## A Remnant Problem

I haven’t posted any physics problems for a while so here’s a quickie involving dimensional analysis. You have to assume that the supernova remnant mentioned in the question is roughly spherical, like the one shown above (SNR 0500-67.5):

As usual, answers and comments through the box below please!

Click on the `continue reading’ thing if you would like to see my worked solution:

This is known as the Sedov solution for a spherical blast wave.

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November 16, 2020 at 1:44 pm

Probably made a mistake, but I get R propto E^0.2 p^-0.2 and t^0.4

November 16, 2020 at 5:10 pm

Typo in your equation for powers of mass! You typed \alpha + \gamma = 0 but you mean \alpha + \beta = 0.

November 16, 2020 at 5:48 pm

Oops. Thank you. Now fixed.

November 16, 2020 at 7:33 pm

The assumption here seems to be that there’s no dependence on the initial mass of the star or mass of the ejecta…is that because it’s assumed that the mass of the star has a one-one correlation (approximately) with the energy of the explosion, and so there’s no need for a separate variable?

November 19, 2020 at 2:03 am

I reached the same result in a rather quick and simple manner:

assume that all the energy (E) of the explosion is turned into kinetic

energy (KE) of the swept-up material.

mass M = (4/3) pi R^3 rho

avg velocity v = (R/t)

KE = (1/2) M * v^2 ~ R^3 rho (R^2 / t^2) ~ R^5 rho t^(-2)

So

E ~ R^5 rho t^(-2)

Re-arrange to solve for R

R ~ E^(1/5) rho^(-1/5) t^(2/5)