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VASIMR Plasma Thruster To Be Tested Aboard ISS
Toren Altair brings news that NASA and the Ad Astra Rocket Company finalized a Space Act Agreement earlier this week to test the Variable Specific Impulse Magnetoplasma Rocket (VASIMR) on the International Space Station. The agreement hinges on a series of requirements for the thruster's performance and efficiency in ground-based tests. "The primary technical objective of the project is to operate the VASIMR VF-200 engine at power levels up to 200 kW. Engine operation will be restricted to pulses of up to 10 minutes at this power level. Energy for these high-power operations will be provided by a battery system trickle-charged by the ISS power system. These tests will mark the first time that a high-power, steady-state electric thruster will be used as part of a manned spacecraft." Reader clarkes1 points out related news of a runway trial for Virgin Galactic's WhiteKnightTwo, the mothership that is designed to carry SpaceShipTwo from the ground to 50,000 feet. A very brief video shows the oddly-shaped plane moving down a runway under its own power.
You get some free energy from the spin, yes, but the main reason you try and launch from lower latitudes is that you want to have the option to get a near equatorial orbit inclination. Basically, you can't launch to an inclination lower than your latitude; if you think about it, launching due east (or west, but that would be going against the spin) would put you in an orbit thats the same inclination. If you aim a little north or south you end being inclined a little bit more... whether its north or south changes the position of the ascending node, but not the overall inclination. If you aim due north or south you get into a polar orbit. As far as the additional altitude... its such a miniscule amount that its not worth worrying about.
The reason you may want to have a low as inclination as possible is because if you're going to GEO or lunar or planetary missions, you want to be near zero inclination. In order to get there, you have to do an expensive plane change maneuver, which has a delta-v=sin(i)*V, so getting that inclination lower means big fuel savings.
As far as calculating the fuel savings, just consider the difference between the rotational speed of the point on the surface (sin(lat2)-sin(lat1))*r_earth*(2*pi/24 hrs) to get the additional velocity you get (and thus less delta-v you need to apply on orbit). Running that between the Russian Star City (45 degrees) and the cape (21 degrees) shows that we get ~150 m/s difference, which is nice but not game changing.
As far as sky launch or mountain launch, I learned a great little rule of thumb here a few weeks ago, the 666 rule. Launching from Mach 6 at 60000 feet (probably much higher than any reasonable air launch system), gives you only a 6% energy savings for orbital systems. So, it really doesn't give you a whole lot for the added complexity, which is why as far as i know theres only one air-launched system, an Orbital Sciences rocket that launches off an L1011. The reason why it works for Virgin/Scaled Composites is that it gives you probably 30% of the energy needed to reach the altitude, but not the orbital velocity.
As far as equations... the atmospheric drag models make launch hard to judge, but what is cleverly called the "Rocket Equation" is a really easy way to look at fuel usage with impulsive delta-vs... usually a more valuable quantity than energy since it directly applies to the amount of fuel needed and used.