Atmospheric heat redistribution effect on Emission spectra of Hot-Jupiters

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Atmospheric heat redistribution eﬀect on Emission

spectra of Hot-Jupiters

Soumya Senguptaa,b, Sujan Senguptaa

aIndian Institute of Astrophysics, Koramangala 2nd Block, Sarjapura Road, Bangalore

560034, India

bPondicherry University, R.V. Nagar, Kalapet, 605014, Puducherry, India

Abstract

Hot Jupiters are the most studied and easily detectable exoplanets for transit

observations. However, the correlation between the atmospheric ﬂow and the

emission spectra of such planets is still not understood. Due to huge day-night

temperature contrast in hot Jupiter, the thermal redistribution through atmo-

spheric circulation has a signiﬁcant impact on the vertical temperature-pressure

structure and on the emission spectra. In the present work, we aim to study the

variation of the temperature-pressure proﬁles and the emission spectra of such

planets due to diﬀerent amounts of atmospheric heat redistribution. For this

purpose, we ﬁrst derive an analytical relation between the heat redistribution

parameter f and the emitted ﬂux from the uppermost atmospheric layers of hot

Jupiter. We adopt the three possible values of f under isotropic approximation as

4,1

2and 2

3for full-redistribution, semi-redistribution and no-redistribution cases

respectively and calculate the corresponding temperature-pressure proﬁles and

the emission spectra. Next, we model the emission spectra for diﬀerent values of

f by numerically solving the radiative transfer equations using the discrete space

theory formalism. We demonstrate that the atmospheric temperature-pressure

proﬁles and the emission spectra both are susceptible to the values of the heat

redistribution function. A reduction in the heat redistribution yields a thermal

inversion in the temperature-pressure proﬁles and hence increases the amount of

∗Corresponding author

Email address: soumya.s@iiap.res.in (Soumya Sengupta)

Preprint submitted to Journal of L

X Templates October 18, 2022

arXiv:2210.08755v1 [astro-ph.EP] 17 Oct 2022

emission ﬂux. Finally, we revisits the hot Jupiter XO-1b temperature-pressure

proﬁle degeneracy case and show that a non-inversion temperature-pressure pro-

ﬁle best explains this planet’s observed dayside emission spectra.

Keywords: planets and satellites: atmospheres, gaseous planets, atmospheric

eﬀects, Hot-Jupiters

1. Introduction

Since the detection of the ﬁrst Jupiter-like extrasolar planet 51-Pegasi-b

around a solar type star [1], the giant planets remain the most observed and

investigated exoplanets till date. For transit observations, hot Jupiters are

the ﬁrst as well as the most easily detectable planets [2], [3] among a large

variety of exoplanets discovered. The atmospheric temperature-pressure pro-

ﬁles of hot Jupiters have been modeled using the radiative equilibrium con-

ditions both analytically [4, 5], as well as numerically [6, 7]. Using these

modelled temperature-pressure proﬁles and the atmospheric chemical compo-

sitions, the atmospheric spectra of such exoplanets can be obtained by solving

the radiative transfer equations (e.g. [8, 9, 10]) with diﬀerent techniques such

as two-stream approximation [11, 12], Feautrier method [13], Discrete Space

Theory[14] etc. Ultimately, by comparing those synthetic spectra with the

extracted spectra from transit observations, one can retrieve the atmospheric

chemical compositions [15, 16, 17, 18, 19, 20, 21] and the temperature-pressure

proﬁles [22, 23, 24, 25, 26].The atmospheric retrieval of a huge number of

Gaseous and hot Jupiter planetary atmospheres are reported in [27, 28].

Three types of spectra that can be obtained during transit and eclipse ob-

servations, e.g., transmission, reﬂection and emission spectra [9] are studied

extensively (e.g. [29, 30, 31, 10, 32, 33, 34, 35]). Among all these, only the plan-

etary emission spectra observed during secondary eclipse observations carry the

full imprint of the atmospheric temperature-pressure proﬁles [9, 36, 37, 38, 39,

40, 41], whereas reﬂection and transmission spectra are sensitive to the upper

atmosphere only [29]. The emission spectra carry the compositional signature

of the atmospheric layers in terms of absorption and scattering as well as the

temperature of each atmospheric layer [42, 43]. Due to the tidal locking of the

hot Jupiters with its host star, there is a huge diﬀerence in the atmospheric tem-

perature at the permanent day-side and the permanent night-side of the planet.

This temperature diﬀerence gives rise to a pressure gradient which results to

the horizontal atmospheric ﬂow [3]. Also the hot Jupiter’s small rotation period

eﬀects the horizontal ﬂow geometry in terms of coriolis force which is very much

diﬀerent than the solar system giants [44]. Now the stellar irradiation ﬂux has

a direct eﬀect on planetary spectra through re-emission [30] and in heat redis-

tribution through equatorial-jet circulation [45]. It was pedagogically shown in

[4] that the planetary atmospheric heat redistribution has a direct eﬀect on the

temperature-pressure proﬁle as well as in the emission spectra. Recently, [46]

demonstrated that due to large scale equatorial waves generated by prominent

day-night temperature contrast, there is a temporal variation in the secondary

eclipse depth ( 2% locally) and the temperature-pressure proﬁles. A number of

studies of atmospheric heat redistribution has been done using diﬀerent available

mechanisms (see [47, 48, 49] and the references therein). Initially the shallow wa-

ter model is used to explain the atmospheric heat redistribution in hot Jupiters

[50]. Recently, [51] investigated the heat transfer from dayside to nightside in

ultra hot jupiters by considering the General Circulation Model with an eﬀect of

molecular hydrogen dissociation in dayside and atomic hydrogen recombination

in nightside. Hence each vertical atmospheric layer in the dayside atmosphere is

cooled due to the heat transfer from substellar to anti-stellar side of the planet.

Although, diﬀerent kind of atmospheric spectra of hot Jupiters has been

studied, the eﬀect of atmospheric heat redistribution on hot Jupiter’s atmo-

sphere remains unresolved. Thus, to get the full realization of the atmospheric

heat redistribution, one needs to study the temperature-pressure proﬁles as well

as the emission spectra for diﬀerent degree of atmospheric heat redistribution.

In this paper, we present the eﬀect of day to night side atmospheric heat redis-

tribution of hot Jupiters. We however, do not address the possible mechanisms

of heat redistribution. The eﬀect is analyzed in terms of temperature-pressure

proﬁle and day-side emission spectrum as suggested by [4]. We use the analyt-

ical formulation of temperature-pressure proﬁles presented by [5] to derive the

analytical relation of redistribution parameter with the emission proﬁles. Using

these proﬁles and solar abundance composition in the atmosphere, the absorp-

tion as well as the scattering co-eﬃcients is estimated by using the publicly

available software package Exo-Transmit [32, 52, 53, 54]. Finally, the modelled

temperature-pressure proﬁle, absorption and scattering co-eﬃcients are used to

calculate the dayside emission spectra numerically by using discrete space the-

ory formalism described in [14], [29]. Thus, from the emission spectra we infer

the role of heat redistribution in the atmosphere of hot-juipters.

In section 2, we present the analytical derivations of the expression that de-

scribe the explicit dependence of atmospheric thermal proﬁle as well as the day-

side emission spectra on the redistribution parameter f. The numerical method

for solving the 1-D radiative transfer equations is described and the solutions

are validated in section 3. The resulting eﬀects of redistribution parameter on

the temperature-pressure proﬁle and the dayside emission spectra are presented

in section 4. Finally we discuss our results and conclude this work in the last

section.

2. Theoretical Models

A hot close-in gas giant planet is tidally locked to its host star. When

the starlight is irradiated on the atmosphere, then two phenomena can occur:

Either the heat can be absorbed and redistributed in the atmosphere or the

heat is reradiated immediately from the atmosphere.

The analytical expression for the atmospheric temperature T under radiative

equilibrium is given by [5]

T4=3T4

int

42

3+τ+3T4

irr

4µ02

3+µ0

γ+γ

3µ0−µ0

γe−γτ

µ0(1)

where,Tint is the internal temperature of the planet, Tirr is temperature due to

the ﬂux irradiated on the planetary atmosphere along the direction cosine µ0.

Here µ0is the direction cosine of the angle made by the irradiated radiation

with the outward normal of the atmospheric layer [5]. γis the ratio between

the optical and the infrared absorption co-eﬃcients, i.e. κvis/κinf and τis the

optical depth deﬁned in terms of pressure (P), density (ρ) and constant surface

gravity g as, dτ =dP κinf

g[5].

Thus the mean intensity expression can be obtained by multiplying equation

(1) by σ

π, where σis the Stefan-Boltzmann constant, on either side as,

J=3

4Fint 2

3+τ+3

4Firr µ02

3+µ0

γ+γ

3µ0−µ0

γe−γτ

µ0(2)

where, we replaced σT 4

int/π and σT 4

irr /π by internal ﬂux Fint and irradiated

ﬂux Firr respectively and make use the fact that J=σT 4

π[55].

Under radiative equilibrium condition, the speciﬁc intensity at τ= 0 and

the mean intensity relation can be written as [4],

I(τ= 0, µ, µ0) = Z∞

J(t)e−t

µdt

µ(3)

Now solving equation (3) by using equation (2) we get,

I(0, µ, µ0) = 3Fint

4(2

3+µ) + 3Firr

4µ02

3+µ0

γ+γ

3µ0−µ0

γ)1

1 + γµ/µ0

] (4)

This is the ﬂux emerging out from the uppermost atmospheric layer of the

planet where τ= 0. For secondary eclipse observation, the total observed ﬂux

is the radiation coming from the substellar point of the planet at full phase with

µ=µ0. Taking the integral over this phase and following the notations of [4],

we can write the full emerging ﬂux as,

Ffull = 2 Z1

µ0I(0, µ0, µ0)dµ0=Fint +3Firr

22

9+1

4γ+γ

6−1

4γ1

1 + γ

(5)

in principle, the comparison of the ﬂux observed during secondary eclipse

to the model ﬂux Ffull can tell us whether the energy is redistributed from

substellar side to anti-stellar side of the planet, because the observed emitted

ﬂux from the substellar side in that case would be less than that expected for

no-redistribution model. Thus dividing the re-radiation term i.e. the second

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AtmosphericheatredistributioneectonEmissionspectraofHot-JupitersSoumyaSenguptaa,b,SujanSenguptaaaIndianInstituteofAstrophysics,Koramangala2ndBlock,SarjapuraRoad,Bangalore560034,IndiabPondicherryUniversity,R.V.Nagar,Kalapet,605014,Puducherry,IndiaAbstractHotJupitersarethemoststudiedandeasilydetectab...

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Atmospheric heat redistribution effect on Emission spectra of Hot-Jupiters

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