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Precise laser ranging to the moon is shown to be capable of measuring the ratio (m₆{m₈}) of gravitational mass to inertial mass of the earth to accuracy sufficient to detect gravitational self-energy contributions. If m₆{m₈} of the earth differs from 1 by an amount of order (earth gravitational self-energy) / (earth total energy), then the lunar orbit will acquire a range oscillation of amplitude about 12 m.
Kenneth Nordtvedt (Tue,) studied this question.