Chapter 10 — Key Takeaways (Orbital Maneuvers)

A one-page reference. Reread this before an exam, or before you plan any orbit change or rendezvous.

The equations that price every maneuver

Equation Use it to find
$\Delta v_1 = \sqrt{\dfrac{\mu}{r_1}}\left(\sqrt{\dfrac{2r_2}{r_1+r_2}} - 1\right)$ Hohmann first burn (leave inner circle $r_1$)
$\Delta v_2 = \sqrt{\dfrac{\mu}{r_2}}\left(1 - \sqrt{\dfrac{2r_1}{r_1+r_2}}\right)$ Hohmann second burn (join outer circle $r_2$)
$t_{\text{transfer}} = \pi\sqrt{a_t^{\,3}/\mu}, \quad a_t = \dfrac{r_1+r_2}{2}$ Hohmann transfer time (half the ellipse period)
$\Delta v_{\text{plane}} = 2v\sin(\Delta i/2)$ cost of a pure plane change of angle $\Delta i$ at speed $v$
$\Delta v_{\text{comb}} = \sqrt{v_a^2 + v_{c2}^2 - 2v_av_{c2}\cos\Delta i}$ combined speed change + plane change (law of cosines)

Each maneuver is ultimately a difference of two vis-viva speeds (Ch. 6): find your speed on the orbit you have and on the orbit you want, at the point they share, and subtract.

What every symbol means (and its units)

Symbol Name Units Notes
$r_1, r_2$ inner / outer circular radii km measured from the body's center, not the surface
$a_t$ transfer-ellipse semi-major axis km $=(r_1+r_2)/2$ for a Hohmann transfer
$v_a$ transfer-orbit speed at apogee km/s slow — the good place to change plane
$\Delta i$ plane-change angle degrees the tilt between old and new orbital planes
$\Delta v$ delta-v of a burn km/s priced into propellant by the rocket equation (Ch. 3)
$r_b$ bi-elliptic intermediate apoapsis km $> r_2$; where the cheap middle burn happens

The four maneuver types at a glance

Maneuver Burns Cost driver When to use
Impulsive burn 1 changes velocity at a point; opposite side of orbit moves the atom of every maneuver
Hohmann transfer 2 (both prograde to raise) tangent ellipse; minimum fuel almost always, for $r_2/r_1 < 11.94$
Bi-elliptic transfer 3 (via high $r_b$) cheap middle burn far out only for $r_2/r_1 \gtrsim 11.94$; costs much time
Plane change 1 (normal) $\propto v$ — brutal avoid; if forced, do it slow (high, at apoapsis)

Read the numbers (LEO→GEO benchmark)

Quantity Value
LEO→GEO Hohmann $2.40 + 1.46 = \mathbf{3.86\ \text{km/s}}$, over $\approx 5.3$ hours
Plane change, per degree in LEO $\approx \mathbf{134\ \text{m/s}}$ (at $v = 7.673\ \text{km/s}$)
$28.5^\circ$ plane change in LEO $3.78\ \text{km/s}$ (nearly as much as reaching orbit!)
$28.5^\circ$ plane change at GEO $1.51\ \text{km/s}$ (slower = cheaper)
Combined circularize + $28.5^\circ$ at GTO apogee $\approx \mathbf{1.83\ \text{km/s}}$ (saves $\approx 1.15$ vs separate)
$60^\circ$ plane change $= v$, the entire orbital velocity

The threshold concepts

  • A transfer is two vis-viva speeds subtracted. Delta-v is a difference of speeds; the rocket equation turns it into propellant. Whole journeys reduce to arithmetic on two orbits sharing a point.
  • Plane changes cost $\propto v$, so change your plane where you are slowest (high up, at apoapsis) — and launch into the right plane in the first place, because fixing it later is unaffordable.
  • Slow down to catch up (Ch. 6): to reach a target ahead, drop into a lower, faster, shorter-period orbit; to fall back to one behind, rise into a higher, slower orbit.
  • Delta-vs add as vectors, not scalars. Combining a speed change and a plane change into one burn is never more expensive, and usually much cheaper, than doing them separately.

Decision aid — "which maneuver do I use?"

You want to… Use Key relation
raise/lower between coplanar circles Hohmann transfer $\Delta v_1, \Delta v_2$ above
change altitude by a huge factor ($>12\times$) consider bi-elliptic 3 burns via high $r_b$
change the orbital plane plane change, done slow $2v\sin(\Delta i/2)$
change your timing/position on an orbit phasing orbit Kepler III: $T = 2\pi\sqrt{a^3/\mu}$
circularize and change plane at once combined burn law of cosines
meet another spacecraft rendezvous = phasing + proximity ops all of the above

Common pitfalls

Pitfall Reality
Adding a combined burn's parts as scalars Delta-vs are vectors; use the law of cosines — the sum overcharges you.
Doing a plane change in LEO to "get it over with" Cost $\propto v$; do it high and slow (apoapsis). LEO is the worst place.
Thrusting toward a target ahead to catch it That raises your orbit and drops you behind. Burn retrograde to catch up.
Thinking bi-elliptic always saves fuel Only for $r_2/r_1 \gtrsim 11.94$, and then barely, at a huge time cost.
Fixing the orbital plane after launch Launch into the plane; a $10^\circ$ LEO fix costs $\approx 1.3\ \text{km/s}$.

Mission / astrotools additions this chapter

  • MDR: added a Transfers section to the delta-v budget — each orbit change priced with this chapter's tools (Track A: LEO→GEO $\approx 4.23\ \text{km/s}$ with the combined apogee burn).
  • maneuvers.py: hohmann(mu, r1, r2) → (dv1, dv2, total); plane_change(v, di_deg) → $2v\sin(\Delta i/2)$; dv_budget(legs) → rolls a list of (label, dv) legs into a total. (Uses speeds from orbits.py, Ch. 6; feeds mission.py, Ch. 29.)