Giant Planets at Small Orbital Distances
astro-ph/9511109 22 Nov 1995 hubbard jlunine dsaumon lpl arizona edu burrows cobalt physics arizona edu Department Departments Hubble Postdoctoral T Received Guillot Giant of of A Burrows Planetary Physics Fellow Planets and Sciences W B Hubbard Astronomy at University Small University accepted Orbital J I Lunine of Arizona of Distances Arizona Tucson and D Saumon Tucson AZ AZ guillot
ABSTRACT
Using Doppler sp ectroscopy to detect the re ex motion of the nearby star
Pegasi Mayor Queloz claim to have discovered a giant planet in a
A U day orbit They estimate its mass to b e in the range M to
J
M but are not able to determine its nature or origin Including the e ects
J
of the severe stellar insolation implied we extend the theory of giant planets
we have recently developed to encompass those at very small orbital distances
Our calculations can b e used to help formulate search strategies for luminous
planets in tight orbits around other nearby stars We calculate the radii and
luminosities of such giants for a variety of comp ositions H He He H O and
olivine evolutionary tracks for solar comp osition gas giants and the geometry
of the Hayashi forbidden zone in the gas giant mass regime We show that such
planets are stable and estimate the magnitude of classical Jeans evaporation
and of photo disso ciation and loss due to EUV radiation Even over the lifetime
of the primary the companion would not have lost a large fraction of its mass
In addition we demonstrate that for the mass range quoted such planets are
well within their Ro che lob es We show that the strong comp osition dep endence
of the mo del radii and distinctive sp ectral signatures provide clear diagnostics
that might reveal Peg B s nature should interferometric or adaptive optics
techniques ever succeed in photometrically separating planet from star
Introduction
As the search for planets and brown dwarfs around nearby stars accelerates we should
exp ect to b e surprised In no instance is this b etter illustrated than in the recent discovery
by Mayor Queloz of a planet orbiting a G star Pegasi parsecs away
With a day p erio d a semi ma jor axis of A U an eccentricity less than and
an inferred mass b etween and Jupiter masses M this ob ject is surely the most
J
problematic nd in recent memory As of this writing the tell tale p erio dic Doppler shift
in the sp ectral lines of the primary had b een con rmed by Marcy Butler and by
Noyes et al A go o d case can b e made for the existence of Peg B simply from
the absence of signi cant photometric variations in V mag Mayor Queloz
Burki Burnet Kuenzli or asymmetries in the line pro les Mayor Queloz
and from the di culty of explaining such a p erio d as pulsation of a near solar analog
One hundred times closer to its primary than Jupiter itself Peg B thwarts
conventional wisdom Boss had argued that the nucleation of a H He rich Jovian
planet around a ro ck and ice core could b e achieved in a protostellar disk only at and
b eyond the ice p oint at Kelvin exterior to A U Walker et al had surveyed
for re ex motion G type stars over years and seen nothing more massive than
M interior to A U Zuckermann Forveille Kastner had measured CO
J
emissions from a variety of near T Tauri disks had extrap olated to H and had concluded
that there may not b e enough mass or time to form a Jupiter around a ma jority of stars
The discovery of Peg B while not strictly inconsistent with any of these pap ers vastly
enlarges the parameter space within which we must now search
Several scenarios for the origin of Peg B are emerging
It could b e a canonical gas giant that formed many A U s from Peg A but
through frictional and tidal e ects spiraled inward during the protostellar phase Lin
Bo denheimer
As ab ove it could have formed conventionally but collided with a massive companion
and lost of its orbital angular momentum and of its orbital energy
It could b e comp osed predominantly of hydrogen and helium accreted from the
protostellar disk but have nucleated in situ around a large ro ck core without ice
It could b e a giant terrestrial planet formed by the accumulation of planetesimals or
It could b e an evaporated ablated or tidally stripp ed brown dwarf or star
However whatever the provenance or evolutionary history of Peg B a knowledge of
the thermal and structural characteristics of giant planets with a variety of comp ositions
and masses M is required to understand it and others like it A chondritic helium
p
or ice planet with a mass of M has a radius R that is signi cantly smaller than
J p
that of a hydrogen rich Jupiter At a given e ective temp erature smaller radii translate
into smaller luminosities L
Recently we have studied the theoretical evolution of gas giants around nearby stars
with masses from through M and of Brown Dwarfs M Dwarfs with masses from
J
through M Saumon et al Burrows et al Saumon et al Burrows
J
et al Burrows Hubbard Lunine Over these three orders of magnitude in
mass the basic input physics is the same Though we had previously considered the e ects
of stellar insolation we had not explored such e ects at separations near those of Peg
B In this pap er we present a theory of extra solar giant planets at small orbital distances
D We calculate the radii and luminosities of planets with a variety of comp ositions
masses and separations Though we fo cus on the Peg A Peg B system our results
can b e extended and scaled to planetary systems with other characteristics and are meant
to aid in the formulation of search strategies around nearby stars In addition we explore
the p ossibilities of tidal truncation and evaporation and conclude with a discussion on the
photometric discriminants of the various theories concerning the nature of Peg B
The Radii of Giant Planets as a Function of Comp osition
The hydrogen helium equation of state that we have employed for this study is
describ ed in Saumon Chabrier Van Horn and incorp orates state of the art
prescriptions for the interactions among H H protons and electrons and for the
metallization of hydrogen helium mixtures at high pressures The evolutionary co des that
we have employed are a Henyey co de Burrows et al and a co de that treats
the quasi static problem as an implicit two point b oundary value problem Guillot Morel
The latter was constructed to address the p ossibility that Jupiter and Saturn may
have non adiabatic structures Guillot et al and allows for the existence of large
radiative zones We have expanded this more general co de to address the evolution of giant
planets very near their primaries for a variety of planet masses For the evolution of the gas
giants we used the opacities of Alexander Ferguson
At the distance of Jupiter from the Sun the di erence b etween convective and
radiative convective mo dels is slight However when the orbital distance of a gas giant
from a G V star for example is smaller than A U external heating by the star
b ecomes imp ortant early in the planet s evolution towards its steady state Due to the very
small heat di usivities in gas giants at high pressure this radiative zone do es not p enetrate
deeply in mass p erhaps encompassing but can p enetrate deeply in radius by as
much as in a Hubble time Convective heat transp ort from the inner convective zone
continues to co ol it The entropy of the radiative zone at the photosphere is maintained at
a higher roughly constant value as the e ective temp erature T of the planet stabilizes
e
Since its interior continues to co ol and lose its thermal pressure supp ort the entire planet
continues to shrink Because the planet s e ective temp erature is stabilizing its luminosity
is decreasing though at a progressively lower rate see x b elow For Peg B the
predicted radii after Gigayear Gyr are b etween R and R for M s from
J J p
M to M These are as much as a factor of two smaller that the corresp onding radii
J J
for fully convective planets After Gyr the estimated age of Peg A the radii for
these same planets are b etween R and R The b ehavior of radius versus mass for
J J
giant planets in the mass range suggested for Peg B is depicted in Figure Radii and
b olometric luminosities not including the re ected comp onent for various representative
mo dels are given in Tables a b We have included on Figure the corresp onding curves
for helium H O and olivine Mg SiO planets as well as that for fully convective planets
The equations of state for H O and olivine as representative of ro ck were taken from the
ANEOS compilation Thompson The large mean molecular weight of giant ro cky
planets ensures that thermal e ects are small in most of the interior Hubbard so
that an olivine or ice planet in close orbit around a star will probably not have a radius
signi cantly larger than predicted by our calculations
Though Peg A has not had enough time to synchronize its spin p erio d with Peg
B s orbital p erio d Peg B s spin p erio d is surely tidally lo cked with its orbit at days
The time for the tidal spin down of the planet is given by
R
M D
p
p
Q
p
GM M R
p p
where Q is the planet s tidal dissipation factor is the planet s primordial rotation
p
rate M is the star s mass and G is the gravitational constant Taking Q and
s Jupiter s values we obtain years For a giant terrestrial
p
planet R is smaller but Q is also smaller and so would not b e very di erent Therefore
p
since Peg A s age is years Peg B should always present the same face to its
primary whatever its comp osition
The equilibrium e ective temp erature of the planet is given by the formula
T T R D f A
eq
and the equilibrium luminosity by
L L A R D
eq p
where R T and L are the primary s radius e ective temp erature and luminosity and
A is the Bond alb edo of the planet which for Jupiter is The re ected luminosity is
L A A The factor f is if the heat of the primary can b e assumed to b e evenly
eq
distributed over the planet and if only one side reradiates the absorb ed heat For
Peg B and an alb edo of T is roughly K an order of magnitude ab ove that of
eq
Jupiter and indep endent of R and M We assumed that the luminosity of Peg A is
p p
higher than that of our Sun Mayor Queloz and that absorb ed heat is quickly
distributed to the night side to b e radiated f The latter assumption is fully justi ed
for a gas giant or for any planet with a thick atmosphere due to rapid zonal and meridional
circulation patterns but may b e problematic for a bare ro ck If Peg B were a giant
terrestrial planet without an atmosphere its temp erature at the sub stellar p oint could b e
as high as K Table b ab ove the melting p oint of many ro cks Needless to say the
planet s L is una ected by tidal lo cking though the phase dep endence of its brightness is
eq
Figure and Tables a b show the pronounced and diagnostic variation of radius
with comp osition Planets comp osed of materials with low electron fraction p er baryon and
high Z are signi cantly more compact Zap olsky Salp eter A giant terrestrial planet
would b e three times smaller than a gas giant of the same mass and its corresp onding
luminosity would b e an order of magnitude lower The latter dep ends up on the alb edos
assumed but only weakly for alb edos b elow If photometry can b e p erformed on Peg
B a measurement of its b olometric luminosity would immediately distinguish the di erent
mo dels
The H R Diagram for Giant Planets
Figure is a theoretical Hertzprung Russell diagram that p ortrays the ma jor results of
y
this study Depicted are L T tracks for the evolution of a M gas giant and L for a
e J
M olivine planet op en triangles all at a variety of orbital distances indicated by the
J
arrows Also shown are the Hayashi track b oundary of the dark shaded region
the Hayashi exclusion zone the dark shaded region itself the Ro che exclusion zone the
lightly shaded region and the classical Jeans evaporation limit dash dotted line The
shap e of the Hayashi exclusion zone and the evolutionary ages dep end slightly up on the
atmospheric mo del employed Figures such as Figure can b e rendered for any sp eci c
planetary mass alb edo and primary but we fo cus here on M M A and
p J
Peg A The dashed lines on Figure are lines of constant radius The numbers on the
tracks are the common logarithms of the ages in years
The evolution of a fully convective planet can b e separated into two phases a rapid
contraction phase with large internal luminosity converted from p otential gravitational
energy and increasing e ective temp erature the Hayashi b oundary from the top right to
the top middle of Figure and a slow co oling phase during which b oth the internal
luminosity and the e ective temp erature decrease the Hayashi b oundary from the top
y
Here again luminosities do not include the re ected comp onent
center to the b ottom right of Figure The transition b etween these two phases o ccurs
at R s around R regardless of the mass of the planet The planet s internal luminosity
p J
tends to zero and its e ective temp erature tends to T The present Jupiter is depicted by
eq
a diamond in the lower right hand corner of Figure Its evolutionary track closely follows
the convective Hayashi track
For a given mass and comp osition every fully convective mo del lies on the same curve
in the H R diagram No mo del can exist to the right of this curve at lower T This
e
region the dark shaded zone to the right in Figure is the Hayashi forbidden zone As
a corollary any planet that lies to the left of the curve is partially radiative conductive
This implies that fully convective mo dels cannot exist for large e ective temp eratures
T K for M M However as T approaches T its internal luminosity
e p J e eq
drops until a radiative zone app ears in the outer region and grows This allows the planet
to co ol and shrink b eyond the limit set by fully convective calculations This b ehavior is
illustrated in Figure by the curling of the lines o of the Hayashi track At A U a
M planet follows the fully convective track for less than years It then has a radius
J
of ab out R At that p oint a radiative outer region app ears and the planet slowly
J
contracts at a nearly constant e ective temp erature After billion years of evolution its
radius is only R and its luminosity is ab out L more than times
J
the present luminosity of Jupiter and only a factor of two b elow that at the edge of the
main sequence The radiative region encompasses the outer in mass and in
radius The temp erature is ab out K at bar and around K at the center of
the planet
The quasi static evolution of partially radiative planets is p ossible even for tiny
star planet separations Such mo dels are not unstable At small orbital separations the
evolution is substantially slowed down by stellar heating The almost vertical evolution
tracks seen in Figure for orbital distances smaller than A U are a consequence of
the fact that the internal luminosity of the planet is constrained to b e small Otherwise it
would b e fully convective which is not p ossible at these e ective temp eratures Similarly
the planet must contract slowly to keep the internal luminosity small However note that
the lines are illustrative evolutionary tracks started for sp eci city at high L s and T
e
T
eq
The Ro che excluded region is b ounded by a line of nearly constant L whose value is
This small but signi cant region constrains mo dels for Peg B s prop ortional to M
p
formation If Peg B were formed b eyond an A U and moved inward on a timescale
greater than years it would closely follow the R R tra jectory to its equilibrium
p J
p osition on Figure
Thermal and Non thermal Evaporation of a Gas Giant
If Peg B is a gas giant is it stable to evaporation and if so what is its current
evaporation rate We consider two p otential loss mechanisms classical Jeans
evaporation and the non thermal pro duction of hot hydrogen atoms and ions by
absorption of ultraviolet radiation from Peg A
The classical Jeans escap e ux is prop ortional to e where GM m k T R
p H p
Chamberlain and Hunten Here m is the mass of the hydrogen atom or molecule
H
k is Boltzmann s constant and T is the temp erature of the planet at the escap e level For
atomic hydrogen if T K R R and M M is close to and
p J p J
Jeans escap e might b e imp ortant The dash dotted line on Figure is the line
However this combination of parameters is unlikely for Peg B see Figures and
Our hydrogen helium giant mo dels at the age of Peg A have radii closer to R
J
and actual s b etween and Hence our mo del planets in the Mayor Queloz mass
range are much to o compact for classical Jeans escap e of any ion or atom to b e signi cant
The pro duction and escap e of hot ions H and H and hot atomic hydrogen by
stellar ultraviolet radiation is much more likely since these fragments obtain a residual
non thermal kinetic energy during their pro duction This leaves them in the fast tail of the
Jeans escap e function Using the estimate of Atreya of cm s for the total
H H and H ux from Jupiter and assuming that the EUV ux from Peg A is the
same as the Sun s we nd that a gas giant at A U with a mass of M would lose
J
H s s or M y r Only of the mass of a M gas giant at the p osition
J
of Peg B would b e lost due to EUV radiation over the main sequence lifetime of Peg
A Since the mechanical luminosity of the solar wind is similar to the Sun s total EUV
luminosity extrap olating the Sun s wind p ower to Peg A implies that wind ablation of
Peg B may b e no more imp ortant Note that a planet comp osed of a higher Z material
would b e much less prone to evaporation or stripping Interestingly the EUV evaporation
rate of Peg B may exceed of the mass loss rate from Peg A itself
While these numbers suggest that a Jupiter type planet at A U is stable they
are only ab out two orders of magnitude away from erosion of the entire planet Close
observation may reveal rather dramatic phenomena asso ciated with the escap e from Peg
B of the disso ciation and ionization pro ducts of H
Mo del Diagnostics
We have shown how luminosity and radius are the primary discriminants b etween
gas giant and giant terrestrial planet mo dels for Peg B However it may someday b e
p ossible to identify sp ectral signatures which can directly characterize the comp osition
and or origin of the ob ject
A primarily silicate but Jovian mass planet is an unusual ob ject which we cannot
rule out As Figures and Tables a b demonstrate its luminosity would b e
one tenth that of a gas giant of the same mass Its sp ectroscopic signature would b e a
strong silicate absorption band in the micron wavelength region A massive water vapor
atmosphere would long ago have b een photo disso ciated into hydrogen and oxygen unless
the abundance of water were a signi cant fraction of the mass of the planet We
consider this unlikely The presence of molten sulfur comp ounds on the surface cannot b e
ruled out and might even obscure completely the silicate sp ectral signature
A predominantly hydrogen helium planet will not app ear like Jupiter even if the
comp osition is similar Ammonia clouds in Jupiter s atmosphere play a signi cant role in
determining the scattering prop erties of the atmosphere and help to shap e and de ne the
micron sp ectral window region Such clouds will b e absent in Peg B as will water
clouds At an e ective temp erature of roughly K the primary cloud forming materials
near the surface are magnesium silicates and other silicate comp ounds However our
mo dels suggest that these clouds will b e b elow the unity optical depth level Because of
this the scattering optical depth of the atmosphere is exp ected to b e small and absorption
features relatively deep and well de ned Collision induced molecular hydrogen opacity will
b e an imp ortant source of absorption in the near infrared and infrared and may provide
detectable features Imp ortantly water and carb on monoxide absorption features should
b e present Lunine et al In contrast to Jupiter methane is exp ected to b e absent
sp ectroscopically b ecause at high temp eratures carb on monoxide is the thermo dynamically
preferred carb on b earing molecule
We have demonstrated in this pap er that gas giants can b e stable even for very
small orbital distances and have explored the structural and thermal consequences of
various mo dels of Peg B Photometry and sp ectrophotometry using very advanced
interferometric and adaptive optics techniques may well b e the key to distinguishing the
di erent theories for the origin and nature of Peg B Angel Kulkarni
However whatever its true nature Peg B has op ened a new chapter in planetary studies
Acknowledgments
The authors would like to thank Willy Benz Peter Hauschildt Jim Lieb ert Doug
Lin Geo Marcy Jay Melosh and Andy Nelson for stimulating conversations numbers
in advance of publication or b oth Gratitude is extended to the N S F and to N A S A
for supp ort under grants AST and NAG resp ectively T G acknowledges
supp ort from the Europ ean Space Agency and D S acknowledges NASA grant HF
A from the Space Telescope Science Institute which is op erated by the Asso ciation of Universities for Research in Astronomy Inc under NASA contract NAS
Table a Radiative convective H He planet orbiting at A U from Peg A
M M R R T K log L L
p J p J e
Table b Olivine planet orbiting at A U from Peg A syn is short for
synchronous and implies that the p eak values for the star facing hemisphere are
given
M M R R T K non syn T K syn log L L
p J p J e e
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This manuscript was prepared with the AAS L T X macros v E
Fig Radius R versus mass M for top to b ottom fully adiabatic gas giants
p p
with surface temp erature determined by radiative equilibrium with Peg A gas giants
with radiative regions near the surface at the age of Peg A realistic gas giant mo del
pure helium giants with radiative convective structure at the same age pure H O mo dels at
zero temp erature pure olivine Mg SiO mo dels at zero temp erature The structures of the
H O and olivine planets were determined using the ANEOS equation of state Thompson
The temp erature corrections to the radii of these compact planet types are small
Fig Hertzprung Russell diagram for M planets orbiting at
J
and A U from a star with the prop erties of Peg A assuming a Bond alb edo of
These can b e scaled for di erent alb edos A mo del with D A U A is identical
to one with D A see Eq Arrows indicate the corresp onding equilibrium
e ective temp erature A Jupiter mo del is also shown the diamond in the b ottom right hand
corner corresp onding to the present day e ective temp erature and luminosity of the planet
Evolutionary tracks for planets of solar comp osition are indicated by lines connecting dots
which are equally spaced in log time The numbers are the common logarithms
of the planet s age Zero temp erature mo dels for M planets made of olivine Mg SiO
J
are indicated by triangles The Hayashi forbidden region which is enclosed by the fully
convective mo del s evolutionary track is shown in dark grey see text Mo dels in the light
grey region assume an alb eb o of and have radii ab ove the Ro che limit and therefore are
tidally disrupted by the star The region where classical Jeans escap e b ecomes signi cant
is b ounded by the dash dotted line Lines of constant radius are indicated by dotted curves
These corresp ond from b ottom to top to radii in units of R in multiples of starting
J
at