Elon Musk recently presented his plans for taking humans to Mars and eventually establishing a colony/city with a million people. Like many things he does, his plan is incredibly ambitious and quick-moving. The first landings are planned to happen in around 8 years.
Currently, the cost to take a person to Mars is enormous enough to be considered to be infinite. The first challenge is to bring the price to a manageable range for the richest of people... in other words, around $10 billion a trip. In the end, to make the idea of regular flights to Mars feasible, the goal is to reduce that price to a few hundred thousand dollars per person which might make it reasonable to travel to Mars and "buy a house" there.
A few of the challenges deal with simply the scale of the ambition. Currently, space vehicles carry only a few people. The goal is to have the Mars transport vessels carry 100-200 people each trip. Unfortunately, even with that larger capacity, around 10,000 trips to Mars will still be required.The sheer number of flights drive many of the engineering challenges which must be resolved to make this vision a reality.
Another challenge is that a window for a short transit to Mars only occurs every two years. Outside of these windows, travel time to Mars becomes prohibitively long. This will force a large number of ships to be built because single ships can only make an efficient round trip once every two years. To achieve the total number of flights planned, a hundred or more ships will need to fly each transit window.
Each of the enormous ships that will make the trip to Mars must be completely reusable. The expectation is that a ship will have a lifespan of 30 years and will make a dozen or more trips. This was compared to modern airliners. If a new airplane needed to be constructed for every flight, then any air travel would be prohibitively expensive. The high cost of building a 747 is spread over hundreds of flights which make the per flight cost of construction much lower. In addition, all of the support vehicles would be reusable in the same way that the current Space-X Falcon-9 rocket lands after each flight and can then be re-used.
The ability to refuel the ship that will make the Earth-Mars transit in orbit is also a necessity. That ship is already very large and if it had to launch from Earth with a full fuel load, it would be too large to be feasible. Instead, the ship will launch with a full payload of persons and cargo and then receive the fuel it needs once it is in Earth orbit. This allows the overall capacity of the ship to be much larger.
Another method which will allow the ship to carry more cargo instead of fuel is refueling in orbit around Mars as well. By refueling at Mars, the ship can carry only enough fuel for a one-way trip, which will leave a lot of space for more people and cargo.
In order to make both of the fueling issues work, the ship needs to use the right propellant. Kerosene based rocket fuels are volatile, corrosive and hard to move into orbit in volume. Further, they are hard to create on Mars. Given that Mars has a large amount of ice which can provide hydrogen and oxygen as well as an atmosphere with carbon dioxide, Mars is a good source for a methane based fuel which is more easily made on Mars and has the right characteristics for energy density and stability for these missions.
All of these items together are planned to make initial Mars trips possible within the decade and eventually to establish a large human habitation on Mars.
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