Question

B.2 This question concerns the possible tidal disruption of a spherical moon on a circular orbit of radius r about a host planet. The planet has mass Mp, radius R and mean density pp; the moon has mass M, radius Rm rand mean density Pm You may ignore any forces beyond the moon-planet system. (i) Show that tidal forces lead to a differential acceleration, between the face of the moon closest to the planet and the moon's centre, of amplitude 2GMpRm Aa p3 [3] (ii) Making the assumption that the moon will be disrupted if the tidal force ex ceeds the moon's gravitational binding force, the criterion for tidal disruption would be GM 2MpR Show that under these assumptions the moon will be disrupted if its distance r from the planet is less than 1/3 Pp idfr Pm Rp. (1) and find the value of the constant fr. [3] (ii) Now suppose that the moon is in synchronous rotation about the planet so that its orbital and rotational periods are equal. Taking into account the additional disruptive forces due to the moon's rotation, show that the criterion for disruption is still given by equation (1), but with a new value of f given by 1/3 Mm fR =3 м, Hint: you may wish to use Kepler's third law. 9]

Answer 1

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Jf nhe mCn is SPinning Caud iho ay is hee s on addilional Contributiun to -ihe differen tiung acela Yaliun The ace laatin at Centre o, due to Yolatiun Centri petal accele Yatiun Suiface But 211 Lere Time period of Yotatium nlow difperential accela vation due to This Contributin - nNow To find ,1 ne Know that T' is cuyo Period f Yotatin of moo aYound Plan, Wrth Rediou sing Reple ry Las. 2. 411 G CmpiMm 2 J mptmm) 2 Y 3 Nowho inenaliy in P roblem (i) hay on additimal facto Contributing 3 2 GmpRn mGCmptmm) Pm. CMm 3 Y3

GMP Qmt 2 Mp Rm 3 MP Y Mom MP Pm 3t m p 2 m Pm > ep 2 mp Same aj 3efere epPe Prm 3+ Mm m p PP Mm 3+ Pm MP 3+ p