Worksheet 8; Problem 5: Kelvin-Helmholtz timescale

The problem: The Virial Theorem states that half of a gravitationally-bound system’s potential energy goes into the kinetic energy of the system’s constituent particles. In the case of a cloud of gas, the cloud can shrink, but only if it loses energy by radiating. This occurs because as the radius shrinks, the potential energy becomes more negative, and therefore the particle motions must increase (higher kinetic energy). Particles moving faster leads to a higher gas temperature, and an increase in thermal emission.

We know that the Sun started from the gravitational collapse of a giant cloud of gas. Let’s hypothesize that the Sun is powered solely by this gravitational contraction, as was once posited by astronomers long ago. As it shrinks, its internal thermal energy increases, increasing its temperature and thereby causing it to radiate. How long would the Sun last if it was thermally radiating its current power output, \( L_{\odot} = 4 \times 10^{33} erg \: s^{-1}\)? This is known as the Kelvin-Helmholtz timescale. How does this timescale compare to the age of the oldest Moon rocks (about 4.5 billion years, also known as Gyr)?

Solve: 

What we know:

• \(M_{\odot} = 2 \times 10^{33} g\)
  
• \(R_{\odot} = 7 \times 10^{10} cm\)
• \( L_{\odot} = 4 \times 10^{33} erg \: s^{-1}\)
• \( G = 6.67 \times 10^{-8} \frac{cm^3}{gs^2}\)

From the information provided in the problem, we can see that the units for luminosity are ergs per second, from this we can conclude that \[ Luminosity\: \times time \: = \: energy \] We also know from the virial theorem that \[K = \ \frac{1}{2}U = - \frac{GM^2}{2R} \] Combining these two equations gives \[L \times t = \frac{GM^2}{R} \] Solving for t gives: \[ t = \frac{GM_{\odot}^2}{R_{\odot}L_{\odot}}\] Now we can plug in the values we know.  \[ t = \frac{6.67 \times 10^{-8} \times (2 \times 10^{33} )^2}{ 7 \times 10^{10} \times4 \times 10^{33}} \approx 9.5 \times 10^{14} s \: or \: 3 \times 10^7 years\] In comparison, the age of the oldest moon rocks is \(4.5 \times 10^9 years\) which is about 300 times older than our estimate, which makes me grateful the sun is not solely powered by gravitational contraction.

Acknowledgements: I worked with Barra in this problem.