Combined maneuvers — optimal split between plane change and velocity change

3.2.23 · Physics › Orbital Mechanics & Astrodynamics

HUM KYA COMBINE KAR RAHE HAIN?

Do velocity vectors jo magnitude aur direction dono mein alag hain:

$v_{1}$ = burn se pehle ki speed
$v_{2}$ = burn ke baad ki speed
$θ$ = dono velocity vectors ke beech ka angle (plane-change angle)

Humein jo maneuver dena hai woh vector $Δ v = v_{2} - v_{1}$ hai.

COMBINED $Δ v$ DERIVE KAISE KAREIN (law of cosines)

$v_{1}$ aur $v_{2}$ ko tail-to-tail rakhein, unke beech angle $θ$ ho:

$Δ v = v_{2} - v_{1}$ $∣Δ v ∣^{2} = Δ v \cdot Δ v = v_{2}^{2} - 2 v_{1} \cdot v_{2} + v_{1}^{2}$

Kyunki $v_{1} \cdot v_{2} = v_{1} v_{2} cos θ$ :

Sanity checks (Forecast-then-Verify):

$θ = 0$ (koi plane change nahi): $Δ v = (v_{1} - v_{2})^{2} = ∣ v_{2} - v_{1} ∣$ . ✅ pure speed change.
$v_{1} = v_{2} = v$ (pure plane change): $Δ v = v 2 - 2 cos θ = 2 v sin (θ /2)$ . ✅ classic plane-change formula.

PLANE CHANGE KO OPTIMALLY SPLIT KAISE KAREIN

Standard problem socho: do circular orbits ke beech Hohmann transfer jahan final orbit starting orbit se total angle $i$ se inclined bhi hai. Aapke paas do burns hain:

burn 1 (departure, speed $v_{p}$ — zyaada, chhoti orbit ke paas),
burn 2 (arrival, speed $v_{a}$ — kam, badi orbit ke paas).

Maano ki plane change $s$ burn 1 pe aur $(i - s)$ burn 2 pe hota hai. Har burn ek combined maneuver hai:

$Δ v_{1} (s) = v_{1, i}^{2} + v_{1, f}^{2} - 2 v_{1, i} v_{1, f} cos s$ $Δ v_{2} (s) = v_{2, i}^{2} + v_{2, f}^{2} - 2 v_{2, i} v_{2, f} cos (i - s)$

Total $Δ v_{tot} (s) = Δ v_{1} (s) + Δ v_{2} (s)$ . Minimize karein:

$\frac{d Δ v _{tot}}{d s} = 0$

Figure — Combined maneuvers — optimal split between plane change and velocity change

Worked Example 1 — LEO→GEO with inclination change

Diya hai: $r_{1}$ (LEO, fast) aur $r_{2}$ (GEO, slow) ke beech transfer, total inclination change $i = 28. 5^{\circ}$ (Cape Canaveral se equatorial GEO).

Approximate speeds (typical): perigee burn $v_{1, i} = 7.73$ aur $v_{1, f} = 10.15$ km/s ke beech; apogee burn $v_{2, i} = 1.61$ aur $v_{2, f} = 3.07$ km/s ke beech.

Step — saara plane change apogee pe karo ( $s = 0$ ): $Δ v_{2} = 1.6 1^{2} + 3.0 7^{2} - 2 (1.61) (3.07) cos 28. 5^{\circ} = 1.83 km/s$ Yeh step kyun? Apogee pe velocities sabse chhoti hain, toh $2 v sin$ ka cost yahaan sabse kam hai.

Step — optimally split karo: $Δ v_{tot} (s)$ numerically minimize karne par $s \approx 2. 2^{\circ}$ perigee pe aur $26. 3^{\circ}$ apogee pe milta hai। Yeh step kyun? Equal-marginal-cost condition; perigee pe thoda sa hissa thoda aur bachata hai.

Step — naive "saab perigee pe" se compare karo: $Δ v_{1} (all 28. 5^{\circ}) = 7.7 3^{2} + 10.1 5^{2} - 2 (7.73) (10.15) cos 28. 5^{\circ} = 5.0 km/s (!)$ Yeh step kyun? Penalty dikhata hai: fast burn pe plane change karna poore mission ki cost ko lagbhag double kar deta hai.

Takeaway: plane change apoapsis ke liye bachao, perigee pe sirf ek chhooti si matra rakho.

Worked Example 2 — Pure equal-speed plane change vs split

Maano $v_{1} = v_{2} = v$ (same circular orbit, sirf use tilt karna hai $θ$ se).

Single burn: $Δ v = 2 v sin (θ /2)$ .

Bi-elliptic trick (conceptual): apoapsis ko bahut door raise karo (sasta prograde burn), wahaan plane change karo jahan $v$ tiny hai (toh $2 v sin (θ /2)$ tiny hai), phir wapas neeche aao. Yeh step kyun? Velocity change (raise/lower) ko plane change se alag karna low speed ka faayda uthata hai — bade $θ$ ke liye yeh single burn se better hai. Yehi principle hai combined-maneuver splitting ka: turn karo jahan slow ho.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tum tez daud rahe ho aur nai direction mein mudna chahte ho. Agar sprint karte waqt mudo, toh bahut zyaada mehnat lagti hai — bahut effort chahiye. Agar tum pehle slow down karo, phir mudo, phir speed up karo, toh mudna aasaan ho jaata hai. Space mein, orbit ki tilt change karna ("turning") sabse zyaada cost karta hai jab tum tez chal rahe hote ho. Toh spacecraft samajhdaari se apna turning orbit ke door, slow part pe karte hain, aur ek hi angled push se speed up aur turning dono ek saath kar lete hain — do jerks ki jagah ek smooth swerve ki tarah.

Flashcards

Combined-maneuver

Δ v

formula

Δ v = v_{1}^{2} + v_{2}^{2} - 2 v_{1} v_{2} cos θ

Yeh law of cosines kyun hai?

Δ v

velocity triangle ki teesri side hai jiska

v_{1}

aur

v_{2}

ke beech included angle

θ

hai.

Pure plane-change cost (equal speeds)

Δ v = 2 v sin (θ /2)

Speed

v

6 0^{\circ}

plane change ki cost

Δ v = v

(poori orbital speed dobara).

Zyaadatar plane change kahaan karna chahiye, aur kyun?

Apoapsis / slower burn pe, kyunki cost wahaan ki velocity ke saath scale hoti hai.

Optimal-split condition (words mein)

Ek degree aur plane change ki marginal

Δ v

cost dono burns pe barabar honi chahiye.

Optimal-split condition (formula)

\frac{v _{1, i} v _{1, f} sin s}{Δ v _{1}} = \frac{v _{2, i} v _{2, f} sin ( i - s )}{Δ v _{2}}

Do alag burns ki jagah combine kyun karein?

Triangle inequality:

v_{1}^{2} + v_{2}^{2} - 2 v_{1} v_{2} cos θ \leq ∣ v_{2} - v_{1} ∣ + 2 v_{2} sin (θ /2)

Sanity check

θ = 0

Δ v = ∣ v_{2} - v_{1} ∣

(pure speed change).

LEO→GEO 28.5° ke liye split roughly kaisi hogi?

Lagbhag saara plane change apogee pe (~26°), sirf ~2° perigee pe.

Connections

Hohmann Transfer Orbit — do-burn backbone jise yeh optimize karta hai.
Plane Change Maneuvers — $2 v sin (θ /2)$ special case.
Bi-elliptic Transfer — extreme "turn where slow" strategy.
Vis-viva Equation — har burn pe speeds $v_{1}, v_{2}$ deta hai.
Delta-v Budget — mission planning mein yeh savings kahaan dikhti hain.
Law of Cosines — derivation ka geometric engine.