Boom: Blog 21, Worksheet 7.1, Problem 4

This week we are going to talk about one of the brightest, yet most transient, objects in our sky: supernovae.

Source

Astronomers believe that most supernovae are the result of 1) the gravitational collapse of a massive star at the end of its main-sequence life or 2) the explosion of one or more white dwarfs, likely caused by the collision between two white dwarfs. In this problem, we’ll focus on the second type of supernovae, or Type Ia supernovae which we are familiar with.

Calculate the total energy output, in ergs, of the explosion, assuming that the white dwarf’s mass is converted to output energy via fusion of carbon into nickel. Note that the process of carbon fusion is not entirely efficient, and only about 0.1% of this mass will be radiated away as electromagnetic radiation (light!). How does this compare to the total binding energy, in ergs, of the original white dwarf? Does the white dwarf completely explode, or is some mass left over in the form of a highly concentrated remnant?

To calculate energy output we can call upon possibly the most famous equation in science: \[ E=mc^2 \] Well, 0.1% of that energy anyway. Because we know the speed of light \( c = 3 \times 10^8)^2 \) and the mass of a white dwarf is \( M_{WD} = 1.4 M_{\odot}\). Plugging everything in gives us: \[E = \frac{1}{1000} (1.4 \times 2 \times 10^{32} ) ( 3 \times 10^{10})^2  \frac{1}{1000} (1.4 M_{\odot} ) ( 3 \times 10^8)^2 = 2.7 \times 10^{51} \: ergs \]

We can find binding energy from the potential energy of the star: \[U = E = \frac{GM^2}{R} \] We also know the radius of the White Dwarf is about twice the radius of the Earth. \[ E_{binding} = \frac{ c_s  M_{WD}^2 }{R_{WD}} = \frac{ (6.67 \times 10^{-8}) (1.4 \times 2 \times 10^{33})^2 }{2 \times 6.4 \times 10^8} \approx 4.1 \times 10^{50} \: ergs \]

There is something off here, the energy output is not equal to the energy input. So how can that be? Thanks to Einstein (e.g. \(E = mc^2 \) ) we know that mass can be converted to energy. In the explosion, some of the mass of the white dwarf gets turned into energy, and the rest is flung out throughout the galaxy. Explosions like these can even help explain why we have heavier elements dispersed through the Milky Way.