Greenhouse effect
Back to Math And Terraforming.
In order to terraform a planet, most important is to change its atmospheric temperature. This is done in different ways: for an Inner Planet you need Anti Greenhouse Technology, including Micro Helium Balloons, for an Outer Planet Greenhouse Gases are the best option, while for an Earth - like planet you will need little adjustments in the composition of the atmosphere.
Please note that the formulas listed here are made by myself, based on many sources and other formulas I've been adapting since 2002, based on various sources. Also, there is no way to take the entire behavior of a planet into a simple equation. Nobody was able to predict exactly the temperature increase and climate changes for Earth, so nobody will be able to predict how an alien planet will behave. There are many unknown parameters that must be analyzed. The formulas that you will encounter on this page are simple answers to questions like How much greenhouse gas is needed to terraform Titan? or How much could we decrease the temperature of Mercury?. The values you will get from these formulas have an error of 33%. They are not good for high-accuracy calculations.
About greenhouse effects
Each atmosphere creates a certain greenhouse effect, as showed below:
Basic atmosphere structure
Each terraformed atmosphere has its own greenhouse effect, because every gas has at least a very low greenhouse effect. It consists of the following:
- Oxygen, with a very low greenhouse effect.
- Nitrogen, a very low greenhouse effect, similar to oxygen.
- Helium, has 4.5 times lower greenhouse effect then nitrogen (if the atmosphere has high amounts of helium as an inert gas)
- Argon, if in high amount, has a 0.75 lower greenhouse effect then nitrogen.
- Carbon dioxide should be in 0.03% of the mixture of gasses, for a standard atmosphere. If the atmosphere is denser, we can afford a lower concentration of CO2, with the condition that the same mass to exist. So, if we have a pressure of 2 atm, we can have 0.015% CO2 for the same result.
On Earth, an increase of CO2 concentration of 0.01% results in a temperature increase of 1 degree C or K.
Natural greenhouse effect
Without added greenhouse gasses, a planet would have a certain temperature. This is influenced by the Solar Constant, which you can calculate for each planet's orbit. Go to Temperature and find the formula for void temperature (the temperature of a surface painted 25% gray, exposed directly to the solar constant). For Earth, it is around +98C (or 371 K). For Earth, without the greenhouse effect of our atmosphere, temperature should be around -11 C (or 262 K).
Every planet has its own albedo (a number below 1, that represents the amount of light it reflects). The albedo is higher for ice caps and deserts, but lower for oceans and forests. However, a terraformed planet should have a similar albedo with Earth. If not, you need to do little adjustments.
Average temperature without any atmospheric greenhouse effect is:
Ta = Tv*0.7062*A
where:
- Ta is the average temperature without atmospheric greenhouse effect;
- Tv is the void temperature (see Temperature for more data);
- A is the albedo correction (if you need small adjustments, for Earth it is 1).
You first need to go to Atmosphere Parameters and see what is the amount of gasses (compared with Earth's) that you will need in your atmosphere. Around a planet with low gravity, the atmosphere will be far larger, to get a significant pressure on the ground. So, the pressure above sea level is not important. What we need is atmospheric msss. Divide the mass to the planet's surface:
M = Am/(D^2)
Here, we have:
- M is the atmospheric mass per surface unit;
- Am is the total atmospheric mass (Earth's is 1);
- D is planet's diameter (or radius), rounded to Earth's (for Earth, it's 1).
The value you will get is not in kg or other mass unit. It is a value that you will need to further use in calculations below.
Now, the temperature formula for a basic atmosphere:
Tnat = Ta + Ta*(mNO + mHe*0.22)*.099*M
where:
- Tnat is the natural temperature (without CO2 or any other greenhouse gas);
- Ta is the average temperature without atmosphere (see formulas above);
- mNO is proportional to the amount of nitrogen and oxygen in the atmosphere (100% is 1, 50% is 0.5, 1% is 0.01);
- mHe is proportional to the amount of helium in the atmosphere (80% is 0.8 and so on);
- M is the atmospheric mass per surface (see formulas above).
After using this formula, you will get the temperature without any amount of carbon dioxide in the air or any added greenhouse gas.
Adding carbon dioxide
Each greenhouse gas has its warming power. The growth in temperature is not proportional to the increase of gas amount, it is rather logarithmic. Still, for small amounts (lesser then 1%), we can use without significant error simple formulas.
Carbon dioxide is found in Earth's atmosphere in 611 parts per million by mass or 6.29 kg per square meter. However, this is now (2017). Pre-industrial values were of 4.8 kg/sqm. This 4.8 accounts for a temperature increase of about 5 degrees. The formula becomes:
Geff = Ta/262*Mg*Ge
where:
- Geff is the greenhouse effect of the gas;
- Ta is the atmosphere temperature without atmosphere (see formulas above);
Mg is the mass of gas in kg per square meter; Ge is the greenhouse effect.
For carbon dioxide, the greenhouse effect is rounded at 1, but many authors place it at a higher or lower value (for our formula, between 0.78 and 1.23). Personally, I use the value of 1.
It is good to firstly include an amount of carbon dioxide in the atmosphere similar to what was on Earth before the industrial era. Then, we can add other gasses.
Earth - like planet
This is how things work for a planet with a Solar Constant close to Earth's (1.98). The formulas would work for Mars and Venus, assuming they have similar atmospheres like Earth. Major changes in the composition of non-greenhouse gasses (nitrogen, oxygen, helium) will not dramatically change the formulas, but massive changes in carbon dioxide compositions are not compatible with this model of formulas.
Calculating
First, you have to calculate the void Temperature based on the Solar Constant. Once you have the void temperature and you know the the Atmosphere Parameters, the following formula is all what's needed:
Teff = Tv*0.7062 + Tv*0.7062*21*dNO*0.099 + TV*0.7062*dHe*0.218 + (TV*0.7062/262)*mCO2 + (TV*0.7062/262)*mGH*C
where:
- Teff is the planet temperature;
- Tv is the void temperature;
- dNO is the amount of nitrogen and oxygen (kg per square cm, for Earth it is 1);
- dHe is the amount of helium (kg per square cm, if the atmosphere has helium as an inert gas);
- mCO2 is the mass of carbon dioxide (kg per square meter);
- mGH is the mass of added greenhouse gas (kg per square meter);
- C is the greenhouse effect (how many times this gas is more potent then carbon dioxide).
Outer planet
For outer planets, the greenhouse effect of an existing atmosphere is too small, compared with the effect created by artificial greenhouse gasses that will be added. Since the amount of added gasses will be too high, formulas for an Earth-like planet will not prove efficient.
Natural greenhouse effect
Once we know the Solar Constant, we can determine the Void Temperature for the specified planet. Then, we need Atmosphere Parameters. With this in mind, we will get two different types of atmospheric temperatures: one without the effect of any atmosphere and one with the tiny effect of the atmosphere (but without added greenhouse gasses):
Ta = Tv*0.7062
where:
- Ta is the average temperature without atmospheric greenhouse effect;
- Tv is the void temperature (see Temperature for more data);
Tadd = Tv*0.7062*21*dNO*0.099 + TV*0.7062*dHe*0.218 + (TV*0.7062/262)*mCO2
where:
- Tadd is the added temperature by atmosphere's natural greenhouse effect;
- Tv is the void temperature;
- dNO is the amount of nitrogen and oxygen (kg per square cm, for Earth it is 1);
- dHe is the amount of helium (kg per square cm, if the atmosphere has helium as an inert gas);
- mCO2 is the mass of carbon dioxide (kg per square meter).
The result should be something like this: for Triton, given an Earth-like albedo, you will get an average surface temperature (Ta = 70 K). The atmosphere's natural greenhouse effect (Tadd = 4 K). Without greenhouse gasses, average temperature will be only 74 K (-199 C), too low for life to survive.
Adding greenhouse gasses
The greenhouse effect is in fact not linear (an increase of 20% of gas will result an increase of temperature of 20%). The effect is following a logarithmic curve. Using a linear increase, one could get for Venus an average temperature higher then the Sun.
A much better way to see this is like an increase of the Solar Constant. In fact, greenhouse gasses are radiating back the infrared towards the planet, like if they were increasing the amount of energy received by the planet.
If we add 1 kg/sqm of carbon dioxide in Earth's atmosphere, it will act like if we increase the solar constant with 0.0114. The planet will behave like if it is receiving more heat, like if it were under a higher solar constant. It appears like this:
Ksg = Ks + Ks*M*G*0.00577
where:
- Ksg is the greenhoused solar constant;
- Ks is the solar constant;
- M is the mass of existing greenhouse gas (kg/square meter);
- G is the greenhouse effect (carbon dioxide = 1).
Now, using the formulas for Temperature, you can find out what would be the temperature for the greenhoused solar constant we calculated so far. Then, we have to add something:
Teff = Tg + Tadd
where:
- Teff is the average temperature;
- Tg is temperature gathered for the greenhoused solar constant;
- Tadd is the increased temperature based on the atmosphere's natural greenhouse effect.
How much greenhouse gas would we need?
Based on this, if we consider the atmosphere's greenhouse effect to be negligible, we can find out the amount of greenhouse gas needed by the formula used for the greenhoused solar constant:
Ksg = Ks + Ks*M*G*0.00577
The desired value for Ksg is 1.98, similar to Earth's, so we can calculate the mass of needed greenhouse gas:
M = (1.98-Ks)/(Ks*G*0.00577)
where:
- M is the mass of greenhouse gas, in kg per square meter;
- Ks is the solar constant for the planet;
- G is the greenhouse effect of the specified gas. For carbon dioxide, G = 1.
Based on this formula, we get the following amounts of CO2 per square meter:
Mars: 206 kg CO2/sqm Ceres: 1157 kg CO2/sqm Jupiter: 4519 kg CO2/sqm Saturn: 15670 kg CO2/sqm Uranus: 63970 kg CO2/sqm Neptune: 157200 kg CO2/sqm Pluto (average distance): 270000 kg CO2/sqm Eris (average distance): 11500000 kg CO2/sqm Sedna (average distance): 88800000 kg CO2/sqm
One can see that even for Mars this is too much. It would mean 1.37% from the volume of an Earth-sized atmosphere. At Jupiter, it will be 30%. By using synthetic greenhouse gasses, we will need far lesser amounts:
Sulfur hexaflouride (23900 times more potent then CO2):
Mars: 0.00862 kg/sqm Ceres: 0.0484 kg/sqm Jupiter: 0.189 kg/sqm Saturn: 0.656 kg/sqm Uranus: 2.68 kg/sqm Neptune: 6.58 kg/sqm Pluto (average distance): 11.3 kg/sqm Eris (average distance): 481 kg/sqm Sedna (average distance): 3750 kg/sqm
Remember that on Earth, we have 6 kg of CO2 per square meter. One can see that, at least to the orbit of Saturn, the amount of sulfur hexafluoride or nitrogen trifluoride (72% as effective as sulfur hexafluoride) is not so high. Fluorine is found through the Solar System not in large quantities, but still enough for this. Between Saturn and Neptune, it will be harder, but not impossible to gather the needed amount of fluorine. Beyond the orbit of Neptune, even if in theory it is possible to terraform a planet, in practice, plant life would be impossible or very hard without the use of artificial light. But even so, for Sedna, at its average distance from the Sun, assuming the atmosphere will have a similar density like Earth's, only 2.5% of the atmosphere will be composed of sulfur hexafluoride.
Inner planet
Terraforming inner planets is far more difficult then terraforming outer planets.
Between Earth and Venus
In case f inner planets, we need to cool them. Classic solutions, which are possible for a planet between Earth and Veenus, include:
- Design an atmosphere with helium as an inert gas, only with limited amount of nitrogen;
- Make a rarefied atmosphere;
- Decrease the amount of carbon dioxide;
- Decrease albedo (for example, paint deserts in white);
- The use of Anti Greenhouse Gasses and hazes, to temporarily decrease temperature.
The first 4 solutions have an effect that can be calculated based on formulas shown for an Earth-like planet. The 5th solution is much more complex, its effects cannot be shown in a single equation.
Hotter then Venus
For an inner planet, closer then Venus, the only possible solution is to decrease the Solar Constant to a value close to 1.98, as it is for Earth. This can be done by placing mirrors and lens in outer space or in the atmosphere. The only feasible solution for a planet like Mercury remains the use of Micro Helium Balloons. They are small (below 1 cm) and coated with a material that reflects light and heat.
For such a planet, it makes more sense to create an atmosphere similar to Earth's. The most important thing is the layer of balloons covering the planet at a specified height. The layer will decrease the solar constant below it.
If the number of balloons is small, all of them will be exposed to light. However, if we need many balloons, some will be shadowed by others and their efficiency will decrease.
For the use of micro helium balloons, the formula is:
Ksl = Ks/(1+S),
where:
- Ksl is the solar constant below the layer of balloons;
- Ks is the solar constant for the planet;
- S is the surface of shadows created by all balloons per surface unit.
For a small number of balloons, S is equal with the shadow created by all balloons. However, if their number is too high, they will be placed in layers and some balloons will be in shadow by the balloons above them. For Mercury, which is 6.67 times more illuminated then Earth, you need to let only a tiny amount of light reach the ground. The surface of aligned balloon shadows will be 6.67 square meters for each square meter of ground.
How could we cool an inner planet?
The simple formula listed above can be used to find out the number of balloons we need. From it, we can get another formula:
S = (Ks-Ksl)/Ksl.
Ks is the solar constant we have for the planet. Ksl is the constant below balloon layer and must be close or similar to Earth's (1.98). So, now we get the next formula:
S = (Ks-1.98)/1.98.
Now, in order to see how many balloons will be needed, we can use the formula for surface. We need the radius of one balloon. Let's consider the diameter of an average balloon to be 1 cm. It will create a shadow of 0.785 square cm, or 0.0000785 sqm. Its volume would be 0.524 cubic cm, or 0.000000524 cubic meters. The layer of balloons will not be close to ground surface, to protect them from impacts, but at a certain altitude. There, air pressure is lower, so the weight of a cubic meter of air is smaller. Balloons will have a similar density with the air they're floating in. Each atmosphere is different, but for inner planets we would like to create similar atmospheres with Earth's. So, we would place the balloons somewhere in the stratosphere. There, a cubic meter of air might weight 0.2 kg, so a balloon would weight 0.000000105 kg. With these values, we can see what would mean to terraform Mercury or Venus.
Venus - solar constant = 3.78
- S = 0.909
- Number of balloons = 11580 per square meter
- Volume of balloons = 0.00607 cubic meters per sqm
- Mass of balloons = 0.00122 kg per square meter.
Mercury - solar constant = 13.20
- S = 5.67
- Number of balloons = 72190 per square meter
- Volume of balloons = 0.0378 cubic meters per sqm
- Mass of balloons = 0.00758 kg per square meter
Vulcan (hypothetical planet) , solar constant = 39.6
S = 19.0
- Number of balloons = 242000 per square meter
- Volume of balloons = 0.127 cubic meters per sqm
- Mass of balloons = 0.0254 kg per square meter.
As one can see, the amount of materials required for balloons would not be so large, but for now, the technology required for manufacturing them in huge amounts is not developed. Still, this is the only way a hot inner planet can be terraformed.