Expected Planets in Globular Clusters
EXPECTED PLANETS IN GLOBULAR CLUSTERS
Noam Soker1 & Alon Hershenhorn 1
ABSTRACT
We argue that all transient searches for planets in globular clusters have
a very low detection probability. Planets of low metallicity stars typically do
not reside at small orbital separations. The dependance of planetary system
properties on metallicity is clearly seen when the quantity Ie ≡ Mp[a(1 − e)]
is considered; Mp, a, e, are the planet mass, semi-major axis, and eccentricity,
respectively. In high metallicity systems there is a concentration of systems at
high and low values of Ie, with a low-populated gap near Ie ∼ 0.3MJ AU
2, where
MJ is Jupiter’s mass. In low metallicity systems the concentration is only at the
higher range of Ie, with a tail to low values of Ie. Therefore, it is still possible
that planets exist around main sequence stars in globular clusters, although at
small numbers because of the low metallicity, and at orbital periods of & 10 days.
We discuss the implications of our conclusions on the role that companions can
play in the evolution of their parent stars in globular clusters, e.g., influencing
the distribution of horizontal branch stars on the Hertzsprung-Russell diagram
of some globular clusters, and in forming low mass white dwarfs.
Subject headings: stars: horizontal-branch globular clusters: general - stars:
rotation - planets
1. INTRODUCTION
Evolved sun-like stars that burn helium in their cores occupy the horizontal branch
(HB) in the Hertzsprung-Russel (HR) diagram. HB stars that have low metallicity and/or
low envelope mass are blue, and are termed blue HB (BHB) stars in globular clusters (GCs),
and sdB or sdO (sdOB together; termed also extreme HB stars or EHB) in the field (not
in GCs). There are strong indications that many of the sdOB stars in the field are in close
binary systems (e.g., Maxted et al. 2001; Maxted 2004), and there is a strong support to
1Department of Physics, Technion−Israel Institute of Technology, Haifa 32000 Israel;
soker@physics.technion.ac.il
http://arxiv.org/abs/0704.1067v2
– 2 –
the idea that the sdOB phenomena is caused by binary companions (e.g., Han et al. 2003;
Maxted 2004). The large fraction of stars in the field that are likely to have planets around
them (Lineweaver & Grether 2003) hint that planets can also lead to the formation of sdOB
stars (Soker 1998). We note the recent tentative claim for a planet orbiting an sdB star at
an orbital separation of a = 1.7 AU (Silvotti et al. 2007), that might hint that closer planets
existed in the system before the progenitor turned into a red giant star.
The distributions of stars on the HB (the HB morphology; also referred to as the color-
magnitude diagram [CMD]) differ substantially from one GC to another. It has long been
known that metallicity is the main, but not sole, factor which determines the HB morphology
(for an historical review see, e.g., Fusi Pecci & Bellazzini 1997). The other factor (or factors)
which determines the HB morphology is termed the ‘second parameter’. There is a debate on
what are the main processes influencing the second parameter (Catelan 2007 and references
therein), with a binary interaction being one of these processes. In the low mass binary
second parameter model the companion is very light (a very low mass main sequence [MS]
star, a brown dwarf, or a massive planet), and in most cases will be destroyed as it falls
deeper into the envelope (Soker 1998; Soker & Harpaz 2007). Therefore, the non-detection
of companions to HB stars in GCs (Moni Bidin et al. 2006) is not in contradiction with the
model.
Although close brown dwarf companions exist (e.g., Zucker & Mazeh 2000), and are
included in the low mass binary second parameter models, they are rare (e.g., Mazeh et al.
2003; Grether & Lineweaver 2006), and so we concentrate on planets and low mass MS stars.
More problematic to the binary second parameter model might seem to be the null
detection of planets to MS stars in GCs (Weldrake et al. 2007b). Weldrake et al. (2007b)
looked for transits in the GCs ω Cen and 47 Tuc, and did not find any close planets, but
did find stellar companions (Weldrake et al. 2007a). The null detection of planets in 47 Tuc
(Gilliland et al. 2000) does not contradict the binary model (Soker & Hadar 2001). The
reason is that this GC has only few stars on its blue HB. If there were many planets in this
GC, then they would be swallowed by the red giant branch (RGB) star progenitor of the
HB star, causing high mass loss rate (Livio & Soker 2002) and the formation of many BHB
stars.
The null detection of planets in the GCs ω Cen (Weldrake et al. 2007b) can be com-
patible with the planet binary model for the second parameter if the planets don’t reside
in close orbits. Namely, orbital periods & 10 days. For that, Soker & Harpaz (2007)
predicted-conjectured that planet companions might exist in GCs, but at orbital separations
of 0.3 AU . a . 3 AU. Planetary systems in the upper orbital separation range will be
destroyed in GCs, but those in the lower range can survive (Fregeau et al. 2006).
– 3 –
Soker & Harpaz (2007) only based their prediction on the requirement of their model,
but did not bring any observations to support this conjecture. Our main goal is to further
explore the binary model for the second parameter, and in particular to examine the possible
role of planets. For that, we turn to examine what can be learned from the wealth of data
acquired in the field of exoplants. We use recent results from the field of exoplants to support
the claim that if planets exist in GCs, they are at larger orbital separations than planets
around stars close to the sun. Unlike planets, low mass MS companions can reside close to
the parent star in GCs (Weldrake et al. 2007a), and be much more significant in influencing
the evolution of their parent star. Namely, the low mass binary second parameter model in
low metallicity GCs must be based mainly on low mass main sequence stellar companions,
but with contribution of massive planets as well.
2. PLANET COMPANIONS
To support the conjecture that if planets exist in GCs they don’t reside at small orbital
separations we present several correlations between properties of known extra solar planets.
We start by presenting the well known distribution of planets by their orbital period P
(Figure 1), but motivated by the recent results of Grether & Lineweaver (2007) we do
so separately for three ranges of metallicity of the parent stars: [Fe/H] < −0.1 (black),
−0.1 ≤ [Fe/H] ≤ 0.1 (gray), and 0.1 < [Fe/H] (white). Here and in the rest of the diagrams
in the paper, each bin shows the number of planets with a period (or other relevant quantities)
greater than the number to the left of the bin and smaller than the number to the right of the
bin. The leftmost bin shows the number of planets with a period smaller than the number
to the right of the bin. The rightmost bin shows the number of planets with a period greater
than the number to the left of the bin. All planets data used here are from the Extrasolar
Planets Encyclopaedia maintained by Schneider (2007, and references therein), as of June
1, 2007. We skip the comparison of the planet hosting star metallicity distribution with that
of other field stars (see, e.g., Santos et al. 2001).
There is a population gap in this histogram, i.e., the planets are concentrated in two
regions. The gap exists only for the high and medium metallicity systems, as marked by
the two thick horizontal arrows in Figure 1. The long period region is limited from above
by selection effects. This gap is well known, e.g., Udry et al. (2003) noted the shortage of
planets in the range 0.06 AU ≤ a ≤ 0.6 AU.
We seek a better quantity to distinguish between close and wide planets, and between
the metallicity ranges. For that, and motivated by known correlations between planets’ mass
and period (e.g., Mazeh et al. 2005), we plot in Figure 2 planets in the Mp − a plane, where
– 4 –
Log[ Period/Day ]
Period/Day
1 2 4 7 10 20 40 70 10
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
Period
Fig. 1.— Histogram of the number of planets as a function of the orbital period P in days,
for three ranges of metallicity as indicated. Each bin shows the number of planets with a
period greater than the number to the left of the bin and smaller than the number to the
right of the bin. The leftmost bin shows the number of planets with a period smaller than
the number to the right of the bin. The rightmost bin shows the the number of planets with
a period greater than the number to the left of the bin. The horizontal thick arrows mark
the gaps in the respective two high metallicity ranges. Data from the Extrasolar Planets
Encyclopaedia maintained by Schneider (2007, June 1, and references therein).
– 5 –
Mp is the minimum planet mass, used here in units of Jupiter mass MJ , and a is the orbital
semi-major axis, used here in units of AU. Filled and empty circles are for systems where
the host star metallicity is below and above solar metallicity, respectively. We note that
there is a morphological structure along a few lines of
α = constant. (1)
Two lines are marked by their α value on Figure 2. These lines bound a low-populated
stripe between them, and emphasize two populated clumps: one consists of planets having
high masses and large orbital separations, and the other consists of planets having low masses
and small orbital separations. Based on this, we take α = 2 as our standard value to further
analyze the correlations.
In Figures 3-9 we show the distribution of the entire sample as a function of the following
quantities: Mpa
2, Mpa
Ie ≡ Mp[a(1− e)]
2, (2)
[a(1 − e)]2, Mp[a(1 − e)]
3, [a(1 − e)]2/Mp, and [a(1 + e)]
2/Mp, respectively. Planets with
unknown eccentricity were calculated with e = 0. The quantity Ie has the dimension of
moment of inertia, and might therefore indicate the importance of some kind of interaction
between the planet and the parent star, as will be discussed in section 4.3. As the strongest
interaction occurs near periastron, the relevant distance is a(1− e) rather than a alone.
From Figures 1-9 we learn the following.
1. As is well known (e.g., Udry et al. 2003; Marcy et al. 2005) there is a concentration of
planets at very short periods of days. Then there is a low-populated range, the gap,
and a rise to a second grouping of planets at hundreds to thousands of days. The
gap exists only at the two higher metallicity ranges, as marked on Figure 1 by the
horizontal thick arrows.
2. At lower metallicities planets tend to have longer orbital periods. The ratio of planets
with P > 100 day to planets with P < 100 day is 28/16 = 1.75 and 31/17 = 1.8 for
[Fe/H] < −0.1 and −0.1 ≤ [Fe/H] ≤ 0.1, respectively, while it is only 64/53 = 1.2 for
0.1 < [Fe/H]. This trend was found also by Marchi (2007) in the C2 and C3 sub
samples defined there. This trend will have to be checked with much larger samples in
the future.
3. We find that the double-peak distribution of planets at high metallicity is more pro-
nounced when instead of the period other quantities are used, e.g., Mpa
2 or [a(1 −
e)]2/Mp, or [a(1 + e)]
2/Mp, and even more so when the quantity Ie = Mp[a(1 − e)]
used.
– 6 –
Log[ a/AU ]
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
1 2 4 7 10
[Fe/H] < 0
[Fe/H] > 0
1.7α =
2.36α =
Fig. 2.— The distribution of all known planets (from The Extrasolar Planets Encyclopaedia)
in the minimum mass (in MJ)−semi-major axis (in AU) plane. Filled and empty circles are
for host stars metallicity below and above solar, respectively. Two lines are drawn to show
morphological features in the distribution, with the value of α marked (eq. 1). The
morphological feature we refer to is a low-populated stripe between the two lines, and two
populated clumps, one above the upper line and one below the lower line.
– 7 –
Log[ (M
p/MJ)(a/AU)
.5 0.
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
Fig. 3.— Histogram of the number of planets as a function of Mpa
2 (in MJ AU
2) for the
three metallicity ranges.
– 8 –
Log[ (M
p/MJ)(a/AU)
1/2  ]
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
Fig. 4.— Histogram of the number of planets as a function of Mpa
1/2 (in MJ AU
1/2) for
the three metallicity ranges.
– 9 –
Log[ (Mp/MJ)([a/AU][1-e])
.5 0.
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
( )[ ]
M a e−
Fig. 5.— Histogram of the number of planets as a function of Ie = Mp[a(1−e)]
2 (inMJ AU
for the three metallicity ranges.
– 10 –
Log[ ([a/AU][1-e])
.5 0.
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
( )[ ]
1a e−
Fig. 6.— Histogram of the number of planets as a function of [a(1 − e)]2 (in AU2) for the
three metallicity ranges.
– 11 –
Log[ (Mp/MJ)([a/AU][1-e])
.5 0.
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
( )[ ]
M a e−
Fig. 7.— Histogram of the number of planets as a function of Mp[a(1 − e)]
3 (in MJ AU
for the three metallicity ranges.
– 12 –
Log[ ([a/AU][1-e])2/(Mp/MJ) ]
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
( )[ ]
a e M−
Fig. 8.— Histogram of the number of planets as a function of [a(1− e)]2/Mp (in AU
2 M−1
for the three metallicity ranges.
– 13 –
Log[ ([a/AU][1+e])2/(Mp/MJ) ]
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
≤ ≤( )[ ]
a e M+
Fig. 9.— Histogram of the number of planets as a function of [a(1 + e)]2/Mp (in AU
2 M−1
for the three metallicity ranges.
– 14 –
4. From the different parameters we have tried, Ie = Mp[a(1 − e)]
2 shows most clearly
the two groups of planets at high metallicity, and the differences between planets with
different metallicity of their parent star. The criteria we use to prefer the parameter Ie
are (a) a smooth variation in the peaks of the histogram, i.e., small fluctuations in peaks
(e.g., the peaks of the graph of [a(1 − e)]2/Mp in Fig. 8 have larger fluctuations than
the peaks of Ie); (b) a sharp jump between the group of planets with large separations
and the deep gap for the two high metallicity ranges (e.g., for Mpa
2, presented in
Fig. 3, the difference between the peaks and gap is not as sharp as for Ie presented
in Fig. 5); (c) A wide gap (e.g., for Mpa
1/2 presented in Fig. 4 the gap is very
narrow); and (d) A clear different behavior of the low metallicity range and the two
high metallicity ranges. In particular, when using Ie such a jump does not exist for
the low metallicity range. For the highest metallicity range used here there is a large
jump at Ie ≃ 0.3MJ AU
2 (log Ie = −0.5), which clearly separates two Ie-populations
of planets. While in high metallicity systems there are two well defined populations of
planets, in low metallicity systems there is only one peak: Planets of low metallicity
stars have typically larger orbital separations. Because Mp is the minimum mass, in
the histogram showing [a(1 − e)]2/Mp (Fig. 8) of the real distribution, systems will
move to the left smearing the peak of planets in the lowest metallicity range. In the
histogram using Ie (Fig. 5), on the other hand, using the real mass will move systems
to the right. This might make the peak on the right for the lowest metallicity range
more pronounced.
5. We have tried to use the same quantities listed above with 1 + e instead of 1− e. For
most cases the separation between close and wide planets is worse than when 1 − e is
used. This implies that the periastron is physically more influential than the apastron
for defining close and wide planets. However, for the quantity [a(1 + e)]2/Mp shown in
figure 9 a partition to two groups is evident. Still, we regard Ie to be the best indicator
of the two planet populations.
Burkert & Ida (2007) find that a gap in the semimajor axis distribution does not exist
if only a sample of systems with hosting stellar mass of M < 1.2M⊙ is used. We find
that when using only hosting stars with masses of M < 1.2M⊙ a gap does exist in the Ie
distribution for the high metallicity range. There is a gap in the medium metallicity range
as well. We also find that ∼ 70% of the hosting stars with masses of M > 1.2M⊙ are
in our high metallicity range. The planets in lower metallicity systems with stellar mass
of M > 1.2M⊙ tend to have larger values of Ie than planets in systems with stellar mass
of M < 1.2M⊙. We therefore propose that both stellar mass (Burkert & Ida 2007) and
metallicity are fundamental quantities influencing the distribution of planets around stars.
– 15 –
Burkert & Ida (2007) also find that the gap in the high hosting stellar mass sample
exists for planets with Mp sin i > 0.8M⊙, but not for M sin i ≤ 0.8M⊙. The dependance
on the planet mass is included together with the semimajor axis and eccentricity in Ie. We
find that the Ie distribution does not have a gap for systems with both Mp > 0.8MJ and
M > 1.2M⊙, unlike the semimajor axis distribution found by Burkert & Ida (2007). We do
see the gap in the Ie distribution of systems with Mp > 0.8MJ and in the Ie distribution of
systems with M < 1.2M⊙.
It is not clear if stars in GCs have planets at all, and in particular close-in planets that
could be found with the transit technique, as the fraction of detected planetary systems
decreases sharply with decreasing metallicity (e.g., Santos et al. 2001; Fischer & Valenti
2005; Grether & Lineweaver 2007). According to Grether & Lineweaver (2007) the most
probable value of this fraction for the metallicity range appropriate for GCs is . 0.1%,
although the uncertainty is large, and values of ∼ 1% are still possible. Contrary to the
general trend is the low metallicity of M-dwarfs (low mass MS stars) hosting planets (Bean
et al. 2006), which still leaves hope for planetary systems in GCs. If some planets do exist
in GCs, the implications of the results presented here are clear: the planets will not be on
short orbital periods. Therefore, we conclude, all transient searches for planets in GCs have
a very low detection probability. (At least in the statistical sense, as rare close planets might
exist.)
However, planets might be detected in metal-rich clusters. The open cluster NGC 6791
has [Fe/H]∼ 0.4, and has a large population of EHB stars and low mass WDs (Kalirai et
al. 2007), both of which were formed by stars having increased mass loss on the RGB.
Stellar companions are not likely to cause this increased mass loss (Kalirai et la. 2007). We
propose that the increased mass loss on the RGB in this cluster is partially caused by planets
swallowed by the RGB progenitors of EHB stars and low mass WDs. We therefore predict
that many transient events can be detected in this cluster.
3. LOW MASS MAIN SEQUENCE COMPANIONS
A similar analysis was conducted as in the previous section but for stellar companions
based on the same sample of 135 systems analyzed by Grether & Lineweaver (2006). We
present two histograms of the period of the secondary. In figure 10 the sample was divided
into three ranges of metallicity as in the previous section. In figure 11 we follow the division
of Grether & Lineweaver (2006) and divide the sample into two ranges of color of the parent
star as marked on the figure.
– 16 –
Log[Period/Day]
Period/Day
1 2 4 7 10 20 40 70 10
           [Fe/H] < -0.1
 -0.1    [Fe/H]    0.1
  0.1 < [Fe/H]
≤≤Period
Fig. 10.— Histogram of the number of stellar companions as a function of the orbital period
P in days, for three ranges of metallicity as indicated. Data from Grether & Lineweaver
(2006, 2007).
– 17 –
Log[Period/Day]
Period/Day
1 2 4 7 10 20 40 70 10
           B-V < 0.75
0.75 < B-V
Period
Fig. 11.— Histogram of the number of Stellar companions as a function of the orbital period
P in days, for two ranges of color as indicated.
– 18 –
From figures 10 and 11 we can deduce the following:
1. As with planets, there are two concentrations of stellar companions: at short and long
orbital periods.
2. Contrary to the behavior of planets, this double-distribution is more pronounced in the
low and medium metallicity ranges, with longer orbital periods at higher metallicities.
The ratio of stellar companions with P > 100 day to P < 100 day is 27/49 = 0.55,
15/27 = 0.56 and 10/7 = 1.43 for [Fe/H] < −0.1, −0.1 ≤ [Fe/H] ≤ 0.1 and 0.1 <
[Fe/H], respectively.
3. Redder systems tend to have a slightly shorter orbital periods. The ratio of stellar
companions with P > 100 to P < 100 day is 34/50 = 0.68 and 18/33 = 0.55 for
B− V < 0.75 and B− V > 0.75, respectively.
4. The width of the gap between long and short orbital periods is ∼ 70 (from ∼ 30
to ∼ 100 day) and ∼ 460 day (from ∼ 100 to ∼ 550 day) for [Fe/H] < −0.1 and
−0.1 ≤ [Fe/H] ≤ 0.1, respectively. The center of the gap is located at P ≈ 56 and
P ≈ 247 day for [Fe/H] < −0.1 and −0.1 ≤ [Fe/H] ≤ 0.1, respectively.
We have tried to use a finer classification of metallicity and color ranges, namely
[Fe/H] < −0.3, −0.3 < [Fe/H] < −0.1, −0.1 ≤ [Fe/H] ≤ 0.1, and 0.1 < [Fe/H], for metal-
licity, and B− V < 0.6, 0.6 < B− V < 0.7, 0.7 < B− V < 0.8, and 0.8 < B−V for the
color (data not shown). These did not add new information. We find that the ratio between
the number of systems with P < 100 days to systems with P > 100 days is 49/27 = 1.8,
33/15 = 2.2, and 28/10 = 2.8, for [Fe/H] < −0.1, [Fe/H] < −0.3, and [Fe/H] < −0.4,
respectively. This shows that the tendency of low metallicity systems to harbor short period
companions is robust.
Although we find that low metallicity stars tend to be slightly closer (shorter orbital
periods) than higher metallicity systems, this does not automatically imply the same trend
to low mass systems. Maxted & Jeffries (2005) find that a large fraction of very low mass
stars seem to be in binary systems, but not very close ones. In addition, a large fraction
of stellar companions to low mass stars can have very low mass M2 < 0.3M⊙ (Mazeh et al.
2003), and low mass stellar companions tend to be at large orbital separation (Grether &
Lineweaver 2007).
– 19 –
4. DISCUSSION AND SUMMARY
4.1. Main Results
As is well known and can be seen in figures 1-9, there are two regions highly-populated
with planets, with a low-populated gap between them. What we have found here (sections
2 and 3) is the following.
1. In planets the partition to two groups is more significant for high metallicity systems.
2. From the different quantities we have tried, the quantity Ie ≡ Mp[a(1 − e)]
2 both
distinguishes between high and low metallicity systems, and shows best this partition
to two planet populations in the high metallicity range.
3. In high metallicity systems planets tend to reside on an average closer orbital periods.
We note that these two properties depend also on the hosting stellar mass (Burkert &
Ida 2007).
4. This trend for metallicity dependance is opposite for stellar companions (section 3).
5. This trend for stellar companions is mainly due to metallicity and not to the parent
star’s mass. There is only a small difference in the ratio of stellar companions with
P > 100 to P < 100 day for the two color ranges used.
We note that there are other properties for which stars with planetary systems and
with stellar companions show opposite behavior. The most important is the finding that as
metallicity decreases the star is much more likely to have a stellar companion than to harbor
a planetary system (Grether & Lineweaver 2007).
Our results have implications for two areas.
4.2. Implications for Globular Clusters
Since in galactic GCs the metallicity is very low, close-in planets are not expected
there. However, if planets do exist around some stars in GCs, they will most likely have
large orbital separations. Therefore, the transit search for planets in GCs has a very low
detection probability, and the non-detection of close-in planets should not be considered as
evidence against the presence of planets in GCs. We should stay open to the possibility that
wide (large orbital periods) planetary systems exist in GCs.
– 20 –
The large fraction of low metallicity stars that have stellar companions (Grether &
Lineweaver 2007) implies that a large fraction of stars in GCs might have formed with
stellar companions around them. It seems that binary systems are indeed common in GCs
(Leigh et al. 2007).
Not only metallicity, but other properties at the formation epoch of globular clusters
(Soker & Hadar 2001) might determine the presence of companions and planets. For example,
density in star forming regions can determine stellar rotation (Wolff et al. 2007).
Put together, our results support the low mass binary second parameter model, but
the companions in low metallicity star clusters are more likely to be very low mass stellar
companions, as observed by, e.g., Weldrake et al. (2007a), rather than massive planets. In
that model a low mass companion (a very low mass main sequence star, a brown dwarf,
or a massive planet) influences the post-main sequence evolution of stars, in particular the
properties of the parent stars on the horizontal branch. (Soker 1998; Soker & Harpaz 2007).
In high metallicity clusters, such as NGC 6791 (Kalirai et al. 2007), planets might be more
important than stellar companions in forming EHB stars and low mass (undermassive) white
dwarfs. We predict that many transient events can be detected in this cluster.
4.3. Implications for Planetary Systems
This topic is beyond our scope. However, our findings suggest that the migration of
planets from large to small orbital separation depends on a combination of parameters ex-
pressed by Ie (eq. 2). This quantity has the dimension of moment of inertia, and may imply
that a process reminiscent of the Darwin instability (e.g., Eggleton 2006) is in operation. In
particular, the strong dependance on eccentricity, in that 1− e is a much better factor than
1 + e in showing the two planet populations, suggests that some sort of tidal interaction is
operating.
Although the situation cannot be simple, let us try the following. The left limit of the
right populated area in figure 5 is Ie ≃ 0.3. Let us substitute this in the condition for the
Darwin instability to occur (Eggleton 2006, sec. 4.2)
> λcrit, (3)
where M1 is the stellar mass, R1 is the stellar radius, kR1 the radius of gyration with
k2 ≃ 0.05 − 0.1 for main sequence stars, Ω is the stellar angular velocity, and ω = 2π/P is
the mean orbital angular velocity. The critical value λcrit rapidly decreases with increasing
eccentricity, with λcrit = 1/3, 0.171, 0.102, and 0.052, for e = 0, 0.3, 0.4, and 0.5, respectively.
– 21 –
As most planets have e . 0.4, in this range we see that
λcrit ≃
(1− e)2 for e . 0.5. (4)
Substituting approximation (4) in equation (3), and taking M1 = 1M⊙, k
2 = 0.075, and
Ω ≃ ω, the condition for the Darwin instability to occur becomes
R1 & 8
0.3MJ AU
R⊙ (5)
This explanation requires that the planets interact with an inflated pre-main sequence star.
This possibility will be studied in a future paper.
We thank Charles Lineweaver and Daniel Grether for giving us their data on stellar
companions. We thank David Weldrake for useful comments, and an anonymous referee for
comments that improved the presentation of our results. This research was supported by a
grant from the Asher Space Research Institute at the Technion.
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This preprint was prepared with the AAS LATEX macros v5.2.
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http://arxiv.org/abs/0705.0910
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	INTRODUCTION
	PLANET COMPANIONS
	LOW MASS MAIN SEQUENCE COMPANIONS
	DISCUSSION AND SUMMARY
	Main Results
	Implications for Globular Clusters
	Implications for Planetary Systems
