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Thus c(new) = c(old)/2 but r_max = c(new)/(1-e) => e = 1/2 in new
orbit since c(old) = R. (Very eccentric!) Or if you prefer to know
the new perigee distance, it is r_min = c(new)/(1+e) = R/3.
Note that a = (r_min + r_max)/2 = 2R/3. (Another nice way to get
this is to use the vis-viva equation: the speed squared at apogee is
GM/R in the original orbit and 2G*2M/R-G*2M/a in the new orbit.
Equate these and solve for a.) Thus ratio of new to old period of
Moon is (a/R)^1.5 * (M_old/M_new)^0.5 = 2/3^1.5 = 10.6 days.