Exercises: Interplanetary Trajectories

Work these with a calculator. Constants (Appendix B): $\mu_\odot = 1.327\times10^{11}\ \text{km}^3/\text{s}^2$, $\mu_\oplus = 3.986\times10^5\ \text{km}^3/\text{s}^2$, $\mu_{\text{Mars}} = 4.283\times10^4\ \text{km}^3/\text{s}^2$, $1\ \text{AU} = 1.496\times10^8\ \text{km}$, Earth mean radius $6{,}371\ \text{km}$, Mars mean radius $3{,}390\ \text{km}$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/answers-to-selected.md — try each cold before you peek. For "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. Remember to always ask "relative to which body?" and to sanity-check every number.

Part A — Warm-ups: convert and compute (⭐)

11.1 † A departure trajectory has a characteristic energy $C_3 = 12\ \text{km}^2/\text{s}^2$. What is its hyperbolic excess velocity $v_\infty$?

11.2 A mission needs a hyperbolic excess of $v_\infty = 3.6\ \text{km/s}$ at departure. What $C_3$ must the launch vehicle deliver?

11.3 † Earth's sphere of influence has radius $\approx 924{,}000\ \text{km}$. Express this in Earth radii ($6{,}371\ \text{km}$) and in Earth–Moon distances ($384{,}400\ \text{km}$).

11.4 Compute Venus's heliocentric orbital speed. Its orbit is at $a = 0.723\ \text{AU}$. Sanity-check against the fact that inner planets move faster than Earth.

11.5 † Find the Earth–Venus synodic period. Venus's orbital period is $224.70\ \text{days}$; Earth's is $365.25\ \text{days}$. Express the answer in days and months.

11.6 A Hohmann transfer to Jupiter has perihelion at $1\ \text{AU}$ and aphelion at Jupiter's orbit, $5.203\ \text{AU}$. Find the transfer ellipse's semi-major axis in AU and in km.

Part B — The interplanetary Hohmann (⭐⭐)

11.7 † Compute the full Earth→Venus Hohmann transfer ($r_1 = 1\ \text{AU}$, $r_2 = 0.723\ \text{AU}$ — an inward transfer). Find the hyperbolic excess velocity at Earth, the hyperbolic excess at Venus, and the transfer time. (Hint: for an inward transfer, Earth is at the ellipse's aphelion and Venus at its perihelion; the spacecraft must end up moving slower than Earth.)

11.8 Compute the full Earth→Jupiter Hohmann ($r_2 = 5.203\ \text{AU}$): the departure $v_\infty$ and $C_3$, the arrival $v_\infty$ at Jupiter, and the transfer time. Compare the departure $C_3$ with Mars's ($\approx 8.7$). What does the comparison say about why we send probes to Jupiter via gravity assists?

11.9 † The Hohmann transfer time is $t_{\text{trans}} = \pi\sqrt{a_t^3/\mu}$. Use it to find the one-way transfer time to a main-belt asteroid at $2.7\ \text{AU}$ (perihelion at $1\ \text{AU}$).

11.10 In one physical sentence, explain why a Hohmann transfer to Jupiter costs so much more departure $v_\infty$ than one to Mars, and state the approximate ratio of their $C_3$ values.

Part C — Launch windows and the synodic period (⭐⭐)

11.11 † Compute the Earth–Saturn synodic period (Saturn's period is $10{,}759\ \text{days}$). Explain why it is only slightly more than one year, even though Saturn's year is nearly 30 of ours.

11.12 A Mars mission slips its launch by two weeks and misses its window. Estimate how long until the next comparable opportunity, and explain in one sentence why a two-week slip can cost far more than two weeks.

11.13 † Compute the phase (lead) angle for an Earth→Venus transfer, using Venus's period ($224.70\ \text{days}$) and the transfer time from 11.7. Is Venus ahead of or behind Earth at departure?

Part D — Departure, arrival, and $C_3$ (⭐⭐)

11.14 † Compute the trans-Mars injection burn from a 200 km parking orbit (for $v_\infty = 2.95\ \text{km/s}$) and compare it with the **300 km** result of §11.4 ($3.59\ \text{km/s}$). Is it larger or smaller? Does your answer contradict the Oberth effect? Explain.

11.15 † Size a Mars orbit-insertion burn into an elliptical capture orbit with periapsis at $400\ \text{km}$ altitude ($r_p = 3{,}790\ \text{km}$) and apoapsis at $30{,}000\ \text{km}$ altitude ($r_a = 33{,}390\ \text{km}$), arriving with $v_\infty = 2.65\ \text{km/s}$. Compare it with the $2.08\ \text{km/s}$ needed to capture directly into a low circular orbit.

11.16 (Design it — conceptual) Above roughly what arrival $v_\infty$ does propulsive Mars capture become painfully expensive, and why does aerocapture become relatively more attractive as $v_\infty$ rises? (No arithmetic required — argue from where each method's cost comes from.)

11.17 † A faster, non-Hohmann Mars trajectory needs $v_\infty = 3.8\ \text{km/s}$ at Earth departure. Find its $C_3$ and its trans-Mars injection burn from a $300\ \text{km}$ parking orbit, and compare with the minimum-energy Hohmann ($v_\infty = 2.95$). What did the shorter trip cost at departure?

Part E — Implement it in Python (⭐⭐)

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

11.18 † Write hohmann_transfer(mu, r1, r2) returning (v_inf_depart, v_inf_arrive, tof_seconds) as in the chapter. Trace it for Earth→Mars: hohmann_transfer(1.327e11, 1.496e8, 2.2799e8).

11.19 Write synodic_period(T1, T2). Trace it for Earth–Jupiter with T1 = 365.25, T2 = 4332.59 (days).

11.20 † Write injection_dv(mu, r_park, v_inf) returning the departure burn $\sqrt{v_\infty^2 + 2\mu/r_{\text{park}}} - \sqrt{\mu/r_{\text{park}}}$. Trace it for injection_dv(3.986e5, 6671, 2.95).

Part F — Gravity assists (⭐⭐ / ⭐⭐⭐)

11.21 † A spacecraft approaches Jupiter ($v_J = 13.06\ \text{km/s}$) with $v_\infty = 8\ \text{km/s}$ directed opposite to Jupiter's motion, so its heliocentric speed is $13.06 - 8 = 5.06\ \text{km/s}$. An (idealized) $180^\circ$ flyby flips $v_\infty$ to point along Jupiter's motion. Find the heliocentric speed afterward and the gain. Show that the gain equals $2v_\infty$.

11.22 Using $e = 1 + r_p v_\infty^2/\mu$ and $\sin(\delta/2) = 1/e$, find the turning angle for a Venus flyby with $v_\infty = 5\ \text{km/s}$ at closest approach $r_p = 6{,}300\ \text{km}$ ($\mu_{\text{Venus}} = 3.249\times10^5\ \text{km}^3/\text{s}^2$).

11.23 † Explain, using the eccentricity and turn-angle formulas, why a closer flyby (smaller $r_p$) produces a larger change in the spacecraft's velocity.

11.24 (⭐⭐⭐, back of the envelope) Estimate the maximum heliocentric speed a spacecraft could reach from a single Jupiter flyby, given $v_J \approx 13\ \text{km/s}$ and an arrival $v_\infty$ up to $\approx 15\ \text{km/s}$. Then explain in one or two sentences why Voyager chained four planets rather than seeking one enormous flyby.

Part G — Find the error, "why can't you just…", and interleaved (⭐⭐⭐)

11.25 † Find the error. A student says: "To reach Mars, add the required $2.95\ \text{km/s}$ to Earth's $29.78\ \text{km/s}$ heliocentric speed, so the spacecraft travels at $32.73\ \text{km/s}$ the whole way to Mars." What is wrong, and what is the spacecraft's speed when it reaches Mars's orbit?

11.26 Why can't you just fly a straight line to Mars — point at it and thrust? Frame your answer in terms of gravity, delta-v, and the fact that Mars is moving.

11.27 † (Interleaved, Ch. 6) Compute the specific orbital energy $\varepsilon = -\mu_\odot/(2a_t)$ of the Earth→Mars transfer ellipse, then use $v = \sqrt{2(\varepsilon + \mu_\odot/r)}$ to recover the perihelion and aphelion speeds. Confirm they match the $32.73$ and $21.48\ \text{km/s}$ from vis-viva.

11.28 (Interleaved, Ch. 10) The interplanetary Hohmann uses identical math to the LEO→GEO Hohmann. Set up the LEO→GEO transfer ($r_1 = 6{,}771\ \text{km}$, $r_2 = 42{,}164\ \text{km}$, $\mu_\oplus$) and list which quantity in it corresponds to which in the Mars transfer (central body, the two "orbits," the burns, the timing law).

11.29 † (Back of the envelope) A crewed mission wants to cut the one-way Mars trip from $\approx 259$ to $\approx 180$ days. Must it spend more or less departure $C_3$? Explain physically, and estimate the direction and rough size of the change.

11.30 (Design it — Mission project) For your chosen track, add the interplanetary transfer block to your MDR: departure and arrival $v_\infty$, departure $C_3$, transfer time, synodic period (your launch cadence), and an arrival delta-v (orbit insertion) or an aerocapture note. Track C uses Mars as computed in the chapter; Track D substitutes your asteroid's semi-major axis for Mars's $1.524\ \text{AU}$. Add the TMI and insertion burns to the delta-v budget you began in Chapter 3.

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: the correct central body and $\mu$, explicit units, an "relative to which body?" check on every velocity, and a sanity check on the final number (e.g., "is $C_3$ in the right ballpark for that destination?").