Appendix G: The Delta-v Map of the Solar System

Distance is the wrong way to think about the solar system. The Moon is $384{,}400\ \text{km}$ away and Mars, at its closest, is roughly a thousand times farther — yet in the sense that decides how much rocket you must build, the two trips are almost the same size once you are off the launch pad. The quantity that actually measures the cost of going somewhere is not kilometers but delta-v ($\Delta v$), the velocity change your propulsion must supply, introduced in Chapter 3. This appendix gathers the delta-v "prices" between the important places in the inner solar system into a single chart: the delta-v map, the closest thing spaceflight has to a subway map.

On a subway map the geographic distances are distorted and beside the point; what matters is which stations connect and what each hop costs. Here the "fare" for each hop is a delta-v in km/s, and a mission is a route across the map whose total cost is the sum of the fares along it — its delta-v budget (see Chapter 3, §3.6 and Chapter 29). The numbers below are consistent with the simplified map in Chapter 3 and are derived from the constants in Appendix B.

A word on the numbers and their honesty. Every value here is quoted in km/s and tagged with a source tier. Tier 2 means a widely reported, approximate figure — the standard delta-v-map value, reproducible from two-body orbital mechanics using Appendix B, but not pinned to one primary source (exact values shift with launch site, orbit altitude, and date). Tier 3 means an illustrative round number chosen for teaching (a nominal margin, a sample multi-year budget). Every leg is an ideal delta-v — the minimum an impulsive, two-body, minimum-energy maneuver would need. Real missions cost more; §"How to use the map" explains how much more, and why.

G.1 The map

  THE DELTA-v MAP OF THE INNER SOLAR SYSTEM
  (each leg is an ideal delta-v in km/s; Tier 2 unless noted otherwise)

   Earth surface
        |
        |  9.4      surface -> LEO   (includes gravity + drag + steering losses)
        v
       LEO ----2.5----> GTO ----1.5----> GEO ---( +0.05 / yr: station-keeping )
        |
        +----3.1----> Trans-lunar injection (TLI)
        |                    |
        |                    |  0.7   insertion into low lunar orbit
        |                    v
        |               Low Lunar Orbit
        |                    |
        |                    |  1.7   powered descent (ascent back up ~1.7)
        |                    v
        |               Moon surface
        |
        +----3.2----> Earth escape (C3 = 0) ......... gateway to the planets
        |
        +----3.6----> Trans-Mars injection --(cruise; date-dependent)--> Mars
        |                                            capture / orbit:
        |                                            ~0.9-2.1  propulsive
        |                                            (~0 with aerocapture)
        +----3.5----> Trans-Venus injection
        +----6.3----> Trans-Jupiter injection
        +----7.3----> Trans-Saturn injection

Every branch that leaves LEO is a burn from low Earth orbit. The trunk (surface → LEO) is the one leg you pay before any of the others become available, and — as the map makes visible — it is by far the most expensive single step in the entire solar system.

G.2 Earth–Moon system legs

Reference altitudes: LEO is taken as a $\sim 200\ \text{km}$ circular orbit (orbital speed $7.79\ \text{km/s}$; see Appendix B); GTO is a geostationary transfer orbit with perigee at LEO and apogee at GEO altitude ($35{,}786\ \text{km}$).

The Earth–Moon block already tells the book's central story — the tyranny of the rocket equation. Reaching LEO costs $9.4\ \text{km/s}$; everything after that, all the way to the lunar surface, adds up to only about $3.1 + 0.7 + 1.7 = 5.5\ \text{km/s}$. Half your delta-v to the Moon is spent in the first eight and a half minutes, fighting Earth's gravity and atmosphere to reach orbit at all.

G.3 Departure (injection) legs from LEO to the planets

These are LEO departure burns — the single impulsive burn from low Earth orbit that places you on a minimum-energy (Hohmann) transfer to each destination. They correspond to a departure hyperbola with excess speed $v_\infty$ and characteristic energy $C_3 = v_\infty^2$; the mechanics are derived in Chapter 11. The arrival column is a separate cost that depends heavily on what you do at the far end.

Two instructive extremes. Escaping the solar system from LEO (a prograde burn, no gravity assist) costs on the order of $8.8\ \text{km/s}$ — barely more than reaching Neptune. Falling into the Sun is far harder: you must cancel most of Earth's $\approx 30\ \text{km/s}$ orbital motion, on the order of $24\ \text{km/s}$ from LEO (Tier 2, strongly model-dependent). Reaching the Sun costs more delta-v than leaving the solar system — the reason the Parker Solar Probe used repeated Venus flybys to shed velocity rather than burning it off.

Notice the pattern that the outer planets reveal: past Jupiter, the Hohmann departure cost flattens near the solar-escape value and the trip times balloon to decades. This is exactly why the elegant trick of the gravity assist — stealing a sliver of a planet's orbital momentum for free — is not a curiosity but a necessity for the outer solar system (Chapters 11 and 15).

G.4 How to read the map

Once in LEO, you are halfway to anywhere. The map makes visible a truth that surprises everyone the first time they meet it: the $9.4\ \text{km/s}$ to reach LEO is larger than the departure burn to any planet. Trans-Mars injection is only $3.6\ \text{km/s}$ on top of it; full Earth escape only $3.2$. In delta-v terms the hardest part of a voyage to Mars is the first two hundred kilometers, not the next two hundred million. (Chapter 3 quotes the line usually attributed to Robert Heinlein — "once you are in orbit you are halfway to anywhere" — and the map is why it is true.)

A route is a sum. To budget a mission, trace its path across the map and add the legs. Example — a cargo lunar lander delivered from LEO by its own stage (Track B of the Mission Design project): the spacecraft's own delta-v is $\mathrm{TLI} + \mathrm{LOI} + \text{descent} = 3.1 + 0.7 + 1.7 = 5.5\ \text{km/s}$ (one way, no return). The launch vehicle separately pays the $9.4$ to reach LEO. Feed that $5.5\ \text{km/s}$ into the rocket equation and you have the propellant fraction the lander must carry — the whole point of building a budget.

The Oberth effect makes some burns cheaper than the map suggests. A given delta-v buys more energy when you are moving fast, because kinetic energy grows as $v^2$: a burn made deep in a gravity well, at periapsis where speed is highest, adds more orbital energy than the same burn made slowly far away. This is why departure burns are done in a single hard push at LEO periapsis rather than spread out, and why a low, fast flyby is the most efficient place to add energy to a trajectory. The energetics behind this are developed in Chapter 6 (specific orbital energy and vis-viva) and applied to injection burns in Chapter 11.

G.5 How to use the map — and how not to

Treat every number here as an ideal floor, not a mission budget. Between the map and a real vehicle sit several unavoidable additions:

• Losses on the way up — gravity, drag, and steering losses. For the surface → LEO leg these are already baked into the $9.4$ (the ideal orbital speed is only $\sim 7.8\ \text{km/s}$; ascent losses account for the rest — see Chapter 4).
• Finite-burn and pointing losses for in-space maneuvers, plus trajectory-correction maneuvers to fix navigation error.
• Plane changes, which are expensive and are not shown as separate legs here; add them where your launch inclination differs from your target orbit's (this is much of why real GTO → GEO exceeds the ideal $1.5$).
• Reserves for disposal, contingency, and unmodeled effects.

A common practice in early design is to carry a margin of 5–10% (Tier 3, illustrative) on the summed ideal budget, tightening it as the design matures; Chapter 29 formalizes how margins are set and tracked. The map gets you the shape of a mission and a first, honest estimate of its size; it does not replace a real trajectory computation.

A note on precision and dates. The Earth–Moon legs are close to fixed, because the Moon's orbit is nearly constant. The interplanetary departure figures are not: they assume a minimum-energy transfer with the planets ideally placed, and real planets move. The actual delta-v to Mars or Venus depends on the launch date, rises and falls over each planet's synodic period (Mars swings noticeably over its $\sim 26$-month cycle, and again over a longer $\sim 15$-year cycle as its orbital eccentricity comes into play), and requires a time-dependent ephemeris to pin down. Computing a specific window — the departure and arrival dates, the transfer time, and the exact $C_3$ — is the work of Chapter 11. Read the planetary rows here as "typical minimum-energy cost," never as a guaranteed fare.