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Psycopathak
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Homework Statement
Find the pressure at the center of a planet
Homework Equations
dP/dr = -ρg (Hydrostatic Equilibrium)
g = GM/r2 (acceleration due to gravity)
The Attempt at a Solution
dP/dr = -ρg
Assume that density is constant.
subsitute GM/r2 for g in the pressure gradient formula
dP/dr = -ρGM/r2
Now we need Mass as a function of radius. Divide the planet up into differential concentric rings
then the differential mass element is related to the surface area.
dM = 4πr2ρdr or dM/dr = 4πr2ρ (conservation of mass)
Then integrating both sides of the the above equation gives.
∫dM = ∫4πr2ρdr
M(r) = ∫4πr2ρdr With limits of integration from 0 to r
and since density is constant
M(r) = 4/3πr3ρ
so the pressure gradient with a constant density gives
dP/dr = (-ρg/r2)(4/3πr3ρ)
Solving the differential equation for P gives
∫dP = ∫-(4πGρ2rdr)/3
with limits of integration from r1 to r2 (radius) and P1 to P2 (pressure)
gives:
P2-P1 = (-4Gπρ2/3)(r22-r12/2)
Solving for P1 gives
P1 = P2+(-2Gπρ2/3)(r22-r12/2)
Setting the boundry condition as
r2 = R1 and P2=0
gives the final equation for Pressure as a function of radius inside the planet with density ρ, planetary radius R and varying radius r as
P(r) = (2πGρ2R2/3)(1-r2/R2)
Is this correct?
Does this mean that inserting 0 for r will give you (1-0) or just 1 and then the pressure at the center of the planet is dependent on just the density of the planet and the radius of the planet?