The Habitable Zone and the Generalized Greenhouse Effect

Abstract

It is generally tacitly assumed that the atmosphere of a habitable planet absorbs in the infrared and is transparent in the visible. We consider the cases in which the optical depth in the visible is not negligible. We show that such planets can harbor life under conditions similar to those found on Earth. Moreover, it is conceivable that planetary evolution starts with phases in which the optical depth in the visible is not negligible and evolves out of it or passes through such a phase during the evolution of the planet.

To this goal, we present a generalized theory of the greenhouse effect (GHE) as a function of the optical depths in the visible and the infrared and solve it under the radiative boundary conditions relevant to Earth. We then investigate the parameter space to search for domains with adequate average surface temperature and show how the optical depths in the visible and infrared combine to create a range of life-supporting temperatures and hence extend the habitable zone significantly.

5. Appendices

1.1 5.1. Appendix A. The Radiative Model Equations

The radiative transfer equation including isotropic scattering is given by

$$ \mu \frac{{\mathrm{d}I\left( {z,\lambda, \propto} \right)}}{{\mathrm{d}z}}=-\kappa \left( \lambda \right)\left( {I\left( {z,\lambda, \mu } \right)-B\left( {T(z),\lambda } \right)} \right)+\sigma \left( \lambda \right)\left( {J\left( {z,\lambda } \right)-I\left( {z,\lambda, \mu } \right)} \right) $$

(A.1)

with

$$ J\left(z,\lambda\right)={\displaystyle {\oint }_{4\pi}I}\left(z,\lambda,\omega\right)d\Omega $$

(A.2)

where $\omega$is the solid angle, $\kappa (\lambda )$ is the pure absorption coefficient, and $\sigma (\lambda )$is the isotropic scattering. $B\left(T(z),\lambda \right)$ is the Planck function, and T(z) $T(z)$is the temperature at height $z$above the surface. The temperature is obtained from the energy conservation equation, namely,

$$ \frac{1}{{C}_{\text{V}}}\frac{dQ(z)}{dt}={\displaystyle {\int} _0^\infty }\left[B\left(T(z),\lambda\right)-J\left(z,\lambda\right)\right]k\left(z,\lambda\right)d\lambda=0.$$

(A.3)

This condition has to be satisfied at all heights z of the atmosphere. The radiative transfer equation is solved in the two-stream approximation. For details and mode of solution, see Shaviv and Wehrse (1991). In this approximation, we have a specific intensity ${I}^{+}$in the outward direction and a specific intensity ${I}^{-}$ in the inward direction. The intensity vector is then written as

$$ I=\left(\begin{array}{c}{I}^{+}\\ {I}^{-}\end{array}\right),$$

(A.4)

and the radiative transfer equation becomes

$$ \pm \frac{{\mathrm{d}{I^{\pm }}}}{{\mathrm{d}z}}=-\left( {\kappa \left( {z,\lambda } \right)+\sigma \left( {z,\lambda } \right)} \right){I^{\pm }}+\sigma \left( {z,\lambda } \right)J\left( {z,\lambda } \right)+\sigma \left( {z,\lambda } \right)B\left( {T(z),\lambda } \right) $$

(A.5)

with the mean intensity defined as

$$ J\left(z,\lambda\right)=\frac{{I}_{+}\left(z,\lambda\right)+{I}_{-}\left(z,\lambda\right)}{2} $$

(A.6)

In the present model we do not discuss scattering, only absorption. We actually show that an aGH is obtained even without scattering.

1.1.1 5.1.1. A.1. Boundary Conditions on the Radiation Transfer Problem

At the top of the atmosphere, we have$z=Z$and${I}_{-}(Z)=\frac{1}{4}\frac{{R}_{*}^{2}}{{d}^{2}}B\left({T}_{*},\lambda\right),$where ${T}_{*}$ is the surface temperature of the sun/central star, and we assumed that the planet is a fast rotator. No condition is imposed on I ₊ (Z) at the top of the atmosphere, but a consistency check of the calculation is the fulfillment of the condition

$$ \frac{{R}_{*}^{2}}{{d}^{2}}{\displaystyle {\int }_{0}^{\infty }B}({T}_{*},\lambda)d\lambda=\frac{{R}_{*}^{2}}{{d}^{2}}\sigma{T}_{*}^{4}={\displaystyle {\int }_{0}^{\infty }{I}_{+}}(Z,\lambda)d\lambda,$$

(A.7)

namely, the planet is in steady state and does not store or lose energy. At the surface we have

$$ {\displaystyle {\int }_{0}^{\infty }\left(1-a(\lambda)\right){I}_{-}\left(0,\lambda\right)d\lambda=\sigma{T}_{\text{surf}}^{4},} $$

(A.8)

where ${T}_{\text{surf}}$is the surface temperature which is unknown and is iterated for.

1.2 5.2. Appendix B. The Transition from Line Absorption to the Semi-gray Approximation

1.2.1 5.2.1. B.1. Transition in the Visible Range

Both the Rosseland and the Planck mean are poor approximations in insolated planetary atmospheres and hence require modifications – the first because of the existence of spectral windows which do not exist in stars and the second because in many wavelength domains the total optical depth is much greater than unity. We observed above that the radiation field changes its nature between the vis and the fir ranges and consequently the behavior of the specific intensity changes, calling for different forms of averaging the molecular line absorption. Consider first the vis range. Since the temperature of the radiation is that of the sun and hence very high relative to the self emission of the atmosphere, the zeroth solution for the transmission of specific intensity$I(z,n)$is given by

$$ I(z,n)={I}_{*,n}^{TOA}{e}^{-k(n)z},$$

(B.1)

where ${I}_{*,n}^{TOA}$is the stellar specific intensity at the top of the atmosphere. To secure the energy flux transfer through the atmosphere, we write therefore that

$$ {\displaystyle {\int }_{{v}_{1}}^{{v}_{2}}{I}_{*,v}^{TOA}}{e}^{-k(v)z}dV={e}^{-\langle {k}_{vis}\rangle z}{\displaystyle {\int }_{{v}_{1}}^{{v}_{2}}{I}_{*,v}^{TOA}}dV. $$

(B.2)

Next we note that the radiation interacts with the atmosphere which is at temperature T _atm and the stellar radiation is to a good approximation that of a blackbody at a temperature T _* , so we have

$$ ln\langle {\tau}_{vis}\rangle =-\frac{{\displaystyle {\int }_{{v}_{1}}^{{v}_{2}}B}\left({T}_{*},v\right){e}^{-\tau\left(v,{T}_{atm}\right)}dv}{{\displaystyle {\int }_{{v}_{1}}^{{v}_{2}}B}\left({T}_{*},v\right)dv},$$

(B.3)

where v ₁ and v ₂ are both in the vis range, T _atm is the temperature of the atmosphere, and τ _vis is the total optical depth for this range. It is important to note that the temperature in the weighting function is not that of the plasma through which the radiation passes but that of the insolating star, the sun in the particular case of the Earth or the star in the general case. The optical depth, ${\displaystyle {\int }_{0}^{Z}k}\left(\lambda,T,P\right)d\lambda$, however, is calculated with the temperature of the atmosphere.

1.2.2 5.2.2. B.2. Transition in the Far Infrared Domain

Consider now the radiative transfer in the fir range. Let τ be measured from the top of the atmosphere downward. If we write I ₊ (τ) as the thermal flux toward larger optical depths (downward) and I   ₋  (τ) as the flux toward smaller optical depths, then the solutions for I _± (τ) under the above approximations are

$$ F(\tau)\equiv {I}_{-}(\tau)-{I}_{+}(\tau)=\text{const}.$$

(B.4)

$$ E(\tau)\equiv {I}_{-}(\tau)+{I}_{+}(\tau)=\left[{I}_{-}(\tau)-{I}_{+}(\tau)\right]+\text{const}.$$

or by comparing the conditions at the top$\left(\tau=0\right)$to the bottom$\left(\tau={\tau}_{\text{tot}}\right)$, we obtain

$$ {I}_{-}\left({\tau}_{\text{tot}}\right)-{I}_{+}\left({\tau}_{\text{tot}}\right)={I}_{-}(0)-{I}_{+}(0),$$

(B.5)

$$ {I}_{-}\left({\tau}_{\text{tot}}\right)+{I}_{+}\left({\tau}_{\text{tot}}\right)=\left[{I}_{-}(0)-{I}_{+}(0)\right]{\tau}_{\text{tot}}+\left[{I}_{-}(0)+{I}_{+}(0)\right].$$

The boundary conditions we have are

$$ {I}_{+}(0)={I}_{\stackrel{\circ }{a},fir}\text{ and }{I}_{-}({\tau}_{\text{tot}})={I}_{p,fir},$$

(B.6)

where ${I}_{\stackrel{\circ }{a},fir}$ is the insolation for $\lambda >{\lambda}_{rad}$ and ${I}_{p,fir}$is the planet’s emission at $\lambda > {\lambda}_{rad}$. It is generally assumed that ${I}_{*,fir}=0$. However, if ${\lambda}_{rad}$ decreases significantly, it may no longer be justified to assume the vanishing of ${I}_{*,fir}$.

Next we consider the thermal equilibrium of the surface, that is, total absorption equals the total emission:

$$ \left(1-a\left(\lambda\right)\right)\text{I}_{\stackrel{\circ }{a},vis}+{I}_{atm,\downarrow }=\left(1-a\left(\lambda\right)\right)\text{I}_{p,vis}+{I}_{p,fir}, $$

(B.7)

where ${I}_{*,vis}$ is the insolation for $\lambda < {\lambda}_{rad}$, ${I}_{atm,\downarrow }$ is the emission of the atmosphere toward the surface (at $\lambda >{\lambda}_{rad}$), ${I}_{p,vis}$ is the planet’s emission at $\lambda < {\lambda}_{rad}$, and $a$ is the albedo at the short wavelengths. Our main point is that ${F}_{p,vis}$must be included at relatively high surface temperatures.

Using the two sets of Eqs. B.5 and B.6, the thermal equilibrium becomes

$$ \left(1-a\left(\lambda\right)\right)\left({I}_{\stackrel{\circ }{a},vis}-{I}_{p,vis}\right)=\frac{2\left({I}_{p,fir}-{I}_{\stackrel{\circ }{a},fir}\right)}{2+{\tau}_{fir,tot}}. $$

(B.8)

From the above set of equations, we can derive an expression for the average net fir flux (per unit frequency) over a finite band $\Delta n$and define an effective opacity through the following:

$$ {\overline{\Delta I}}_{fir}=\frac{1}{\Delta v}{\displaystyle {\int }_{{v}_{1}}^{{v}_{2}}\frac{\left[{I}_{p,fir}\left({T}_{p}\right)\text-{I}_{*,fir}\right]}{1+3\tau(v)/4}}\text dv $$

$$ \approx \frac{\left[{\overline{I}}_{p,fir}\left({T}_{p}\right)-{\overline{I}}_{*,fir}\right]}{\Delta v}{\displaystyle {\int }_{{v}_{1}}^{{v}_{2}}\frac{dv}{1+3\tau(v)/4}}$$

$$ \equiv \frac{\left[{\overline{I}}_{p,fir}\left({T}_{p}\right)-{\overline{I}}_{*,fir}\right]}{\Delta v}\frac{1}{1+3{\tau}_{fir,tot}/4},$$

(B.9)

that is,

$$ {\tau}_{fir,tot}=\frac{4}{3}\left[\frac{{\displaystyle {\int }_{{v}_{1}}^{{v}_{2}}B}\left({T}_{atm},v\right)\text dv}{{\displaystyle {\int }_{{v}_{1}}^{{v}_{2}}\frac{B\left({T}_{atm},v\right)\text dv}{1+3{\tau}_{tot}(v)/4}}}-1\right]. $$

(B.10)

This expression for the gray absorption has several advantages besides conserving energy. If the optical depth is constant as a function of frequency, we find τ _fir  →  τ. However, when τ « 1, we also find that τ _{fir =} τ. Windows in the absorption coefficient, which would cause vanishing values of the absorption and explosion in the mean Rosseland, cause no problem. Similarly, the vanishing of the Planck absorption is taken care of properly.

1.3 5.3. Appendix C. Calculation of the Effective Optical Depths

The HITRAN 2008 (Rothman and Gordon, 2009) compilation provides a list of all molecular lines in the range of interest. The range of interest is determined by the effective temperature of the insolating star and by the temperature of the atmosphere. In our particular case, we considered the range 10³–10⁶ Å. We checked that extending the integration to 2  ×  10⁶ Å affected the results for the surface temperature only in the forth significant figure.