Appendix E: Orbital Mechanics Formula Reference
This appendix gathers, in one scannable place, the orbital-mechanics formulas the book builds — grouped by the job they do. Each entry gives the equation, a one-line statement of what it is for, and a link to the chapter where the book derives it. Go there for the physical reasoning, the assumptions behind the formula, and a worked example with real numbers; this page is for when you already understand a relation and just need to reach for it again. Symbols follow Appendix A; the numerical constants that go into these formulas — Earth's gravitational parameter $\mu_\oplus$, equatorial radius $R_\oplus$, the oblateness coefficient $J_2$, and planetary data — are the teaching values in Appendix B. Everything here assumes the two-body (Keplerian) model — one central body of gravitational parameter $\mu = GM$ — except the final section, which adds the leading perturbation.
Units and conventions. Keep $\mu$, $r$, $a$, and $v$ in one consistent set of units; this book works in kilometres and km/s, so $\mu$ is in $\text{km}^3/\text{s}^2$ and speeds come out in km/s. The radius $r$ is the distance from the centre of the central body, not the altitude; the semi-major axis $a$ is the orbit's size — the two are equal only for a circle. Specific energy $\varepsilon$ and $C_3$ are quoted per unit mass ($\text{km}^2/\text{s}^2 = \text{MJ/kg}$). Angles inside a trig function are in radians unless a degree symbol is shown. A formula written with $\mu_\oplus$ or $R_\oplus$ is Earth-specific but holds for any body with that body's constants.
Two-body basics
Everything in a single gravity field reduces to a handful of relations tying together where you are ($r$), how big your orbit is ($a$), how fast you are moving ($v$), and how much energy you carry ($\varepsilon$). The star of the group is vis-viva: hand it $r$ and $a$ and it returns your speed anywhere on any conic — every impulsive maneuver later is just two vis-viva speeds subtracted.
| Formula | What it's for | Derived in |
|---|---|---|
| $v_{\text{circ}} = \sqrt{\mu/r}$ | speed on a circular orbit of radius $r$ — the baseline "orbit speed" at that altitude | Ch. 2 · Ch. 6 |
| $v^2 = \mu\!\left(\dfrac{2}{r}-\dfrac{1}{a}\right)$ | vis-viva — speed at radius $r$ on any orbit of semi-major axis $a$ (the workhorse) | Ch. 6 |
| $\varepsilon = \dfrac{v^2}{2}-\dfrac{\mu}{r}$ | specific orbital energy from a state $(v,r)$; conserved as you coast | Ch. 6 |
| $\varepsilon = -\dfrac{\mu}{2a}$ | the same energy from the orbit's size; invert for $a = -\mu/(2\varepsilon)$ | Ch. 6 |
| $v_{\text{esc}} = \sqrt{2\mu/r} = \sqrt{2}\,v_{\text{circ}}$ | escape speed from radius $r$ — the marginal ($\varepsilon = 0$) parabola | Ch. 2 · Ch. 6 |
| $v_\infty = \sqrt{2\varepsilon} = \sqrt{C_3}$ | leftover speed at infinity on an escape hyperbola | Ch. 6 · Ch. 11 |
| $\Delta v = \sqrt{v_\infty^2 + \dfrac{2\mu}{r}} - \sqrt{\dfrac{\mu}{r}}$ | burn from a circular parking orbit (radius $r$) to reach hyperbolic excess speed $v_\infty$ | Ch. 6 · Ch. 11 |
| $\Delta\varepsilon = v\,\Delta v + \tfrac12(\Delta v)^2$ | energy a burn $\Delta v$ adds at speed $v$ — the Oberth effect (burn deep and fast) | Ch. 6 |
| $a_g = \mu/r^2$ | local gravitational acceleration (m/s²); the falling body's mass cancels | Ch. 2 |
| $U = -\mu m/r$ | gravitational potential energy of a mass $m$ (J); zero at infinity, negative in the well | Ch. 2 |
Here $\mu = GM$ is the central body's gravitational parameter; $v$ is speed and $r$ the radius from the body's centre. Escape speed is always $\sqrt{2}\approx1.41$ times circular speed at the same radius — at Earth's surface, $11.19$ km/s versus about $7.9$ km/s. The sign of $\varepsilon$ is the single most useful number in this table: $\varepsilon < 0$ is a bound ellipse, $\varepsilon = 0$ a parabola, and $\varepsilon > 0$ a hyperbola that escapes with speed $v_\infty$ to spare.
Kepler's laws and orbital elements
Kepler's three laws are consequences of inverse-square gravity, not axioms; all three fall out of $\ddot{\mathbf{r}} = -\mu\mathbf{r}/r^3$. The six orbital elements turn a trajectory into six numbers plus a named primary. In the two-body model the first five are constant and only the true anomaly $\nu$ moves.
| Element | Symbol | Sets |
|---|---|---|
| Semi-major axis | $a$ | size (⇒ period and energy) |
| Eccentricity | $e$ | shape (circle → hyperbola) |
| Inclination | $i$ | tilt of the plane from the equator |
| Right ascension of the ascending node (RAAN) | $\Omega$ | swivel of the plane about the pole |
| Argument of periapsis | $\omega$ | orientation of the ellipse within its plane |
| True anomaly | $\nu$ | where the body is now — the only element that moves |
| Formula | What it's for | Derived in |
|---|---|---|
| $T = 2\pi\sqrt{a^3/\mu}$ | orbital period (Kepler's third law); depends on $a$ alone | Ch. 8 |
| $n = \dfrac{2\pi}{T} = \sqrt{\mu/a^3}$ | mean motion — the orbit's average angular rate (rad/s) | Ch. 8 |
| $r = \dfrac{p}{1 + e\cos\nu}$ | the orbit (conic) equation — radius at true anomaly $\nu$ | Ch. 8 |
| $p = a(1-e^2) = h^2/\mu$ | semi-latus rectum — the conic's width parameter (km) | Ch. 8 |
| $r_p = a(1-e), \quad r_a = a(1+e)$ | periapsis and apoapsis radii from size and shape | Ch. 8 |
| $a = \dfrac{r_p+r_a}{2}, \quad e = \dfrac{r_a-r_p}{r_a+r_p}$ | size and shape recovered from the two apsidal radii | Ch. 8 |
| $h = \sqrt{\mu a(1-e^2)} = \sqrt{\mu p}$ | specific angular momentum (km²/s); constant on the orbit | Ch. 8 |
| $M = E - e\sin E$ | Kepler's equation — links geometry ($E$) to time ($M$) | Ch. 8 |
| $M = n(t - t_p)$ | mean anomaly from time since periapsis passage $t_p$ | Ch. 8 |
| $r = a(1 - e\cos E)$ | radius from the eccentric anomaly $E$ | Ch. 8 |
| $\tan\dfrac{\nu}{2} = \sqrt{\dfrac{1+e}{1-e}}\,\tan\dfrac{E}{2}$ | convert eccentric anomaly $E$ ↔ true anomaly $\nu$ | Ch. 8 |
Time → position (the three-step procedure): (1) $M = n(t - t_p)$; (2) solve $M = E - e\sin E$ for $E$ numerically (Newton's method, starting from $E_0 = M + e\sin M$); (3) convert $E \to \nu$ and take $r = a(1 - e\cos E)$. The three anomalies — mean $M$ (grows uniformly with time), eccentric $E$ (geometric middleman), and true $\nu$ (the physical angle at the focus) — coincide only at periapsis and apoapsis; for a near-circular orbit they converge and position advances almost uniformly.
Orbital maneuvers
An impulsive maneuver is a difference of two vis-viva speeds at the point the two orbits share, turned into propellant by the rocket equation (Chapter 3). Two prograde burns raise you between coplanar circles (a Hohmann transfer); changing the orbital plane is separately and brutally expensive, because its cost scales with your speed.
| Formula | What it's for | Derived in |
|---|---|---|
| $\Delta v_1 = \sqrt{\dfrac{\mu}{r_1}}\left(\sqrt{\dfrac{2r_2}{r_1+r_2}} - 1\right)$ | Hohmann first burn — leave the inner circle $r_1$ onto the transfer ellipse | Ch. 10 |
| $\Delta v_2 = \sqrt{\dfrac{\mu}{r_2}}\left(1 - \sqrt{\dfrac{2r_1}{r_1+r_2}}\right)$ | Hohmann second burn — circularize at the outer radius $r_2$ | Ch. 10 |
| $\Delta v_{\text{Hohmann}} = \Delta v_1 + \Delta v_2$ | total two-burn cost between coplanar circles | Ch. 10 |
| $t_{\text{tr}} = \pi\sqrt{a_t^3/\mu}, \quad a_t = \dfrac{r_1+r_2}{2}$ | transfer time — half the transfer-ellipse period | Ch. 10 |
| $\Delta v_{\text{plane}} = 2v\sin(\Delta i/2)$ | cost of a pure plane change of angle $\Delta i$ at speed $v$ | Ch. 10 |
| $\Delta v_{\text{comb}} = \sqrt{v_1^2 + v_2^2 - 2v_1v_2\cos\Delta i}$ | combined speed change and plane change in one burn (law of cosines) | Ch. 10 |
Here $r_1$ and $r_2$ are the inner and outer circular radii (from the body's centre), $a_t$ the transfer-ellipse semi-major axis, and $v$ the speed at which the plane change is made. Do plane changes slow: a $60^\circ$ change costs a full $\Delta v = v$ (the entire orbital speed), so change plane high up at apoapsis where $v$ is small, or — better — launch directly into the right plane. Combined beats separate: because delta-vs add as vectors, folding a plane change into a speed change (the law-of-cosines form) is never dearer, and usually much cheaper, than doing the two in turn — which is why GTO-to-GEO circularization and the final inclination fix are performed together at apogee.
Bi-elliptic crossover. For a very large radius ratio, a three-burn bi-elliptic transfer — coast out to an intermediate apoapsis $r_b > r_2$, make a small raising burn there, then drop back down — can beat the Hohmann. The break-even is $r_2/r_1 \approx 11.94$: above it the bi-elliptic can win (and only with patience, since it takes far longer); below it the two-burn Hohmann is optimal. See Chapter 10.
Interplanetary trajectories
A whole interplanetary mission is stitched from three two-body conics — patched conics — handing the spacecraft off between gravity fields at each planet's sphere of influence:
Earth departure heliocentric target arrival
HYPERBOLA --> ELLIPSE --> HYPERBOLA
(inside Earth SOI) (Sun's domain) (inside target SOI)
| Formula | What it's for | Derived in |
|---|---|---|
| $r_{\text{SOI}} \approx a_{\text{pl}}\left(\dfrac{m_{\text{pl}}}{m_\odot}\right)^{2/5}$ | radius of a planet's sphere of influence — where to switch central bodies | Ch. 11 |
| $v^2 = \mu_\odot\!\left(\dfrac{2}{r} - \dfrac{1}{a_t}\right)$ | heliocentric speed on the transfer ellipse (vis-viva about the Sun) | Ch. 11 |
| $v_\infty = \lvert v_{\text{transfer}} - v_{\text{planet}}\rvert$ | hyperbolic excess speed — speed relative to the planet at its SOI | Ch. 11 |
| $C_3 = v_\infty^2 = 2\varepsilon = -\dfrac{\mu}{a}$ | characteristic energy — the launch energy a mission demands (km²/s²) | Ch. 11 |
| $\dfrac{1}{T_{\text{syn}}} = \left\lvert\dfrac{1}{T_1} - \dfrac{1}{T_2}\right\rvert$ | synodic period — how often the launch window recurs | Ch. 11 |
| $t_{\text{tr}} = \pi\sqrt{a_t^3/\mu_\odot}$ | one-way Hohmann cruise time (half the transfer-ellipse period) | Ch. 11 |
| $\Delta v_{\text{inj}} = \sqrt{v_\infty^2 + \dfrac{2\mu}{r_p}} - \sqrt{\dfrac{\mu}{r_p}}$ | departure (or capture) burn from a parking orbit of radius $r_p$ | Ch. 11 |
| $e = 1 + \dfrac{r_p v_\infty^2}{\mu}, \quad \sin\dfrac{\delta}{2} = \dfrac{1}{e}$ | gravity-assist turn angle $\delta$ for a flyby at closest approach $r_p$ | Ch. 11 |
Here $\mu_\odot = 1.327\times10^{11}\ \text{km}^3/\text{s}^2$ is the Sun's gravitational parameter, $a_t = (r_1+r_2)/2$ the heliocentric transfer semi-major axis, $a_{\text{pl}}$ and $m_{\text{pl}}$ the planet's Sun-distance and mass, $m_\odot$ the Sun's mass, and $T_1, T_2$ the two bodies' orbital periods (Appendix B). Watch the frame: heliocentric speeds are measured relative to the Sun, but $v_\infty$ is relative to the planet — confusing the two is the classic interplanetary error, since at arrival the two large heliocentric velocities nearly cancel. $C_3$ is the quantity a launch vehicle's performance curve is plotted against, and a single well-aimed flyby can swing heliocentric speed by up to $2v_\infty$ for free.
Perturbations: J2 and secular drift
Real orbits are not perfect Kepler ellipses. Add a small perturbing acceleration $\mathbf{a}_p$ to the two-body equation of motion and track which contributions accumulate (secular) rather than average away over each revolution (periodic). Earth's equatorial bulge — captured by the oblateness coefficient $J_2$ — is the dominant perturbation at low altitude.
$$\ddot{\mathbf{r}} = -\frac{\mu}{r^3}\mathbf{r} + \mathbf{a}_p, \qquad \mathbf{a}_p = \mathbf{a}_{J2} + \mathbf{a}_{\text{drag}} + \mathbf{a}_{\text{3-body}} + \mathbf{a}_{\text{SRP}} + \cdots$$
| Formula | What it's for | Derived in |
|---|---|---|
| $\dot\Omega = -\dfrac{3}{2}\dfrac{n J_2 R_\oplus^2}{(1-e^2)^2 a^2}\cos i$ | nodal regression — secular drift of the RAAN $\Omega$ (rad/s) | Ch. 12 |
| $\dot\omega = \dfrac{3}{4}\dfrac{n J_2 R_\oplus^2}{(1-e^2)^2 a^2}(5\cos^2 i - 1)$ | apsidal precession — secular rotation of the argument of periapsis $\omega$ (rad/s) | Ch. 12 |
| $i = 63.4^\circ \ \ (5\cos^2 i = 1)$ | critical inclination — $\dot\omega = 0$, so the apsidal line freezes (Molniya) | Ch. 12 |
| $\cos i = -\dfrac{\dot\Omega_{\text{req}}}{\frac{3}{2}\,n J_2 (R_\oplus/a)^2}$ | sun-synchronous inclination — tune $i$ so the plane tracks the Sun | Ch. 12 |
In these, $n = \sqrt{\mu/a^3}$ is the mean motion, $J_2 = 1.0826\times10^{-3}$ is Earth's oblateness coefficient, and $R_\oplus = 6{,}378\ \text{km}$ is Earth's equatorial radius (Appendix B); $i$, $e$, and $a$ are the inclination, eccentricity, and semi-major axis. J2 reorients the orbit but does not change $a$ or $e$ secularly — it only swivels the plane ($\Omega$) and rotates the ellipse ($\omega$). Both rates scale as $a^{-7/2}$, roughly $600\times$ stronger at LEO than at GEO. Nodal regression is zero at $i = 90^\circ$ (a polar orbit's plane is inertially fixed) and largest near the equator. For a sun-synchronous orbit the plane must precess eastward at Earth's mean orbital rate, $\dot\Omega_{\text{req}} = +1.991\times10^{-7}\ \text{rad/s} = +0.9856^\circ/\text{day}$; that forces $\cos i < 0$, i.e. a retrograde $i \approx 98^\circ$ at 700–800 km altitude.
On using these formulas. Every relation above is the two-body teaching form: idealized, impulsive burns and point-mass gravity, plus the single $J_2$ term in this last section. They are accurate enough to design a mission and size a delta-v budget to a percent or two — which is their job — but real trajectory work integrates the full equation of motion numerically. Keep units consistent (this book stays in km, km/s, and $\text{km}^3/\text{s}^2$), watch which body each speed is measured against, and read the sign of $\varepsilon$ (or equivalently $C_3$) to know at a glance whether an orbit is bound or escaping. For the meaning of every symbol see Appendix A; for the constants that feed these formulas see Appendix B; and for representative delta-v costs across the solar system see the delta-v map in Appendix G.