In order to carry the already existing ISS into a geostationary orbit, i.e. approximately one-tenth of the distance between the Earth and the Moon, scientists will require approximately half of all the fuel that was spent on the assembly of the ISS with all the rockets for all time.
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To send the ISS into lunar orbit, they will require even more fuel. And we do not even take into account the significant change in the orbital plane of the ISS. Now, the ISS has an orbital inclination of 51.6 degrees, the lunar orbital inclination is 28 degrees, and quite a lot of fuel will also be required to change the inclination of the ISS orbit.
On the ISS itself, there is no place for such an amount of fuel. So, in order to accommodate such an amount of fuel and oxidizer, tanks would be required that are larger than the ISS itself; therefore, to send the ISS to the Moon, we would either need to attach fuel tanks of the required size to it or refuel the ISS in flight as soon as all the fuel is consumed.
The ISS is refueling via the Zvezda module, so the only way to refuel the ISS at the moment is to use the Russian spacecraft Progress. The problem is that the more the ISS moves away from the Earth, the more difficult it will be for Progress to reach it, and the less fuel it will be able to bring in one time.
At the same time, Progress is certified for a maximum flight altitude of 460 kilometers, and there are 384,000 kilometers more to the moon. Hypothetically, the SpaceX Dragon can be modified to make it possible to refuel the ISS with its help, however, the flight altitude of the SpaceX Dragon can also not exceed 1200 kilometers, so obviously, Dragon cannot be used to refuel the ISS on the way to the Moon as well.
We currently do not have the technology to transport the ISS to the moon. In order to solve this problem, we need to either get involved in the incredibly expensive construction of giant fuel tanks, or in the development of an extremely expensive ISS refueling system during the flight.
And even if we somehow manage to deliver the ISS to the moon, the station will turn into a useless piece of metal. The thing is that the ISS was designed to be at an altitude of 280 to 460 kilometers above the Earth.
The ISS does not have the radiation protection equipment that will be needed for a long stay of the crew in the orbit of the moon.
The navigation system is based on the use of GPS satellites and is based on the assumption that the station is always located under satellites (the GPS satellite’s orbit is about 20,000 km).
For communication with the Earth, the ISS uses TDRSS satellite in geostationary orbit (35,800 km).
The ISS power supply and heating system are designed with the expectation that the ISS orbital period will be 90 minutes.
The problems listed above, as well as the extremely high cost of any of the possible projects for transporting the ISS to the Moon make the issue of such transportation itself inappropriate. Assembling a new station in orbit of the Moon is incomparably cheaper.