Why doesn't the solar wind push the Earth out of orbit?

Because the amount of mass that the earth intercepts from this wing from moment to moment is so little that it transfers virtually no momentum to the earth at all. With an average density of 5 atoms per cubic centimeter and traveling at 450 kilometers per second, even spread out over the surface of the earth facing the sun, the earth only intercepts about

                  density   x   velocity   x   area of earth
                         -24                                      2
    M  =   5 x 1.6 x 10   grams  x  450 km/sec  x pi x (6500 km)

or about 500 grams per second of mass. If you assume this has been going on for 4 billion years, this only amounts to 6 x 10^19 grams. This is about equal to the mass of an asteroid about 10 kilometers in radius! The earth has a mass of something like a 100 million times more than this so the effect is not very significant dynamically.
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All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the
NASA IMAGE/POETRY Education and Public Outreach program.