Subfield: Mechanics

2. State secrets and the stars 

In 1949, the physicist G. I. Taylor used publicly released images of the Trinity nuclear test to estimate the yield of the world's first atomic bomb (i.e., the energy released by the detonation). His estimate was stunningly accurate, despite the fact that the number itself was still classified by the government at the time. In this problem, we will follow a simplified version of his estimate, and use the procedure to estimate the energy released by the supernovae which end the lives of massive stars.

<img_4341>

Figure 1: A snapshot of the Trinity blast wave 15 milliseconds after the explosion, reproduced from Taylor's original paper. A helpful scale bar is included.
Context question:
(a) After a very short moment, the shock front caused by an explosion sweeps up the material around it, increasing the mass which is blasted outwards. At early times, the energy $E \sim M v^{2} / 2$ can be thought of as roughly conserved.

Let $R$ be the radius of the explosion at some time $t$. Estimate the speed of the shock front as $v \sim R / t$ and the swept up mass $M \sim \rho R^{3}$, where $\rho$ is the density of the surrounding material (don't bother to include factors like $4 \pi / 3$ ).

Estimate the radius of the explosion $R(t)$ as a function of the energy yield $E$, elapsed time $t$, and the density of the surrounding medium $\rho$.
Context answer:
\boxed{$\left(\frac{E}{\rho}\right)^{1 / 5} t^{2 / 5}$}


Context question:
(b) Air has a density $\rho \simeq 1 \mathrm{~kg} \mathrm{~m}^{-3}$. Using the provided image, roughly estimate the yield of the Trinity explosion in kilotons (of TNT).

Note that 1 kiloton is equal to approximately $4.2 \times 10^{12} \mathrm{~J}$.
Context answer:
\boxed{$11$}


Question: (c) In 1054, Chinese astronomers observed and documented a supernova which was bright enough to be visible during the day for around a month. The rubble left behind is a rapidly expanding supernova remnant called the Crab Nebula, which is intensely studied today.


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Figure 2: Photographs of the Crab Nebula taken by the Hale Telescope in 1950 (top left) and the Hubble Space Telescope in 2000 (bottom right).

The surrounding interstellar medium contains about 1 proton per cubic meter-protons have a mass $m_{p} \approx 1.7 \times 10^{-27} \mathrm{~kg}$. While not perfectly spherical, Crab Nebula has a radius of roughly $1.7 \mathrm{pc}\left(1 \mathrm{pc}=3.1 \times 10^{16} \mathrm{~m}\right)$. From this information and the given images of the Crab Nebula from 1950 and 2000, estimate the energy released by the initial supernova.

Your answer should be close to the typical energy scales of supernovae, which briefly outshine their host galaxies and can be seen from across the universe.

Solution:
The density of the interstellar medium is

$$
\rho \sim \frac{m_{p}}{\left(1 \mathrm{~cm}^{3}\right)} \sim 1.7 \times 10^{-21} \mathrm{~kg} \mathrm{~cm}^{-3}
\tag{5}
$$

We are given that the Crab Nebula has a radius of approximately 1.7 pc. We see that the explosion seems to expand by $\sim 25 \%$ or so during the fifty years of observation along the long axis, which we take to be a very rough measure of its velocity. In particular,

$$
v \sim \frac{25 \% \times 1.7 \mathrm{pc}}{50 \mathrm{yr}} \sim 8.3 \times 10^{6} \mathrm{~m} \mathrm{~s}^{-1}
\tag{6}
$$

We can use $E \sim M v^{2}$ and $M \sim \rho R^{3}$ to write

$$
E \sim \rho R^{3} v^{2} \sim 1.7 \times 10^{43} \mathrm{~J}
\tag{7}
$$

where we have used $t \sim 1000 \mathrm{yr}$ (the age of the supernova remnant). This is $E \sim$ $1.7 \times 10^{50} \mathrm{erg}$, which is in the neighborhood of true supernovae which are typically at $\sim 10^{51} \mathrm{erg}$ (a quantity of energy which has been affectionately named the "f.o.e." by astrophysicists because of this fact).

Final answer: $1.7 \times 10^{43}$ J