Infrared observations of Saturn indicate that it is radiating L IR = 2×10 24 erg
ID: 2305287 • Letter: I
Question
Infrared observations of Saturn indicate that it is radiating LIR = 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?
Explanation / Answer
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