Bi-elliptic transfer — when it wins over Hohmann

3.2.21 · Physics › Orbital Mechanics & Astrodynamics

Bi-elliptic transfer KYA hota hai?

Detour kabhi fuel kyun bachata hai?

Key physics hai vis-viva equation — kisi bhi orbit par speed:

$v^{2} = μ (\frac{2}{r} - \frac{1}{a})$

Total $Δ v$ KAISE derive karein (scratch se)

Sab kuch vis-viva se derive karte hain. Define $μ = GM$ .

Circular orbits par speeds. Vis-viva mein $a = r$ set karke: $v_{c} (r) = \frac{μ}{r}$ Toh $v_{c 1} = μ / r_{1}$ , $v_{c 2} = μ / r_{2}$ .

Ellipse 1 $r_{1}$ (periapsis) aur $r_{b}$ (apoapsis) ko connect karta hai: $a_{1} = \frac{r _{1} + r _{b}}{2}$

Kyun? Semi-major axis do apsidal radii ka average hota hai.

Ellipse 2 $r_{2}$ (periapsis) aur $r_{b}$ (apoapsis) ko connect karta hai: $a_{2} = \frac{r _{2} + r _{b}}{2}$

Ab har burn point par vis-viva apply karo.

Burn 1 — $r_{1}$ par, circular ( $a = r_{1}$ ) se ellipse 1 ( $a = a_{1}$ ) par jump: $Δ v_{1} = μ (\frac{2}{r _{1}} - \frac{1}{a _{1}}) - \frac{μ}{r _{1}}$

Burn 2 — $r_{b}$ par, ellipse 1 se ellipse 2 par jump (dono apna opposite side uthate hain): $Δ v_{2} = μ (\frac{2}{r _{b}} - \frac{1}{a _{2}}) - μ (\frac{2}{r _{b}} - \frac{1}{a _{1}})$

Yeh chota kyun hai: bade $r_{b}$ par, $\frac{2}{r _{b}}$ term dono roots mein dominate karta hai aur nearly cancel ho jaate hain.

Burn 3 — $r_{2}$ par, ellipse 2 se circular ( $a = r_{2}$ ) par drop. Yeh ek braking burn hai, isliye magnitude lete hain: $Δ v_{3} = μ (\frac{2}{r _{2}} - \frac{1}{a _{2}}) - \frac{μ}{r _{2}}$

Crossover rule (80/20 takeaway)

Maano $R = r_{2} / r_{1}$ . Bi-parabolic limit mein ( $r_{b} \to \infty$ ) classic result yeh hai:

Agar $R < 11.94$ → Hohmann hamesha jeetat hai.
Agar $R > 15.58$ → bi-elliptic (suitable $r_{b}$ se) hamesha jeetat hai.
$11.94$ aur $15.58$ ke beech → depend karta hai chosen $r_{b}$ par.

Figure — Bi-elliptic transfer — when it wins over Hohmann

Worked examples

Worked example Example 1 — LEO se ek badi orbit tak (

$R = 20$ ) Lo $μ = 1$ , $r_{1} = 1$ , $r_{2} = 20$ , aur choose karo $r_{b} = 100$ .

Circular speeds: $v_{c 1} = 1/1 = 1$ , $v_{c 2} = 1/20 = 0.2236$ . Kyun? $v_{c} = μ / r$ .

Ellipse axes: $a_{1} = (1 + 100) /2 = 50.5$ , $a_{2} = (20 + 100) /2 = 60$ . Kyun? apsidal radii ka average.

Burn 1: $2/1 - 1/50.5 - 1 = 1.9802 - 1 = 0.4072$ . Kyun? vis-viva at $r_{1}$ minus circular speed.

Burn 2: $2/100 - 1/60 - 2/100 - 1/50.5 = 0.003333 - 0.000198 = 0.0577 - 0.0141 = 0.0437$ . Kyun? door-wala burn — notice karo kitna tiny hai.

Burn 3 (brake): $2/20 - 1/60 - 1/20 = 0.08333 - 0.2236 = 0.2887 - 0.2236 = 0.0651$ .

Total bi-elliptic: $0.4072 + 0.0437 + 0.0651 = 0.516$ .

Hohmann: $a_{H} = (1 + 20) /2 = 10.5$ . Burn A $= 2/1 - 1/10.5 - 1 = 1.9048 - 1 = 0.3801$ . Burn B $= 1/20 - 2/20 - 1/10.5 = 0.2236 - 0.00476 = 0.2236 - 0.0690 = 0.1546$ . Total Hohmann $= 0.5347$ .

Result: bi-elliptic $0.516 < 0.535$ Hohmann → bi-elliptic jeeta ✅ (kyunki $R = 20 > 15.58$ ).

Common mistakes

Feynman check

Recall Ek 12-saal ke bachche ko samjhao

Socho tum ek merry-go-round par ho aur ek bahut bade, dheemar wale par jump karna hai jo door hai. Jab tum tez ghoom rahe ho toh apne aap ko turn karna mushkil hota hai. Toh instead tum pehle apne aap ko wahaaan tak throw karte ho jahan sab kuch slow aur lazy drift karta hai — wahan tum gently apne aap ko nudge kar sakte ho bade ride ke saath line up karne ke liye, kyunki kuch tezi se nahi guzar raha. Phir tum coast karke wapas aate ho aur hop on karte ho. Lamba raasta silly lagta hai, par us calm far-away zone mein gentle nudge itni energy bacha sakta hai jo cost se zyada ho — lekin sirf tab jab bada ride bahut, bahut door ho (lagbhag 12× bada). Warna seedha do-jump path lo.

Flashcards

Bi-elliptic transfer mein kitne burns hote hain, Hohmann ke mukable?

Bi-elliptic 3 burns aur 2 ellipses use karta hai; Hohmann 2 burns aur 1 ellipse use karta hai.

Kis radius ratio

R = r_{2} / r_{1}

se neeche Hohmann hamesha bi-elliptic ko beat karta hai?

R < 11.94

Kis radius ratio ke upar bi-elliptic (suitable

r_{b}

se) hamesha better hota hai?

R > 15.58

Bi-elliptic transfer ka middle (second) burn itna cheap kyun hota hai?

Yeh bade radius

r_{b}

par hota hai jahan orbital speeds tiny hoti hain, isliye periapsis uthane ke liye velocity change chhota hota hai (vis-viva:

2/ r_{b}

term dominate karta hai aur nearly cancel ho jaata hai).

Burn 3 (

r_{2}

par circularization) prograde hai ya retro burn?

Retro (brake) —

r_{2}

par tum ellipse 2 ke periapsis par ho, circular speed se faster move kar rahe ho.

Vis-viva equation kya hai?

v^{2} = μ (2/ r - 1/ a)

Pehle bi-elliptic ellipse ka semi-major axis kya hai?

a_{1} = (r_{1} + r_{b}) /2

Bi-parabolic limit kya hai aur uski cost kya hai?

r_{b} \to \infty

wala case jo minimum bi-elliptic

Δ v

deta hai, lekin infinite transfer time ki cost par.

Bi-elliptic ka main hidden downside jab bhi fuel bachata hai?

Bahut zyada transfer time (badi orbits → lambe periods).

Connections

Hohmann transfer — do-burn baseline jisse yeh compete karta hai
Vis-viva equation — ek hi tool jisse saare $Δ v$ aate hain
Oberth effect — burns local speed ke hisaab se zyada/kam effective hoti hain
Semi-major axis and orbital energy — kyun $a$ speed set karta hai
Plane change maneuvers — aksar bi-elliptic ke saath combine hoti hain (door par saste)
Transfer time vs delta-v tradeoffs — time penalty
Bi-parabolic transfer — limiting best case