Here is a celestial mechanics problem I can't seem to solve:To escape the solar system, an interstellar spacecraft must overcome the gravitational attraction of both the Earth and Sun. Ignore the effects of the other bodies in the solar system. Show that the escape velocity is(adsbygoogle = window.adsbygoogle || []).push({});

[tex]v = \sqrt{v_E^2 + (v_S - v_0)^2}[/tex]

where [itex]v_E[/itex] is the escape velocity from the Earth; [itex]v_S = \sqrt{2GM_S/r_{SE}}[/itex] is the escape velocity from the gravitational field of the Sun at the orbit of the Earth but far from the Earth's influence ([itex]M_S[/itex] is the mass of the Sun and [itex]r_{SE}[/itex] is the Sun-Earth distance); and [itex]v_0[/itex] is the Earth's orbital velocity about the Sun.

The thing I don't understand here is how the orbital velocity of the Earth plays a role. Suppose I took the Earth out of the picture and the spacecraft is orbitting the sun with orbital velocity equal to that of Earth's. Isn't the escape velocity from the sun just [itex]v_S[/itex]. What do I need [itex]v_0[/itex] for?

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# Escape Velocity from Solar System

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