Back to Math And Terraforming

The behavior of an atmosphere is very important for terraforming. There are a few formulas that can calculate if a planet is suitable to support an atmosphere and under what conditions.

You can determine average gas speed with the formula [1]:

**Gs = 3*0.017077*(T(C)+273.15)/m**

This formula is adapted by myself, to fit better. **Gs** is gas speed (in km/s), **T(C)** is temperature in Celsius degrees and **m** is the molecular mass of the gas (1 for hydrogen, 16 for atomic oxygen, 32 for molecular oxygen). By using it, you will get for oxygen at 31 C, the speed of 0.4869 km/s. Gs is the average speed of a gas molecule, but some will move faster and some far slower.

The second thing we need to consider is escape velocity, to see if the gas will escape or not the planet. The formula, adapted for our simulation and our measure units, is:

**Ev = ((1601*mass)/diam)^0.5**

Here, **Ev** is the escape velocity in km/s, **mass** is the planet's mass (Earth's mass = 1) and **diam** is the diameter, in thousands km.

By combining both formulas, we get the next value:

**xxx = ((3*0.017077*(T(C)+273.15)/m))/(((1601*mass)/diam)^0.5)*100**

The symbol *xxx* is again my invention, there is no such thing. This formula allows you to quickly see if a planet will be able to sustain an atmosphere or not. Here, *T(C)* is the temperatuire (in Celsius), *m* is gas molecular mass, *mass* is planet's mass (Earth = 1) and *diam* is planet's diameter in thousands km. The numbers you will get for Earth, at 15 C, are as follows:

- Hydrogen: 131
- Helium: 33
- Nitrogen (molecular): 4.7
- Oxygen (molecular): 4
- Carbon dioxide: 3
- Methane: 8

The bigger the number, the lower the chance for specified gas to remain in the atmosphere. If we compute the same data for Earth's moon, we get:

- Hydrogen: 653
- Oxygen (molecular): 20
- Carbon dioxide: 15.

Basically, I would say that a gas is safe if the calculated value is less then 10. Also, if value is below 100, the gas can still remain in the atmosphere for a human lifetime.

Gerald Kuiper designed a planetary constant which he named **k**. This constant shows the probability for a celestial body to host an atmosphere (the probability that majority of molecules will not be in escape velocity). For him, if **k** had a higher value then 5, the planet or moon could host atmosphere. This way, he proved a long time ago that Titan has an atmosphere. His formulas are more complex, but more accurate.

## Atmosphere Size Edit

Small bodies have a weaker gravity. Their atmospheres will be really huge, extending perhaps hundreds of km. The following is what you need to write a Microsoft excel table, where you can simulate atmosphere parameters around any planet or moon:

- Column A: write
**temperature**in degrees Celsius. - Column B: write
**atmospheric mass**(Earth's is 1). - Column C: write
**planet's mass**(Earth's is 1). See Planetary Parameters for more. - Column D: write
**planet's diameter**(thousands km). - Column E: write (for cell E2):
**=C2/(D2/12.756)^2**to get**gravity**(Earth's is 1). - Column F: write (for cell F2):
**=(0.029846*(A2+273.15))/(B2*E2)**to get**atmosphere height**in km. - Column G: write (for cell G2):
**=D2+(F121/1000*2)**to get**diameter + atmosphere**in thousands km. - Column H: write (for cell H2):
**=(G2^3)-(D2^3)**to get**atmosphere volume**(about 8 for Earth's atmosphere). - Column I: write (for cell I2):
**=B2/H2*8.40757**to get**surface pressure**(Earth's is 1).

If you play a bit with these values, you will get interesting results. For example, if Earth's temperature were -200 C, you would get an atmosphere of only 2 km thick, but with a density of 3.9. For a small object like the Moon, you would in fact need only 88% of Earth's atmosphere to get on the ground a pressure of 1 atm. The gas layer will reach about 72 km high above surface. For Pluto, things are even stranger. To get a pressure of 1 atm. on the ground, you will need a gas layer higher then the planet's radius itself. This proves why Atmosphere around small bodies is difficult to create and maintain.

### Greenhouse Effects Edit

By adding greenhouse gasses to a planet, two things will happen. Firstly, the lower part of the atmosphere will heat-up and its gasses will move faster. Secondly, the upper part will remain cold, perhaps even colder since it will not receive the same amount of heat radiated from lower layers. So, the lower layer will expand a bit, while the upper one might in fact contract.

## Minimum Safe Mass Edit

The table below shows for given temperatures and given planet densities, what is the minimum diameter (in km) for a safe atmosphere.

Temperature
| Density 1.5 | Density 3 | Density 4.5 | Density 6 |

-200 C
| 2 600 | 1 900 | 1 400 | 1 300 |

-180 C
| 3 400 | 2 300 | 1 900 | 1 700 |

-150 C
| 4 300 | 3 100 | 2 500 | 2 200 |

-100 C
| 6 100 | 4 300 | 3 500 | 3 100 |

-50 C
| 7 800 | 5 600 | 4 500 | 3 900 |

15 C
| 10 500 | 7 200 | 5 900 | 5 100 |

150 C
| 15 000 | 11 000 | 8 600 | 7 400 |

300 C
| 21 000 | 15 000 | 12 000 | 10 000 |

500 C
| 27 000 | 20 000 | 16 000 | 14 000 |

1000 C
| 45 000 | 32 000 | 26 000 | 23 000 |

The table include *safe* values, since molecules have a low chance of reaching escape velocity. Atmospheres will still be affected by solar winds. Still, the atmosphere will be *safe* relative to a human lifetime even at smaller diameters.