What Is the Kinetic Energy Generated by a White Dwarf Collapse?

[HawkEye5220]

Homework Statement

Calculate the kinetic energy generated when a white dwarf in a binary system accretes enough material to increase its mass beyond the Chandrasekhar limit (1.4 solar masses) and collapse to the radius of a neutron star.

radius of white dwarf- 5520 km
radius of neutron star- 12 km

Homework Equations

U=−$\frac{3}{5}$ $\frac{GM^{2}}{R}$

The Attempt at a Solution

It has been a while since I have done a problem like this but I was thinking that if I found the gravitational potential energy at both radii and then found the difference, this difference would be the kinetic energy generated in the collapse. Am I on the right track?

 

[mfb]

Right. You have the right formula there, so you just have to plug in numbers.

 

[HawkEye5220]

Thank you, it is one of those problems where it seemed too simple.

 

[HawkEye5220]

Ok, after calculating. I get (-5.6*10^43 Joules) for the white dwarf and (-2.82*10^46 Joules) for the neutron star. This is a difference of 2.814*10^46 Joules, is this the amount of kinetic energy generated? Also, when I compare this number to the 10^44 Joules released by a supernova explosion it is 280 times greater. Why is this? Is it due to the fact that it requires more energy for atoms to attract to each other than it does for them to separate? Also what would the required efficiency of energy conversion be? I can't find a formula for the efficiency in our book anywhere.

 

[mfb]

I get slightly different values with WolframAlpha, but that could be a rounding error.

Core collapse supernovae release much more than 10^44 J. Looking at this wikipedia table and reference 101 there the emitted energy is of the order of 10^46 J. Some other fraction of the energy gets used for endothermic fusion reactions. Looks good.

 

[HawkEye5220]

I should have specified that I am comparing it to the kinetic energy of the supernova explosion which is on the order of 10^44 J. This is where I am confused.

 

[mfb]

The kinetic energy of the fragments that get ejected? There is no reason to assume it would be similar to the total energy that gets released in the supernova.