccentricities varies with the period, and the distribution of P is not
the same for planets as for SB. Fortunately, another approach may be used
to visually compare these objects, which is to derive the original
distributions of eccentricities corrected for the bias coming from tidal
circularization.
The low significance values of the Spearman correlation tests performed above
allows us to assume that, apart from the area affected by circularization,
the eccentricity distribution does not vary with the period. Therefore, it
is possible to derive the intrinsic distribution of the eccentricities for
planets and for SB, using the method of the "nested boxes''.
For that purpose, we use Eq. (3) to compute, for each system,
the maximum eccentricity unaffected by tides,
.
We then apply the method given in the Appendix.
The results are shown in Fig. 7.
![]() |
Figure 7: Distribution of the eccentricities of SB and planets corrected for tidal circularization. |
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The frequency of planets with low eccentricities is mainly due to the planets
with masses below 2 Jupiter. The largest difference between the two groups
of planets is obtained for e=0.20 exactly: 40% of the planets below 2
Jupiter masses have orbits with
,
instead of 18% for the others.
However, a Smirnov test indicates that this discrepancy is far from
sufficient to reject the hypothesis that all planets obey the same
distribution, since the rejection threshold is as large as 38%.
An anonymous referee wondered whether this test could be affected by a bias
related to the radial velocity semi-amplitude. However, this bias is
unfavorable to the planets with the lowest masses, since they are the
most difficult to detect, and its efficiency is maximum when they have
eccentric orbits (see Sect. 5.3 hereafter). As a consequence,
correcting this bias would slightly decrease
the proportion of
among the planets below 2 Jupiter, and
then still increase the threshold of the test.
We conclude again that we can assume that the eccentricity distribution of the
planets does not depend on mass.
The differences pointed out by the tests and by the comparison of the medians are clearly visible in this figure. The maximum of the distribution is around 0.4 or 0.5 for the non-twin SB, and between 0.3 and 0.4 for the planets. Moreover, the distribution of e decreases rapidly after the maximum for the planets, but it is rather flat until 0.8 for the non-twins SB. A comparison between the median e of the exoplanets and of the twins could suggest that these two kinds of objects have similar distributions of eccentricities. However, differences appear in Fig. 7: instead of exhibiting a maximum like the exoplanets, the distribution of eccentricities of twins is nearly flat over a wide range, from e=0.1 to e=0.7.
We now want to check that the lack of planets with large eccentricities is
not due to a selection effect against the detection of these systems.
In contrast to the SB,
the detection of planets is far from complete, and our sample is
obviously biased in favor of those which are easiest to detect. The most
obvious bias
is against the detection of long period systems, but this does not affect
the reliability of our test based on the median e;
it just decreases the contribution of the long-period planets.
However, another bias is directly related to
the eccentricity: a large eccentricity increases the semi-amplitude in RV,
but, at the same time, it decreases the rms of the RV measurements.
Therefore, the detection of a system close to the limit of the
instrument is more difficult when the eccentricity is large.
Another consequence of this effect is a bias in the distribution of the
periastron longitude, .
The orbits with
around 0 or
are more difficult to detect than those with
around
or
.
This is visible, although not very significant, in our sample of
planets: we count only 19
orbits with
among 43 planets with
(the orbits with e<0.1 are
not taken into account since
is then not reliable).
Simulations have been performed to investigate if this bias
may explain the discrepancy
between planets and SB. Each planet receives the eccentricity
of a SB, randomly taken among the 10 SB with periods closest to that of
the planet. The periastron longitude of the planet is randomly generated, and
3 radial velocity measurements are produced for 3 epochs randomly chosen,
adding errors drawn from the residual rms of the true orbit;
(in reality, each star observed for planet detection receives much more
than 3 observations, but our aim is to derive an upper limit to the bias).
When the standard deviation of
the simulated RV is larger than the threshold corresponding to
%, the planet is counted as detected; if not, another
eccentricity is generated, and the simulation of the detection is performed
again, until the planet satisfies the detection condition. When the
complete sample has thus been detected by the simulation, the test of the median
eccentricity in the (P - e) diagram is performed. The simulation of the
diagram is repeated 50 000 times.
It appears from this calculation that the effect of the bias is to
shift on average 0.6 more planets below the common median.
Assuming that the number of planets below the median would be 32 in the
absence of bias (instead of 33, see Sect. 5.1), the rejection
threshold of H0 becomes 4.7%.
This is still small enough to maintain rejection. At the same time, we count
the planets with
which
are detected in the simulation. Among the planets with e>0.1, their
proportion is 45.5%, in very nice agreement with the observed one, which
is
19/43 = 44%. We conclude then that the bias against detection of
orbits with large eccentricities cannot explain the excess of planets with
e smaller than the common median in the (P - e) diagram.
We have found some relevant features in comparing the eccentricities of the SB to those of the exoplanets :
Acknowledgements
We are grateful to Piet Hut and Jean-Paul Zahn for their explanations and for their valuable comments on the draft version of the paper. Douglas Heggie carefully read a preliminary version and added relevant corrections. An anonymous referee made valuable comments. The A&A language editor, Jet Katgert, corrected the English. The selection of the exoplanets was partly based on data taken from Simbad, the database of the Centre de Données astronomiques de Strasbourg.
This method was initially dedicated to the derivation of a bias-free distribution of mass ratios of visual binaries (Halbwachs 1983; Halbwachs et al. 1997), as is explained in detail in Halbwachs (2001). It is adapted hereafter to the derivation of the intrinsic distribution of the eccentricities.
We consider a sample with periods P larger than
,
the period
corresponding to tidal circularization, as explained in Sect. 2.1.
For each system, P and
are used to derive
,
the maximum
eccentricity of the
orbits unaffected by tidal effects. The systems having eccentricities
,
if any, are discarded from the sample.
The principle of method is as follows:
The distribution is normalized after adding the last box.