Equation 1: Escape Velocity


where:
MMass of the planet (kg)
rPlanetary radius (m)
hHeight of the exobase, use h = 0 if unknown (m)
LLuminosity of the star, sol: 3.86 x1026 (W)
Ro For a planet: Minimum orbital radius of the planet (m)
For a moon: Minimum orbital radius of the planet - Maximum orbital radius of the moon (m)

This equation reveals a minimum value. Below this value, Hydrogen will escape from the atmosphere into space. Without Hydrogen there is no water, and without liquid water the planet cannot support life as we know it. Hydrogen escapes from an atmosphere slowly at first. As Hydrogen is lost gravity decreases slightly, increasing the rate of loss. On a gas giant, atmosphere is lost until a balance is reached with heavier gases. On a wet terrestrial planet, the increasing Oxygen concentration will determine what happens next. If there is enough Carbon then CO2 will increase atmospheric temperature, increasing the rate of loss. This will produce a runaway greenhouse effect, like Venus. If there is not enough Carbon then the planet will cool. If it cools enough then the atmosphere could collapse, like Mars. A collapsing atmosphere will also increase the rate of loss.

This equation can also be used for other gases using different k values. The k value is inversely proportional to the mass of the gas. Hydrogen, being the lightest gas, will escape the fastest.

H-valuePass/Fail
Sol222.227Pass
Jupiter67.341Pass
Saturn32.262Pass
Neptune25.728Pass
Uranus16.551Pass
Earth1.062Pass
Venus0.777Fail
Mars0.253Fail
Titan0.179Fail
Ganymede0.143Fail
Io0.125Fail
Callisto0.113Fail
Triton0.098Fail
Mercury0.087Fail
Europa0.078Fail
Pluto0.070Fail
Luna0.048Fail
Titania0.022Fail
Charon0.020Fail
Oberon0.019Fail
Rhea0.011Fail
Ariel0.011Fail
Umbriel0.011Fail
Iapetus0.009Fail
Dione0.006Fail
Tethys0.005Fail
1 Ceres0.004Fail
2 Pallas0.002Fail
4 Vesta0.002Fail
Miranda0.001Fail
Enceladus0.001Fail
Hyperion0.001Fail
Mimas0.001Fail

Earth can only just hold Hydrogen, it's right near the crossover point in the equation. The amount of Hydrogen it retains is stable over geological timescales, so it will not turn into a gas giant.