Patched conic method — interplanetary trajectory design

WHAT is being patched?

We divide the whole trip into three conic patches:

Departure hyperbola — inside Earth's SOI, Earth is central body. Spacecraft leaves on a hyperbola.
Heliocentric transfer ellipse — between the SOIs, the Sun is central body. Usually a Hohmann-type ellipse.
Arrival hyperbola — inside the target's SOI, the target planet is central body.

Figure — Patched conic method — interplanetary trajectory design

WHY does the SOI radius have that $2/5$ power? (Derivation)

Formula Deriving

$r_{SOI}$ Consider the spacecraft near a planet at distance $r$ from it, and distance $\approx a_p$ from the Sun. Compare gravity of Sun vs planet as primary vs perturbing in two frames.

Frame A (planet-centered): planet is primary → its acceleration $\sim \dfrac{Gm_p}{r^2}$ ; the Sun is a perturbation → tidal-type disturbing acceleration $\sim \dfrac{Gm_\odot\, r}{a_p^3}$ . Perturbation ratio: $\displaystyle A_A = \frac{Gm_\odot r / a_p^3}{Gm_p/r^2}=\frac{m_\odot}{m_p}\frac{r^3}{a_p^3}$ .

Frame B (Sun-centered): Sun is primary → $\sim \dfrac{Gm_\odot}{a_p^2}$ ; the planet is the perturbation → $\sim \dfrac{Gm_p}{r^2}$ . Perturbation ratio: $\displaystyle A_B = \frac{Gm_p/r^2}{Gm_\odot/a_p^2}=\frac{m_p}{m_\odot}\frac{a_p^2}{r^2}$ .

The SOI boundary is where treating the planet as primary is equally as good as treating the Sun as primary — set the two disturbance ratios equal: $A_A = A_B$ . $\frac{m_\odot}{m_p}\frac{r^3}{a_p^3}=\frac{m_p}{m_\odot}\frac{a_p^2}{r^2}$ $r^5 = a_p^5\left(\frac{m_p}{m_\odot}\right)^2 \;\Rightarrow\; r = a_p\left(\frac{m_p}{m_\odot}\right)^{2/5}\quad\blacksquare$

WHY it feels right: near the planet ( $r$ small) $A_A\to 0$ (planet clearly wins); far away $A_B\to 0$ (Sun clearly wins). The crossover is the SOI.

HOW to design the transfer: the heliocentric leg

Formula Hohmann transfer ellipse (Sun-centered)

Take circular, coplanar planet orbits of radii $r_1$ (Earth) and $r_2$ (Mars). The cheapest ellipse touches both:

perihelion at $r_1$ , aphelion at $r_2$ → semi-major axis $a_t=\frac{r_1+r_2}{2}$

Speeds come from the vis-viva equation, derived from energy conservation $\tfrac12 v^2-\tfrac{\mu}{r}=-\tfrac{\mu}{2a}$ (here $\mu=Gm_\odot$ ): $v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)}$

Circular planet speeds: $v_{c,i}=\sqrt{\mu/r_i}$ . Transfer speed at Earth: $v_{t,1}=\sqrt{\mu\left(\dfrac{2}{r_1}-\dfrac{1}{a_t}\right)}$ .

The heliocentric excess the spacecraft must have relative to Earth is $v_\infty^{dep}=|v_{t,1}-v_{c,1}|.$ This is what the departure hyperbola must supply at the SOI edge.

HOW to design the departure/arrival hyperbola

Formula Escape from parking orbit

Inside Earth's SOI ( $\mu_E=Gm_E$ ), the craft is on a hyperbola with excess speed $v_\infty$ . On a hyperbola the energy is $\varepsilon=\frac{v^2}{2}-\frac{\mu_E}{r}=\frac{v_\infty^2}{2}\;(>0).$ At perigee radius $r_p$ (parking orbit), the required speed is $v_p=\sqrt{v_\infty^2+\frac{2\mu_E}{r_p}}.$ The injection burn (departure $\Delta v$ ) from a circular parking orbit of speed $v_{circ}=\sqrt{\mu_E/r_p}$ is $\boxed{\Delta v = v_p - v_{circ}=\sqrt{v_\infty^2+\frac{2\mu_E}{r_p}}-\sqrt{\frac{\mu_E}{r_p}}}$

WHY the $2\mu_E/r_p$ ? Escaping to infinity from $r_p$ needs $v_{esc}=\sqrt{2\mu_E/r_p}$ (that gives $v_\infty=0$ ). To also have leftover speed $v_\infty$ at infinity, energies add: $v_p^2=v_{esc}^2+v_\infty^2$ . That's just energy bookkeeping — the "Oberth effect" is baked in: burning deep in the well ( $r_p$ small) makes $v_\infty$ cheap.

Worked Example 1 — Earth→Mars Hohmann heliocentric speeds

Use $\mu_\odot=1.327\times10^{20}\ \text{m}^3/\text{s}^2$ , $r_1=1.496\times10^{11}$ m, $r_2=2.279\times10^{11}$ m.

$a_t=\frac{r_1+r_2}{2}=1.888\times10^{11}$ m. Why? ellipse touches both orbits.
$v_{c,1}=\sqrt{\mu_\odot/r_1}=\sqrt{1.327\times10^{20}/1.496\times10^{11}}\approx 2.978\times10^4$ m/s. Why? Earth's circular speed sets the baseline.
$v_{t,1}=\sqrt{\mu_\odot(2/r_1-1/a_t)}\approx 3.279\times10^4$ m/s. Why? vis-viva at perihelion of transfer.
$v_\infty^{dep}=v_{t,1}-v_{c,1}\approx 2.94\times10^3$ m/s ≈ 2.94 km/s. Why? speed the departure hyperbola must deliver.

Worked Example 2 — Departure $\Delta v$ from a 200 km LEO

$\mu_E=3.986\times10^{14}$ m³/s², $r_p=R_E+200\text{ km}=6.578\times10^6$ m, $v_\infty=2.94$ km/s.

$v_{circ}=\sqrt{\mu_E/r_p}=\sqrt{3.986\times10^{14}/6.578\times10^6}\approx 7.784$ km/s. Why? parking-orbit speed.
$v_p=\sqrt{v_\infty^2+2\mu_E/r_p}=\sqrt{2940^2+2(3.986\times10^{14})/6.578\times10^6}$ $=\sqrt{8.64\times10^6+1.212\times10^8}\approx 1.140\times10^4$ m/s ≈ 11.40 km/s. Why? hyperbolic perigee speed.
$\Delta v = 11.40-7.784\approx 3.62$ km/s. Why? the injection burn.

Worked Example 3 — Transfer time

Half the ellipse's period: $t=\pi\sqrt{a_t^3/\mu_\odot}$ . Why? Hohmann is half an orbit. $t=\pi\sqrt{(1.888\times10^{11})^3/1.327\times10^{20}}\approx 2.24\times10^7$ s ≈ 259 days.

Common mistake Steel-manning the classic errors

Error 1: "Just add $v_\infty$ to $v_{circ}$ to get $\Delta v$ ." Why it feels right: velocities in the same direction should add. The fix: $v_\infty$ is measured at infinity, but you burn at perigee where the craft is fast. Energy (not velocity) is conserved along the coast: $v_p=\sqrt{v_\infty^2+v_{esc}^2}$ , so $\Delta v = v_p-v_{circ}$ , which is less than $v_\infty+(v_{esc}-v_{circ})$ . Deep burns are cheap (Oberth).

Error 2: " $v_\infty$ = spacecraft's heliocentric speed." Why it feels right: both are "the speed leaving." The fix: $v_\infty$ is relative to the planet. The heliocentric transfer speed is $v_{c,1}\pm v_\infty$ — you must vector-add the planet's orbital velocity.

Error 3: Treating the SOI as a real hard wall. Why it feels right: the math switches central bodies there. The fix: it's an approximation; the true field is continuous. Patched conics give a great first guess, then you refine numerically.

Recall Feynman: explain to a 12-year-old

Imagine throwing a ball from a fast-moving merry-go-round (Earth) to another merry-go-round (Mars) circling a lamppost (the Sun). It's too hard to think about the lamppost, your merry-go-round, and the other one all at once. So we cut the trip into chapters: Chapter 1 — only your merry-go-round matters, you run and let go (a curved escape path). Chapter 2 — now only the lamppost matters, the ball swings on a long looping arc toward Mars. Chapter 3 — only Mars's merry-go-round matters as you catch it. Each chapter is an easy shape we already know how to draw. Then we glue the chapters together so the ball never jumps. That gluing is the "patch."

Connections

Vis-viva equation — the engine for every speed here.
Hohmann transfer orbit — the cheapest heliocentric ellipse.
Hyperbolic escape trajectories — departure/arrival legs.
Oberth effect — why deep burns are efficient.
Sphere of influence — the boundary that defines each patch.
Kepler's laws — guarantee each patch is a conic.
Launch windows & synodic period — when Mars is in the right place.

Patched conic method's core assumption

At any instant only one body's gravity dominates, so motion is a two-body conic.

Formula for sphere-of-influence radius

r_{SOI}=a_p (m_p/m_\odot)^{2/5}

Why the 2/5 power?

Setting the planet-frame and Sun-frame perturbation ratios equal gives

r^5=a_p^5(m_p/m_\odot)^2

Vis-viva equation

v=\sqrt{\mu(2/r-1/a)}

, from energy

\tfrac12 v^2-\mu/r=-\mu/2a

Hohmann semi-major axis

a_t=(r_1+r_2)/2

Hyperbolic excess speed

v_\infty

meaning

Speed relative to the planet at the SOI boundary (as

r\to\infty

in the planet frame).

Perigee speed on departure hyperbola

v_p=\sqrt{v_\infty^2+2\mu_E/r_p}

Departure injection

\Delta v

\sqrt{v_\infty^2+2\mu_E/r_p}-\sqrt{\mu_E/r_p}

Why is

\Delta v < v_\infty + (v_{esc}-v_{circ})

Oberth effect: energy conservation means burning deep in the well makes

v_\infty

cheap.

Heliocentric departure excess for Earth→Mars

v_\infty=|v_{t,1}-v_{c,1}|\approx 2.94

km/s.

Hohmann transfer time

t=\pi\sqrt{a_t^3/\mu_\odot}

(half the ellipse period), ~259 days for Earth→Mars.

Is the SOI a real physical wall?

No — it's an approximation; the true gravity field is continuous.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Earth se Mars tak spacecraft bhejna asli mein ek "many-body problem" hai — Sun, Earth, Mars sab ek saath gravity laga rahe hain, aur iska koi clean formula nahi milta. Patched conic method ek smart shortcut hai: hum maan lete hain ki kisi bhi ek time par sirf ek body ki gravity important hai. Isse pura safar teen simple pieces mein tut jaata hai — Earth ke paas ek hyperbola (nikalne ka rasta), beech mein Sun ke around ek ellipse (Hohmann transfer), aur Mars ke paas ek aur hyperbola (pahunchne ka rasta). In teeno ko ek continuous trajectory mein "jod" dete hain — isliye naam "patched".

Har planet ke around ek Sphere of Influence (SOI) hoti hai — ek imaginary bubble jiske andar us planet ki gravity boss hai, bahar Sun boss hai. Uska radius $r_{SOI}=a_p(m_p/m_\odot)^{2/5}$ hota hai. Yeh $2/5$ power aise aata hai: hum planet-frame aur Sun-frame dono mein perturbation ka ratio nikaalte hain aur unhe barabar rakh dete hain — bas, formula nikal aata hai.

Speeds ke liye vis-viva equation use karte hain: $v=\sqrt{\mu(2/r-1/a)}$ , jo simply energy conservation se aati hai. Transfer ke liye Earth se jo extra speed chahiye planet ke sapeksh, use $v_\infty$ (hyperbolic excess) kehte hain. Ek important baat — parking orbit se jo burn ( $\Delta v$ ) chahiye woh $\sqrt{v_\infty^2+2\mu_E/r_p}-\sqrt{\mu_E/r_p}$ hota hai, seedha $v_\infty$ add mat karna! Yeh Oberth effect hai: gehre gravity well mein burn karo to $v_\infty$ sasta padta hai. Yeh method exact nahi hai, par first design ke liye zabardast — baad mein computer se fine-tune kar lete hain.

Patched conic method — interplanetary trajectory design

WHAT is being patched?

WHY does the SOI radius have that $2/5$ power? (Derivation)

HOW to design the transfer: the heliocentric leg

HOW to design the departure/arrival hyperbola

Worked Example 1 — Earth→Mars Hohmann heliocentric speeds

Worked Example 2 — Departure $\Delta v$ from a 200 km LEO

Worked Example 3 — Transfer time

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Orbital Mechanics & Astrodynamics

Connections