Paul Hewitt’s Figuring Physics in the Feb. 2016 issue asked whether it would take a larger velocity change to stop a satellite in a circular orbit or to cause it to escape. An extension of this problem asks: What minimum velocity change is required to crash a satellite into the planet, and how does that compare with the velocity change required for escape? The solution presented here, using conservation principles taught in a mechanics course, serves as an introduction to orbital maneuvers, and can be applied to questions regarding the removal of objects orbiting Earth, other planets, and the Sun.

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To find the exact values, set the slope of Eq. (5) to zero and solve the resulting cubic equation. The existence of a maximum can be explained thus: For close orbits, |Δvin| increases from zero with rS as the planet’s angular size decreases, requiring a larger distortion of the satellite’s original circular orbit. For rS >>R, where the planet appears almost “point-like,” |Δvin| tends to the speed required to stop the satellite, vC, which is a decreasing function of rS. Therefore, a maximum must exist in between.
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A treatment that includes the Earth-to-heliocentric orbit transfer finds a total specific impulse of 16.7 km/s. See
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8.
If you really want to crash a high-orbit satellite into the planet using minimal fuel, execute a two-impulse biparabolic transfer as follows: First send the spacecraft very far from the planet at the escape speed, then (much later) apply a tiny retrograde impulse, causing it to fall back towards the planet on a nearly radial path. See
A.
Prado
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9.
For information on these and other missions, see http://solarsystem.nasa.gov/missions/type/orbiter.
10.
To obtain free educational licenses for Systems Tool Kit (STK), visit https://www.agi.com/resources/educational-alliance-program.
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