Question

Hydrostatic Balance equation applied to planet interior structure: For lower mass rocky planets up to a few times the mass of the Earth, one can assume that the density p of the material making the planet is constant at any radius through the planet's interior. We want to evaluate the pressure at each radius r in the interior of such a planet. Using the hydrostatic balance equation we derived in class, we can slightly rewrite this relation for any interior radius r as: (note ris spherical radius and increases) dp = – g(r)p; dr gr) is the gravitational acceleration due to all the mass M(r) below the layer at radius r; g(r) = G M(r) / p2 Show that the pressure at the centre of the planet is given by the expression (Hint, a simple Integration needed): Po ~ 240 GP2R2 where is the planet's radius.

Answer 1

To calculate the pressure at the center of a planet

- with g(P)2 G Mor) 4 пта - , (r) : 2 17 62 pr dep 2 - T6 pm . .. Integrating : P 2-9 TIGA2 rde . . 2-4T GZplx tc... () . where is the constant of integration. At rar, P20 1 So, 02-21T GP?R?+ c :: or C 2.3TG2 PRE P2 - 7 tr G prx 2 + 2 GP?R?. (2) - (Putting the value of c in C")) : at center (120) the pressure ,. Poon 3 Tap?R? (from (2) Shown putting rzo) 2