Thermal Tides in Short Period Exoplanets

0.63 MB·

by  Phil Arras

Draftversion January21,2009
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
THERMAL TIDES IN SHORT PERIOD EXOPLANETS
Phil Arras1 and Aristotle Socrates2
Draft version January 21, 2009
ABSTRACT
Time-dependent insolation in a planetary atmosphere induces a mass quadrupole upon which the
stellar tidal acceleration can exert a force. This “thermal tide” force can give rise to secular torques
on the planet and orbit as well as radial forces causing eccentricity evolution. We apply this idea to
9
theclose-ingasgiantexoplanets(“hotJupiters”). Theresponseofradiativeatmospheresiscomputed
0
in a hydrostatic model which treats the insolation as a time-dependent heat source, and solves for
0
thermal radiation using flux-limited diffusion. Fully nonlinear numerical simulations are compared
2
to solutions of the linearized equations, as well as analytic approximations, all of which are in good
n agreement. We find generically that thermal tide density perturbations lead the semi-diurnal forcing.
a
As a result thermal tides can generate asynchronous spin and eccentricity. Applying our calculations
J
to the hot Jupiters, we find the following results: (1) Departure from synchronous spin is significant
7
for hot Jupiters, andincreases with orbitalperiod. (2) Ongoinggravitationaltidaldissipation in spin
equilibrium leads to steady-state internalheating rates up to 1028 erg s−1. If deposited sufficiently
] ∼
P deep, these heating rates may explain the anomalously large radii of many hot Jupiters in terms of
E a “tidal main sequence” where cooling balances tidal heating. At fixed stellar type, planet mass and
tidalQ,planetaryradiusincreasesstronglytowardthestarinsideorbitalperiods.2weeks. (3)There
.
h existsanarrowwindowinorbitalperiodwheresmalleccentricities,e,growexponentiallywithalarge
p
rate. This window may explain the 1/4 of hot Jupiters which should have been circularized by the
- ∼
o gravitational tide long ago, but are observed to have significant nonzero e. Conversely, outside this
r window, the thermal and gravitationaltide both act to damp e, complicating the ability to constrain
t the planet’s tidal Q.
s
a Subject headings: planets – tides
[

1
v 1. INTRODUCTION arapidlyrotatingyoungstar(Dobbs-Dixon et al.2004),
perturbationsfromotherplanets,oralargerangeoftidal
5 A number of puzzles have arisen for the gas giant ex-
3 oplanets orbiting close to their parent stars, the “hot Q = 105−109 (Matsumura et al. 2008). In the contin-
7 Jupiters.” ued absence of a detectable perturber with appropriate
0 mass and orbit to account for the observed eccentric-
AlargefractionoftransitinghotJupitersareobserved
. ity, a mechanisminvolving an isolatedstar and planet is
1 to have radii far larger than the radius of Jupiter, RJ,
needed. Itisalsonotclearwhyplanetswithpresumably
0 implying high temperatures deep in the planetary inte-
similar internal structure and orbits should have such
9 rior (Burrows et al. 2000). Strong irradiation has been
0 found to have an insulating effect on planets, slowing different levels of internal friction.
: theircoolingandcontraction(Burrows et al.2000). This In the absence of direct observations, it is commonly
v
assumed that the rotation of present day hot Jupiters is
effect can explain radii R (1.0 1.2) R , but is in-
i p J
X sufficient to explain the rad∼ii of a−signifi×cant fraction of highlysynchronous. Thisassumptionismotivatedbythe
short( Myr)synchronizationtime usingQ=105 106
r the population with Rp (1.2 1.8) RJ (see Fortney ∼ −
a 2008forarecentreview).∼Apow−erfulin×ternalheatsource for the gas giant. However, despite the short expected
synchronization time, hot Jupiters are particularly sus-
mustbeactingtopreventtheseplanetsfromcontracting.
ceptible to develop asynchronous rotation. Gas giants
Circularization of the planet’s orbit by dissipation of
are relatively frictionless compared to terrestrial plan-
the gravitational tide has been invoked to explain the
ets with Q 10 100. Even weak opposing external
small eccentricities of most hot Jupiters (Rasio et al.
∼ −
torques may compete with the gravitationaltide to pro-
1996; Marcy et al. 1997). Using Jupiter’s inferred tidal
Q=105 106 (Yoder & Peale1981),orbitsareexpected duce asynchronous spin.
− The proximity of hot Jupiters to their parent star,
tobecircularouttoorbitalperiodsP .1week. Curi-
orb 102 times closer than Solar System giants, implies
ously,alargefractionofthe populationthatshouldhave a∼n insolation stronger by 104 and stellar tidal forces
been circularized in Gyrs is not (Dobbs-Dixon et al.
∼ stronger by 106. Insolat∼ion and tidal forces may then
2004; Matsumura et al. 2008). The zero and nonzero ∼
play a far more important role for close-in planets. In
eccentricity planets occupy the same orbital period
this paper we discuss “thermal tide torques”, which are
range. Suggested explanations are tidal interaction with
created through the interplay of stellar irradiation and
Electronicaddress: [email protected],[email protected] gravitationaltidal forces.
1Department of Astronomy, University of Virginia, P.O. Box Gold & Soter (1969)first appliedthermaltide torques
400325, Charlottesville,VA22904-4325 to explain the observed rotation rate of Venus. A num-
2Institute for Advanced Study, Einstein Drive, Princeton, NJ
ber of more detailed studies followed (e.g. Ingersoll and
08540
Dobrovolskis 1981). Venus resides in state of slow ret-
2
rograderotation. Gravitationaltides acting alone would Semi-Diurnal(m=2)
havesynchronizedVenus’spintohighprecision. Further-
more,Venus’ spin seems to be in resonancewith Earth’s thermaltide
orbit. TidesfromEarthactingonpermanentquadrupole
inVenus shouldhavea negligibleeffectincomparisonto sunset HOT
solar gravitational torques. The possibility that torques
fromsolargravitationalandthermaltidesnearlybalance
that allowsa smalleffect such as tidal forces fromEarth ǫ
to have an observable effect.
Previous studies of the thermal tide have assumed a
star
thin atmosphere on top of a solid surface, as appropri-
ate for Venus and Earth. In addition, they have as-
sumed that the gravitational tide is strongly dissipative sunrise
(Q 10 100). In applying the thermal tide to close-in ǫ 1/2Q
∼ − gravitationaltide COLD ≃
gas giant planets, a difference between our model and
those for Venus and Earth is that we assume an opti- Fig. 1.— Geometry of the semi-diurnal (m = 2) thermal and
gravitationaltidaldensityperturbations. Theplanetrotatescoun-
cally thick atmosphere, rather than ground, completely
terclockwise. Inorderforthetorquetopushtheplanetawayfrom
absorbsthe stellarradiation. We include flux-limitedra- synchronousspin,thedensityperturbationmustleadthelinejoin-
diative diffusion of heat above and below this absorbing ingtheplanetandstar.
layer. We find thermal diffusion is crucial in order to tude estimates are given below.
accurately estimate the magnitude of the thermal tide,
2.1. Induced Quadrupoles
particularly for long forcing periods. In applying our re-
sults, we find that the thermal tide can affect not only First we summarize the gravitationaltide. The stellar
the planet’sspin,butalsotheeccentricityandradiusfor tidal force raises a bulge on the planet, quantified to
close-in planets. lowest order by the quadrupole moment
Theplanofthepaperisasfollows. 2containsaqual-
3
§ h R
itative discussion of why thermal tide effects are impor- (grav) t M R2 M R2 p . (1)
tant. There, we argue that spin, eccentricity and global Q ∼(cid:18)Rp(cid:19) p p ∼ ⋆ p(cid:18) a (cid:19)
energetics of hot Jupiters are determined by the compe-
Here R and M are the radius and mass of the planet,
p p
tition between the thermal and gravitational tides. 3
M is the mass of the star, a is the semi-major axis of
§ ⋆
containsadetailedtreatmentofatmosphericresponseto the orbit, and h R (M /M )(R /a)3 is the height of
t p ⋆ p p
time-dependentinsolation,andthe resultantquadrupole ∼
thetide. Inthe absenceoffrictionwithinthe planet,the
moments. Readers wishing to ignore these technical de-
gravitationaltide does notcausesecularevolutionofthe
tails should skip this section. Basic equations and ge-
orbital elements or spin. The presence of friction causes
ometry are introduced in 3.1, the heating function is
hightide tolagmaximumtidalaccelerationatnoonand
§
expanded in a Fourier series in 3.2, temperature pro-
§ midnight. Forthecaseofasynchronousspin,thisimplies
filesfromthenonlinearsimulationsarediscussedin 3.3,
thatthetidalbulgeismisalignedwiththelinejoiningthe
§
quadrupole moments are computed in 3.4, linear per-
planet and star,as pictured in figure 1. The stellar tidal
§
turbation equations are derived in 3.5, and analytic so-
forcethentorquesthe planetby“pulling”onthe projec-
§
lutionsinthehighandlowfrequencylimitsarepresented
tion of the quadrupole moment (grav) perpendicular to
in 3.6and3.7, respectively. Applications tohotJupiter Q
§ the star-planet line. This small lag in phase or time is
spin,radiusandeccentricityareinthefollowingsections.
convenientlyparametrizedbythetidalQparameter(e.g.
Equilibriumspinfrequenciesarecomputedin 4. 5con-
§ § GoldreichandSoter1965). Theeffectivequadrupolemo-
tainstidalheatingrates,andadiscussionoftheenerget-
ment contributing to secular evolution is
ics required to power the observed radii. 6 contains
eccentricity growth/decayrates due to both t§he thermal (grav) σ
(grav) Q (2)
andgravitationaltides. Ourmostdetailedresultsforthe Qsec ∼ Q n
simultaneousequilibriaofspinrate,planetaryradiusand (cid:18) (cid:19)(cid:16) (cid:17)
eccentricity are contained in 7. In 8 we qualitatively whereσ isthetidalforcingfrequencyandnistheorbital
discuss how thermal and grav§itation§al tidal forces may frequency. The factor σ/n ensures the misalignment of
drive differential rotation which, in itself, may lead to the tidal bulge vanishes for synchronousand circular or-
dissipation. Summary and conclusions are presented in bits (see 3.4). The torque,N, on the planet due to this
§
9. The Appendices contains a derivation of the torque, quadrupole moment scales as N n2 (grav).
§orbitalevolutionrates,andtidalheatingratesduetothe Jupiter’s Q is thought to be in∼theQrsaencge Q = 105
presence of mass quadrupoles of arbitrary form, as well 106 based onexpansion of Io’s orbit to formthe Laplac−e
as the mass-radius relation and core luminosity. resonance (e.g. Yoder & Peale 1981). Several studies
havealsofoundthathotJupitersshouldhaveQ=105
2. BASICIDEAANDINITIALESTIMATES 106 to explain the circularization of their orbits on Gy−r
Tidal forces become increasingly important with de- timescales (e.g. Ogilvie & Lin 2004; Wu 2005). These
creasing orbital radius. Acting alone, dissipation of the studiesassumethatthegravitationaltideisactingalone.
gravitationaltide leads to synchronousspin and circular Here,weshowthatconstraintsonQofextrasolarplanets
orbit for the planet. Thermal tide torques complicate are complicated by the influence of the thermal tide.
this simple model, leading to qualitatively different evo- Anotherpossibilityfortidalphenomenacomesfromin-
lution. A conceptual introduction and order of magni- tense time-dependent insolation. Time-dependence may
3
arise either throughasynchronousrotationor orbital ec- alone acts, the end state is synchronous spin and a
centricity. Forcircularorbitsandasynchronousspin,the circular orbit. As torque equilibrium is realized in
∼
diurnal forcing frequency is σ = n Ω, where Ω is the Myr, over Gyr timescales the only source of tidal heat-
−
planet’sspinfrequency. Thereisalsopoweratthehigher ing is finite eccentricity. In the absence of an exter-
harmonics σ = m(n Ω), where m is the azimuthal nal perturbation continually pumping the eccentricity,
−
wavenumber. Sincethediurnalcomponentdoesnotgen- tidal heating is limited by the initial energy reservoir in
erateaquadrupolemomentwhichcancoupletothestel- the orbit GM M e2/2a, which may be comparable
⋆ p
∼
lartidalfield,thesemi-diurnalcomponent(m=2)dom- to planet’s binding energy for initially eccentric orbits
inates the torque on the planet. The magnitude of the (Bodenheimer et al. 2001).
thermal tide quadrupole is roughly (see 3.6 and 3.7) Planets with anomalously large radii but small ec-
§
centricity, such as HD 209458b with e 0.01
∆MR2 ≃
(th) p (3) (Laughlin et al. 2005), are difficult to explain with the
Q ∼ (σt )δ gravitationaltidealonesincethecurrenteccentricityim-
th
plies negligible tidal heating. In order to explain the
where ∆M is the mass of the atmospheric layer heated
large radius with transient tidal heating one has to in-
by time-dependent insolation,
voke that heating ended recently, and we are observing
∆MCpT the planet before it has had time to contract. Since con-
t , (4)
th ∼ Rp2F⋆ tractiontimesforradii(1.2−1.8)×RJ aremuchshorter
thanaGyr,thisargumentrequiresthatweobservethese
the exponent δ 0.5 1.0 and F⋆ is the flux at the sub- systems during a special time in their evolution.
∼ −
stellar point. The thermaltime, tth, is the time required Inclusionofthethermaltidequalitativelychangesthis
for the layer of mass ∆M to absorb/emit its thermal picture. Asteadystatemaybereachedinwhichthermal
energy. andgravitationaltideeffectsbalanceoneanother. Inthis
One of our principal goals is to show that for the at- scenario, gravitational tidal dissipation continues to act
mospheres of interest, the thermal tide bulge leads the
intheequilibriumstate,duetoasynchronousspinand/or
sub-stellarpoint,oppositethebehaviorofthedissipative eccentricity. The thermal tide forces are not inherently
gravitationaltide. Ingersoll & Dobrovolskis(1978)made
dissipative. However differential rotation set up in the
similar arguments for Venus based on two assumptions. atmosphereby opposing torques atdifferent depths may
First, the temperature of the absorbing layer lags the cause significant heating.
timeofmaximumheating,similartoourdailyexperience Tangential forces applied to the thermal tide
onEarth. Second, that fluid elements remainatroughly quadrupole torque the planet away from synchronous
constant pressure. Regions of lower temperature then
spin. In equilibrium, the thermal and gravitational tide
implyhigherdensityandviceversa,andthehighdensity torques on the planet are in balance, setting the equi-
regions(“tidalbulges”)leadthesub-stellarpoint. Inour
librium spin rate. The timescale to attain equilibrium
analysis,weadoptthissimplifying“constantpressureap- spin is shorter than the Gyrs age for orbital periods up
proximation.” The horizontalreadjustment of mass nec- to several months. Radial forces from the thermal tide
essarily requires zonal winds. In this paper we do not quadrupole act to alter the orbital eccentricity. If both
solveforthewindstructure;seeDobrovolskis& Ingersoll the thermaland gravitationaltides actto circularizethe
(1980)forthewindstructureresultingfromasimilarcal-
orbit, the equilibrium state is again zero eccentricity.
culation applied to Venus. However, the thermal tide can also pump eccentricity,
Boththegravitationalandthermaltidalbulgescanbe
opposing the gravitational tide. In that case, a balance
torquedbythestellartidalfield. Theangularmomentum ofthermalandgravitationaltidesimpliesnonzerovalues
evolves due to the net torque. In spin equilibrium, the for the equilibrium eccentricity. The timescale to attain
thermalandgravitationaltidetorquesbalance,ingeneral equilibrium eccentricity is shorter than the age for plan-
leading to an asynchronous spin state. We estimate the etswithorbitalperiodsshorterthan 1week. Lastly,in
ratio of quadrupole moments to be (eq.2 and 3) ∼
the absence of tidal heating, planets cool and contract.
(th) n 1+δ ∆M a3 Q Iftidesdepositheatdeepintheconvectiveinteriorofthe
Q planet, a thermal equilibrium is possible in which heat-
(grav) ∼ σ M R3 (nt )δ
Qsec (cid:16) (cid:17) (cid:18) ⋆ p(cid:19)(cid:18) th (cid:19) ing of the core is balanced by outward heat loss at the
n 1+δ ∆M M radiative-convectiveboundary. We find thermal equilib-
p
rium is achieved on a timescale shorter than the age for
≃ σ 10−8M 10−3M
(cid:16) (cid:17) (cid:18) p(cid:19)(cid:18) ⋆(cid:19) orbital periods .1 week (see section 7).
3
a Q 1 The thermal tide torque is exerted on a layer near the
. (5)
×(cid:18)100Rp(cid:19) (cid:18)105(cid:19)(ntth)δ pdhenocteosopfhtehreegorfatvhiteatsitoenllaalrtridadeitaotrioqnu.e iTshuencdeerpttahindaesptehne-
This order of magnitude estimate shows that for param- nature of gravitational tidal dissipation is complicated
eters characteristic of the hot Jupiters the thermal and and not yet fully understood. For instance, turbulent
gravitationaltideeffectscanbecomparable. Settingeq.5 viscosity damping would mainly occur just below the
equaltounityallowsonetoestimatetheequilibriumspin radiative-convective boundary (Goldreich & Nicholson
frequency since σ n Ω. 1977; Wu 2005), shear layers due to “wave attractors”
∝ −
reside deep in the convective core (Ogilvie & Lin 2004;
2.2. Orbit, spin, and thermal equilibrium
Goodman & Lackner 2008), and upward propagating
Gravitationaltidal dissipation converts rotationaland wavesgenerated at the tropopause may break above the
orbital energy into heat. When the gravitational tide photosphere (Ogilvie & Lin 2004). If the thermal and
4
gravitationaltorques are not exerted at the same depth, Firstconsider the changein the lightcurvefor thermal
differential rotation is induced, distinct from the usual radiation. For small flux from the core, and short ther-
thermally-drivencirculation patterns. For simplicity, we mal time in the absorbing layer, there will be a large
defer effects due to differential rotation until 8. day-night temperature difference. If, for example, 1%
§
of the insolation is deposited in the core by the grav-
2.3. Energetics, and implications for the lightcurve and itational tide, then the day-night temperature ratio is
spectrum (F /F )1/4 0.011/4 0.3, where F is the flux
core ⋆ core
∼ ∼
The stellar tidal acceleration creates regions of high emerging from the convective core. For small Fcore, the
andlowgravitationalpotentialinthe longitudinaldirec- only possibility of high temperature on the night side is
tion. Insolation heats the fluid, and this heat can be redistribution of heat by zonal winds. Tidal dissipation
tapped to perform work against the stellar tidal field by generating a large Fcore provides an alternative means
causingpressuregradientswhichmovefluidfromregions to reduce the day-night temperature contrast(see figure
of low to high potential. The ultimate source of energy 11 for the luminosity exiting the core versus planetary
for tidal heating driven by the thermal tide is then the radius).
stellarradiationfield. Therateofworkdonebythether- A change in the vertical distribution of heat sources
mal tide is alsoaffectstheverticaltemperatureprofileatthedepths
where the spectrum is formed. Depositing heat in the
σ σ
E˙(TT) ∆M ( ∇U) dl N(TT), (6) core rather that at the photosphere leads to a radia-
work ≃− m − · ≃−m
I tivefluxmoreconstantwithdepthnearthephotosphere.
whereU is the tidalpotential, σ =2(n Ω), m=2,and This alterationof the temperature profile may in princi-
−
theintegralistakenalongafluidtrajectorydl=R dφφˆ. ple be imprinted on the spectrum.
p
Here N(TT) ∆M ( ∇U) dl is the thermal tide
≃ − · 3. TEMPERATUREPROFILESANDQUADRUPOLE
torque on the heated layer. In torque equilibrium, the
MOMENTS
thermal and gravitatHional tide torques balance giving
In this section we describe the model for time-
N(GT) = N(TT). The heating rate due to the gravi-
− dependent thermal forcing of hot Jupiter atmospheres
tational tide is then
due to asynchronous rotation and/or an eccentric orbit,
σ σ
E˙(GT)= N(GT) = N(TT) =E˙(TT), (7) as well as the the resultant gravitational forces on the
heat m −m work
perturbed atmosphere. The full nonlinear equations are
showingexplicitlythattheworkdoneontheatmosphere presented,andthenapproximatedasatime-independent
by the thermal tide is converted into heat by the gravi- background and harmonic perturbations. Analytic solu-
tational tide in steady state. tions are developed for the background and perturba-
Since the work done on the atmosphere is ultimately tions, and compared to the full nonlinear solutions.
powered by the stellar radiation, the maximum power
input is given by the rate of absorption of stellar flux, 3.1. Equations and geometry
E˙max=L⋆ R2ap 2 1−e2 −1/2 (MWp,eRcpo)nasindder(Ma⋆p,Rla⋆n)e,treasnpdecsttivaerlyo.f Tmhaessplaanndetroardbiiutss
(cid:18) (cid:19) with separation D(t) and true anomaly Φ(t) which we
(cid:0) (cid:1) 2
=9 1028 erg s−1 Rp L⋆ will treat as a nearly Keplerian orbit with semi-major
× (cid:18)RJ(cid:19) (cid:18)L⊙(cid:19) aMxis)/aa,3]e1c/c2e.nWtriecictyonesidaenrdamneaatnmomspothieorne nin=un[iGfo(rMm⋆ro+-
2/3 4/3 p
M⊙ 4 days 1 e2 −1/2 (8) tationwith rateΩ,andthe spinandorbitalangularmo-
×(cid:18)M⋆(cid:19) (cid:18) Porb (cid:19) − mentum aligned. Differential rotation will be discussed
(cid:0) (cid:1) in 8. We will work in a non-rotating coordinate system
where L⋆ is the stellar luminosity and Porb = 2π/n is wh§ose origin is at the center of the planet, and whose
the orbitalperiod. We willshowin 5thatthe efficiency
§ axes are fixed with respect to distant observers. Spheri-
of converting stellar flux to heat by gravitational tidal
calpolarcoordinates(θ,φ)areused,wherethestarorbits
dissipationisashighas 1 10%,andincreasestoward
∼ − at the equator with a colatitude θ = π/2 and longitude
the star.
φ=Φ(t),andafixedpointontheplanetrotateswithan-
Some of the heat deposited by insolation is converted
into work, leading to less thermal energy immediately gularfrequencyφ˙ =Ω. Thederivativecomovingwiththe
re-radiated back out into space. In the (unphysical) planet is then d/dt = ∂/∂t+Ω∂/∂φ. The cosine of the
limitof100%efficientconversionofheattowork,andno angleχbetweentheverticalandthevectortothestaris
subsequent conversion of kinetic energy back into heat, cosχ=sinθcos(φ Φ). Thedaysideisovertheangular
−
the atmospherewouldbe farcoolerthanthe equilibrium range π/2 φ Φ π/2. Let z be the altitude above
temperature, and the thermal emission from the planet some a−pprop≤riate−refe≤rence level and y(z)= ∞dz′ρ(z′)
z
wouldbesolelyduetothefluxcomingoutfromthecore. the mass column above that altitude. The atmosphere
In torque equilibrium, the work done on the atmosphere is treatedas being both thin and inhydrostaRtic balance,
bythethermaltideisconvertedintoheatbythegravita- hence the pressure P = gy, where the surface gravity is
tionaltide. Wepresumethegravitationaltidedissipation g =GMp/Rp2.
occurs in the convective core, and entropy is efficiently We use an approximate treatment of radiative trans-
mixedthroughout. Thetemperatureprofileasafunction port in which the time-dependent insolation is treated
ofdepth,latitudeandlongitudemaythendiffermarkedly as a specified heat source. We approximate the trans-
from the case with no tidal heating. fer of thermal radiation by flux-limited diffusion. For
5
the thermal radiation, we employ the solar composition Thetime-dependentheatingrateperunitmassisgiven
“condensed” phase Rosseland opacities of Allard et al. by Beer’s Law (Liou 2002)
(2001), whichinclude the effect ofgrainsinthe equation
κ y
ofstate,butignorestheiropacity,asisappropriateifthe ǫ(y,θ,φ,t)=κ F (D)exp ⋆ Θ(cosχ) (12)
⋆ ⋆
grainshaverainedouttohigherdepth. Thereisapromi- −cosχ
(cid:18) (cid:19)
nentdipinκalongisobarscenteredaroundT =2000K.
where the constant κ specifies the column (y κ−1)
This feature will be apparent in our numerical results. ⋆ ∼ ⋆
at which the stellar radiation is absorbed, and F (D) =
We use the equation of state from Saumon et al. ⋆
σ T4(R /D)2 is the bolometric stellar flux at the sub-
(1995), with 70% hydrogen and 30% helium by mass. sb ⋆ ⋆
solar point on the planet. We present numerical results
At low density, this is an ideal gas equation of state
only for a solar-like star with M = M , T = 5780 K
P = ρk T/µm , where ρ is the mass density, T is the ⋆ ⊙ ⋆
b p and R =R . However, our analytic analysis allows for
temperature, m is the proton mass, k is Boltzmann’s ⋆ ⊙
p b
rescaling to stars of different mass and luminosity. The
constant, and µ 2.4 is the mean molecular weight for
≃ step function Θ(cosχ) enforces heating only on the day
a mixture of molecular hydrogen and helium.
side. In radiative equilibrium, the thermal flux exiting
The temperature andflux arefound by solutionof the
the top of the atmosphere is
heat equation
∞
dT ∂F F(y =0,θ,φ,t)=F + dy ǫ(y,θ,φ,t)
C = +ǫ (9) core
p
dt ∂y Z0
=F +cosχF Θ(cosχ). (13)
core ⋆
and the equation for flux-limited diffusion
Wetaketheupperlimitofintegrationtobey = since
16σsbT3Λ∂T y = κ−1 is a shallow layer in the atmosphere, a∞nd the
F= . (10) ⋆
κ ∂y integral converges exponentially.
The gravitational tide can only couple to fluid per-
Here C 7k /2µm is the specific heat per gram, and
p b p turbations with quadrupolar angular dependence and a
≃
ǫ is the heating rate per gram. Since column, or equiv-
specific harmonic time dependence. Therefore, we now
alently pressure, is assumed constant for fluid elements,
expand the insolationin terms of a Fourier series in lon-
the thermodynamic relation ds/C =dT/T dP/P
p ad gitude and time. Since cosχ depends on only the com-
−∇
is simplified to Tds = C dT. We ignore vertical fluid
p bination of longitudes ψ =φ Φ, we write
motion relative to constant pressure surfaces, as well as −
horizontal motions relative to the mean rate Ω. The ∞ κ y
quantity Λ is the flux limiter, which allows a smooth ǫ(y,θ,φ,t)=κ⋆F⋆(D) gm ⋆ eim(φ−Φ)(14)
sinθ
transition between the optically thick regime, Λ = 1/3, mX=−∞ (cid:16) (cid:17)
andtheopticallythinregimeΛ 0. Forconvenience,we
where the integral
→
use the limiter prescription of Levermore & Pomraning
(1981) 1 π/2
g (s) dψ e−imψ−ssecψ (15)
m
2+R ≡2π Z−π/2
Λ= (11)
6+3R+R2 containsacontributiononlyfromthedayside. We com-
pute the integrals g (s) numerically. The m = 0 term
where R = (4/κ)∂lnT/∂y is the ratio of photon mean m
| | is the average over longitude. For eccentric orbits, us-
free path to temperature scale height. The boundary
ing F (D) = F (a)(a/D)2, the time-dependence can be
conditionsare(1)the radiationfreestreamsatsmallop- ⋆ ⋆
tical depths, F 4σ T4, and (2) F F at large expanded as a sum of harmonic terms using the Hansen
sb core
≃ → coefficients defined in eq.A13 of appendix A
y, where F is the thermal flux that emerges from the
core
convective core. ∞
a 2
Werefertodirectsolutionsofeq.9and10as“nonlinear e−imΦ = X1m(e)e−iknt. (16)
D k
solutions”, as temperature changes are not assumed to
(cid:16) (cid:17) k=X−∞
be small. The nonlinear solutions will be compared to
Theheatingrateineq.12canthenbewritteninaFourier
solutionsofthe linearizedequationswhicharevalidonly
series in both longitude and time as
for small temperature changes.
Tonumericallysolveeq.9and10,weadaptedthe code κ y
ǫ(y,θ,φ,t)=κ F (a) X1m(e)g ⋆ eimφ−ik(n1t7.)
of Piro et al. (2005). We use second-order finite differ- ⋆ ⋆ k m sinθ
encein y andbackwarddifference int forstability. Typ- Xmk (cid:16) (cid:17)
ically 128 y grid points were used, but selected results The (m,k)’th term in the series has a forcing frequency
were checked at higher resolution and found to be accu- σ = kn mΩ in the co-rotating frame and kn in the
mk
rate. Thetimestepissetsothattheaveragetemperature inertial fra−me. The (m,k) = (0,0) term is the time and
ateachgridpointchangesby.10−3 overthe time step. longitude averagedheating rate, which we denote as
A background model, which we discuss below, is used
as an initial condition, and the initial time taken to be ǫ(y,θ)=F⋆(a)κ⋆X010(e)g0(κ⋆y/sinθ). (18)
noon (φ = Φ). Transients due to the initial conditions
Since X1m(e) = δ (1 e2)−1/2, there are no time-
are observed to relax over a few forcing periods. 0 m0 −
independent (k = 0) non-axisymmetric (m = 0) heating
6
terms. Axisymmetric (m = 0) time-dependent heating
3.2. Time-dependent insolation and Fourier components
does occur for eccentric orbits, due to the radial force.
6
Fig. 2.— Temperatureversuspressureoveronediurnalforcing Fig. 3.— Same as figure 2 but with “shallow” forcing κ⋆ =
periodfor“deep”forcingattheequator(θ=π/2). Notethelagbe- 10−0.5 cm2 g−1. The base of the heated layer is now at Pbase =
tweenmaximumforcing(atnoon)andmaximumtemperature(be- g/κ⋆ = 103.5 dynecm−2. The temperature perturbations diffuse
tween noon and 3pm), and alsothe pronounced “bump” near the significantlybelow the baseof the layer directlyheated. Herethe
baseoftheheatedlayeratPbase=g/κ⋆=105.5 dynecm−2. This forcing period, 2π/σ, is large in comparison to the thermal time
plotwasproducedusingthenonlinearsimulationsforacircularor- tth at y . κ−⋆1 giving temperature maximum in phase with the
bitwithPorb=4days,super-synchronousrotationPspin=3days, maximum heating at noon. A thermal diffusion wave penetrates
κ⋆=10−2.5 cm2 g−1,g=103 cms−2,Fcore=104 ergcm−2 s−1 tolargedepths,andlagsinphasewithrespecttothetemperature
andasolartypestar. Noonisthetimeofmaximu∞mheating. at y .κ−⋆1. This lag inphase leads to atorque that attempts to
For the time-independent background, dx g (x) = increasethespinoftheplanet.
0 0
1/π leads to a backgroundsurface flux
R is P = 4 days and the spin period P = 3 days,
∞ orb spin
F(y =0,θ)=F + dy ǫ(y,θ) giving a diurnal forcing period of 12 days. Each curve
core
is labeled by phase in the heating cycle, where noon
Z0
1 means heating is maximum, etc. Figure 2 illustrates a
=F + F (a)X10(e)sinθ (19)
core π ⋆ 0 heating function which extends “deep”, down to a base
pressure P = g/κ = 105.5 dyne cm−2, while fig-
and thermal luminosity base ⋆
ure 3 shows “shallow” heating down to P = g/κ =
base ⋆
103.5 dyne cm−2. Naively one may have expected the
L=R2 dΩF(y =0,θ)
p temperature to change significantly only above P .
base
Z Thisisapproximatelytrueforthedeepheatingcase. The
X10(e)
=4πR2 F + 0 F (a) (20) shallow heating case, however, shows large temperature
p core 4 ⋆ perturbations extending a factor of 30 deeper in pres-
(cid:18) (cid:19)
∼
in thermal radiation from the planet. We will use eq.18 sure than Pbase. Heating below Pbase is due to diffusion,
to define background models, on top of which we solve rather than direct absorption of stellar radiation. We
forperturbations. We define the time-dependent pertur- willshowthattherearetwolimitsfortheforcingperiod.
bation to the heating rate as In the “highfrequency limit”, appropriatefor planets at
large separation from the star, the temperature pertur-
κ y
δǫ(y,θ,φ,t)=κ⋆F⋆(a) Xk1m(e)gm sin⋆θ eimφ−i(k2n1t). bations are large only for depths P .Pbase. In the “low
frequency limit”, appropriate for close-in planets, tem-
mX,k6=0 (cid:16) (cid:17)
perature perturbations have time to diffuse down below
Fore=0,Xkℓm(0)=δmk enforcesk =m,andtheforcing Pbase, so that the star torques a layer deeper than that
frequency becomes σmm = m(n Ω), where 2π/n Ω directly heated by the stellar radiation.
− | − |
is the diurnal forcing period. The “bump” in the temperature profiles in figure 2 at
The values of gm(s) in the heating function decrease P = 105.0 106.2 dyne cm−2 is evidence of an inwardly
with increasing m. While the diurnal (m = 1) terms propagatin−g thermal diffusion wave. This bump is also
±
dominate the observable flux variation,they do not con- seenintemperatureprofilesinfigure5ofShowman et al.
tribute to the torque since the tidal potential has no (2008), althoughthe authors do not identify this feature
dipole component. The quadrupole moments responsi- as such. Showman et al. (2008) havea far more detailed
ble for spin and orbital evolution are dominated by the solution method, including nonlinear three-dimensional
semi-diurnal (m=0, 2) components. hydrodynamics, angle and frequency dependent radia-
±
tive transfer, and detailed chemical abundances and
3.3. Discussion of temperature profiles
monochromaticopacities. Thatoursolutionsareinqual-
Figures 2 and 3 show temperature versus pressure itativeandroughquantitativeagreementintheregionof
over one diurnal forcing period for a planet in circu- interest lends confidence to our results.
lar orbit around a solar-type star. The orbital period Another important comparison to the more rigorous
7
Fig. 4.— Thermal fluxatthe surfaceversus timeforthecases Fig. 5.— Effective column Σ (eq.24), converted into units of
shown in figures 2 and 3. These simulations are at the equator fractionalmassoftheplanet,forthedeepheatingcase(red,figure
(θ = π/2), flux is normalized by the average surface flux Fcore+ 2) and the shallow heating case (blue, figure 3). The solid lines
F⋆(a)/4, and time is in units of diurnal forcing period (12 days). are the full numerical result. Fourier transform of the solid lines
Thesolidblacklineistheoutgoingfluxifnonetheatisabsorbed gives the diurnal component (dotted lines) and the semi-diurnal
or emitted by the atmosphere (eq.13). The blue long-dashed line component(dashedlines).
istheshallowheatingcase(figure3). Theredshort-dashedlineis
thedeepheatingcase(figure2). heating is due to κ κ; stellar irradiation deposits
⋆
≪
heat at large optical depth, leading to a temperature
results of Showman et al. (2008) concerns the lag in the increase by a factor (κ/κ )1/4 above the skin tem-
⋆
temperature profile, shown in their figure 3. Although ∼
perature (F /4σ )1/4. There has been recent interest
theirsimulationisofasynchronousplanet,theygenerate ⋆ sb
in shallow heating, κ κ, due to TiO/VO absorption
a super-rotating equatorial jet which mimics the super- ⋆ ≫
at P mbar, producing an inwardly decreasing tem-
synchronous rotation in our results. Those authors in- ∼
perature profile, the “stratosphere” case (Hubeny et al.
deed find the temperature reaches a maximum eastward
2003; Fortney et al. 2008). To model this effect would
of the subsolar point; maximum temperature lags max-
require angle and frequency-dependent solution of the
imum heating. This lag is especially prominent at large
transfer equations, beyond the scope of this paper. We
depth (their bottom panel). We regard this as strong
note, however,that below P , the stratospherecase is
evidence that, while inclusion of fluid motion may al- base
expected to have temperature (F /σ )1/4, similar to
ter our results quantitatively, the qualitative result that ∼ ⋆ sb
our results with flux-limited diffusion. Since the torque
temperature lags, due to thermal inertia, is in agree-
is applied well below P , we expect our results to be
ment with Showman et al. (2008). A crucial omission base
qualitatively, and perhaps quantitatively correct.
in their simulations is that the tidal force from the star
is notincluded. We argue,basedonthe temperature lag
3.4. Quadrupole moments
seen in their figure 3, that the torque on this thermally-
generated quadrupole will act to generate asynchronous Appendix A contains a derivation of the secular
spin. changes in e, a and Ω due to a quadrupole moment.
Figure 4 shows the thermal flux exiting the surface, In addition, we derive the change in total spin plus or-
in units of the average thermal flux, for the two cases bital energy due to this quadrupole moment. Here, we
showninfigures2and3. Theshallowheatingcaseshows outline the method of computation for the thermal tide
largerfluxvariationsandsmallerlagtime incomparison quadrupole moment, as well as review Darwin’s theory
to the deep heating case. In both cases, the maximum of secular evolution for gravitational tides.
in thermal emission lags the maximum in heating (at The “thermal tide” force results from the mass mul-
noon)duetothermalinertia. TheSpitzerlightcurvesfor tipole moments induced in the atmosphere by time-
HD189733bshowmaximumthermalemissionjustbefore dependent insolation. The multipole moments are de-
secondary eclipse. Using the geometry from figure 1, fined in terms of the density field that is the solution to
maximum emission occurs just before secondary eclipse eqs. 9 and 10. We have
for a planet with super-synchronous rotation, in order
that the temperature reaches maximum to the east of (TT)(t)= d3x ρ(x,t)rℓ Y∗ (θ,φ). (22)
Qℓm ℓm
the subsolar point, after it has passed through noon. A Z
similar effect would occur for a super-synchronous zonal The density ρ(x,t) is computed from the temperature
wind, as in the model of Showman et al. (2008). profile T(x,t) using the ideal gas law and hydrostatic
Onetechnicalpointaboutflux-limiteddiffusionisthat balance P = gy. Since rℓρ(x,t) is a real quantity,
itproducestemperatureprofileswhichincreaseinwardat and Y∗ = ( 1)mY , the moments must satisfy
smallopticaldepth. The“greenhouse”caseshowninfig- ℓm∗ − ℓ,−m
ure2iscapturedinoursolutions. Recallthatgreenhouse Q(ℓTmT) =Q(ℓT,−Tm)(−1)m. Asthestaticandazimuthally
(cid:16) (cid:17)
8
symmetric background model has zero quadrupole mo- τ as it is in phase with the stellar gravitational tidal
lag
ment, it can be subtracted off the integrand. We then forcing and therefore, cannot lead to a torque. We find
make a change of integration variable from radius r to the effective gravitationaltidal quadrupole moment
column dy = ρdr, and approximate r R , yielding
− ≃ p R 3
the expression (GT)(t)=iτ k M R2Y (π/2,0) p
Q2m lag p ⋆ p 2m a
(cid:18) (cid:19)
Q(ℓTmT)(t)=Rp2+ℓZ dΩ Yℓ∗m(θ,φ) × (kn−mΩ)Xk2m(e)e−iknt, (27)
∞ ρ(x,t) ρ(y,θ) Xk
dy − . (23)
× ρ(y,θ) which is defined in the inertial frame. Rewriting
Z0 (cid:18) (cid:19) this expression in terms of Q′ = (3/2k )(1/nτ )
p p lag
Inspection of eq.23 shows that the tide couples to an
(Mardling & Lin 2002) and by taking the Fourier trans-
“effective column”
form in time, we find the imaginary component
∞ ρ(x,t) ρ(y,θ)
Σ(θ,φ,t)≡ dy ρ(y−,θ) . (24) Im (GT) = 3 M⋆Rp5 kn−mΩ
Figure 5 shows nZu0meric(cid:18)al examples of (cid:19)the time- (cid:16)Q2mk (cid:17) (cid:18)2Q′p(cid:19) a3 !(cid:18) n (cid:19)
dependent column Σ for the same simulations as in fig- ×Y2m(π/2,0)Xk2m(e). (28)
ures2and3. Theeffectivecolumn(solidlines)ingeneral
As a check of this result, we compared eq.28, A16, A17
leadstheheatingfunction,aconsequenceofthetempera-
and A18 against Hut’s analytic formulas for a range of
ture lag and the constantpressure assumption. We have
orbital period, spin period and eccentricity.
taken a Fourier transform of the time-series for Σ, and
Note that Hut’s “constant time lag” prescription for
show the diurnal and semi-diurnal components, which
orbitandspinevolutiondisagreeswiththe“constantlag
also lead. The diurnal component dominates the tem-
angle” approach of Goldreich & Soter (1966). The use
perature and density response, but produces no torque
of a constantlag angle, with a constant Q, is unphysical
since the tidal force has no ℓ = 2, m = 1 components.
± as the torque is discontinuous as the spin changes from
The semi-diurnal response is smaller, by a factor 2 in
∼ sub- to super-synchronous and vice-versa. By adopting
the present example, but dominates the torque.
thelagintimeapproachofHut(1981)orMardling & Lin
The expressions for the secular change in the planet’s
(2002) with the above choice for Q, we find agreement
orbit require the Fourier transform in time of the
with the results of Goldreich & Soter (1966) when the
quadrupole moments, which we perform numerically
planet is far from a synchronous spin state, as in the
fromoursimulationoutput. Wedefinethedesiredtrans-
case of the Jupiter-Io system. Therefore,the constraints
forms
on Q inferred by Goldreich & Soter (1966) may still be
∞ applied. Note, however, that Ingersoll & Dobrovolskis
(TT)(t) (TT)e−iknt (25) (1978) and Dobrovolskis & Ingersoll (1980) utilize the
Qℓm ≡ Qℓmk
m=−∞ constant lag angle approach. We find both qualitative
X
and quantitative differences with their work.
where
n 2π/n 3.5. Linear Perturbation Theory
(TT)= dt eiknt (TT)(t), (26)
Qℓmk 2π Qℓm Forlargeforcing frequencies,the time-dependent tem-
Z0
perature and flux changes are small compared to the
⋆
and satisfy (TT) = ( 1)m (TT) . For the non- time-average values. One may then treat the time-
Qℓ,−m,−k − Qℓmk
dependent changes δT(y,θ,φ,t) and δF(y,θ,φ,t) as lin-
linearsimulations,weevolvedth(cid:16)eatmos(cid:17)pherefor10forc-
ear perturbations about the time-independent back-
ingperiods,andcomputedtheFouriertransformsnumer-
ground values T(y,θ) and F(y,θ).
ically using orbits 9 and 10.
The background model is computed by integrating
Wenowturntothegravitationaltide. Thestellargrav-
itationaltidalacceleration ∇U inducesfluidflowinthe dT κF
planet. Dissipation causes−a phase lag 1/Q between = (29)
dy 16σ T3Λ
∼ sb
theaccelerationandhightide,whereQisthe tidalqual-
ity factor. This phase lag is the origin of the “gravita- with flux found by integrating eq.18
tionaltide torque”,whichattempts tomakethe planet’s ∞
spin synchronous. Hut (1981) contains a clear review of F(y,θ)=Fcore+ dy′ ǫ(y′,θ)
Darwin’s theory of tidal evolution. There it is assumed Zy
tqhuaatdrtuhpeofllauridflrueisdporensspeolnasgescfaonrcibnegvbiyewaedtimasetτwlaog.pToihnet =Fcore+ F⋆(a)sinθX010(e) π/2dφcosφ e−(κ⋆y/sinθ)s(e3c0φ),
π
masseseachofmass(k /2)M [R /D(t τ )]3,withlon- Z0
p ⋆ p lag
gitude Φ(t τ ) Ω(t τ ) and Φ−(t τ ) Ω(t and subject to the boundary condition T
τ )+π. H−erelagk −is the−apslaidgal motion−conlsatgan−t of th−e (F(0,θ)/4σ )1/4 at small optical depth. We construc≃t
lag p sb
planet, which takes into account the internal structure this model by integrating inward from the surface. An
in the external potential perturbation produced by the example is given in figure 6. Our treatment of radia-
quadrupole moment. Under the assumption that τ is tivetransferissimplified,butneverthelessreproducesthe
lag
small in comparisonto the forcing period, we expand to main features of full solutions to the transfer equations.
first order in τ . We ignore the term independent of At small optical depth the solution becomes isothermal
lag
9
Fig. 6.— Background model constructed using time- Fig. 7.— Semi-diurnal (m = 2), linear temperature pertur-
a1v0e4raegrgedcmh−ea2tisn−g1,rgat=e,10u3sincgmκs−⋆2,=a s1o0la−r2-t.5ycpme2stga−r1a,ndFcPoorerb == braatteioonfsPfosrpina =pla1n.e0tdaaty,Pogribvin=g4adsaemysi-wdiiuthrnaalpfroorgcrinagdepreoritoadtioonf
4days. 2/3 day. Surface gravity is g = 103 cms−2, θ = π/2, and
κ⋆ = 10−2.5 cm2 g−1. The high frequency limit from eq.38 is
at the “skin” temperature. Heating by inward-going agoodapproximationforP &104dynecm−2.
stellar photons, here through the heating function, gen-
erates an equal outward flux of thermal photons (for
the time-independent case). At a column y κ−1,
≃ ⋆
theheatingfunctionandoutwardfluxdecreaseexponen-
tially,andthetemperatureprofilebecomesisothermalat
T(κ−1,θ) (F(0,θ)/σ )1/4(κ/κ )1/4. Lastly, the flux
⋆ ≃ sb ⋆
F from deep in the atmosphere eventually causes the
core
temperature to increase inward, the gradient becoming
steep enough to cause convection.
The linearized equations are found by perturbing eq.9
and 10 while enforcing strict hydrostatic balance for
the perturbations, so that each fluid element remains
at constant y. We eliminate dependence on φ and t
by expanding δT and δF in in a Fourier series (e.g.,
eq.17). Thetimedependenceoftheperturbationsisthen
d/dt= iσ . Wewillsuppressmandkinthefollowing
mk
−
equations. The linearized form of eq.9 and 10 are
∂δT dT/dy δF δT
= + κ 3 Λ (κ +1) (31)
T R T
∂y 1 Λ F { − − } T
− R (cid:20) (cid:21)
and
Fig. 8.— Same as figure 7 but with a spin period Pspin =
dδF 3daysandshallowheatdepositionκ⋆=10−0.5cm2 g−1. Thelow
= iσCpδT δǫ, (32) frequency limit is a good approximation for P 106dynecm−2,
dy − − below which the oscillations in the real and im≤aginary part are
evidenceofadiffusionwave.
where κ =∂lnκ/∂lnT and
T P Given the temperature perturbation, the density per-
|
R2(R+4) turbation is δρ/ρ = δT/T at constant pressure. To
−
Λ = . (33) linear order, the effective column is given by
R (6+3R+R2)(2+R)
∞ δT(y,θ)
At large (small) optical depth, Λ 0 (1). We solve Σ(θ)= dy , (34)
eq.31and32subjecttotheboundaRry→conditionsδF/F = Z0 (cid:18)− T(y,θ) (cid:19)
4δT/T at the top of the grid and δT,δF 0 at the implyingthat the temperatureperturbationslagging the
→
base. The equations are solved by finite difference in heatingtendtotorquetheplanetawayfromsynchronous
y to obtain a banded matrix equation which is readily rotation.
invertedtofindtheresponseδT andδF totheheatingδǫ. Examplesofthesolutionofthelinearizedequationsare
Thetopofthegridissetatacolumn10−2 timessmaller showninfigures7and8,alongwithanalyticapproxima-
than the thermal photosphere, and the base of the grid tions to be discussed presently. The difference between
issetattheradiative-convectiveboundary,typicallywell thesetwoplotsismainlyduetothedepthofheating,i.e.
below y =κ−1. deep in figure 7 and shallow in figure 8.
⋆
10
Forthedeepheatingcaseinfigure7,theforcingperiod torqueissmall,theequilibriumspinmustbeintheshort
(=2/3 day) was chosen to be shorter than the diffusion forcingperiodlimit. Thedividinglinebetweenthesetwo
time for the base of the heated layer. There is no time limits depends on factors such as stellar flux, opacities
fordiffusiontotransportheat,the∂F/∂y termineq.9is and tidal Q.
small, and the temperature fluctuates solely due to the Unless ∂T/∂y = T/y, the “diffusion depth”, y ,
diff
time-dependent heating. A longer forcing period leads differs from the often-used “cooling depth”, y =
cool
to larger input of heat, and effective column Σ P , F/(C T σ ). See the lower panel in figure 6 for a nu-
f p
∝ | |
where P = 2π/σ is the forcing period. An analytic mericalexample. In the problem at hand, y and y
f cool diff
| |
solution will be presented in 3.6. Since the imaginary can differ by orders of magnitude, especially at small
§
partofthe temperature perturbationis largeonly above optical depths and below the base of the heating layer
P =g/κ , the torques will extend down to P . where the flux drops exponentially. Physically, y is
base ⋆ base cool
The shallow case in figure 8, on the other hand, the depth down to which the heat content can be ra-
has a sufficiently long forcing period (6 days) that the diated in the timescale 2π/σ . In the absence of heat-
| |
perturbations extend deeper than P by some 2 or- ing, the flux tends to become constant with depth for
base
ders of magnitude in pressure. For pressures P . y . y (e.g. Piro et al. 2005). As we will show in the
cool
106 dyne cm−2, the firsttermineq.9canbe ignoreddue nextsections,y isthedepthdowntowhichsmallper-
diff
to the small forcing frequency. In this limit, the atmo- turbationscanpropagate,andhenceisthemorerelevant
sphere responds in phase to the forcing, changing quasi- for heating deep in the atmosphere.
statically from one equilibrium to another. No torque is
exertedontheatmosphereinthislimitsincethetemper- 3.6. Analytic solution in the high frequency limit:
atureperturbationisinphasewiththe heatingfunction. σtth 1
≫
Thatis,forthesemi-diurnalmode,temperaturemaxima The thermal time at the base of the absorbing layer,
occur at noon and midnight, while temperature minima t C T/κ F , is the timescale over which the layer
th p ⋆ ⋆
take place at sunrise and sunset. In this “σ = 0” limit, can∼heat or cool. For large forcing frequencies σt 1,
th
we can solve for the (real) temperature perturbation by heatcannotdiffuse oversignificantdistances inafor≫cing
integratingeq.9withdT/dt=0. Inthisregime,thether- periodandthetermdδF/dycanbeignored. Inthislimit,
mal inertia of the material is small is comparison to the thetemperatureperturbationisdeterminedsolelybythe
amount of photon energy absorbed per cycle and there- local heating function (eq.32), i.e.,
fore, the temperature is “locked” to the heating func-
δǫ
tion. For the forcing frequencies of interest, this limit δT =i mk , (38)
mk
will always be invalid sufficiently deep in the envelope, σmkCp
at which point the solution is well described by a diffu- whereδǫ istheappropriateFouriercoefficientineq.17.
mk
sionwave. Thiswaveisapparentinfigure8intheregion This expression shows that the temperature perturba-
P =106 107dyne cm−2, where the real and imaginary tions are small in the high frequency limit, δT decreases
−
parts are comparable, and exhibit oscillations. The first exponentially below y = κ−1 and its magnitude in-
⋆
and second terms in eq. 9 are comparable for diffusion creases with forcing period since there is more time to
waves below the base of the heating layer. The decay of absorbheat. With respectto the forcing frequency σ ,
mk
the envelope of this wave into the planet determines the the temperature perturbation lags the forcing by 90◦ in
column to which torque is applied. phase. It is compared to the solution of the linearized
Using dimensional analysis on eq.9 and 10, heat can boundary value problem in figure 7.
diffuse down to a column The effective column is then
ydiff=s136κσCspb|Tσ3| (35) Σmk(θ)≃−σmkiCp Z0∞dy(cid:18)δǫTm(ky(,yθ,)θ)(cid:19). (39)
inaforcingperiodPf =2π/σ . Thisdepthistobecom- Since T(y,θ) varies relatively slowly for y .κ−⋆1, we ap-
pared to the base of the hea|ti|ng layer at κ−1. Heating proximateitasaconstantandpullitoutoftheintegral.
⋆ The remaining integral is
can diffuse below the base of the heating layer when
∞
3π κC
Pf&Pdiff = 8 σsbT3pκ2⋆ Z0 ds gm(s)≡Gm, (40)
κ where G0 = 1/π and G2 = 1/3π. Plugging eq.40 into
=3.6 days
0.1 cm2 g−1 eq.39 we find
(cid:18) (cid:19)
0.01 cm2 g−1 2 2000 K 3 Σ (θ) iF⋆(a)Xk1m(e)Gmsinθ. (41)
. (36) mk ≃− σ C T
× κ T mk p
(cid:18) ⋆ (cid:19) (cid:18) (cid:19)
For the diurnal (semi-diurnal) component, eq.41 implies
In the low frequency limit, the effective column
Σ leads the heating function by π/2 (π/4) in longitude.
Σ y P1/2, (37) This phase lead corresponds to 6am (9am) for the max-
∼ diff ∝ f
ima of the diurnal (semi-diurnal) components in figure
which increases more slowly as P 0 than the high
f → 5.
frequency limit.
Thequadrupolemomentsrequiretheangularintegrals
Close to the star,where the gravitationaltide is large,
π
the equilibrium spin must be in the long forcing pe-
F =2π dθsin2θY (θ,0). (42)
riodlimit, andfar fromthe star,where the gravitational ℓm ℓm
Z0