Sidebar: It takes much more energy to get to Mars than to the Moon
A rough approximation to the energy needed to travel from Earth to another body in the Solar System just accounts for the changes in gravitational energy. That is, we ignore the work done in traversing the Earth’s atmosphere. That adds about 20% to Moon shots. We also ignore retrorocketing energy needed for landing but also the modest energy saving (0.3%) in using the rotational velocity of the Earth by judicious aiming. Pure gravitational potential energy at any point relative to a body we’re leaving is given by Newton’s formula
Here, G is the universal gravitational constant, 6.67408 × 10-11 m3 kg-1 s-2. Its actual magnitude and units are not relevant, since we’re doing comparisons; M is the mass of the body that we’re escaping to some distance far greater than its radius; m is the mass of the body we’re on (the final stage, the payload with the crew as part of it).
Some terminology:
- ME is the mass of the Earth, 5.97×1024 kg
- rE is the radius of the Earth, 6380 km or 6.38×106 m
- MM is the mass of the Moon, 7.34767309 × 1022kg, 1.23% that of the Earth
- rM is the radius of the Moon, 1738 km or 1.738×106, 27% that of the Earth
- MS is the mass of the Sun, 1.99×1030 kg, 333,000 times that of Earth
- rE-S is the radial distance of the Earth from the Sun, 150×106 km
- MA is the mass of Mars (“A” is for Ares),
Earth to the Moon: To leave the Earth as the start of the journey, the mass M is that of the Earth, mE and the radius is that of the Earth, rE. The potential energy per unit mass is then -62.4 million joules per kg. relative to an infinite distance from Earth. We didn’t have to escape the Earth’s gravity completely, going out to 60 times the Earth radius; we “save” 1/60th of the energy, so the cost is 61.4 million joules per kg. At the Moon, we gain energy by falling into its gravitational field. The downhill trip in potential energy is GMMm/rM, which is 2.8 million joules per kg or about 4.5% of the cost of getting off the Earth. Total: 61.4-2.8 = 58.6 million joules per kg.
A quick note of comparison: the Saturn V rocket’s 3 stages for Trans-lunar injection in the Apollo program contained 2,738 tonnes of propellant at liquid oxygen + RP-1 (kerosene). This mixture has a specific impulse of 304 s in weight units or 2979 meters per second in mass units. That is just the exhaust velocity. The energy content per mass is 44 megajoules per kg of RP-1. The liquid oxygen is needed in the ratio 48 g per 14 g of hydrocarbon (CH2 + 1.5 O2 CO2 + H2O), so the energy content per mass of total propellant is 9.94 MJ per kg. The total energy in the propellant in all stages is then 2.72×1013 J. The payload at 48,600 kg gained 2.9×1012 J, about 11% of the propellant energy – not bad, given all the stage masses that had to be accelerated and then jettisoned.
We ignore the change in energy in the gravitational field of the Sun, since the Earth and the Moon are very nearly at the Sun distance from the Sun at all times.
Earth to Mars: To leave the Earth and go far away again has a cost of 62.4 million joules per kg. Falling into the gravitational field of Mars recoups an amount GMAm/rA, which is 12.6 million joules per kg. The net cost for the planetary parts is 62.4-12.6 = 49.8 million joules per kg. However, we have to account for the shift in the Sun’s gravitational field. We started at Earth’s distance from the Sun. Here the gravitational potential is -885 million joules per kg. At Mars it is -582 million joules per kg. The net gain in energy needed is 885-582 = 306 million joules per kg. Add that to the planetary energy shifts and we get 356 million joules per kg.
That’s a factor 356/58.6 or 6.07 times greater than what’s needed to go to the Moon. It’s bigger yet, since the rocket itself has to weigh more to provide all that energy. If it’s bigger in mass by the same ratio because the mass of propellant dominates, the cost goes as the square of that factor, or more than 36 times more. That’s for the same final payload. We’ve never had a rocket 6 times larger than Saturn V; we never tried to land a payload on Mars that’s as big as what we put on the Moon. The Curiosity lander had a mass of 1000 kg. The Apollo lunar module had a mass of 15,200 kg with its fuel, crew, oxygen, etc.
A rocket to carry even a modest crew, their equipment, and their supplies will be at the limit of chemical rocket technology. A rocket to carry many colonists will have to have a barebones payload, mostly crew, with an established infrastructure already on Mars to receive them. Colonization of Mars would have to be done in many stages, at a correspondingly massive expense. Getting 100 people on Mars will take many flights and much time. Getting 1000 people on Mars won’t happen.