Blog #31 - Supernovae

Supernovae

(a) Let's estimate the energy output of a supernova. In a core-collapse supernova, the star collapses to a neutron star. A neutron star is a star/stellar remnant supported by neutron

degeneracy pressure (in contrast with white dwarfs, which are held by electron degeneracy pressure) with a diameter of ∼ 10 km and mass of ∼ 1.4 M. Using conservation of energy, we know that the energy in a supernova explosion must come from the liberation of gravitational potential energy. Estimate the energy available in a supernova explosion. Assume that the star's radius prior to collapse is 10 km.

For the first part we assume all of the kinetic energy is converted from potential energy upon explosion: \[KE = PE \rightarrow KE = \frac{3GM^2}{5R}\] Plugging in the numbers given in the problem we get \[KE = \frac{3*6.67*10^{-11}*(1.4*2*10^{30})^2}{5*10^3} = 2.1*10^{46} \space joules \]

(b) Most of the energy from part (a) is carried away by neutrinos, tiny charge less particles that rarely interact with other matter. A portion of it, however, is injected directly into the gas of the interstellar medium. When an energy E is suddenly injected to a background gas of density ρ, a strong blast wave that travels outwards is created. Using an order of magnitude estimate,show that the radius of the blast wave R scales with time t and density as R ∼ t^2/5 ρ^-1/5.
Hint: Assume that the Kinetic Energy of the blast wave is constant. Why might this be a good assumption?

Now we convert the kinetic energy of the blast into the radius that it scales to. \[KE = \frac{1}{2} m v^2 \approx mv^2\] The kinetic energy can be extrapolated by order of magnitude approximation into its components of density and radius as \[mv^2 \approx R^3 \rho (\frac{R}{t})^2 \] Which simplifies into \[KE \approx \frac{R^5 \rho}{t^2}\]Assuming the kinetic energy is constant considering that the losses in free space are negligible, we can rearrange to get [R^5 = \frac{t^2}{rho} \] or simplified to the scaling relation we expected \[R \approx \frac{t^{\frac{2}{5}}}{\rho^{\frac{1}{5}}} \]

(c) The blast wave from part (b) is known as the Sedov-Taylor blast wave, and was studied during the Second World War in the context of trying to estimate the destructive potential of the (then theoretical) atomic bomb. Use your order-of-magnitude estimate from (b) to estimate the velocity v(t) of the blast wave. Hint: Write the velocity v(t) in terms of the derivative of the radius R(t).

We know from (b) that \[R \approx \frac{t^{\frac{2}{5}}}{\rho^{\frac{1}{5}}} \] from this we can use the differential form of \(\frac{dR}{dt} = v(t) \) to solve for v(t). \[\frac{dR}{dt} = \frac{d}{dt}(t^{\frac{2}{5}}\rho^{-\frac{1}{5}}) \] Taking the derivative of the right side we get \[v(t) = \frac{2}{5} t^{-\frac{3}{5}}\rho^{-\frac{1}{5}} \]