I agree with Tim B's assessment that this is a complicated question. However, I disagree on just why this is the case. The characteristics of the star are well defined, given that it's a solar analog. From this, we immediately have mass, luminosity, and other characteristics. However, the reason I think that this is not a simple question is that there are a lot of conflicting models of the effects of different greenhouse gases and of the circumstellar habitable zone. I've tried to choose the best ones, but they're only approximations; the exact fate of Epimetheus will require some deeper thinking.

The Greenhouse Effect

^{Disclaimer: This section is mostly me muddling through a topic I'm not too familiar with.}

Radiative forcing

I'll start by assuming that the main source of atmospheric warming comes from radiative forcing because of the various greenhouse gases in the atmosphere. Radiative forcing is the difference between the amount of sunlight absorbed by a planet and the amount of sunlight emitted back into space. The difference between absorption and emission is primarily due to the atmosphere.

Radiative forcing causes a change in surface temperature, which is proportional to the radiative forcing: $$\Delta T_s=\lambda\Delta F\tag{1}$$ where $T_s$ is surface temperature, $\Delta F$ is radiative forcing, and $\lambda$ is the climate sensitivity of the planet.¹ For Earth, $\lambda\approx0.8$, and without more detailed data on Epimetheus, I'll have to assume that it's about the same there.

Each greenhouse gas contributes differently to radiative forcing. The IPCC Second Assessment Report^{2, 3} gives a chart (Table 6.2) of approximations of expressions for radiative forcing contributions from several of the major greenhouse gases (CO₂, CH₄, N₂O, CFC-11, and CFC-12): $$\begin{array}{|c|c|c|} \hline \text{Trace Gas} & \text{Simplified Expression} & \text{Constants}\\& \text{Radiative Forcing, }\Delta F \left(\text{Wm}^{-2}\right)\\ \hline \text{CO}_2 & \Delta F=\alpha\ln\left(\frac{C}{C_0}\right) & \alpha=5.35\\\text{} & \Delta F=\alpha\ln\left(\frac{C}{C_0}\right)+\beta\left(\sqrt{C}-\sqrt{C_0}\right) & \alpha=4,841,\beta=0.006\\\text{} & \Delta F=\alpha(g(C)-g(C_0) & \alpha=3.35\\ \hline \text{CH}_4 & \Delta F=\alpha(\sqrt{M}-\sqrt{M_0})-(f(M,N_0)-f(M_0,N_0)) & \alpha=0.036\\ \hline \text{N}_2\text{O} & \Delta F=\alpha(\sqrt{N}-\sqrt{N_0})-(f(M_0,N)-f(M_0,N_0)) &\alpha=0.12\\ \hline \text{CFC-11} & \alpha(X-X_0) & \alpha=0.25\\ \hline \text{CFC-12} & \alpha(X-X_0) & \alpha=0.32\\ \hline \end{array}$$ The two functions are given by $$g(C)=\ln(1+1.2C+0.005C^2+1.4\times10^{-6}C^3),$$ $$$f(M,N)=0.47\ln(1+2.01\times10^{-5}(MN)^{0.75}+5.31\times10^{-15}M(MN)^{1.52})$$ $C$, $M$, $N$, and $X$ denote the concentrations of CO₂, CH₄, N₂O, and the CFCs, respectively; the subscript $_0$ denotes a reference concentration. All gases are given in parts per billion, except CO₂, which is given in parts per million.

The total radiative forcing is the sum of the contributions of each greenhouse gas. If the atmosphere is mostly made of carbon dioxide, then, using the first-order approximation: $$\Delta T_s=\lambda\times5.35\times\ln\left(\frac{C}{C_0}\right)\tag{2}$$ It's easy to see from this that doubling the concentration, for example, leads to a temperature increase of about 3°C.

Different models and feedback mechanisms

Before I go further, I should address my choice of equations. Some would argue that it would be a lot simpler and much more intuitive to use the idealized greenhouse model⁴ (another application of radiative forcing), which can be derived from the Stefan-Boltzmann law and a few other assumptions. In fact, after exploring some of the above calculations, I began to wonder the same thing myself, because the idealized greenhouse model can be adjusted to produce different warming effects by simply changing one parameter: the emissivity, $\varepsilon$.

I eventually did decide to work with the original table from the IPCC because it allows us to play around with the specific gases a little bit. I can attack the problem quite specifically directly from concentrations, rather than having to figure out the deceptively simple $\varepsilon$. Another place where the IPCC model has have an edge is through feedback mechanisms, which is what I'll explore next.

The runaway greenhouse effect is perhaps best known for its drastic effects on Venus, turning what was once a world similar to Earth into a hellhole beyond belief. What happened was that carbon dioxide and water vapor began warming the planet (yes, H₂O is also a greenhouse gas, and an important one). The water vapor upset the balance just enough so that the oceans that once covered part of the planet began to evaporate, creating more water vapor, which intensified the problem more and more.

There are other types of feedback mechanisms at work on Earth, involving carbon dioxide, greenhouse gases being released from permafrost, changes in albedo from melting ice, and much more. However, the dominant one is still feedback from water vapor, which is actually more important than CO₂. Water vapor alone can double the effects of warming, according to the IPCC and other models, and can then help start other feedback pathways that will do further damage.

The reason that we can't just double our value of $\Delta T_s$ to compensate for this is that $\lambda$ already takes water vapor and the other feedback mechanisms into account. The reason that I said that the IPCC model has an advantage here is that all of the feedback mechanisms can be encoded into the parameter, which describes the planet as a whole. $\varepsilon$ really only describes part of the atmosphere.

Putting it all together

The above argument shows that our first-order approximation can give us a reasonable approximation of changes in temperature, given an appropriate $\lambda$. It's true that this value will have to be adjusted; models of Earth use different $\lambda$s. However, if we assume that Epimetheus is roughly Earth-like, with the exception of the heavy concentration of CO₂, then we don't have to change much.

The Habitable Zone

I actually think that defining where the circumstellar habitable zone is without taking the characteristics of Epimetheus into account is more challenging than modeling its atmosphere. The reason for this is that models of the inner and outer radii of the habitable zone of the Solar System differ, as a quick glance at Wikipedia shows. Inner estimates range from 0.75 AU to 0.99 AU, while outer estimates range from 1.01 AU to 3.0 AU. So which one do we use?

I'm going to instead reference Kasting et al. (1993), because, while it may be a little out of date, it bases its conclusions primarily on the effects of climate, which obviously play a major role in the situation at hand.

The authors rely on something called the carbonate-silicate cycle. The important total reaction is in their Equation 3: $$\text{CaSiO}_3+\text{CO}_2\to\text{CaCO}_3+\text{SiO}_2\tag{3}$$ In the biotic version⁵, organisms use CO₂ to create shells, which then are buried in the seafloor; Equation 3 gives the net reaction. Eventually, carbonate metamorphism reverses the reaction, putting CO₂ back into the atmosphere. This acts to keep the climate of a planet in the habitable zone in something of an equilibrium, because, as I've discussed before, carbon dioxide plays a major role in regulating surface temperature.

The reason that this reaction leads to equilibrium is that both the biotic and abiotic (generally involving weathering, primarily from liquid water) require liquid water. Without liquid water, carbon dioxide levels rise, melting ice and restoring liquid water levels, and with them, equilibrium.

This cycle determines the outer edge of the habitable zone because at low enough temperatures, CO₂ can condense, forming clouds. The clouds then alter the planet's albedo by reflecting substantial amounts of solar radiation, drastically cooling the planet. Additionally, the latent heat amplifies the greenhouse effect even more. The authors use this to estimate a conservative outer limit of approximately 1.77 AU, assuming non-negligible amounts of greenhouse gases.

The authors then created a more complicated and detailed climate model using different planetary masses. They found that the greenhouse effect could let the habitable zone extend to about 1.67 AU for a planet of one Earth mass, and to about 1.64 AU for a planet ten times that mass. These outer boundaries take versions of radiative forcing into account, but are limited by the formation of CO₂ clouds. The authors' calculations indicate that Epimetheus should reside somewhere in that vicinity, at maximum.

Some things to note: - The authors' models ignore the formation and effects of clouds, as they say. - The drastically high levels of CO₂ on Epimetheus will result in different partial pressures, meaning that the outer edge will be affected. - CO₂ levels may be affected by the amount stored in the crust.

That said, newer models have replicated the results of Kasting et al. Kopparapu (2013) came up with a nearly identical result for the outer edge, also assuming greenhouse effects, and with models partially based on the older ones.

Conclusion

We can make some basic approximations of temperature changes if we choose the correct values for several constants and use simple models of radiative forcing, primarily of CO₂, but with feedback from water vapor taken into account. We can actually do this with any greenhouse gas, but these two are the primary culprits.

Carbon dioxide actually turns out to also be a good indicator of the outer edge of the habitable zone, because lower temperatures cause CO₂ condensation, even accounting for the greenhouse effect. We can place an outer limit on Epimetheus' orbit somewhere around 1.67 AU, assuming an atmosphere similar to the ones investigated by Kasting et al.

^{1 Note that this is not the same as the equilibrium climate sensitivity (ECS), which is currently thought to lie around 3°C on Earth.
2 The linked section is on radiative forcing; all IPCC reports (including the second, from 2001) can be found here. The equations are taken from or derived from Hansen et al. (1988), IPCC (1990), Shi (1992), and IPCC (1999).
3 The first equation for CO₂ is the first-order approximation of a Taylor series; see Lam (2007), referenced here.
4 You can read more about it here.
5 There is, of course, an abiotic version, which plays a bigger role in planets without life or without life that can carry out these processes.}

posted over 8 years ago

HDE 226868‭

601 reputation 61 463 80 49