[Physics] Could the end cap of the Pascal B (1) survive its trip through the atmosphere

atmospheric scienceexplosionshistorynuclear-physicsthermodynamics

In 1957, the US had a nuclear test where a shaft was dug 152 meters into the ground. A 100 mm thick, 900 kg steel plug was installed and welded at the top of the shaft. Under it was 2 feet of concrete.

The bomb was detonated created a blast of less than 1 kt. The concrete was vaporized and the steel plug shot into the air at approximately 66 km / second (125,000 – 150,000 mph).

Some believe it became the first man made item to reach orbit, after all, all you need is approximately 7 km/sec to reach orbit, and this greatly exceeded that.

Others believe it was vaporized by the intense pressure of traveling though the atmosphere at this speed.

No one ever reported it coming down anywhere…

Is there a physics / mathematical way to determine what happened to this steel plug?

Best Answer

Building off of akhmeteli's excellent answer, I implemented the differential equations from the asteroid webpage in Mathematica. I then tried to tweak the numbers, within realistic bounds, to get the thing into space. In no realistic case was I able to get the thing more than a few hundred meters up before it completely burned away.

To maximize the distance travelled, we want $\Lambda$ and $A$ to be as small as possible; respectively, these correspond to the rate at which heat is transferred to the "asteroid" and the effective cross-sectional area of the object (taking into account turbulence). In addition, we want the heat of ablation $\zeta$ (the amount of heat require to vaporize a certain mass of the substance) to be relatively high, since this will reduce the rate at which mass is lost.

The parameter $\Gamma$ also has an effect; it describes the amount of drag experienced with the atmosphere. Interestingly, one can actually get the projectile higher by increasing the drag: a higher drag means the projectile slows down faster, but that means that the projectile can slow down enough enough that it doesn't burn up immediately.

My optimistic estimates are $\Lambda \approx 0.15$ (note that this number is used in the code example on the page) and $A = 1$ (which would be more streamlined than a sphere). I also used $\zeta = 10\times 10^6$ J/kg, since it was the highest "typical" value in the table.1 Finally, I used $\Gamma = 0.5$, an estimate given on that webpage for the lower atmosphere.

Here's the result of the simulation, with the parameters given above. The vaporization of the plug is complete at a height of 312 meters.

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And here is the simulation for akhmeteli's parameters, with $\Gamma = 0.5$. The plug does not significantly change its velocity before it burns up; the final height is a little over 6 meters. As would be expected, this is within an order of magnitude of akhmeteli's back-of-the-envelope estimate.

enter image description here enter image description here

If you tweak the unknown parameters of my "optimistic" case above, you can attain a height of 1 km if:

  • $\Lambda \approx 0.064$ (more than twice as small)
  • $A \approx 0.031$ (more than three times smaller)
  • $\zeta \approx 23.5 \times 10^6$ J/kg (over twice as large.)
  • $\Gamma \approx 1.6$ (much more drag—this slows it down sufficiently before too much of it burns away)

All in all, it seems unlikely that the plug got anywhere near space.

Mathematica Code:

Feel free to tweak this code as you see fit. The code stops integration when either the mass of the steel falls below 1 gram, or the speed falls below 1 m/s. The code does implement a height-dependent atmospheric density via a simple exponential model, though it turns out not to be all that relevant for realistic parameters. The acceleration due to gravity is assumed to be constant.

Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
Λ = 0.15;(*heat transfer *)
A = 1;(*shape factor *)
Γ = 0.5; (*drag coefficient*)
ρa0 = 1.25 ;(*atmo. density *)
v0 = 66000; (*initial velocity*)
m0 = 900 ;(*initial mass*)
ζ = 7*10^6;(*heat of ablation*)
ρm = 7800;(*steel density*)
h = 7000; (*atmospheric "height"*)
a = Γ A ρa0 / ρm^(2/3);
b = Λ A ρa0/(2 ζ ρm^(2/3));
soln = NDSolve[{x''[t] == - a Exp[-x[t]/h] x'[t]^2/m[t]^(1/3) - 9.8 m[t], 
   m'[t] == -b Exp[-x[t]/h] x'[t]^3 m[t]^(2/3), x[0] == 0, 
   x'[0] == v0, m[0] == m0, 
   WhenEvent[{m[t] < 0.001, x'[t] < 1}, "StopIntegration"]}, 
    {x, m}, {t, 0, 1000}]
{ti, tf} = First[InterpolatingFunctionDomain[x /. First[soln]]]
Plot[x'[t] /. First[soln], {t, ti, tf}, PlotRange -> {0, 66000}, 
 AxesLabel -> {"Time (s)", "Velocity (m/s)"}]
Plot[m[t] /. First[soln], {t, ti, tf}, PlotRange -> {0, 900}, 
 AxesLabel -> {"Time (s)", "Mass (kg)"}]
x[tf] /. First[soln]

1 It is not clear to me whether these are the appropriate units for $\zeta$; the page is unclear. They're dimensionally correct, though.

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