Exercises: Atmospheric Re-Entry

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, Earth $\mu = 3.986\times10^5\ \text{km}^3/\text{s}^2$, $R_\oplus = 6{,}371\ \text{km}$, air $c_p \approx 1{,}005\ \text{J/(kg·K)}$, and scale height $H \approx 7\ \text{km}$ unless told otherwise. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each cold first. For the "implement it" problems, do not run the code: hand-trace it and write the result in an # Expected output: comment, exactly as the chapter does.

Part A — Warm-ups: energy, temperature, and blackout (⭐)

7.1 † A spacecraft crosses the entry interface at $v = 7.6\ \text{km/s}$. Compute its kinetic energy per kilogram, in MJ/kg.

7.2 A capsule returns from the Moon and enters at $11\ \text{km/s}$. Compute its kinetic energy per kilogram, and state how many times larger it is than a $7.8\ \text{km/s}$ LEO entry.

7.3 † Estimate the ideal stagnation-temperature rise $\Delta T \approx v^2/(2c_p)$ for a vehicle at $v = 6\ \text{km/s}$. Comment on whether real air actually reaches that temperature, and why.

7.4 A deorbit maneuver needs $\Delta v = 100\ \text{m/s}$. With a thruster of exhaust velocity $v_e = 3\ \text{km/s}$, what mass ratio does the burn require? Roughly what fraction of the vehicle is that propellant?

7.5 † Compute the plasma frequency $f_p = 8.98\sqrt{n_e}\ \text{Hz}$ for an electron density $n_e = 10^{18}\ \text{m}^{-3}$. Is a spacecraft's S-band link at $2.2\ \text{GHz}$ blacked out? Justify.

7.6 The Space Shuttle's black tiles tolerated a surface temperature of about $1{,}260\,^\circ\text{C}$. Convert that to kelvin. How does it compare with the $6{,}000$–$11{,}000\ \text{K}$ of the shock-layer gas, and what keeps the tile so much cooler than the gas a few centimetres away?

Part B — Compression heating and the shock layer (⭐⭐)

7.7 † Using the stagnation heating scaling $\dot q \propto \sqrt{\rho/R_n}\,v^3$, by what factor does the peak convective heat flux change if the entry speed rises from $7.8\ \text{km/s}$ (LEO) to $11\ \text{km/s}$ (lunar return), holding $\rho$ and nose radius $R_n$ fixed?

7.8 Two candidate capsules have nose radii $R_n = 0.3\ \text{m}$ and $R_n = 3\ \text{m}$, identical in every other respect. Using the same scaling, find the ratio of their stagnation heat fluxes. Which is the "blunt body," and by what factor is its stagnation-point heating reduced?

7.9 † In one short paragraph, explain why the naive estimate $\Delta T = v^2/(2c_p)$ predicts an absurd $\sim 30{,}000\ \text{K}$ for a LEO entry, yet the real shock-layer gas settles near $10{,}000\ \text{K}$. What physical processes absorb the missing energy, and how do they connect to the blackout of §7.6?

Part C — Ballistic coefficient and deceleration (⭐⭐)

Use the Allen–Eggers peak deceleration $a_{\max} = v_E^2 \sin\gamma_E / (2eH)$ with $e = 2.718$ and $H = 7{,}200\ \text{m}$.

7.10 † Compute the peak deceleration, in Earth g's, for a ballistic entry with $v_E = 7.8\ \text{km/s}$ and $\gamma_E = 2^\circ$.

7.11 Repeat 7.10 with $\gamma_E = 5^\circ$. By what factor did the peak g change, and does that match the prediction that $a_{\max}\propto \sin\gamma_E$?

7.12 † A dense reentry test article (high ballistic coefficient $\beta$) and a light broad capsule (low $\beta$) enter at the same speed and angle. Which pulls the higher peak g, and at which altitude does each reach its peak? Explain using $a_{\max}$ and the deceleration relation $D/m = \rho v^2/(2\beta)$.

Part D — Implement it in Python (⭐⭐)

Write each function, then hand-trace it for the given inputs and record the result in an # Expected output: comment. Do not run it.

7.13 † Write peak_g(v_entry, gamma_deg, H=7000) returning the Allen–Eggers peak deceleration in g's. Trace it for peak_g(7800, 6, 7000).

7.14 Write specific_ke(v) returning the kinetic energy per kilogram, $\tfrac12 v^2$ (J/kg), for a speed v in m/s. Trace it for specific_ke(7800).

7.15 † Write plasma_frequency(n_e) returning $8.98\sqrt{n_e}$ Hz. Trace it for plasma_frequency(1e16), and state whether a $400\ \text{MHz}$ UHF link would be blacked out.

Part E — Design it (⭐⭐ / ⭐⭐⭐)

7.16 † Design a deorbit burn. A vehicle in a $500\ \text{km}$ circular orbit must lower its perigee to $60\ \text{km}$ altitude to begin re-entry. Using vis-viva from Chapter 6, $v = \sqrt{\mu(2/r - 1/a)}$, compute the retrograde $\Delta v$ required at the burn point. (Hint: the burn is at the high point of the new ellipse; its apogee radius equals the original circular radius.)

7.17 (Mission project) Write the Entry / Disposal note for your chosen mission track (A/B/C/D): does any part enter an atmosphere, at what speed, and what thermal-protection strategy and ballistic- coefficient class does that imply? If it never enters an atmosphere, state why and what the disposal plan is instead. Save it to your MDR.

7.18 † A crewed capsule returns from Mars at $v_E = 12\ \text{km/s}$. Argue quantitatively whether it should fly a ballistic or a lifting entry, using the peak-g scaling and the corridor idea, and name the thermal-protection strategy you would choose and why.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

7.19 † Estimate the total heat energy a $5{,}000\ \text{kg}$ capsule must dissipate returning from LEO ($v \approx 7.8\ \text{km/s}$). Then estimate how many kilograms of water that energy could boil from room temperature (take $\sim 2.6\ \text{MJ/kg}$ to heat and boil water). Compare the result to the capsule's own mass and comment.

7.20 Why can't you just fire retro-rockets to brake gently from orbit and land without a heat shield? Frame your answer in delta-v and mass ratio (use $v_e = 3\ \text{km/s}$), and connect it to the tyranny of the rocket equation.

7.21 † Why can't you just enter as shallowly as possible to keep the g-forces low? Explain the two things that go wrong at the shallow extreme, using the re-entry corridor.

Part G — Interleaved & synthesis (⭐⭐ / ⭐⭐⭐)

7.22 † (Ch. 6) A vehicle is on an elliptical orbit whose perigee just dips into the atmosphere. Using vis-viva, is it moving faster at perigee or apogee? Since heating scales as $\rho v^3$, near which point of the orbit does the heating peak, and why do both $\rho$ and $v$ conspire to put it there?

7.23 (Ch. 4) Ascent drag scales as $\rho v^2$ and re-entry heating as $\rho v^3$; both contain the air density $\rho$, which falls off exponentially with altitude. Explain why the peak of each occurs at a middle altitude rather than at the ground or in vacuum — and name the ascent phenomenon that is the mirror of re-entry's peak heating.

7.24 † (Ch. 3) A returning vehicle proposes to cancel its $7.8\ \text{km/s}$ of orbital speed purely with rockets ($v_e = 3.4\ \text{km/s}$). Compute the mass ratio the maneuver would require and comment on what fraction of the returning vehicle would have to be propellant. What does this tell you about why the atmosphere is used instead?

7.25 (Ch. 2) A capsule returning from the Moon arrives at Earth near escape speed, about $11\ \text{km/s}$, versus $7.8\ \text{km/s}$ from LEO. Using the energy relation $E_k\propto v^2$ and the heating relation $\dot q \propto v^3$, explain quantitatively why a lunar return is so much more punishing, and connect it to the $\sqrt{2}$ relation between escape and circular velocity.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: correct set-up (with units), a stated model and its assumptions, and a sanity check on the final number.