Question

Infrared observations of Saturn indicate that it is radiating L_IR = 2×10²⁴ erg/s. However, it is absorbing only about 1.1×10²⁴ erg/s of sunlight. Assume the difference is the result of the release of gravitational potential energy as Saturn contracts. Treat Saturn as a uniform density sphere.

a) What is the rate at which Saturn is shrinking (give dR/dt in cm/s)?

b) At this rate, how long would it take Saturn to contract to a point (in years)?

c) On this basis, is this a tenable theory of the interior heating of Saturn?

Answer 1

given

power of radiation of saturn, LIR = 2*10^24 erg/s

power of absorption of saturn from sun, A = 1.1*10^24 erg/s

hence

net power loss = LIR - A = 0.9*10^24 erg/s

a. let density of saturn be rho

radius r

then

power loss = c^2*mass loss rate

mass loss rate = m'

then

m'*c^2 = 0.9*10^24 erg/s = 0.9*10^17 J/s

m' = 1 kg/s

now, rho = 3m/4*pi*r^3

4*pi*r^3*rho = 3m

4*pi*rho*3r^2*r' = 3m' = 3

r' = 0.0795774715459/rho*r^2

radiius of saturn r = 58,232 km

and density of saturn rho = 687 kg/m^3

hence

r' = 1.9849*10^-12 m/s = 1.9849*10^-10 cm/s

b. hecne at this rate, tiem taken to shrink to a point = T

dr/dt = -0.0795774715459/rho*r^2

r^2*dr = -0.0795774715459*dt/rho

integrating from t = 0 to t = T

(R^3/3) = 0.0795774715459*T/rho

hence

T = R^3*rho/3*0.0795774715459 = 568238251709007067993566188.88168 s= 0.1800638361944810^20 years

c. as extra energy is being radiated and that must be generated by vanishing mass that vanished because of mass loss in fission / fusion reactions in the core of the saturn, providing evidence of interior heating of saturn