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The solar electric ion propulsion engine of the Dawn spacecraft to Vesta and Ceres used Mars for gravity assist. Would it be gainful to complement an ion engine with a high thrust (solid) chemical rocket to fire only during such a gravity assist in order to make maximum use of the Oberth effect? Or can it be indicated that the burden of launching and carrying its mass would be difficult to make up for in that way?

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Solid rocket motors have been used successfully to exploit the Oberth effect during the most critical phase of an interplanetary mission - target body orbit insertion. NASA's Magellan mission to Venus used a standard Thiokol/ATK Star 48 propulsion module.

This same module was employed as an upper stage in the New Horizons flight to Pluto and beyond.

A mass-conscious mission designer is always aware of solids' low specific impulse, yet may use them for their reliability in single-burn maneuvers. Powered gravity assists are mostly done in several burns (Venus-Earth-Earth, for instance), and require very high precision as far as burn $\Delta V$ is concerned. It may be also desirable to re-program burn length in-flight, which is not possible with solids. Hypergolic-based thrusters are much more convenient, well-tested, and flexible.

To sum it up, a mission with an ion thruster could use a solid rocket motor for orbit insertion, if the approach velocity is simply too high for a low-thrust capture. Multiple powered gravity assists - not really. One should note that powered assists are much less desirable mass-wise than unpowered ones.

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