Giant planet formation around M dwarfs
Nearly 150 planets have been detected orbiting nearby stars
(e.g., Marcy et al. 2003; Udry, Mayor & Queloz 2003), but
these planet searches have thus far concentrated on solar-type
host stars. The leading mechanism for explaining the origin
of both these extrasolar planets and the giant planets in our
solar system is the so-called core-accretion process. In this
paradigm, icy planetesimals collide and accumulate until they
build up planetary cores with mass Mc ≈ 5–15 M⊕. These
cores then accrete nebular gas and eventually reach masses
comparable to Jupiter.
A long-standing problem with the core-accretion hypothesis
was that the estimated time required for the core to grow
and subsequently accrete ∼ 1MJ of gas exceeded the observed
lifetimes of circumstellar disks (Pollack et al. 1996).
[We note that the process of gravitational instability does not
suffer from this shortcoming and has been seriously discussed
as an alternate mechanism (Boss 2000).] However, updated
estimates for the opacity in protoplanetary envelopes indicate
that Jupiter-mass planets can readily form through the
core-accretion mechanism in solar-metallicity disks around
solar-mass stars, with a time scale of 2–3 Myr at orbital radii
a ∼ 5 AU (Hubickyj, Bodenheimer & Lissauer 2004). To
achieve this relatively short formation time, the surface density
of solid material in the disk must be somewhat larger
a factor of ∼ 3) than that of the minimum-mass solar nebula
(MMSN). In one calculation, e.g., the resulting coremass was
about 16 M⊕, modestly larger than the core mass (10 M⊕)
deduced for Jupiter (Wuchterl, Guillot & Lissauer 2000). To
account for the low core mass of Jupiter, a cutoff of solid accretion
beyond a certain core mass is required, and can be explained
by nearby planetary embryos that compete for available
solid material. In an independent calculation with similar
opacities (Inaba & Ikoma 2003; Inaba,Wetherill & Ikoma
2003) and a disk with eight times the solid surface density of
the MMSN, a core of 25 M⊕ can form in 1 Myr at 5 AU, so
that the total formation time is 2–3 Myr. In this latter model,
the fragmentation of planetesimals and the enhancement of
the solid accretion rate due to the gaseous envelope (primarily
from gas drag) are taken into account, although the main gas
accretion phase is not calculated. If the solid surface density
is reduced to four times that of the MMSN, an 8 M⊕ core can
still form in 5 Myr. Taken together, these results imply that
giant planets can readily form, but somewhat special circumstances
are required for the core accretion model to explain
the particular properties of our Jupiter.
The ongoing observational surveys are shifting our view
of extrasolar planets from a disorganized collection of individual
systems (e.g., 51 Peg, υ And, or 47 UMa) to a more
robust statistical census. Different categories and populations
of planets can now be delineated (e.g., Marcy & Butler 1998;
Marcy, Cochran&Mayor 2000;Udry et al. 2003; Marcy et al.
2003). This emerging statistical view is important for improving
our understanding of the planet formation process, and to
learn how our own solar system fits into the galactic planetary
census. One of the cleanest statistical results to emerge
from extant planet searches is that stars with observed extrasolar
planets tend to have high metallicities, typically twice
that of the average Population I star in the solar neighborhood
(Fischer & Valenti 2005; Santos et al. 2003; Butler et
al. 2000). In addition, low metallicity stars are observed to be
deficient in currently detectable giant planets (withP <8 yr;
Sozzetti et al. 2004). This connection between planets and
host-star metallicity can be interpreted as evidence in favor
of the core accretion hypothesis (although it can also be interpreted
as evidence in favor of migration – see Sigurdsson
et al. 2003).Metal-rich circumstellar disks have a higher surface
density of solids and growing cores can easily reach the
Mc ≈ 5−−15M⊕ threshold required for rapid gas accretion.
In this work, we assume that the core accretion model
can explain the formation of Jovian planets within disks orbiting
solar-mass stars with solar metallicity. We then address
the question of whether or not Jovian planets can form
within disks orbiting around M dwarfs (M <∼
0.4M ). We
find that the core-accretion process makes a clear prediction
for the relative frequency of Jovian-mass planets as a
function of stellar mass: The circumstellar disks orbiting red
dwarfs are significantly less efficient in producing Jupitermass
planets than disks around solar-mass stars (Sect. 2.1),
although smaller, rockier planets (like Neptune) can readily
form.Moreover, this prediction is immediately testable (Sect.
2.2).
2.1. Theoretical model of planet formation
In the theoretical model explored here, giant planets form
within a circumstellar disk with the following properties: The
surface density σ(r) = σin(rin/r)3/2, where σin is the normalization
factor required to obtain a total disk mass Md(t)
within inner and outer disk radii, rin and rd. The time dependence
of the disk surface density is given by a depletion
function of the form fσ(t) = 1/(1 + t/t0) so that the
disk mass decreases according to Md(t) = Md(0)fσ(t). Observations
of circumstellar disks (e.g., Briceno et al. 2001)
suggest that t0 = 105 yr and Mid(0) = 0.05M are reasonable
benchmark values. The temperature distributions for
both viscously evolving accretion disks and flat, passively irradiated
disks have nearly the same power-law form, Td(r) =
Td (R /r)3/4, where Td is related to the stellar surface temperature
by a geometrical factor (e.g., Td /T ≈ [2/3π]1/4
for a flat disk – see Adams & Shu 1986). This model uses
disks that are flat and passive, and assumes that the disk is
isothermal in the vertical direction. The effective temperature
T of the star is related to the stellar radius R and luminosity
L (t,M ) through T (t,M ) = [L (t,M )/4πR2
σ]1/4.
We adopt T (t,M ) and L (t,M ) from published pre-mainsequence
stellar evolution tracks (D’Antona & Mazzitelli
1994).
We use a Henyey-type code (Henyey, Forbes & Gould
1964; see also Sect. 3) to compute the buildup and contraction
of gaseous envelopes surrounding growing protoplanetary
cores embedded within the evolving disk. This method
(Kornet, Bodenheimer& R´o˙zyczka 2002; Pollack et al. 1996)
adopts recent models for envelope opacity (Podolak 2003)
which include grain settling (see also Hubickyj et al. 2004).
The calculation is simplified in that it uses a core accretion
rate of the form dMc/dt = C1πσsRcRhΩ (Papaloizou &
Terquem 1999; compare with Pollack et al. 1996), where σs
is the surface density of solid material in the disk, Ω is the
orbital frequency, Rc is the effective capture radius for the
accretion of solid particles, Rh = a[Mp/(3M )]1/3 is the
tidal radius of the protoplanet, and C1 is a constant of order
unity. An important feature of this present model is that the
outer boundary conditions for the planet include the decrease
in the background gas density and temperature with time.
The results of this planet formation calculation are illustrated
in Fig. 1. The first simulation shown here corresponds
to a disk orbiting a 1 M star with an initial solid surface
density σs = 11.5 g cm−2 at a = 5 AU, about four times
that of the MMSN. This value is based on an initial gas-tosolid
ratio of 70 in the disk; as the disk evolves, σs decreases
with time because mass accretes onto the growing planet. A
Jupiter-mass planet forms in 3.25Myr with a core mass of 18
M⊕ (see also Hubickyj, Bodenheimer & Lissauer 2004). The
second calculation is for a disk surrounding an M star with
mass M = 0.4 M , where the initial solid surface density is
σs = 4.5 g cm−2 at a = 5 AU (the disk surface density scales
with stellar mass). In this case, the disk is not able to produce
a Jupiter-mass planet: The growing planet has reached a mass
of only MP = 14M⊕ at t = 10 Myr. No additional growth is
expected at later times because the mass of the entire disk is
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
F.C. Adams, P. Bodenheimer & G. Laughlin:M dwarfs: planet formation and long term evolution 915
0 2 4 6 8 10 12
0
10
20
30
40
50
60
time (Millions of Years)
Mass (Earth Masses)
Fig. 1. Growth of the core and envelopes of planets at 5.2 AU in
disks orbiting stars of two different masses (from LBA04). The
curves in the upper left part of the graph show the time-dependent
core mass (dotted curve) and total mass (solid curve) for a planet
forming in a disk surrounding a 1M star. The curves in the lower
right part of the graph show the time dependence of the core mass
(dotted curve) and total mass (solid curve) for a planet forming in
a disk around a 0.4M star. After 10 Myr, the disk masses become
extremely low, which effectively halts further planetary growth. The
planet orbiting the M star gains its mass more slowly and stops its
growth at a much lower mass M ≈ 14M⊕.
less than 1 MJ. The resulting planet is similar in mass, size,
and composition to Uranus and Neptune.
The aforementioned calculation shows that the formation
of giant planets is difficult at a ∼ 5 AU (for M stars).
Could other locations in the disk be viable? Additional calculations
(see LBA04) demonstrate that Jovian planet formation
around a 0.4M star is also compromised at radii of 1 AU
and 10 AU. The lower surface densities (σs) and longer orbital
timescales (Ω = GM /a3) ofM-star disks more than
offset the increase in tidal radius Rh and lead to a greatly reduced
capacity for forming Jovian-mass planets within the
standard core-accretion paradigm. In addition, the reduced
core mass found in the M-star disk results in much longer
times for the accretion of the gaseous envelope. Although red
dwarfs have a hard time forming Jupiter-mass planets, the
formation of Neptune-like objects and terrestrial-type planets
should be common around these low-mass stars. Indeed,
our failed attempts at giant planet formation around M stars
produced bodies with masses 14, 2.0, and 4.3 M⊕ after 10
Myr. Furthermore, the final sizes/masses of these objects correlates
with the surface density of solids (dust and ice) in the
precursor protoplanetary disk and should thus depend on the
metallicity of the host star.
In addition to the effects discussed above, planets forming
around red dwarfs face other problems. Most stars form
within groups and clusters, where external radiation from
other nearby stars can efficiently drive mass loss from disks
around M stars (Adams et al. 2004). M dwarfs are nearly
as bright as solar-type stars in their youth, but their gravitational
potential wells are less deep; this combination of properties
allows the inner disks to be more readily evaporated.
For M stars, photoevaporation due to both external radiation
fields and radiation from the central star can be more effective
than for solar-type stars by a factor of 10–100, depending on
the environment, so the gas supply for planet formation can
be much shorter lived. Star forming regions are dynamically
disruptive due to passing binary stars and background tidal
forces; these influences affect circumstellar disks and planetary
systems around M stars more effectively than in systems
anchored by larger primaries. Because these difficulties
affect not only planet formation, but also planet migration,
short-period Jovian-mass planets (hot Jupiters) should be especially
rare near M stars.
Although large planets like Jupiter should be rare near M
stars, it remains possible for small icy planets like Neptune
to form around solar type stars. The existing data base on extrasolar
planets now includes some examples. An interesting
challenge for the future is to observationally determine the
distribution of planet masses for stars of varying mass, and to
theoretically account for the observed distributions.
2.2. Potential observational tests
A number of observational programs can confirm or falsify
this predicted paucity of Jovian planets orbitingMstars; these
searches and can also detect Neptune-mass objects if they
are present. Given an adequate time baseline, the Doppler radial
velocity method (e.g., Marcy & Butler 1998) will determine
whether Jupiter-mass planets are common around red
dwarfs. A Neptune-mass planet in a circular orbit with semimajor
axis a = 3 AU around a 0.4M star has a period P
= 8.2 yr, and induces a stellar radial velocity half-amplitude
K = 1.6ms−1 (for inclination angle i = 90◦). This type of
planet is marginally detectable using the current RV precision
of 3ms−1, provided that one can attain a high sampling rate.
To date, ongoing planet searches have detected a few examples
of planets orbiting M dwarf stars, and a large collection
(∼ 100) red dwarfs are currently under surveillance.
The GJ876 system (Marcy et al. 2001), with M ∼
0.3 M , contains two Jovian planets (Mc sin i = 0.6MJ,
Mb sin i = 1.9MJ) with periods Pc ∼ 30 d and Pb ∼ 60 d.
Because of its resonant configuration, the system is thought
to have undergone migration (Lee & Peale 2002), which
suggests that an abundant gas supply was present when the
planets formed. The calculations presented here suggest that,
within the standard core-accretion paradigm, systems such as
GJ 876 are intrinsically rare; such systems can form, but they
must be drawn from the extreme high-mass end of the circumstellar
disk mass distribution (LBA04). In the alternative
formation scenario via gravitational instabilities (Boss 2000),
the growth rate depends primarily on the ratio of disk mass to
star mass; instabilities are not suppressed in disks surrounding
low mass stars (compared to disks orbiting solar-type
stars) as long as the disks are sufficiently massive. Systems
like GJ 876 may turn out to be examples of giant planet formation
through gravitational instability. On the other hand,
gravitational fragmentation will not generally produce ice giants
such as Neptune, so the discovery of ice giants in extrasolar
systems provides an important confirmation of the coreaccretion
hypothesis. The recent detection of a Neptune-mass
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
916 Astron. Nachr. / AN 326, No. 10 (2005) / www.an-journal.org
planet orbiting the M star GJ 436 (Butler et al. 2004) is the
first such example.
Microlensing is another promising technique for determining
the census of planets orbitingM dwarfs. Recent work
(Bond et al. 2004) reports an unusual light-curve for a Gtype
source star in the direction of the galactic bulge. The
observed light curve is consistent with the passage of an optically
faint binary lens with mass ratio μ = 0.0039+11
−07 for the
lensing system. Models of the galactic disk (Han & Gould
1996) indicate a ∼ 90% a-priori probability that the optically
faint lens primary is an M dwarf (in which case the
planet mass MP ∼ 1.5MJ) and a ∼ 10% probability that
the primary is a white dwarf (implying a somewhat higher
planet mass). In either case, the projected sky separation of
the primary-planet system is ∼ 3 AU. Our calculations imply
that M dwarfs should rarely harbor Jupiter-mass planets and
favor the possibility that the lensing system has a white dwarf
primary. This prediction can be tested within ∼ 10 years as
propermotion separates the source star from the lens.We also
predict that the mass spectrum of microlensed planets (drawn
largely from M dwarf primaries) should be shifted dramatically
toward Neptune-mass objects compared to the mass
spectrum of planets found by the ongoing RV and Transit surveys
(which draw predominantly from primaries of roughly
solar-mass).
Transits provide yet another method for detecting planets.
A Neptune-mass object in central transit around a 0.3
M M dwarf produces a photometric dip of approximately
1.5%. Such events are easily observed from the ground using
telescopes of modest aperture (Henry et al. 2000; Seagroves
et al. 2003). The forthcoming Kepler mission (Koch et al.
1998) will monitor ∼3600M dwarf stars with mV < 16 over
its 4-year lifetime, and will easily detect transits of objects
of Earth size or larger in orbit aroundM dwarf stars. Assuming
that icy core masses of M ∼ 1M⊕ can accrete at a ∼ 1
AU, the Kepler sample size will be large enough to provide a
statistical test of our hypothesis.
3. Long term evolution of M dwarfs
Solar-type stars have main sequence lifetimes that are comparable
to the current age of the universe, with the latter age
now known to be approximately 13.7 Gyr, and their long
term (post-main-sequence) development is well-studied (Iben
1974). In contrast, smaller stars live much longer than their
larger brethren, and hence red dwarfs live much longer than
a Hubble time. As a result, the post-main-sequence development
of these small stars had not been previously calculated.
This section describes stellar evolution calculations that follow
the post-main-sequence development of M dwarfs (Sect.
3.1). Since the smallest stars do not become red giants at the
end of their lives, this work also shed light on the question of
why larger stars do become red giants (Sect. 3.2).
3.1. Stellar evolution calculations
To follow the long term evolution of red dwarfs, we use not
only the now-standard Henyey method, but also the actual
Henyey code. After making updates of the opacities (Weiss
et al. 1990; Alexander et al. 1983; Pollack et al. 1985) and
equations of state (Saumon et al. 1995), we have carried out
a study of the long term development of red dwarfs (LBA97;
see also Adams & Laughlin 1997, hereafter AL97).
The basic trend for M dwarf evolution is illustrated in
Fig. 2, which shows evolutionary tracks in the H-R diagram
for stars in the mass range M = 0.08-0.25 M . Although it
has long been known that these small stars will live for much
longer than the current age of the universe, these stellar evolution
calculations reveal some surprises. For example, the
star with initial massM = 0.1 M remains nearly fully convective
for 5.74 trillion years. As a result, the star has access
to almost all of its nuclear fuel for almost all of its lifetime.
Whereas a 1.0 M star only burns about 10 percent of its hydrogen
on the main sequence, this star, with 10 percent of a
solar mass, burns nearly all of its hydrogen and thus has about
the same main sequence fuel supply as the Sun.
One of the most interesting findings of this work is that
small red dwarfs do not become red giants in their post-mainsequence
phases. Instead they remain physically small and
grow hotter to become blue dwarfs. Eventually, of course,
they run out of nuclear fuel and are destined to slowly fade
away as white dwarfs. As shown by the evolutionary tracks
in Fig. 2, the smallest star (in mass) that becomes a red giant
hasM = 0.25M . We return to this issue in Sect. 3.2.
The inset diagram in Fig. 2 shows that the stellar lifetimes
for these small stars measure in the trillions of years. A red
dwarf with massM = 0.25M has a main sequence lifetime
of about 1 trillion years, whereas the smallest stars withM =
0.08M last for 12 trillion years. Note that all of these calculations
are performed using solar metallicities. The metals in
stellar atmospheres act to keep a lid on the star and impede the
loss of radiation. As metallicity levels rise in the future, these
small stars can live even longer (AL97). Given that most stars
are red dwarfs, and that these small stars can live for trillions
upon trillions of years, the galaxy (and indeed the universe)
has only experienced a tiny fraction f ≈ 0.001 of its stelliferous
lifetime. Most of the stellar evolution that will occur is
yet to come.
Another interesting feature in Fig. 2 is the track of the star
with M = 0.16M . Near the end of its life, such a star experiences
a long period of nearly constant luminosity, about
one third of the solar value. This epoch of constant power
lasts for ∼ 5 Gyr, roughly the current age of the solar system
and hence the time required for life to develop on Earth.
Any planets in orbit about these small stars can, in principle,
come out of cold storage at this late epoch and can, again in
principle, provide another opportunity for life to flourish.
The galaxy continues to make new stars until it runs out
of gas, both literally and figuratively. With the current rate
of star formation, and the current supply of hydrogen gas,
the galaxy would run of gas in only a Hubble time or two.
Fortunately, this time scale can be extended by several conservation
practices, including recycling of gas due to mass
loss from evolved stars, infall onto the galactic disk, and the
reduction of the star formation rate as the gas supply dwindles.
With these effects included, the longest time over which
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
F.C. Adams, P. Bodenheimer & G. Laughlin:M dwarfs: planet formation and long term evolution 917
Fig. 2. The H-R diagram for red dwarfs with masses in the range M = 0.08 − 0.25M (from LBA97). Stars with mass M = 0.25M
are the least massive stars that can become red giants. The inset diagram shows the hydrogen burning lifetime as a function of stellar mass.
Note that these small stars live for trillions of years.
the galaxy can sustain normal star formation measures in the
trillions of years (AL97).
As the stellar population ages, the more massive stars die
off. Their contribution to the galactic luminosity is nearly
compensated by the increase in luminosity of the smaller
stars. The resulting late-time light curve for a large galaxy is
thus remarkably constant (Adams, Graves & Laughlin 2004).
3.2. Why stars become red giants
All astronomers know that our Sun is destined to become a
red giant (e.g., Iben 1974). On the other hand, a simple “first
principles” description of why stars become red giants is notable
in its absence (see also Renzini et al. 1992; Whitworth
1989). Through this study of low mass stars, which do not
become red giants, we can gain insight into this issue. The
details are provided in LBA97; here we present a simplified
argument that captures the essence of why stars become red
giants at the end of their lives.
This analytic argument begins with the standard relation
connecting the stellar luminosity L , radius R , and photospheric
temperature T , i.e.,
L = 4πR2
σT4
. (1)
Stars become more luminous as they age. This power increase
represents a “luminosity problem”, which can be solved in
one of two ways: The star can either become large in size,
so that R increases and the star becomes “giant”. Alternately,
the star can remain small and increase its temperature,
thereby becominG a BLUE DWARF
Nearly 150 planets have been detected orbiting nearby stars
(e.g., Marcy et al. 2003; Udry, Mayor & Queloz 2003), but
these planet searches have thus far concentrated on solar-type
host stars. The leading mechanism for explaining the origin
of both these extrasolar planets and the giant planets in our
solar system is the so-called core-accretion process. In this
paradigm, icy planetesimals collide and accumulate until they
build up planetary cores with mass Mc ≈ 5–15 M⊕. These
cores then accrete nebular gas and eventually reach masses
comparable to Jupiter.
A long-standing problem with the core-accretion hypothesis
was that the estimated time required for the core to grow
and subsequently accrete ∼ 1MJ of gas exceeded the observed
lifetimes of circumstellar disks (Pollack et al. 1996).
[We note that the process of gravitational instability does not
suffer from this shortcoming and has been seriously discussed
as an alternate mechanism (Boss 2000).] However, updated
estimates for the opacity in protoplanetary envelopes indicate
that Jupiter-mass planets can readily form through the
core-accretion mechanism in solar-metallicity disks around
solar-mass stars, with a time scale of 2–3 Myr at orbital radii
a ∼ 5 AU (Hubickyj, Bodenheimer & Lissauer 2004). To
achieve this relatively short formation time, the surface density
of solid material in the disk must be somewhat larger
a factor of ∼ 3) than that of the minimum-mass solar nebula
(MMSN). In one calculation, e.g., the resulting coremass was
about 16 M⊕, modestly larger than the core mass (10 M⊕)
deduced for Jupiter (Wuchterl, Guillot & Lissauer 2000). To
account for the low core mass of Jupiter, a cutoff of solid accretion
beyond a certain core mass is required, and can be explained
by nearby planetary embryos that compete for available
solid material. In an independent calculation with similar
opacities (Inaba & Ikoma 2003; Inaba,Wetherill & Ikoma
2003) and a disk with eight times the solid surface density of
the MMSN, a core of 25 M⊕ can form in 1 Myr at 5 AU, so
that the total formation time is 2–3 Myr. In this latter model,
the fragmentation of planetesimals and the enhancement of
the solid accretion rate due to the gaseous envelope (primarily
from gas drag) are taken into account, although the main gas
accretion phase is not calculated. If the solid surface density
is reduced to four times that of the MMSN, an 8 M⊕ core can
still form in 5 Myr. Taken together, these results imply that
giant planets can readily form, but somewhat special circumstances
are required for the core accretion model to explain
the particular properties of our Jupiter.
The ongoing observational surveys are shifting our view
of extrasolar planets from a disorganized collection of individual
systems (e.g., 51 Peg, υ And, or 47 UMa) to a more
robust statistical census. Different categories and populations
of planets can now be delineated (e.g., Marcy & Butler 1998;
Marcy, Cochran&Mayor 2000;Udry et al. 2003; Marcy et al.
2003). This emerging statistical view is important for improving
our understanding of the planet formation process, and to
learn how our own solar system fits into the galactic planetary
census. One of the cleanest statistical results to emerge
from extant planet searches is that stars with observed extrasolar
planets tend to have high metallicities, typically twice
that of the average Population I star in the solar neighborhood
(Fischer & Valenti 2005; Santos et al. 2003; Butler et
al. 2000). In addition, low metallicity stars are observed to be
deficient in currently detectable giant planets (withP <8 yr;
Sozzetti et al. 2004). This connection between planets and
host-star metallicity can be interpreted as evidence in favor
of the core accretion hypothesis (although it can also be interpreted
as evidence in favor of migration – see Sigurdsson
et al. 2003).Metal-rich circumstellar disks have a higher surface
density of solids and growing cores can easily reach the
Mc ≈ 5−−15M⊕ threshold required for rapid gas accretion.
In this work, we assume that the core accretion model
can explain the formation of Jovian planets within disks orbiting
solar-mass stars with solar metallicity. We then address
the question of whether or not Jovian planets can form
within disks orbiting around M dwarfs (M <∼
0.4M ). We
find that the core-accretion process makes a clear prediction
for the relative frequency of Jovian-mass planets as a
function of stellar mass: The circumstellar disks orbiting red
dwarfs are significantly less efficient in producing Jupitermass
planets than disks around solar-mass stars (Sect. 2.1),
although smaller, rockier planets (like Neptune) can readily
form.Moreover, this prediction is immediately testable (Sect.
2.2).
2.1. Theoretical model of planet formation
In the theoretical model explored here, giant planets form
within a circumstellar disk with the following properties: The
surface density σ(r) = σin(rin/r)3/2, where σin is the normalization
factor required to obtain a total disk mass Md(t)
within inner and outer disk radii, rin and rd. The time dependence
of the disk surface density is given by a depletion
function of the form fσ(t) = 1/(1 + t/t0) so that the
disk mass decreases according to Md(t) = Md(0)fσ(t). Observations
of circumstellar disks (e.g., Briceno et al. 2001)
suggest that t0 = 105 yr and Mid(0) = 0.05M are reasonable
benchmark values. The temperature distributions for
both viscously evolving accretion disks and flat, passively irradiated
disks have nearly the same power-law form, Td(r) =
Td (R /r)3/4, where Td is related to the stellar surface temperature
by a geometrical factor (e.g., Td /T ≈ [2/3π]1/4
for a flat disk – see Adams & Shu 1986). This model uses
disks that are flat and passive, and assumes that the disk is
isothermal in the vertical direction. The effective temperature
T of the star is related to the stellar radius R and luminosity
L (t,M ) through T (t,M ) = [L (t,M )/4πR2
σ]1/4.
We adopt T (t,M ) and L (t,M ) from published pre-mainsequence
stellar evolution tracks (D’Antona & Mazzitelli
1994).
We use a Henyey-type code (Henyey, Forbes & Gould
1964; see also Sect. 3) to compute the buildup and contraction
of gaseous envelopes surrounding growing protoplanetary
cores embedded within the evolving disk. This method
(Kornet, Bodenheimer& R´o˙zyczka 2002; Pollack et al. 1996)
adopts recent models for envelope opacity (Podolak 2003)
which include grain settling (see also Hubickyj et al. 2004).
The calculation is simplified in that it uses a core accretion
rate of the form dMc/dt = C1πσsRcRhΩ (Papaloizou &
Terquem 1999; compare with Pollack et al. 1996), where σs
is the surface density of solid material in the disk, Ω is the
orbital frequency, Rc is the effective capture radius for the
accretion of solid particles, Rh = a[Mp/(3M )]1/3 is the
tidal radius of the protoplanet, and C1 is a constant of order
unity. An important feature of this present model is that the
outer boundary conditions for the planet include the decrease
in the background gas density and temperature with time.
The results of this planet formation calculation are illustrated
in Fig. 1. The first simulation shown here corresponds
to a disk orbiting a 1 M star with an initial solid surface
density σs = 11.5 g cm−2 at a = 5 AU, about four times
that of the MMSN. This value is based on an initial gas-tosolid
ratio of 70 in the disk; as the disk evolves, σs decreases
with time because mass accretes onto the growing planet. A
Jupiter-mass planet forms in 3.25Myr with a core mass of 18
M⊕ (see also Hubickyj, Bodenheimer & Lissauer 2004). The
second calculation is for a disk surrounding an M star with
mass M = 0.4 M , where the initial solid surface density is
σs = 4.5 g cm−2 at a = 5 AU (the disk surface density scales
with stellar mass). In this case, the disk is not able to produce
a Jupiter-mass planet: The growing planet has reached a mass
of only MP = 14M⊕ at t = 10 Myr. No additional growth is
expected at later times because the mass of the entire disk is
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
F.C. Adams, P. Bodenheimer & G. Laughlin:M dwarfs: planet formation and long term evolution 915
0 2 4 6 8 10 12
0
10
20
30
40
50
60
time (Millions of Years)
Mass (Earth Masses)
Fig. 1. Growth of the core and envelopes of planets at 5.2 AU in
disks orbiting stars of two different masses (from LBA04). The
curves in the upper left part of the graph show the time-dependent
core mass (dotted curve) and total mass (solid curve) for a planet
forming in a disk surrounding a 1M star. The curves in the lower
right part of the graph show the time dependence of the core mass
(dotted curve) and total mass (solid curve) for a planet forming in
a disk around a 0.4M star. After 10 Myr, the disk masses become
extremely low, which effectively halts further planetary growth. The
planet orbiting the M star gains its mass more slowly and stops its
growth at a much lower mass M ≈ 14M⊕.
less than 1 MJ. The resulting planet is similar in mass, size,
and composition to Uranus and Neptune.
The aforementioned calculation shows that the formation
of giant planets is difficult at a ∼ 5 AU (for M stars).
Could other locations in the disk be viable? Additional calculations
(see LBA04) demonstrate that Jovian planet formation
around a 0.4M star is also compromised at radii of 1 AU
and 10 AU. The lower surface densities (σs) and longer orbital
timescales (Ω = GM /a3) ofM-star disks more than
offset the increase in tidal radius Rh and lead to a greatly reduced
capacity for forming Jovian-mass planets within the
standard core-accretion paradigm. In addition, the reduced
core mass found in the M-star disk results in much longer
times for the accretion of the gaseous envelope. Although red
dwarfs have a hard time forming Jupiter-mass planets, the
formation of Neptune-like objects and terrestrial-type planets
should be common around these low-mass stars. Indeed,
our failed attempts at giant planet formation around M stars
produced bodies with masses 14, 2.0, and 4.3 M⊕ after 10
Myr. Furthermore, the final sizes/masses of these objects correlates
with the surface density of solids (dust and ice) in the
precursor protoplanetary disk and should thus depend on the
metallicity of the host star.
In addition to the effects discussed above, planets forming
around red dwarfs face other problems. Most stars form
within groups and clusters, where external radiation from
other nearby stars can efficiently drive mass loss from disks
around M stars (Adams et al. 2004). M dwarfs are nearly
as bright as solar-type stars in their youth, but their gravitational
potential wells are less deep; this combination of properties
allows the inner disks to be more readily evaporated.
For M stars, photoevaporation due to both external radiation
fields and radiation from the central star can be more effective
than for solar-type stars by a factor of 10–100, depending on
the environment, so the gas supply for planet formation can
be much shorter lived. Star forming regions are dynamically
disruptive due to passing binary stars and background tidal
forces; these influences affect circumstellar disks and planetary
systems around M stars more effectively than in systems
anchored by larger primaries. Because these difficulties
affect not only planet formation, but also planet migration,
short-period Jovian-mass planets (hot Jupiters) should be especially
rare near M stars.
Although large planets like Jupiter should be rare near M
stars, it remains possible for small icy planets like Neptune
to form around solar type stars. The existing data base on extrasolar
planets now includes some examples. An interesting
challenge for the future is to observationally determine the
distribution of planet masses for stars of varying mass, and to
theoretically account for the observed distributions.
2.2. Potential observational tests
A number of observational programs can confirm or falsify
this predicted paucity of Jovian planets orbitingMstars; these
searches and can also detect Neptune-mass objects if they
are present. Given an adequate time baseline, the Doppler radial
velocity method (e.g., Marcy & Butler 1998) will determine
whether Jupiter-mass planets are common around red
dwarfs. A Neptune-mass planet in a circular orbit with semimajor
axis a = 3 AU around a 0.4M star has a period P
= 8.2 yr, and induces a stellar radial velocity half-amplitude
K = 1.6ms−1 (for inclination angle i = 90◦). This type of
planet is marginally detectable using the current RV precision
of 3ms−1, provided that one can attain a high sampling rate.
To date, ongoing planet searches have detected a few examples
of planets orbiting M dwarf stars, and a large collection
(∼ 100) red dwarfs are currently under surveillance.
The GJ876 system (Marcy et al. 2001), with M ∼
0.3 M , contains two Jovian planets (Mc sin i = 0.6MJ,
Mb sin i = 1.9MJ) with periods Pc ∼ 30 d and Pb ∼ 60 d.
Because of its resonant configuration, the system is thought
to have undergone migration (Lee & Peale 2002), which
suggests that an abundant gas supply was present when the
planets formed. The calculations presented here suggest that,
within the standard core-accretion paradigm, systems such as
GJ 876 are intrinsically rare; such systems can form, but they
must be drawn from the extreme high-mass end of the circumstellar
disk mass distribution (LBA04). In the alternative
formation scenario via gravitational instabilities (Boss 2000),
the growth rate depends primarily on the ratio of disk mass to
star mass; instabilities are not suppressed in disks surrounding
low mass stars (compared to disks orbiting solar-type
stars) as long as the disks are sufficiently massive. Systems
like GJ 876 may turn out to be examples of giant planet formation
through gravitational instability. On the other hand,
gravitational fragmentation will not generally produce ice giants
such as Neptune, so the discovery of ice giants in extrasolar
systems provides an important confirmation of the coreaccretion
hypothesis. The recent detection of a Neptune-mass
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
916 Astron. Nachr. / AN 326, No. 10 (2005) / www.an-journal.org
planet orbiting the M star GJ 436 (Butler et al. 2004) is the
first such example.
Microlensing is another promising technique for determining
the census of planets orbitingM dwarfs. Recent work
(Bond et al. 2004) reports an unusual light-curve for a Gtype
source star in the direction of the galactic bulge. The
observed light curve is consistent with the passage of an optically
faint binary lens with mass ratio μ = 0.0039+11
−07 for the
lensing system. Models of the galactic disk (Han & Gould
1996) indicate a ∼ 90% a-priori probability that the optically
faint lens primary is an M dwarf (in which case the
planet mass MP ∼ 1.5MJ) and a ∼ 10% probability that
the primary is a white dwarf (implying a somewhat higher
planet mass). In either case, the projected sky separation of
the primary-planet system is ∼ 3 AU. Our calculations imply
that M dwarfs should rarely harbor Jupiter-mass planets and
favor the possibility that the lensing system has a white dwarf
primary. This prediction can be tested within ∼ 10 years as
propermotion separates the source star from the lens.We also
predict that the mass spectrum of microlensed planets (drawn
largely from M dwarf primaries) should be shifted dramatically
toward Neptune-mass objects compared to the mass
spectrum of planets found by the ongoing RV and Transit surveys
(which draw predominantly from primaries of roughly
solar-mass).
Transits provide yet another method for detecting planets.
A Neptune-mass object in central transit around a 0.3
M M dwarf produces a photometric dip of approximately
1.5%. Such events are easily observed from the ground using
telescopes of modest aperture (Henry et al. 2000; Seagroves
et al. 2003). The forthcoming Kepler mission (Koch et al.
1998) will monitor ∼3600M dwarf stars with mV < 16 over
its 4-year lifetime, and will easily detect transits of objects
of Earth size or larger in orbit aroundM dwarf stars. Assuming
that icy core masses of M ∼ 1M⊕ can accrete at a ∼ 1
AU, the Kepler sample size will be large enough to provide a
statistical test of our hypothesis.
3. Long term evolution of M dwarfs
Solar-type stars have main sequence lifetimes that are comparable
to the current age of the universe, with the latter age
now known to be approximately 13.7 Gyr, and their long
term (post-main-sequence) development is well-studied (Iben
1974). In contrast, smaller stars live much longer than their
larger brethren, and hence red dwarfs live much longer than
a Hubble time. As a result, the post-main-sequence development
of these small stars had not been previously calculated.
This section describes stellar evolution calculations that follow
the post-main-sequence development of M dwarfs (Sect.
3.1). Since the smallest stars do not become red giants at the
end of their lives, this work also shed light on the question of
why larger stars do become red giants (Sect. 3.2).
3.1. Stellar evolution calculations
To follow the long term evolution of red dwarfs, we use not
only the now-standard Henyey method, but also the actual
Henyey code. After making updates of the opacities (Weiss
et al. 1990; Alexander et al. 1983; Pollack et al. 1985) and
equations of state (Saumon et al. 1995), we have carried out
a study of the long term development of red dwarfs (LBA97;
see also Adams & Laughlin 1997, hereafter AL97).
The basic trend for M dwarf evolution is illustrated in
Fig. 2, which shows evolutionary tracks in the H-R diagram
for stars in the mass range M = 0.08-0.25 M . Although it
has long been known that these small stars will live for much
longer than the current age of the universe, these stellar evolution
calculations reveal some surprises. For example, the
star with initial massM = 0.1 M remains nearly fully convective
for 5.74 trillion years. As a result, the star has access
to almost all of its nuclear fuel for almost all of its lifetime.
Whereas a 1.0 M star only burns about 10 percent of its hydrogen
on the main sequence, this star, with 10 percent of a
solar mass, burns nearly all of its hydrogen and thus has about
the same main sequence fuel supply as the Sun.
One of the most interesting findings of this work is that
small red dwarfs do not become red giants in their post-mainsequence
phases. Instead they remain physically small and
grow hotter to become blue dwarfs. Eventually, of course,
they run out of nuclear fuel and are destined to slowly fade
away as white dwarfs. As shown by the evolutionary tracks
in Fig. 2, the smallest star (in mass) that becomes a red giant
hasM = 0.25M . We return to this issue in Sect. 3.2.
The inset diagram in Fig. 2 shows that the stellar lifetimes
for these small stars measure in the trillions of years. A red
dwarf with massM = 0.25M has a main sequence lifetime
of about 1 trillion years, whereas the smallest stars withM =
0.08M last for 12 trillion years. Note that all of these calculations
are performed using solar metallicities. The metals in
stellar atmospheres act to keep a lid on the star and impede the
loss of radiation. As metallicity levels rise in the future, these
small stars can live even longer (AL97). Given that most stars
are red dwarfs, and that these small stars can live for trillions
upon trillions of years, the galaxy (and indeed the universe)
has only experienced a tiny fraction f ≈ 0.001 of its stelliferous
lifetime. Most of the stellar evolution that will occur is
yet to come.
Another interesting feature in Fig. 2 is the track of the star
with M = 0.16M . Near the end of its life, such a star experiences
a long period of nearly constant luminosity, about
one third of the solar value. This epoch of constant power
lasts for ∼ 5 Gyr, roughly the current age of the solar system
and hence the time required for life to develop on Earth.
Any planets in orbit about these small stars can, in principle,
come out of cold storage at this late epoch and can, again in
principle, provide another opportunity for life to flourish.
The galaxy continues to make new stars until it runs out
of gas, both literally and figuratively. With the current rate
of star formation, and the current supply of hydrogen gas,
the galaxy would run of gas in only a Hubble time or two.
Fortunately, this time scale can be extended by several conservation
practices, including recycling of gas due to mass
loss from evolved stars, infall onto the galactic disk, and the
reduction of the star formation rate as the gas supply dwindles.
With these effects included, the longest time over which
c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
F.C. Adams, P. Bodenheimer & G. Laughlin:M dwarfs: planet formation and long term evolution 917
Fig. 2. The H-R diagram for red dwarfs with masses in the range M = 0.08 − 0.25M (from LBA97). Stars with mass M = 0.25M
are the least massive stars that can become red giants. The inset diagram shows the hydrogen burning lifetime as a function of stellar mass.
Note that these small stars live for trillions of years.
the galaxy can sustain normal star formation measures in the
trillions of years (AL97).
As the stellar population ages, the more massive stars die
off. Their contribution to the galactic luminosity is nearly
compensated by the increase in luminosity of the smaller
stars. The resulting late-time light curve for a large galaxy is
thus remarkably constant (Adams, Graves & Laughlin 2004).
3.2. Why stars become red giants
All astronomers know that our Sun is destined to become a
red giant (e.g., Iben 1974). On the other hand, a simple “first
principles” description of why stars become red giants is notable
in its absence (see also Renzini et al. 1992; Whitworth
1989). Through this study of low mass stars, which do not
become red giants, we can gain insight into this issue. The
details are provided in LBA97; here we present a simplified
argument that captures the essence of why stars become red
giants at the end of their lives.
This analytic argument begins with the standard relation
connecting the stellar luminosity L , radius R , and photospheric
temperature T , i.e.,
L = 4πR2
σT4
. (1)
Stars become more luminous as they age. This power increase
represents a “luminosity problem”, which can be solved in
one of two ways: The star can either become large in size,
so that R increases and the star becomes “giant”. Alternately,
the star can remain small and increase its temperature,
thereby becominG a BLUE DWARF
