by hstarbuck » Thu Mar 11, 2010 1:01 am
I thought I would do a simple calculation of a hypothetical satellite in the Lagrangian point between the two larger bodies. This would start off as FE- FM= FC --or--G Force of Large mass(like Earth) - G Force of smaller mass (like the Moon) = Centripetal force of satellite (satellite masses cancel from each term so becomes acceleration problem). Then using three distances: Earth to satellite , Moon to satellite, and Earth to Moon as R, r, R+r = D, respectively. Next writing the velocity in the centripetal acceleration as a function of the Moon's period T (same as satellite)--and rearranging/canceling a bit I got: ME/R2 - mM/(D-R)2 = 4pi2R/GTM2. Here G is the gravitational constant and TM is the period of both the Moon and the satellite. I got this far, but I'll be darned if this equation doesn't blow up when trying to solve for R in terms of D, TM, mE, and mM--all of which are given in initial parameters of a problem like this. I don't feel like expanding terms--it makes my brain hurt--and I never know when to do it anyway. Maybe using Lagrangian mechanics this would be ironically easier. Any suggestions--did I miss something obvious?
I thought I would do a simple calculation of a hypothetical satellite in the Lagrangian point between the two larger bodies. This would start off as F[sub]E[/sub]- F[sub]M[/sub]= F[sub]C[/sub] --or--G Force of Large mass(like Earth) - G Force of smaller mass (like the Moon) = Centripetal force of satellite (satellite masses cancel from each term so becomes acceleration problem). Then using three distances: Earth to satellite , Moon to satellite, and Earth to Moon as R, r, R+r = D, respectively. Next writing the velocity in the centripetal acceleration as a function of the Moon's period T (same as satellite)--and rearranging/canceling a bit I got: M[sub]E[/sub]/R[sup]2[/sup] - m[sub]M[/sub]/(D-R)[sup]2[/sup] = 4pi[sup]2[/sup]R/GT[sub]M[/sub][sup]2[/sup]. Here G is the gravitational constant and T[sub]M[/sub] is the period of both the Moon and the satellite. I got this far, but I'll be darned if this equation doesn't blow up when trying to solve for R in terms of D, T[sub]M[/sub], m[sub]E[/sub], and m[sub]M[/sub]--all of which are given in initial parameters of a problem like this. I don't feel like expanding terms--it makes my brain hurt--and I never know when to do it anyway. Maybe using Lagrangian mechanics this would be ironically easier. Any suggestions--did I miss something obvious?