A gravity-well maneuver involved what appears to be a contradiction in the law of conservation of energy. A ship leaving the Moon or a space station for some distant planet can go faster on less fuel by dropping first toward Earth, then performing her principal acceleration while as close to Earth as possible. To be sure, a ship gains kinetic energy (speed) in falling toward Earth, but one would expect that she would lose exactly the same ammount of kinetic energy as she coasted from Earth.The verification of this assertion is a simple aplication of the Vis-Viva Equation in the Hohmann transfer orbit and in the hyperbolic escape orbit:
|V2 = G M (||¾||-||¾||)|
(in the hyperbolic case, consider a < 0 in the Vis-Viva Equation)
Let's consider, as in the book, a spaceship going from Moon to Mars. In the first approximation, the ship is in the same orbit of the Moon, but sufficiently far from the Moon that it's attraction can be neglected. Let's also consider that all orbits (Earth and Mars around the Sun, and Moon around the Earth) are circular and coplanar.
Since the radius of Moon's orbit (average 384 401 km) is much smaller than the radius of Earth's orbit (149 million km) and Mars's orbit (1.6 times Earth's orbit, or 228 million km), it's possible to consider that the direct trip from the Moon to Mars uses the Hohmann transfer orbit calculated with the radius of Earth and Mars's orbits. In this case, it's necessary to add to Earth's orbital speed a DV of 3259 m/s
This speed of 3259 m/s must be achieved as soon as the ship leaves the sphere of influence of the Earth, or, approximately, at infinity. Therefore, to pass from the Moon's orbit to this hyperbolic orbit (relative to the Earth!) it's necessary to apply a DV equal to 2547 m/s. The computation of the DVs for the sequence of two maneuvers using the "gravity well" is similar: a Hohmann transfer from the Moon's orbit to an Earth-razing orbit (let's fix it at 1000 km height), followed by a change from this orbit to the hyperbolic orbit (relative to the Earth).
The DV to change from Moon's orbit to the transfer orbit is 815 m/s.The DV to change from this transfer orbit to the hyperbolic orbit is 600 m/s. So, the DV necessary to go directly from Moon's orbit to the Earth-Mars transfer orbit is 2547 m/s. If we use the two-maneuver sequence, the DV is 815 + 600 = 1415 m/s, a very big fuel saving.
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