Question

Assume that the rate of accretion of material to a spherical planet is given by

dV/dt = A+BV

Where V is volume, t is time and A and B are constants. Neglecting compression of the material that accretes with a density p:

(a) Write an expression for the radius of the planet with time if B=0.

(b) Write an expression for the radius of the planet with time if A=0 and at time t=0 the volume of the planet is V₀.

(c) For case (b) write an expression for the average temperature of the planet with time if the specific heat of the material forming the planet is C_P.

Answer 1

a)

$$\frac{dV}{dt}=A\\V=\frac{4\pi}{3}r^3=\frac{4\pi}{3}r^3_0+At\\r=\left(r^3_0+\frac{3}{4\pi}At\right)^{\frac{1}{3}}$$

b)

$$\ln{\frac{r^3}{r_0^3}}=B\\r=r_0\left(e^{Bt}\right)^{\frac{1}{3}}$$

c)

$$1+\alpha(T-T_0)=\left(e^{Bt}\right)^{\frac{1}{3}}$$