Estimate the Debye length D , the Debye number N D , the plasma frequency fp

Estimate the Debye length λ_D, the Debye number N_D, the plasma frequency fp ≡ ω_p/2π, and the electron deflection timescale t_D^ee ∼ t_D^ep, for the following plasmas.

(a) An atomic bomb explosion in Earth’s atmosphere 1ms after the explosion.

(b) The ionized gas that enveloped the Space Shuttle (Box 17.4) as it reentered Earth’s atmosphere.

(c) The expanding universe during its early evolution, just before it became cool enough for electrons and protons to combine to form neutral hydrogen (i.e., just before ionization “turned off ”). 

Box 17.4

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BOX 17.4. SPACE SHUTTLE NASA's (now retired) Space Shuttle provides many nice examples of the behavior of supersonic flows. At launch, the shuttle and fuel had a mass ~2 x 106 kg. The maximum thrust, 7 ~ 3 × 107 N, occurred at lift-off and gave the rocket an initial acceleration relative to the ground of ~0.5g. This increased to ~3g as the fuel was burned and the total mass diminished. Most of the thrust was produced by two solid-fuel boosters that burned fuel at a combined rate of m~ 10,000 kg s-¹ over a 2-min period. Their combined thrust was T~2 x 107 N averaged over the 2 minutes, from which we can estimate the speed of the escaping gases as they left the nozzles' skirts. Assuming this speed was quite supersonic (so P << Pev), we estimate that ve~T/m ~2 km s. The combined exit areas of the two skirts was A~20 m², roughly four times the combined throat area, A.. Using Eq. (17.9) with y~ 1.29, we deduce that the exit Mach number was Me~ 3. From T pA, and Pe= Cepe/y, we deduce the exit pressure, Pe~T/(M²A)~8 × 104 N m2, about atmospheric. The stagnation pressure in the combustion region was [combine Eqs. (17.2a), (17.2d), and (17.8)] ~ B~B [1 + 0²-DIME 2 35 atmospheres. (1)