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In what way does the concept of 'C3' relate to 'Delta-V' in interplanetary missions?

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That depends on how and where the $\Delta V$ is applied. It is usually applied as an impulsive burn in low-Earth orbit. In that case (assuming a circular LEO, $\mu$ is GM, $r$ is distance from Earth center):

$$\Delta V=\sqrt{C_3+{2\mu_{Earth}\over r_{LEO}}} - \sqrt{\mu_{Earth}\over r_{LEO}}$$

Ah, looks like I've answered this one before.

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