DETAILING APPROACHES TO CLIMATE AND ATMOSPHERE MODELING IN HARD SCI-FI
Over the course of this blog's history, I've made several attempts to tackle climate and atmospheric modeling, often involving some Greenhouse Gas considerations and other comparatives with Earth.
Basic Climate Model (comparing CO² levels)
Energy Budget (comparing energy availability)
Venus Zone (comparing atmosphere escape)
Atmospheric Modeling (trying to dimension an atmosphere)
So today, and over the course of next weeks, I'll try rectify and sum up everything tackled in those posts in a more formal, scientific and less speculative way. Which means this post may get long.
Alright, let's SCIENCE properly this time.
ENERGY BUDGET PROBLEM
Imagine
an aquarium, the fish inside can only grow as large as the amount of
food you give them, or a garden, where plants will only grow as large as
the minerals available in the soil allow.
No sunlight over Earth? No temperature gradient, no wind, no air, no circulation, and a lot of other processes which life needs to exist - ceases to exist, for the exception of the Earth's internal temperature, of course.
That means that in high energy places life would be more active, and in low energy places, less active. For an example of that, is the antarctic ice sheets or anaerobic environments, where only few species of microbes live of very low energy chemical reactions, compared to the African savanna or amazon rainforest, where the abundance of sunlight and carbon compounds allow for a very high biodiversity to be sustained.
This, of course would imply that the relatively closer to the star or the more output light a star produces, the better the planet, but, there are other variables to that, which we will explore later.
Let's just consider what Earth receives of sunlight a being 1E (Earth energy unit).
As sunlight spreads according the Inverse Square law, a planet twice the distance Earth-Sun would not receive 0,5E, but about 0,25E:
1solarLuminosity / 2AU^2 = 1/4 = 0,25
1solarLuminosity / 2AU^2 = 1/4 = 0,25
Planets at varying distances to their suns and varying suns would render very different available energies to their systems.
This 1E is equal to ~1.368 W/m² at the top of atmosphere, considering a rotating spherical Earth's surface - it is about 1/4th that, or about ~342 W/m².
Several parts of the Earth reflects and absorbs this energy in different ways, this is called Albedo, or how much light/energy does an object reflect back.
Earth's average albedo is about 0,30~0,31 - which means that Earth reflects 30~31% of sunlight back to space, and the closer this value approaches 1, or 100%, the more light it reflects back, and thus the colder it will be.
Even though the amount of energy available is similar to Earth's, the real availability of this energy mostly dependent on the existence of an atmosphere able to trap heat properly.
Considering Earth had no atmosphere, we could use Earth's distance to the Sun, and the Sun's output, to figure out Earth's temperature.
Simplifying this math results in a Temperature formula like this:
T = 279*(1-a)^(1/4) * 1/√d
Where a and d are Albedo and Distance to the Sun, T then is given in Kelvin, this render us about 254~255K, which is about between -19ºC and -18ºC.
Simplifying further, and assuming the planet has this 0,3 bond albedo, we get
T = L^(1/4)
D^(1/2)
Where L is the luminosity of your star, and D is the distance between your planet and star, and the final T is a multiple of 254,5K.
The atmosphere creates a greenhouse effect, being made of gases that are transparent to visible light, but pretty much opaque to infrared.
To understand that, we will recur to a simplified model of heat interactions, so, consider that:
- Hot objects lose heat faster than cold objects. And that happens to the 4th power of the temperature. Double the temperature, the rate at which heat is lost is 16x greater than before.
- Planets are found in their equilibrium temperature. They reached a point where the amount of heat energy lost is roughly equal to the energy they receive from their parent star.
Considering no atmosphere, only oceans, grasslands and forests, and deserts albedo (~0,3), and a rotating sphere, we get ~240W/m².
The relationship (experimentally) between heat loss and temperature can be described by the equation:
T= (F/σ)^1/4
Where F is the rate of heat
loss (heat flux), and σ is a fundamental physical
constant (Stephan-Boltzmann constant) with a value
of 5.67 x 10-8 Watts/meter2 Kelvin4s.
Using these values, we also get T=255K, or -18ºC
Now, let's consider a layer of atmosphere that's completely opaque to infrared light, in which case, when it absorbs sunlight, it re-emits it above and bellow, ie, back to the planet and back to space.
In this ideal case, when light enters the system, it warms the planet a little bit, it then also warms the atmosphere, the atmosphere, being this ideal opaque to IR substance re-emits it's heat to space above, and to the planet bellow. The total heat reaching the planet then is twice as before, half coming from sunlight, and half coming from the atmosphere.
T= (480/σ)^1/4 = 303K or 29,85ºC
Of course, this ideal model is an overestimation of the greenhouse effect on Earth, because the atmosphere elements themselves, like clouds contribute to the atmosphere being slightly reflective, that's why we see this blue haze around Earth for instance.
Bellow, a table of different surfaces's Albedo [source]
| SURFACE | ALBEDO % |
| Ocean | 2~10 |
| Forest | 6~18 |
| City | 14~18 |
| Grassland | 7~25 |
| Soil | 10~20 |
| Desert / Sand | 16~20 |
| Ice | 35~45 |
| Cloud cover (Thin, Thick) | 20~70 |
| Snow (new) | 30, 60~70 |
| Snow (old) | 75~95 |
Insolation can also be as low as 37,5% the Equatorial insolation, the further poleward you go.
For an extremely Earth-like atmosphere, the heat energy the atmosphere reflects back to Earth is roughly between +62,3% and +62,5% (assuming perfect black bodies).
The more Earthly your planet appear to be, the more closely it will distribute its energy like Earth does:
1/3 reflected back to space.
1/6 absorbed by the high atmosphere.
1/3 used by the water cycle.
1/6 directly absorbed and radiated to space by ground.
The balance between albedo and atmospheric composition will keep your planet Earth-like in a general way, still, it would be needed to pay further inspection to other aspects such as the...
GREENHOUSE PROBLEM
On Earth, the greenhouse effect is mostly caused by 80~60% H2O, 26% CO2, 5% CH4, 4% O3, 4% CFC/HFC, <1% NO2 and other trace pollutants.
Besides a number of other positive feedbacks, the water cycle is the worst.
Increase the temperature, more water in the atmosphere, the water will increase atmospheric pressure and trap more heat in the atmosphere, leading to more water evaporation, if this happened to Earth, and all of Earth's water evaporated into the atmosphere, the pressure would be over ~358atm (given average ocean depth), because all the Earth's oceans are above you in the atmosphere - and Earth would very probably turn into a planet like Venus this way.
However we should acknowledge that climate change is still debated, and we probably haven't seen it's effects to a full extent, so our current estimations are in risk of being rather underestimates of the real thing. Assuming our early/current estimates are somewhat correct, this means the world warms about 2~3ºC per doubling of CO2, however in times like the Carboniferous period, where CO2 levels were 2x what they are today, global averages were about 20ºC, in fact 1,3x hotter when it should theoretically be ~18ºC, of course, at the time Earth would have a lower albedo due large rainforests and large oceans, but it stops making that much direct sense when you consider that since the Cretaceous period, CO2 levels been decreasing drastically, and even so, temperatures are equivalent to that of the late Devonian when the CO2 were 5~6x current levels.
It would take about 15~16 doublings to make an entire Earth atmosphere out of just CO², with the 3ºC per 2xCO2 rule that's a 42~45ºC increase to about 65ºC avg temp - of course if such a disaster happened we would have water cycle feedback loop set long before it reaches 40ºC.
Talking about water cycle, there is this paper published in Nature that considered an oceanic planet like Earth in different scenarios. And despite the water vapor absorption bands overlap CO2 emission, thus negating some of the climatic change, they still managed to set a moist-greenhouse effect (1,10x Earth's current level) to heat up the planet to ~67ºC.
They've found that exists this threshold between 300K and 330K (26ºC and 56ºC) where the climate enters a rather unstable regime, ie, minor increases in the radiative intake can lead to drastic decreases and increases in temperature, and above 330K exists this warm regime, where for example at 340K it would be needed a full drop in the radiative index back to current Earth levels for temperatures to stabilize back at ~292K, but over a period of 120 years, 80 of which, things could have gone wrong again while in the unstable zone.
Things can be relatively safe up the mark of 1,05x current radiative intake, whereas it would take over 90~100yrs for it to reach an equilibrium at ~62ºC.
However, a minor 1,03x radiative intake is still safe inside the cold regime zone, bellow 300K, despite the temperature going up to 298K (~25 ºC).
This aspect of the thing leads us to the...
Catching back what we talked previously on that matter, I kicked that extra 10% radiative intake could indeed push Earth into a runaway greenhouse effect - the Nature's paper just confirmed that point of view, however, it is fair to admit it didn't really turned Earth into a Venusian planet, in any case, a paper published by James F Kasting and his team in 1993, concluded that a conservative estimate for a continuously habitable Habitable Zone over the course of 4,6Gyr would be squeezed between 0,93AU and 1,37AU, surely between 0,95AU and 1,15 AU, but could as well be larger with different atmospheric conditions than those of Earth.
VENUS ZONE PROBLEM
The Venus Zone
is related to a theoretical area around a star where Earth-like planets
would eventually turn into Venus-relative planets through various
processes, but mainly by triggering a runaway greenhouse effect.
This could also be related to the finding of a Mercury Zone,
which is the area around a star where planet's are unable to retain an
atmosphere due solar wind blowing it away from the planet.
For them, the Venus Zone starts at 0,84AU, with a radiative intake of 1,41x.
Water loss threshold is on that 1,10x RI mark at 0,95AU.
Greenhouse effect can be very helpful up to the 1,67AU mark, where the RI is 0,36x that of Earth's.
However, if greenhouse effect is not addressed, the 1st condensation of atmospheric CO2 happens at the 1,37AU mark, with RI 0,53x.
Greater changes in albedo, atmospheric water and CO2 are not needed at all if the planet orbits a low-mass star, like a K or M class star, as their output light is redhshifted - H2O and CO2 proved to be better suited to absorb IR light, so the more red-shifted the light from the star, the more of it (proportionally) they will absorb.
They also calculated different planet sizes.
The minimum distance for a Mars-like planet is 0,88AU for runaway greenhouse, and 0,98 for water loss.
The minimum distance for a planet with 2,55 Earth's gravity, is 0,81AU for greenhouse, and 0,91 for water loss.
Funfact: They also believe that 5% of double S-type systems (planet orbits 1 of 2 binary stars) could be habitable, whereas 50% of P-type systems (planet orbits a binary star system) could be habitable, in which case, has great implications for SETI.ATMOSPHERIC MODELING
Atmospheric modeling is something I must admit that I wasn't so confident about when doing my stuff, if I could say I really knew anything about it, it would mainly be something similar to what we would normally see in a 5~6th grade natural sciences book - and later on specific one about the atmosphere structure, in the latter, only the very basic.
It isn't something we are usually taught about for real, and it isn't really something I got to study in the college's library because of the Coronavirus Pandemic (2020) so far.
All I have looked for so far, is a peek in climate simulation / greenhouse correction, scale height, and habitability.
However after further research, I had worked on a general spreadsheet for the past 3 days non-stop, so you can derive basic data about your star, planet, planet's climate and atmosphere, all in one place - it can only do 1 star and 1 planet at once, but it was built over these research topics I have been discussing for long here on this blog, and updated.
Don't be shy to report bugs, corrections, or suggest more items to it in future versions.
- M.O. Valent, 24/05/2020
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