# Atmospheric Thermal Emission Effect on Chandrasekhar’s Finite Atmosphere
Problem

Soumya Sengupta Indian Institute of Astrophysics, Koramangala 2nd Block,
Sarjapura Road, Bangalore 560034, India  Pondicherry University, R.V. Nagar,
Kalapet, 605014, Puducherry, India

###### Abstract

The solutions of the diffuse reflection finite atmosphere problem are very
useful in the astrophysical context. Chandrasekhar was the first to solve this
problem analytically, by considering atmospheric scattering. These results
have wide applications in the modeling of planetary atmospheres. However, they
cannot be used to model an atmosphere with emission. We solved this problem by
including thermal emission effect along with scattering. Here, our aim is to
provide a complete picture of generalized finite atmosphere problem in
presence of scattering and thermal emission, and to give a physical account of
the same. For that, we take an analytical approach using the invariance
principle method to solve the diffuse reflection finite atmosphere problem in
the presence of atmospheric thermal emission. We established the general
integral equations of modified scattering function
$S(\tau;\mu,\phi;\mu_{0},\phi_{0})$, transmission function
$T(\tau;\mu,\phi;\mu_{0},\phi_{0})$ and their derivatives with respect to
$\tau$ for a thermally emitting atmosphere. We customize these equations for
the case of isotropic scattering and introduce two new functions $V(\mu)$, and
$W(\mu)$, analogous to Chandrasekhar’s $X(\mu)$, and $Y(\mu)$ functions
respectively. We also derive a transformation relation between the modified
S-T functions and give a physical account of $V(\mu)$ and $W(\mu)$ functions.
Our final results are consistent with those of Chandrasekhar at low emission
limit (i.e. only scattering). From the consistency of our results, we conclude
that the consideration of thermal emission effect in diffuse reflection finite
atmosphere problem gives more general and accurate results than considering
only scattering.

Diffuse radiation, Atmospheric effects, Radiative transfer equation, Radiative
transfer

##  1 Introduction

Chandrasekhar did the pioneering work on the process of radiative transfer
which is the heart of the observations as well as modeling in astrophysical
context Chandrasekhar (1960). One of his most interesting and useful method is
the Invariance Principle technique, which has a great deal of applications in
atmospheric modeling. Although this principle was first introduced by
Ambartsumian (1943, 1944), Chandrasekhar (1960) used this theory to solve the
semi-infinite and finite atmosphere problem in its most elegant way by
introducing the scattering function $S(\tau;\mu,\phi;\mu_{0},\phi_{0})$ and
transmission function $T(\tau;\mu,\phi;\mu_{0},\phi_{0})$. The final results
of those treatments can be represented in terms of H-function (semi-infinite
case) Chandrasekhar (1947a) and X-, Y- functions (finite case) Chandrasekhar
(1948). The values of the H-function (Chandrasekhar & Breen, 1947) and X-,Y-
functions (Chandrasekhar et al., 1952; Chandrasekhar & Elbert, 1952) in the
case of isotropic scattering are directly used in atmosphere modeling. Even a
simple transformation rule between S and T was established by Coakley Jr
(1973).

Although the results provided in Chandrasekhar (1960) have direct applications
in stellar and planetary problems, the treatment is not complete in some sense
as it does not consider the atmospheric emission and scattering
simultaneously. Bellman et al. (1967) include the thermal emission in the
planetary atmosphere problem and started a new technique called invariant
embedding (Bellman & Wing, 1992). In the context of exoplanetary transmission
spectra modeling, Sengupta et al. (2020) and Chakrabarty & Sengupta (2020)
showed the crucial effect of scattering and atmospheric re-emission
respectively. Recently, Sengupta (2021) considered scattering and atmospheric
emission simultaneously to study the modifications in Chandrasekhar’s semi-
infinte atmosphere problem. However, the effect of emission on the finite
atmosphere problem, which is more general than the semi-infinite one, remains
unsolved.

In this work we solve the finite atmosphere problem in case of isotropic
scattering and emission by the same analytical procedure as shown in Sengupta
(2021). For that we consider local thermodynamic equilibrium condition in
vertical atmospheric layers, which ensures the fact that each layer
contributes the Blackbody emission according to Kirchoff’s law (Chandrasekhar,
1960; Seager, 2010). We used the invariance principle method (Ambartsumian,
1944; Chandrasekhar, 1960) to derive the modified scattering, transmission
function and final radiation to show that our results are more general than
Chandrasekhar’s results. This treatment is also free from the isothermal
atmosphere condition, which was a limitation of the work in Sengupta (2021).

In section 2 we state the mathematical formula of invariance principles for
finite atmosphere following Chandrasekhar (1960). Section 3 is devoted to
deriving the general integral equations of the Scattering function (S) and
Transmission function (T) in case of thermal emission with scattering. The
modified form of these functions specifically for the isotropic scattering
case is shown in section 4. Then we establish a simple transformation rule
between the $S(\mu)$-$T(\mu)$ in section 5 and give their physical
interpretations in section 6. The consistency of our new results with the
literature is discussed in section 7 and concluded with an elaborated
discussion in section 8.

##  2 Invariance Principle for Finite atmosphere

The radiative transfer equation in case of plane-parallel approximation can be
written as,

$\mu\frac{dI_{\nu}(\tau_{\nu},\mu,\phi)}{d\tau_{\nu}}=I_{\nu}(\tau_{\nu},\mu,\phi)-\xi_{\nu}(\tau,\mu,\phi)$
(1)

Here $I_{\nu}(\tau_{\nu},\mu,\phi)$ is the specific intensity at a particular
frequency $\nu$, direction cosine $\mu$, angle of azimuth $\phi$ and optical
depth range between $\tau_{\nu}$ to $\tau_{\nu}+d\tau_{\nu}$. With these same
parameters the source function is written as,
$\xi_{\nu}(\tau_{\nu},\mu,\phi)$.

For an atmosphere with simultaneous scattering and absorption, the optical
depth can be defined as (Domanus & Cogley, 1974; Sengupta et al., 2020;
Sengupta, 2021),

$d\tau_{\nu}=-[\kappa_{\nu}(z)+\sigma_{\nu}(z)]dz=-\chi_{\nu}(z)dz$ (2)

Here $\kappa_{\nu}(z),\sigma_{\nu}(z)$ and $\chi_{\nu}(z)$ are the volumetric
absorption co-efficient, scattering co-efficient and extinction co-efficient
at a particular frequency $\nu$ and depth z respectively.

Note that, for the sake of simplicity in further calculations we suppress the
subscript $\nu$ by considering all the calculations at a particular frequency.
It should not be confused with the Grey atmosphere approximation as there is
no such assumption in the present work.

Now finite atmosphere is bounded by optical depth $\tau=0$ to $\tau=\tau_{1}$
(Chandrasekhar, 1960). To provide a solution of the problem of only diffuse
reflection from such an atmosphere, Chandrasekhar (1947b) used the invariance
principle method. We will use the same methodology following (Chandrasekhar,
1960) to get a solution of the more general problem where atmospheric thermal
emission is also included with the diffuse scattering.

When a radiation of light $\pi F$ incident on an atmosphere of optical
thickness $\tau_{1}$ along the direction ($-\mu_{0},\phi_{0}$), then the
diffusely reflected and transmitted intensities can be represented as,

$I(0,\mu,\phi)=\frac{F}{4\mu}S(\tau_{1},\mu,\phi;\mu_{0},\phi_{0});\hskip
14.22636ptI(\tau_{1},-\mu,\phi)=\frac{F}{4\mu}T(\tau_{1},\mu,\phi;\mu_{0},\phi_{0})$
(3)

respectively, where $S(\tau_{1},\mu,\phi;\mu_{0},\phi_{0})$ and
$T(\tau_{1},\mu,\phi;\mu_{0},\phi_{0})$ are the scattering and transmission
functions. Note that, these two intensities refer only that light which has
suffered at least one scattering process and do not include any direct
transmission along $(-\mu_{0},\phi_{0})$ direction. For a detailed discussion
on this, we refer Radiative Transfer by Chandrasekhar (1960).

The four mathematical expressions of invariance principle in finite atmosphere
problem can be written as (Chandrasekhar, 1960),

Principle I

$I(\tau,+\mu,\phi)=\frac{F}{4\mu}e^{-\tau/\mu_{0}}S(\tau_{1}-\tau,\mu,\phi;\mu_{0},\phi_{0})+\frac{1}{4\pi\mu}\int_{0}^{1}\int_{0}^{2\pi}I(\tau,-\mu^{\prime},\phi^{\prime})S(\tau_{1}-\tau,\mu,\phi;\mu^{\prime},\phi^{\prime})d\mu^{\prime}d\phi^{\prime}$
(4)

Principle II

$I(\tau,-\mu,\phi)=\frac{F}{4\mu}T(\tau,\mu,\phi;\mu_{0},\phi_{0})+\frac{1}{4\pi\mu}\int_{0}^{1}\int_{0}^{2\pi}I(\tau,+\mu^{\prime},\phi^{\prime})S(\tau,\mu,\phi;\mu^{\prime},\phi^{\prime})d\mu^{\prime}d\phi^{\prime}$
(5)

Principle III

$\begin{split}\frac{F}{4\mu}S(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})=\frac{F}{4\mu}S(\tau;\mu,\phi;\mu_{0},\phi_{0})+e^{-\tau/\mu}I(\tau,+\mu,\phi)+\frac{1}{4\pi\mu}\int_{0}^{1}\int_{0}^{2\pi}I(\tau,+\mu^{\prime},\phi^{\prime})T(\tau,\mu,\phi;\mu^{\prime},\phi^{\prime})d\mu^{\prime}d\phi^{\prime}\end{split}$
(6)

Principle IV

$\begin{split}\frac{F}{4\mu}T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})=&\frac{F}{4\mu}e^{-\tau/\mu_{0}}T(\tau_{1}-\tau;\mu,\phi;\mu_{0},\phi_{0})+e^{-(\tau_{1}-\tau)/\mu}I(\tau,-\mu,\phi)\\\
&+\frac{1}{4\pi\mu}\int_{0}^{1}\int_{0}^{2\pi}I(\tau,-\mu^{\prime},\phi^{\prime})T(\tau_{1}-\tau,\mu,\phi;\mu^{\prime},\phi^{\prime})d\mu^{\prime}d\phi^{\prime}\end{split}$
(7)

These equations are derived and diagramatically shown in Chandrasekhar (1947b,
1960); Peraiah (2002). The boundary conditions used to calculate
$S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})$ and
$T(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})$ are,

$I(0,-\mu,\phi)=0\hskip 28.45274pt\text{and}\hskip
28.45274ptI(\tau_{1},+\mu,\phi)=0$ (8)

Chandrasekhar (1960) used these boundary conditions in eqn.(1) and derived the
four invariance principles (4)—(7) in terms of source functions
$\xi(0,\mu,\phi)$ and $\xi(\tau_{1},\mu,\phi)$ . We directly use those
relations in this paper.

##  3 The general integral equations for Scattering and Thermally Emitting
Atmosphere

When there is atmospheric emission as well as scattering, the source function
$\xi$ can be written as (Sengupta, 2021),

$\begin{split}\xi(\tau,\mu,\phi)=\beta(\tau,\mu,\phi)+\frac{1}{4}Fe^{-\tau/\mu_{0}}p(\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{-1}^{1}\int_{0}^{2\pi}p(\mu,\phi;\mu^{\prime\prime};\phi^{\prime\prime})I(\tau,\mu^{\prime\prime},\phi^{\prime\prime})d\phi^{\prime\prime}d\mu^{\prime\prime}\end{split}$
(9)

Here $p(\mu,\phi;\mu^{\prime\prime};\phi^{\prime\prime})$ and
$\beta(\tau,\mu,\phi)$ are the phase function and atmospheric emission
respectively. This atmospheric emission $\beta$ can be expanded (Bellman et
al., 1967; Sengupta, 2021) as follows,

$\beta(\tau;\mu;\phi)=\sum_{m=0}^{N}\beta^{m}(\tau,\mu)\cos m(\phi-\phi_{0})$
(10)

For planetary atmosphere, the emission can be caused by different mechanisms
(for example, see Bellman et al. (1967); Chakrabarty & Sengupta (2020);
Malkevich (1963); Sengupta (2021); Seager (2010)). In the current study, we
consider an atmosphere, where each horizontal layer is in Local Thermodynamic
Equilibrium and emits only in terms of Planck Emission (Seager, 2010), as
shown in figure. 1. Hence, considering m=0 with no $\mu$ dependencies,
eqn.(10) will reduce into

$\beta(\tau,\mu,\phi)\approx B(T_{\tau})$ (11)

Here $T_{\tau}$ represents the absolute temperature of that particular
atmospheric layer which has an optical depth $\tau$. It is worth noting that
in case of thermal emission the exact expression of $\beta$ is
$\frac{\kappa}{\chi}B(T_{\tau})$. But in case of low scattering limit (i.e.
$\kappa>>\sigma$), the $\kappa\approx\chi$ and eqn. (11) is valid (Sengupta,
2021).

Assuming low scattering approximation, eqn.(9) will become,

$\begin{split}\xi(0,\mu,\phi)=B(T_{0})+\frac{1}{4}F[p(\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu,\phi;\mu^{\prime\prime};\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\end{split}$
(12)

$\begin{split}\xi(\tau_{1},\mu,\phi)=B(T_{\tau_{1}})+\frac{1}{4}F[e^{-\tau_{1}/\mu_{0}}p(\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu,\phi;-\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\end{split}$
(13)

$\tau=0$$\tau$$\tau=\tau_{1}$$\pi F$B($T_{0}$)B($T_{\tau}$)B($T_{\tau_{1}}$)
Figure 1: This figure depicts the total effect of diffuse scattering and
thermal emission B for finite atmosphere. Considering an arbitrary layer with
optical depth $\tau$ ($0<\tau<\tau_{1}$ ) we show the effect of scattering and
emission, while B is Isotropic in nature and depends only on the temperature
of the emitting layer.

Now using eqns. (12) and (13) in the eqns. 23—26; pg:168 in Chandrasekhar
(1960) we will get (for a detailed derivation see Appendix A ) ,

$\begin{split}&[(\frac{1}{\mu}+\frac{1}{\mu_{0}})S(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})+\frac{\partial
S(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau_{1}}]=4U(T_{0})[1+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\\\
&+p(\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu,\phi;\mu^{\prime\prime};\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})[p(-\mu^{\prime},\phi^{\prime};-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(-\mu^{\prime},\phi^{\prime};\mu^{\prime\prime};\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}\end{split}$
(14)

$\begin{split}&[\frac{\partial
S(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau_{1}}]=4U(T_{\tau_{1}})[e^{-\tau_{1}/\mu}+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}T(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\\\
&+[exp\\{-\tau_{1}(\frac{1}{\mu_{0}}+\frac{1}{\mu})\\}p(\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}e^{-\tau_{1}/\mu}\int_{0}^{1}\int_{0}^{2\pi}p(\mu,\phi;-\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}T(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})[e^{-\tau_{1}/\mu_{0}}p(\mu^{\prime},\phi^{\prime};-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu^{\prime},\phi^{\prime};-\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}\end{split}$
(15)

$\begin{split}&[\frac{1}{\mu}T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})+\frac{\partial
T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau_{1}}]=4U(T_{\tau_{1}})[1+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\\\
&+[e^{-\tau_{1}/\mu_{0}}p(-\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(-\mu,\phi;-\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})[e^{-\tau_{1}/\mu_{0}}p(\mu^{\prime},\phi^{\prime};-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu^{\prime},\phi^{\prime};-\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}\end{split}$
(16)

$\begin{split}&[\frac{1}{\mu_{0}}T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})+\frac{\partial
T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau_{1}}]=4U(T_{0})[e^{-\tau_{1}/\mu}+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}T(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\\\
&+[p(-\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(-\mu,\phi;\mu^{\prime\prime};\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]e^{-\tau_{1}/\mu}\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}T(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})[p(-\mu^{\prime},\phi^{\prime};-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(-\mu^{\prime},\phi^{\prime};\mu^{\prime\prime},\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}\end{split}$
(17)

where, $U(T_{0})=\frac{B(T_{0})}{F}$,
$U(T_{\tau_{1}})=\frac{B(T_{\tau_{1}})}{F}$.

Equations. (14) — (17) represent the integral equations governing the problem
of diffuse reflection and trasmission in presence of atmospheric thermal
emission of a plane-parallel atmosphere with finite optical depth.

##  4 The integral equations in isotropic scattering

It is evident that these four integral equations have an explicit dependency
on the phase function $p(\mu,\phi;\mu^{\prime},\phi^{\prime})$. The different
types of phase functions are discussed in Chandrasekhar (1960); Sengupta
(2021). Here, we specifically study the effect of thermal emission in
isotropic scattering case only. It can be treated in terms of single
scattering albedo $\tilde{\omega}_{0}$ (Sengupta, 2021) as,

$p(\mu,\phi;\mu_{0},\phi_{0})=\tilde{\omega_{0}}$

This axial symmetry in phase function is also a property of scattering and
transmission functions and they can be expressed in axisymmetric terms as,
$S(\tau_{1};\mu;\mu^{\prime})$ and $T(\tau_{1};\mu;\mu^{\prime})$

Then, eqn.(15) will become,

$\begin{split}&[\frac{\partial
S(\tau_{1};\mu,\mu_{0})}{\partial\tau_{1}}]=4U(T_{\tau_{1}})[e^{-\tau_{1}/\mu}+\frac{1}{2}\int_{0}^{1}T(\tau_{1};\mu;\mu^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}]\\\
&+[exp\\{-\tau_{1}(\frac{1}{\mu_{0}}+\frac{1}{\mu})\\}\tilde{\omega_{0}}+\frac{1}{2}e^{-\tau_{1}/\mu}\int_{0}^{1}\tilde{\omega_{0}}T(\tau_{1};\mu^{\prime\prime},\mu_{0})\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]+\frac{1}{2}\int_{0}^{1}T(\tau_{1};\mu,\mu^{\prime})[e^{-\tau_{1}/\mu_{0}}\tilde{\omega_{0}}+\frac{1}{2}\int_{0}^{1}\tilde{\omega_{0}}T(\tau_{1};\mu^{\prime\prime},\mu_{0})\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\frac{d\mu^{\prime}}{\mu^{\prime}}\end{split}$
$\boxed{[\frac{\partial
S(\tau_{1};\mu,\mu_{0})}{\partial\tau_{1}}]=4U(T_{\tau_{1}})W(\mu)+\tilde{\omega_{0}}W(\mu_{0})W(\mu)}$
(18)

Here we define two new functions as,

$\begin{split}V(\mu)&=1+\frac{1}{2}\int_{0}^{1}S(\tau_{1};\mu,\mu^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}\end{split}$
(19)

and

$\begin{split}W(\mu)&=e^{-\tau_{1}/\mu}+\frac{1}{2}\int_{0}^{1}T(\tau_{1};\mu,\mu^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}\end{split}$
(20)

Similarly, eqns.(14), (16), (17) can be expressed in terms of V and W
functions as follows,

$\boxed{(\frac{1}{\mu}+\frac{1}{\mu_{0}})S(\tau_{1};\mu,\mu_{0})=4[U(T_{0})V(\mu)-U(T_{\tau_{1}})W(\mu)]+\tilde{\omega_{0}[}V(\mu)V(\mu_{0})-W(\mu)W(\mu_{0})]}$
(21)

$\begin{split}[\frac{1}{\mu_{0}}T(\tau_{1};\mu,\mu_{0})+\frac{\partial
T(\tau_{1};\mu,\mu_{0})}{\partial\tau_{1}}]=4U(T_{0})W(\mu)+\tilde{\omega_{0}}V(\mu_{0})W(\mu)\end{split}$
(22)

$\begin{split}[\frac{1}{\mu}T(\tau_{1};\mu,\mu_{0})+\frac{\partial
T(\tau_{1};\mu,\mu_{0})}{\partial\tau_{1}}]=4U(T_{\tau_{1}})V(\mu)+\tilde{\omega_{0}}W(\mu_{0})V(\mu)\end{split}$
(23)

Subtracting eqn.(23) from eqn.(22) gives,

$\boxed{(\frac{1}{\mu_{0}}-\frac{1}{\mu})T(\tau_{1};\mu,\mu_{0})=4[U(T_{0})W(\mu)-U(T_{\tau_{1}})V(\mu)]+\tilde{\omega_{0}}[V(\mu_{0})W(\mu)-W(\mu_{0})V(\mu)]}$
(24)

Subtracting eqn.(22)*$\frac{1}{\mu}$ from eqn.(23)*$\frac{1}{\mu_{0}}$ gives,

$\boxed{(\frac{1}{\mu_{0}}-\frac{1}{\mu})\frac{\partial
T(\tau_{1};\mu,\mu_{0})}{\partial\tau_{1}}=4[\frac{1}{\mu_{0}}U(T_{\tau_{1}})V(\mu)-\frac{1}{\mu}U(T_{0})W(\mu)]+\tilde{\omega_{0}}[\frac{1}{\mu_{0}}W(\mu_{0})V(\mu)-\frac{1}{\mu}V(\mu_{0})W(\mu)]}$
(25)

Thus, the functional form of V and W functions (eqns.(19) and (20)) can be
modified as,

$\begin{split}V(\mu)=1+\frac{1}{2}\mu\int_{0}^{1}\\{4[U(T_{0})V(\mu)-U(T_{\tau_{1}})W(\mu)]+\tilde{\omega_{0}[}V(\mu)V(\mu_{0})-W(\mu)W(\mu_{0})]\\}\frac{d\mu^{\prime}}{\mu^{\prime}+\mu}\end{split}$
(26)

and

$\begin{split}W(\mu)=e^{-\tau_{1}/\mu}+\frac{1}{2}\mu\int_{0}^{1}4[U(T_{0})W(\mu)-4U(T_{\tau_{1}})V(\mu)]+\tilde{\omega_{0}}[V(\mu_{0})W(\mu)-W(\mu_{0})V(\mu)]\frac{d\mu^{\prime}}{\mu-\mu^{\prime}}\end{split}$
(27)

The final emitted radiation from $\tau=0$ and $\tau=\tau_{1}$ can be expressed
from eqn. (3) as,

$\begin{split}I(0,\mu)=\frac{\mu_{0}}{\mu+\mu_{0}}[B(T_{0})V(\mu)-B(T_{\tau_{1}})W(\mu)]+\frac{F}{4}\frac{\mu_{0}}{\mu+\mu_{0}}\tilde{\omega_{0}}[V(\mu)V(\mu_{0})-W(\mu)W(\mu_{0})]\end{split}$
(28)

and,

$\begin{split}I(\tau_{1},-\mu)=\frac{\mu_{0}}{\mu-\mu_{0}}[B(T_{0})W(\mu)-B(T_{\tau_{1}})V(\mu)]+\frac{F}{4}\frac{\mu_{0}}{\mu-\mu_{0}}\tilde{\omega_{0}}[V(\mu_{0})W(\mu)-W(\mu_{0})V(\mu)]\end{split}$
(29)

Eqns.(28) and (29) can be expressed as,

$\begin{split}\begin{bmatrix}(\mu+\mu_{0})I(0,+\mu)\\\
(\mu-\mu_{0})I(\tau_{1},-\mu)\end{bmatrix}&=\begin{bmatrix}V(\mu)&W(\mu)\\\
W(\mu)&V(\mu)\end{bmatrix}\begin{bmatrix}\frac{F}{4}\tilde{\omega_{0}}\mu_{0}V(\mu_{0})+\mu_{0}B(T_{0})\\\
-\frac{F}{4}\tilde{\omega_{0}}\mu_{0}W(\mu_{0})-\mu_{0}B(T_{\tau_{1}})\end{bmatrix}\end{split}$
(30)

##  5 A simple transformation rule

Chandrasekhar (1960) introduced two crucial functions
$S(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})$ and
$T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})$ while considering diffuse scattering
in finite atmosphere. The transformation rule between these two functions was
established by Coakley Jr (1973) as,

$\begin{split}&S(\tau_{1},\mu,\phi;-\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}=T(\tau_{1},\mu,\phi;\mu_{0},\phi_{0})\\\
&T(\tau_{1},\mu,\phi;-\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}=S(\tau_{1},\mu,\phi;\mu_{0},\phi_{0})\end{split}$
(31)

Here we will show that these rules are indeed true while the thermal emission
is included in this problem under some circumstances. We replace $\mu_{0}$ by
$-\mu_{0}$ in eqn.(14) and multiplying both sides by $e^{-\tau_{1}/\mu_{0}}$
and get,

$\begin{split}\therefore&[(\frac{1}{\mu}-\frac{1}{\mu_{0}})S(\tau_{1};\mu,\phi;-\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}+\frac{\partial
S(\tau_{1};\mu,\phi;-\mu_{0},\phi_{0})}{\partial\tau_{1}}e^{-\tau_{1}/\mu_{0}}]\\\
&=4U(T_{0})[1+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]e^{-\tau_{1}/\mu_{0}}\\\
&+p(\mu,\phi;\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu,\phi;\mu^{\prime\prime};\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};-\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})[p(-\mu^{\prime},\phi^{\prime};\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(-\mu^{\prime},\phi^{\prime};\mu^{\prime\prime};\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};-\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}\end{split}$
(32)

Now if we make use of eqn.(31) and the symmetric properties of the phase
function $p(\mu,\phi;-\mu_{0},\phi_{0})=p(-\mu,\phi;\mu_{0},\phi_{0})$ and
$p(-\mu,\phi;-\mu_{0},\phi_{0})=p(\mu,\phi;\mu_{0},\phi_{0})$ then equation
(32) will be,

$\begin{split}\therefore&\frac{1}{\mu}T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})+\frac{\partial
T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau_{1}}=4U(T_{0})e^{-\tau_{1}/\mu_{0}}[1+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\\\
&+p(-\mu,\phi;-\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu,\phi;\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})[p(\mu^{\prime},\phi^{\prime};-\mu_{0},\phi_{0})e^{-\tau_{1}/\mu_{0}}\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu^{\prime},\phi^{\prime};-\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}]\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}\end{split}$
(33)

Observing the similarity of this equation with eqn.(16), we can write
condition on thermal emission,

$U(T_{\tau_{1}})=U(T_{0})e^{-\tau_{1}/\mu_{0}}\hskip 28.45274pt\hskip
28.45274ptB(T_{\tau_{1}})=B(T_{0})e^{-\tau_{1}/\mu_{0}}$ (34)

So the transformation rule between S- and T-function eqn.(31) will be valid
when the thermal emission from different atmospheric layers are connected by
eqn.(34). This blackbody emission can be named as the reduced thermal
emission.

For isotropic scattering case the transformation rules of V and W can be
obtained using eqns.(31), (19), (20) as,

$V(-\mu)e^{-\tau_{1}/\mu}=W(\mu)\hskip 28.45274pt\hskip
28.45274ptW(-\mu)e^{-\tau_{1}/\mu}=V(\mu)$ (35)

##  6 The Physical meaning of V and W functions:

We introduced two new functions $V(\mu)$ and $W(\mu)$ which resembles to
Chandrasekhar’s $X(\mu)$\- and $Y(\mu)$\- functions respectively in the
presence of thermal emission with diffusely reflecting finite atmosphere
problem . The physical meaning of X- and Y- functions has been discussed in
Chandrasekhar (1960); Van de Hulst (1948); Peraiah (2002). Here we will
discuss the additional effects introduced in V and W functions.

V($\mu$)W($\mu$)$\tau=0$$\tau=\tau_{1}$B($T_{0}$)B$(T_{\tau_{1}})$ Figure 2:
This figure explains the physical meaning of $V(\mu)$ and $W(\mu)$ functions.
The total brightness observed by a distant observer will be $V(\mu)$ (for the
observer above $\tau=0$) and $W(\mu)$ (for the observer below $\tau=\tau_{1}$)
times the brightness of the point source alone. Hence all the scattering,
transmission and thermal emission from the atmosphere contributes in terms of
V and W functions.

Let there be a point source above the layer $\tau=0$ which has unit brightness
(with the total emitting flux $4\pi$ from the point source). Now the flux will
be scattered and transmitted multiple times by both the atmospheric layers at
$\tau=0$ and $\tau=\tau_{1}$ (see fig 2). In addition to it there is a
contribution of thermal emission $B(T_{0})$ and $B(T_{\tau_{1}})$ respectively
from those layers. Now for an observer at large distance, the combination of
the same point source and the illuminated atmosphere will again appear as a
point source and only the combined effect can be observed. If that distant
observer is in $(+\mu,\phi)$ direction from the atmosphere (i.e. above the
atmospheric layer $\tau=0$ in fig. 2), then $V(\mu)$ will be the total
observed brightness. In the same way, if the observer is in $(-\mu,\phi)$
direction from the atmosphere (i.e. below the atmospheric layer
$\tau=\tau_{1}$ in fig. 2), then $W(\mu)$ will be the total observed
brightness. In both the cases, the factor $\frac{1}{\mu}$ is positive.

In other words, $V(\mu)$ and $W(\mu)$ represents the relative change of the
incident and transmitted flux along $(+\mu)$ and $(-\mu)$ direction
respectively due to the presence of the atmosphere. This relative change shows
the combined effect of scattering, transmission and thermal emission by the
atmospheric layers.

Clearly in the absence of thermal emission, the contribution of thermal
emission will be removed and the observed brightness will be a combination of
atmospheric scattering and transmission of the point source flux only. In such
circumstances, $V(\mu)$ and $W(\mu)$ functions will reduce into the
Chandrasekhar’s $X(\mu)$ and $Y(\mu)$ functions only (see section 7 for more
discussion). Also figure 2 will reduce into the figure given in Van de Hulst
(1948).

In case of semi-infinite atmosphere, the bottom layer will be extended at
$\tau_{1}\to\infty$ as shown in figure 3. In such circumstances, the distant
observer can observe the combined effect from $(+\mu,\phi)$ direction only.
Hence $W(\mu)$ function will vanish and $V(\mu)$ function will give the
combined effect of scattering and thermal emission. In such case
$V(\mu)$-function will reduce into the well known $M(\mu)$-function as
introduced by Sengupta (2021) for semi-infinite atmosphere case.

$M(\mu)$$\tau=0$$\tau=\tau_{1}=\infty$B($T$) Figure 3: A pictorial view of the
M($\mu$)-function derived in Sengupta (2021) for semi-infinite atmosphere. The
total radiation observed from $(+\mu,\phi)$ direction will be the addition of
the light from the source, scattered by the atmosphere and the thermally
emitted radiation from the atmospheric layer. Here all the layers are at same
temperature T.

Hence the V and W functions represents the relative change of the flux from
point source due to the presence of atmospheric scattering, transmission and
thermal emission.

##  7 Consistency Check

In this section we will show that how our results reduce into previous
literature results at specific boundary conditions. It is expected that, when
the atmospheric thermal emission is very less than the incident flux (i.e.
$B(T_{0}),B(T_{\tau_{1}})<<F$) then our solutions should match with the
results of only scattering case as derived by Chandrasekhar (1960).

In case of no thermal emission limit, $U(T_{0}),U(T_{\tau_{1}})\to 0$ and thus
equations (26) and (27) will reduce into that of Chandrasekhar’s X and Y
functions as shown in Chandrasekhar (1960) (pg.181; eqns.84-85)

$\lim_{U\to 0}V(\mu)\to X(\mu)$

and

$\lim_{U\to 0}W(\mu)\to Y(\mu)$

.

Hence in the limit of $U\to 0$, eqn.(21) and (24) will become,

$\begin{split}&(\frac{1}{\mu}+\frac{1}{\mu_{0}})S(\tau_{1};\mu,\mu_{0})=\tilde{\omega_{0}}[X(\mu)X(\mu_{0})-Y(\mu)Y(\mu_{0})]\\\
&\text{and}\\\
&(\frac{1}{\mu_{0}}+\frac{1}{\mu})T(\tau_{1};\mu,\mu_{0})=\tilde{\omega_{0}}[X(\mu_{0})Y(\mu)-Y(\mu_{0})X(\mu)]\end{split}$
(36)

These equations are the same as given in Chandrasekhar (1960) (pg.181; eqns.
80-81) The no thermal emission limit will also affect the final radiation
coming out from both the boundaries at $\tau=0$ and $\tau=\tau_{1}$. Hence
equations (28) and (29) will reduce into the scattering only case. At no
thermal emission case we can write the matrix equation (30) as follows,

$\begin{split}\begin{bmatrix}(\mu+\mu_{0})I(0,+\mu)\\\
(\mu-\mu_{0})I(\tau_{1},-\mu)\end{bmatrix}&=\begin{bmatrix}X(\mu)&Y(\mu)\\\
Y(\mu)&X(\mu)\end{bmatrix}\begin{bmatrix}\frac{F}{4}\tilde{\omega_{0}}\mu_{0}X(\mu_{0})\\\
-\frac{F}{4}\tilde{\omega_{0}}\mu_{0}Y(\mu_{0})\end{bmatrix}\end{split}$ (37)

This is the same form as derived by (Chandrasekhar, 1960) (pg. 201; eqns 108
and 109) in case of diffuse scattering.

Now we will show the two limiting cases of optical depth.

  1. 1.

Semi-Infinite optical depth $(\tau_{1}\to\infty)$: In this condition the
expression of the function $V(\mu)$ will reduced into ,

$V(\mu)=1+2U(T)M(\mu)\mu\log(1+\frac{1}{\mu})+\frac{\tilde{\omega_{0}}}{2}\mu
M(\mu)\int_{0}^{1}\frac{M(\mu^{\prime})}{\mu+\mu^{\prime}}d\mu^{\prime}=M(\mu)$
(38)

This expression is same as of the M-function derived in (Sengupta, 2021) in
the context of semi-infinite atmosphere. Hence we can say that the finite
atmosphere problem boils down to semi-infinite atmosphere problem at this
limiting value.

Now using the transformation rule (35), the W-function can be represented as,

$W(\mu)=\lim_{\tau_{1}\to\infty}V(-\mu)e^{-\tau_{1}/\mu}\to 0$ (39)

  2. 2.

Small optical depth $(\tau_{1}\to 0)$: In such case the W function will be,

$W(\mu)\to e^{-\tau_{1}/\mu}$ (40)

and using the transformation rule we get,

$V(\mu)=W(-\mu)e^{-\tau_{1}/\mu}\to 1$ (41)

These values are same as shown in (Peraiah, 2002) in the case of Y and X
functions respectively.

##  8 Discussion

The finite atmosphere diffuse reflection problem was first introduced by
Chandrasekhar (1960) for scattering only atmosphere where no atmospheric
emission was considered. Here for the first time we include the thermal
emission effect simultaneously with the isotropic scattering from each
atmospheric layers in finite atmosphere diffuse reflection problem. The
thermal emission modifies Chandrasekhar’s results in terms of the factor
$U(T)$, where U is the ratio of blackbody emission (B) and irradiation flux
(F) (see section 3 and 4). Moreover, the modified scattering and transmission
functions obey the same transformation rules as established by Coakley Jr
(1973) (as shown in sec. 5). Then, we show that our results are consistent
with that of the Chandrasekhar (1960) results in the limit of low atmospheric
thermal emission (i.e. $B<<F$). Hence, it can be said that our treatment of
thermal emission and scattering for finite atmosphere problem is more general
than Chansdrasekhar’s one. In exoplanetary context Chandrasekhar’s results are
used to model the reflection, transmission and emission spectra for highly
irradiated low emitting planets Madhusudhan & Burrows (2012). But as it is
evident that when the thermal emission and scattering occurs comparably in a
planetary atmosphere (e.g. low irradiating ultra-hot jupiters), then our
results will provide more accurate modeling than Chandrasekhar’s results.

The thermal emission from each atmospheric layer will travel through other
layers as well and will face the scattering and transmission. For example, the
emission from the layer at $\tau=0$ is $B(T_{0})$ which scattered along the
direction $(\mu,\phi)$ and contribute to the final radiation $I(0,\mu)$ in
terms of $B(T_{0})V(\mu)$ (see eqn.(28)). In the same way it contributes along
the direction $(-\mu,\phi)$ in the radiation in terms of $B(T_{0})W(\mu)$.
Thus, the flux F irradiated on the atmosphere and the atmospheric thermal
emission will follow the same rule of scattering and transmission. So this
theory is applicable to those planetary atmosphere where, (1) the atmospheric
thermal emission is comparable to the irradiated stellar flux and (2) the
atmosphere gives infrared scattering effects.

Here we have revisited the connection relation between scattering and
transmission functions (see (31)) as established by Coakley Jr (1973). This
transformation rule describes the interchange between S- and T- functions
depending on the orientation of the incident beam in case of only diffuse
scattering in finite atmosphere (Coakley Jr, 1973). In this work, we first
show that this relation is indeed true in case of thermally emitting
atmosphere. Secondly, a transformation rule for the V and W functions (see
eqn. (35)) as well as a connection relation between the thermal emission flux
at different atmospheric layers, which can be named as reduced thermal
emission, is established. It ensures that if a beam incident from the upper
side of the layer $\tau=0$, then V and W functions can be represented as shown
in the figure 2. But while the light beam incident from the lower side of
$\tau=\tau_{1}$ layer then the position of V and W functions in figure 2 will
interchange. It shows the symmetry of the solutions provided here. Although
the transformation rules are applicable only if the atmospheric emission at
different optical depths are connected by the relation given in eqn. (34).

The applicability of the transformation rule in case of thermal emission, also
ensures the fact that in case of comparable emission and scattering from an
atmosphere, these both effects are in equal footing. Hence, both effects
should be considered for a full-proof modeling.

The $V(\mu)$ and $W(\mu)$ functions are analogous to Chandrasekhar’s X and Y
functions mentioned in Chandrasekhar (1960). They represent the relative
changes of the radiation from the layer $\tau=0$ along the direction
$(\mu,\phi)$ and from $\tau=\tau_{1}$ along $(-\mu,\phi)$ respectively due to
the presence of the atmosphere. Hence, the atmospheric presence can be
realized in terms of diffuse reflection and atmospheric thermal emission from
the corresponding layer. In other words it can be said that they act as the
source function for the direction $(\mu,\phi)$ at $\tau=0$ and the direction
$(-\mu,\phi)$ at $\tau=\tau_{1}$ respectively.

We showed (in section 7) that at the semi-infinite limit
(i.e.$\tau_{1}\to\infty$), our finite atmosphere results will reduce into that
of semi-infinite results obtained in Sengupta (2021). Hence we can say that
the $M(\mu)$-function (see Sengupta (2021) for detail) is semi-infinite
counterpart of the more general $V(\mu)$-function as shown in figure 3. It
reveals the semi-infinite limiting case of $V(\mu)$-function.

The work presented by (Sengupta, 2021) considered atmospheric thermal emission
in semi-infinite atmosphere case and thus limited by the condition of
translational invariant thermal emission in the atmosphere. For Planetary
atmosphere it means that such theory is applicable only for those planets
which has an isothermal atmosphere. In this work we removed that limitation by
considering the finite atmosphere problem which does not need a translational
invariant thermal emission and in such case the scattering function
$S(\tau,\mu,\phi;\mu_{0},\phi_{0})$ varies with the optical depth of the
atmosphere. It will provide the opportunity to model the atmospheric spectra
for any type of atmospheric temperature structure with simultaneous emission
and scattering.

In this work we considered only the isotropic scattering case which is indeed
the first step to modify the finite atmosphere scattering problem in the
presence of thermal emission. This work can be expanded for the general cases
of scattering with the same recipe and the modifications will follow
accordingly. To include the numerical approach, one should use the Heyney-
Greenstein phase function Henyey & Greenstein (1941),

$p(\cos\Theta)=\frac{1-g^{2}}{(1+g^{2}-2g\cos\Theta)^{\frac{2}{3}}}$

where, $g\in[-1,1]$ is the asymmetry parameter, as shown in Bellman et al.
(1967); Batalha et al. (2019).

The atmospheric thermal emission is the simplest possible emission process
considered in our work. However, we assumed low scattering limit
$\sigma<<\kappa$ to use the simple planck function as the atmospheric emission
term (see eqn. (11)). It simplifies the mathematical derivations
significantly. However, this restriction can be removed by replacing
$B(T_{\tau})$ with $\frac{\kappa}{\chi}B(T_{\tau})$ and the results will
follow accordingly. Although the physical interpretations remain unaltered.

In case of exoplanetary atmosphere modeling, the atmospheric emission can not
always be simplified by the Planck emission. The upper atmosphere of
exoplanets does not hold the local thermodynamic equilibrium condition
(Seager, 2010; Sengupta, 2021). In such cases different types of atmospheric
emission can be considered by the general atmospheric emission function
$\beta$ as shown in (10). By appropriate choice of $\beta$-parameter thermal
re-emission, anisotropic emission can be considered as discussed in Sengupta
(2021).

Finally, the polarization effect is not considered in this work. That can be
included for finite atmosphere in the same way as in the semi-infinite
atmosphere case discussed in Sengupta (2021).

##  Appendix A DERIVATION OF THE SCATTERING FUNCTION and its derivative

In this section we will show the derivation of the scattering function
$S(\tau_{1},\mu,\phi;\mu_{0},\phi_{0})$ and its derivative $\frac{\partial
S(\tau,\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau}|_{\tau=\tau_{1}}$ which has
directly been written in eqns.(14) and (15).For the simplicity of calculations
we write $\frac{\partial
S(\tau_{1},\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau_{1}}$ instead of
$\frac{\partial
S(\tau,\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau}|_{\tau=\tau_{1}}$ in the main
text of section 3. To start with we will use the scattering function relation
with the source function equation given in Chandrasekhar (1960) (page: 168;
eqns: (23) and (25)) as follows,

$\frac{1}{4}[(\frac{1}{\mu}+\frac{1}{\mu_{0}})S(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})+\frac{\partial
S(\tau;\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau}|_{\tau=\tau_{1}}]=\xi(0,+\mu,\phi)+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\xi(0,-\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}$
(A1)

and

$\frac{1}{4}F\frac{\partial
S(\tau;\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau}|_{\tau=\tau_{1}}=exp(-\tau_{1}/\mu)\xi(\tau_{1},+\mu,\phi)+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}T(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\xi(\tau_{1},+\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}$
(A2)

Now the source functions $\xi(0,\mu,\phi)$ and $\xi(\tau_{1},\mu,\phi)$ in
thermal emission case is given in eqns. (12) and (13). Making use of them with
the boundary conditions (eqns.(8)) the above two equations can be written as,

$\begin{split}&\frac{F}{4}[(\frac{1}{\mu}+\frac{1}{\mu_{0}})S(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})+\frac{\partial
S(\tau;\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau}|_{\tau=\tau_{1}}]=B(T_{0})[1+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\\\
&+\frac{1}{4}F[p(\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu,\phi;\mu^{\prime\prime};\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}S(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\\{p(-\mu^{\prime},\phi^{\prime};-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(-\mu^{\prime},\phi^{\prime};\mu^{\prime\prime};\phi^{\prime\prime})S(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\\}\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\end{split}$
(A3)

and

$\begin{split}&\frac{F}{4}[\frac{\partial
S(\tau;\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau}|_{\tau=\tau_{1}}]=B(T_{\tau_{1}})[e^{-\tau_{1}/\mu}+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}T(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\\\
&+\frac{1}{4}F[exp\\{-\tau_{1}(\frac{1}{\mu_{0}}+\frac{1}{\mu})\\}p(\mu,\phi;-\mu_{0},\phi_{0})+\frac{1}{4\pi}e^{-\tau_{1}/\mu}\int_{0}^{1}\int_{0}^{2\pi}p(\mu,\phi;-\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\\\
&+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}T(\tau_{1};\mu,\phi;\mu^{\prime},\phi^{\prime})\\{e^{-\tau_{1}/\mu_{0}}p(\mu^{\prime},\phi^{\prime};-\mu_{0},\phi_{0})+\frac{1}{4\pi}\int_{0}^{1}\int_{0}^{2\pi}p(\mu^{\prime},\phi^{\prime};-\mu^{\prime\prime};\phi^{\prime\prime})T(\tau_{1};\mu^{\prime\prime},\phi^{\prime\prime};\mu_{0},\phi_{0})d\phi^{\prime\prime}\frac{d\mu^{\prime\prime}}{\mu^{\prime\prime}}\\}\frac{d\mu^{\prime}}{\mu^{\prime}}d\phi^{\prime}]\end{split}$
(A4)

Hence, multiplying eqns.(A3) and (A4) by $\frac{4}{F}$ and replacing the
quantities $\frac{B(T_{0})}{F},\frac{B(T_{\tau_{1}})}{F}$ by $U(T_{0})$ and
$U(T_{\tau_{1}})$ respectively, we will get the equations (14) and (15)
respectively. In the similar fashion the Transmission function
$T(\tau_{1};\mu,\phi;\mu_{0},\phi_{0})$ and its derivative $\frac{\partial
T(\tau;\mu,\phi;\mu_{0},\phi_{0})}{\partial\tau}|_{\tau=\tau_{1}}$ can be
derived.

## References

  * Ambartsumian (1943) Ambartsumian, V. 1943, Sci. URSS, 38, 257
  * Ambartsumian (1944) —. 1944, J. Phys. USSR, 8, 65
  * Batalha et al. (2019) Batalha, N. E., Marley, M. S., Lewis, N. K., & Fortney, J. J. 2019, The Astrophysical Journal, 878, 70
  * Bellman et al. (1967) Bellman, R., Kagiwada, H., Kalaba, R., & Ueno, S. 1967, Icarus, 7, 365
  * Bellman & Wing (1992) Bellman, R., & Wing, G. M. 1992, An introduction to invariant imbedding (SIAM)
  * Chakrabarty & Sengupta (2020) Chakrabarty, A., & Sengupta, S. 2020, The Astrophysical Journal, 898, 89, doi: 10.3847/1538-4357/ab9a33
  * Chandrasekhar (1947a) Chandrasekhar, S. 1947a, The Astrophysical Journal, 105, 164
  * Chandrasekhar (1947b) —. 1947b, Astrophysical Journal, 105, 441
  * Chandrasekhar (1948) —. 1948, Astrophysical Journal, 107, 188
  * Chandrasekhar (1960) —. 1960, Radiative transfer (New York: Dover Publications)
  * Chandrasekhar & Breen (1947) Chandrasekhar, S., & Breen, F. H. 1947, The Astrophysical Journal, 106, 143
  * Chandrasekhar & Elbert (1952) Chandrasekhar, S., & Elbert, D. 1952, The Astrophysical Journal, 115, 269
  * Chandrasekhar et al. (1952) Chandrasekhar, S., Elbert, D., & Franklin, A. 1952, The Astrophysical Journal, 115, 244
  * Coakley Jr (1973) Coakley Jr, J. A. 1973, Journal of Quantitative Spectroscopy and Radiative Transfer, 13, 1237
  * Domanus & Cogley (1974) Domanus, H. M., & Cogley, A. C. 1974, Journal of Quantitative Spectroscopy and Radiative Transfer, 14, 705
  * Henyey & Greenstein (1941) Henyey, L. G., & Greenstein, J. L. 1941, The Astrophysical Journal, 93, 70
  * Madhusudhan & Burrows (2012) Madhusudhan, N., & Burrows, A. 2012, The Astrophysical Journal, 747, 25
  * Malkevich (1963) Malkevich, M. 1963, Planetary and Space Science, 11, 681
  * Peraiah (2002) Peraiah, A. 2002, An introduction to radiative transfer : methods and applications in astrophysics (Cambridge New York: Cambridge University Press)
  * Seager (2010) Seager, S. 2010, Exoplanet atmospheres : physical processes (Princeton, N.J: Princeton University Press)
  * Sengupta (2021) Sengupta, S. 2021, The Astrophysical Journal, 911, 126
  * Sengupta et al. (2020) Sengupta, S., Chakrabarty, A., & Tinetti, G. 2020, The Astrophysical Journal, 889, 181
  * Van de Hulst (1948) Van de Hulst, H. 1948, The Astrophysical Journal, 107, 220