Question

a) At what distance from a 2 solar mass neutron star would a planet like the Earth be tidally disrupted (that is, literally pulled apart)? That is, how close would the planet need to be to the NS for the difference between the NS’s gravity at the center of the planet and at the surface of the planet to be greater than the gravity holding the planet together?

b) Would the asteroid Pallas be able to get any closer? (Pallas has a radius of 256 km and a mass of 2.1 x 1023 gm.)

Answer 1

a. given mass of Neutron Star, NS M = 2 *1.99*10^30 kg ( 2 solar masses)
   distance of the planet like earth from the NS = d
   radius of planet like earth = r
   mass of planet like earth = m
   then
   gravity at the near surface of the planet due to planet = g
   g = Gm/r^2

   gravity at the center of the planet due to the NS = g'
   g' = GM/d^2

   for the planet to be puller apart
   g' > g
   M/d^2 - m/r^2 > 0
   d = 149.6*10^9 m
   r = 6400,000 m
   m = 5.972*10^24 kg
   hence
   M/d^2 - m/r^2 = -1.456229*10^11 < 0
   hence a planet like earth shall not be puller apart by NS double the mass of sun

   so let the distance d be unknonw
   then
   d^2 = Mr^2/m
   d = r*sqrt(M/m) = 5.2247*10^9 m
   hence if the distance between the NS and earthlike planet is less than 5.2247*10^9 m = 0.0349 times the distance between earth and sun, then the plant must get torn apart

b. for pallas, m = 2.1*10^20 kg
   r = 256,000 m
   hence
   d = 35.24291*10^9 m
   this distance is more than the distance we got for the earth like planet
   hence pallas will not be able to get any closer