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Example · The Trinity Blast Wave (Sedov–Taylor)

Problem statement

A point explosion releases energy \(E\) into a uniform atmosphere of density \(\rho_0\). Find how the blast-wave radius \(R\) grows with time — and use a declassified photograph to estimate the yield of the 1945 Trinity test, as G. I. Taylor famously did.

Given information

  • Instantaneous point release of energy \(E\); ambient density \(\rho_0\)
  • Strong-shock regime: ambient pressure negligible compared to post-shock pressure

Solution strategy

Route 1 — pure dimensional analysis. Route 2 — physical energy accounting with strong-shock jump conditions, yielding the same scaling with its structure exposed.

Route 1 · Dimensional analysis

Only \(E\,[ML^2T^{-2}]\), \(t\,[T]\), \(\rho_0\,[ML^{-3}]\) can matter. The unique combination with dimensions of length:

\[R \sim \left(\frac{E t^2}{\rho_0}\right)^{1/5}\]

Route 2 · Energy accounting

  1. Swept-up mass: \(M(t) = \frac{4\pi}{3}\rho_0 R^3\).
  2. Post-shock pressure (strong shock): \(p_2 = \frac{2}{\gamma+1}\rho_0 \dot R^2\).
  3. Total (kinetic + thermal) energy of the shell: $\(E = C_s\, M(t)\,\dot R^2, \qquad C_s = \frac{\gamma^2 + 3}{2(\gamma^2 - 1)}\)$
  4. \(E\) constant ⇒ \(M\dot R^2 = \text{const}\) ⇒ \(R^{3/2}\dot R = \text{const}\)
  5. Power law \(R \sim t^\alpha\): \(R^{3/2}\cdot R/t = \text{const} \Rightarrow R^{5/2} \propto t\):
\[\boxed{\,R \sim E^{1/5}\rho_0^{-1/5}\, t^{2/5}\,}\]

Final answer — Taylor's estimate

From the published Trinity photo at \(t = 25\) ms, \(R \approx 140\) m, \(\rho_0 = 1.2\ \text{kg/m}^3\):

\[E \sim \frac{\rho_0 R^5}{t^2} = \frac{1.2 \times (140)^5}{(0.025)^2} \approx 10^{14}\ \text{J} \approx 25\ \text{kt TNT}\]

The classified official yield was ~21 kt. One photograph, one exponent, embarrassingly close — the security establishment was not amused.

Key takeaways

  • Self-similarity: with no intrinsic length scale, the blast has the same shape at every moment, just rescaled — the hallmark of scale-free dynamics.
  • The same \(t^{2/5}\) law governs supernova remnants in their adiabatic (Sedov) phase — 20 orders of magnitude away in energy.
  • \(R \propto E^{1/5}\): yield estimates from blast radius are extremely robust (a factor 10 in energy moves \(R\) by only 1.6×) — and conversely, blast damage radius scales weakly with warhead size.

Dimensional analysis · Shock waves · Rankine–Hugoniot conditions