Now, the heat equation is not only important for wine cellars (see previous article). We’ll look at a much larger-scale and very faraway situation where it plays a key role.
Mercury, the little planet first from the Sun, is an extreme place. Mercury has almost no axial tilt: it orbits upright. So it has no seasonal variations to worry about, only day and night.
But what a day and night! Mercury rotates so slowly that a solar day on Mercury lasts 176 of our Earth days, which is twice a Mercury year of 88 days!
This is no coincidence. Mercury is locked in a spin-orbit resonance with the Sun: it rotates three times on its axis for every two orbits. Therefore, the same longitudes always face the Sun every other orbit: 0°W and 180°W on the equator directly face the sun at perihelion (when Mercury is closest to the Sun), and 90°W and 270°W at aphelion (when it is farthest). In the diagram below, the red pole is at 0°N, 0°W. That is why a Mercury day is exactly twice a Mercury year.
Mercury’s radius is 38% of Earth’s (below picture to scale), making it the smallest of all the planets. Its interior has long since cooled down; it is geologically as dead as our Moon. Its gravity is likewise 38% of Earth’s (and the same as Mars’). Its orbital semi-major axis is about 39% of Earth’s; depending on where it is in its elliptical orbit, the Sun looms between 4.6 and 10.6 times as bright as it is on Earth. Thus, Mercury is too small and too close to the burning Sun to retain an atmosphere that could hold in any heat and give the surface any protection.
Therefore, during the day its surface temperatures rise very high as it bears the Sun’s onslaught for weeks on end; but in the night it drops very low as it freezes into the blackness of space.
However, this is only true of the surface.
As we showed earlier with wine cellars, the temperature of the underground depends mostly on the average surface temperature: the temperature oscillations on the surface are exponentially attenuated as we proceed downwards.
When we average “extremely high” with “extremely low”, the extremes cancel! So, surprisingly for a place so close to the Sun, Mercury gives an opportunity for naturally occurring comfortable Earth-like temperatures off Earth!
In order to check this, we need to find the maximum and minimum temperatures, and then average them. For that we must consider the heat flux through the surface. The analysis that below closely follows Dr. James Shifflett’s 2011 analysis of Mercurian subsurface temperatures; but rephrased in the language used above.
There are three sources of heat flow we have to consider. First, we have the conducted heat flux that enters the surface: by Fourier’s law, that is
At noon (t = 0), this is
and at midnight (t = π/ω), this is
Then we have the radiated heat flux: this is the heat that radiates from the planet’s surface into space.
Here we refer to physics again. By the Stefan-Boltzmann law, it is σT4, where σ = 5.67 x 10-8 W m-2 K-4 is Boltzmann’s constant.
Finally we have the absorbed heat flux from the Sun, which is
This is essentially like the radiated heat flux (as heat is transferred from the Sun via radiation), but with some correction factors.
A = 0.088 is Mercury’s albedo, the fraction of power falling on it that is reflected back into space. So we multiply by (1 – A) to get only the absorbed heat flux.
θ is the latitude. We count only the normal component of the Sun’s rays, hence the cosine.
Tsun is 5778 K (5505°C), the temperature on the Sun’s surface.
Finally, we use the inverse square law to correct for the distance d of Mercury from the Sun compared to rsun = 6.96 x 105 km, the radius of the Sun. The Sun–Mercury distance is 4.60 x 107 km at perihelion and 6.98 x 107 km at aphelion.
As Mercury is geologically dead, it has no internal heat source to factor in; and as it has no atmosphere, there is no greenhouse effect either. (These would have to be considered for the other planets, making the situation more complex.)
To find the maximum surface temperature, we analyse the system at noon, and note that it is in thermal equilibrium. Therefore: the heat gained from the Sun, plus that conducted into the surface, must equal the heat lost by radiation into space. Or more simply: absorbed minus radiated heat flux equals conducted heat flux at noon!
As this is a complicated equation, we use an important technique in applied mathematics: we approximate to obtain a simpler problem.
Since the Sun is so hot, the amount of heat gained from the Sun completely dominates the amount conducted into the surface. Thus, we equate absorbed to radiated heat flux:
At Mercury’s perihelion, it is:
and at Mercury’s aphelion, it is:
To find the minimum surface temperature, we do the same thing at midnight. Now the Sun is not contributing, because it’s night! So we get that the heat flux conducted into the surface exactly compensates that radiated into space. That is:
This uses the approximation (1 + x)a = 1 + ax for small x. This comes from the binomial theorem, which lets us expand the left hand side as an infinite sum. Then we neglect quadratic and higher-order terms, because if is small, then surely x2 is negligible.
To finish, we note that Tmin must be small compared to 4Tmax, so the factor 1 – Tmin/4Tmax is roughly 1. This would give us a rough approximation of
But we can do even better by substituting this into the factor 1 – Tmin/4Tmax, getting:
using the binomial approximation again for the penultimate step.
Let’s substitute in the values. On Mercury, ρc = 106 J m-3 K-1, κ = 10-9 m2 s-1, and ω = 2π/176 days = 4.13 x 10-7 s-1.
At perihelion, we have
which is 94 K (−179°C) on the equator (θ = 0°). At aphelion, we have
which is 88 K (−185°C) on the equator.
Finally, we need to solve for θ such that the average is room temperature: 298 K (25°C). At this point, the equation is simple enough to solve numerically. We could plot a graph, or just try out lots of guesses, repeatedly bisecting to hone in on the solution.
At perihelion:
This is the relevant latitude for the meridians 0°W and 180°W.
And at aphelion:
This is the relevant latitude for the meridians 90°W and 270°W.
By symmetry and continuity, there is thus an elliptical ring around each pole where the underground temperature is 25°C! At a depth of only 19π (2κ/ω)1/2 = 4.15 m, the huge surface temperature fluctuations are damped to just +/- 3.3°C, and the damping improves even further below that!
But the surprises don’t end there!
At latitudes beyond the habitable ring, the noontime Sun appears at an ever lower angle above the horizon. The noontime temperature drops, and so does the average temperature. Eventually, near the poles, the noontime Sun is so low that the walls of a crater suffice to block it from every direction, and the temperatures become constantly low.
In fact, they go even lower than the midnight temperatures calculated above, because the Sun never contributes there! Temperatures below 50 K (−223°C) can persist in such craters!
The interiors of such craters are cold traps: within them lies a polar night, lasting not for months as on Earth, but for billions of years.
Water molecules can be created by the action of the solar wind on Mercury’s surface, or delivered by comets impacting the planet. They are then sent flying by Mercury’s magnetic field. But if they reach the poles, they are trapped to freeze in eternal darkness. And this has been going on for aeons.
A blanket of over a hundred trillion kilograms of water ice now lurks in these recesses – and the same process has likely trapped a vast cache of ammonia and hydrocarbons.
So, the heat equation has shown us that there is more to Mercury than a barren, boiling rock! The proximity of the Sun grants it immense energy stores indeed, but it also has vast underground regions at constant comfortable temperatures, and even permanently cold polar craters that house frozen lakes.
Allowing ourselves some momentary futuristic fantasy, one could thus imagine a Mercurian settlement. It could start from the polar craters, using the frozen lakes as a reservoir of the light elements needed for breathing, drinking, agriculture, and rocketry. Then it could go through winding mining tunnels to the habitable regions, stretching down for kilometres to create a “city-cellar” rather than a wine cellar! Such an underground city could then prosper with its massive energy reserves, fuelling a Solar-System-wide industry of manufacturing and infrastructure mega-projects.
Observe how far mathematical modelling has taken us…
We started with a problem about wine cellars on Earth. Yet because our approach was abstract, we could use it to ride to the heavens.
“To most people, Mercury was a fairly good approximation of Hell … And yet this world had turned out to be, in many ways, the key to the solar system. This seemed obvious in retrospect, but the Space Age had been almost a century old before the fact was realized. Now the Hermians never let anyone forget it.”
Arthur C. Clarke, Rendezvous with Rama (1973)
References and further reading
Jones, B. M., Sarantos, M., Orlando, T. M. (2020). A New In Situ Quasi-continuous Solar-wind Source of Molecular Water on Mercury. The Astrophysical Journal Letters, 891(2). https://iopscience.iop.org/article/10.3847/2041-8213/ab6bda
Lawrence, D. J. (2016). A tale of two poles: Toward understanding the presence, distribution, and origin of volatiles at the polar regions of the Moon and Mercury. JGR: Planets, 122(1), 21–52. https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/2016JE005167
Lewis, J. S. (2004). Physics and Chemistry of the Solar System, Second Edition. Elsevier. pp. 257–258.
“Matter Beam”. (2016, October 13): How to live on Other Planets: Mercury. ToughSF. http://toughsf.blogspot.com/2016/10/how-to-live-on-other-planets-mercury.html
Nittler, L. R., Chabot, N. L., Grove, T. L., & Peplowski, P. N. (2018). The Chemical Composition of Mercury. In Solomon, S. C., Nittler, L. R., & Anderson, B. J. (Eds.), Mercury: The View after MESSENGER (pp. 30–51). Cambridge University Press.
Plait, P. (2012, November 29): New Data Show Mercury Almost Certainly Has Buried Ice at Its North Pole. https://slate.com/technology/2012/11/mercury-has-ice-buried-beneath-its-surface-in-craters-at-its-north-pole.html
Shifflett, J. (2011, November 12): A Mercury Colony? http://einstein-schrodinger.com/mercury_colony.html
Vasavada, A. R., Paige, D. A., & Wood, S. E. (1999). Near-Surface Temperatures on Mercury and the Moon and the Stability of Polar Ice Deposits. Icarus, 141, 179–193. http://luna1.diviner.ucla.edu/~dap/pubs/019.pdf
Williams, M. (2016, August 3). How Do We Colonize Mercury? Universe Today. https://www.universetoday.com/130109/how-do-we-colonize-mercury/
Image credits
Size comparison of Mercury and Earth: in public domain. https://en.wikipedia.org/wiki/File:Mercury,_Earth_size_comparison.jpg
Diagram of Mercury’s orbit: Tos, CC-BY-SA 3.0 Unported. https://en.wikipedia.org/wiki/File:Mercury%27s_orbital_resonance.svg
Poles of Mercury: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie Institution of Washington (MESSENGER). https://messenger.jhuapl.edu/Explore/Science-Images-Database/gallery-image-1187.html
TOM ROCKS MATHS 