Bi-elliptic transfer — when it wins over Hohmann: Foundations

3.2.21 · D1 · Physics › Orbital Mechanics & Astrodynamics › Bi-elliptic transfer — when it wins over Hohmann

Yeh page kuch bhi assume nahi karta. Bi-elliptic transfer — when it wins over Hohmann padhne se pehle, wahan use hone wala har letter aur squiggle yahan ground up se build kiya gaya hai, ek aisi order mein jahan har idea sirf pehle wale ideas par rely karta hai.

1. Central body, aur letters $G$ , $M$ , $μ$

Socho ek planet (ya Sun) still baitha hai, aur ek tiny spacecraft uske around loop kar raha hai. Planet bahut bada hai; spacecraft ek speck hai. Saari gravity planet se aati hai.

$M$ = central body ki mass (planet ya star). Bada $M$ → zyada strong pull.
$G$ = gravitational constant, nature ka ek fixed number jo yeh set karta hai ki gravity har jagah kitni strong hai.
$μ$ (Greek letter "mu") = shorthand $μ = GM$ , jise standard gravitational parameter kehte hain.

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

Picture: planet beech mein, spacecraft ek ring par uske around, aur ek arrow inward ki taraf point karta hua jis par likha hai "gravity, strength set by $μ$ ".

2. Radius $r$ — tum kitne door ho

$r$ = planet ke centre se spacecraft tak ki distance. Surface se nahi — centre se.

Hum $r_{1}$ use karte hain chhote starting orbit ke radius ke liye, $r_{2}$ bade target orbit ke radius ke liye, aur baad mein $r_{b}$ detour ke ek chosen far-away point ke liye.

3. Circular orbit aur uski speed $v_{c}$

Circular orbit mein spacecraft constant distance $r$ par ghoomta hai — ek perfect ring. Har point par woh same speed se move karta hai. Hum us speed ko $v_{c} (r)$ kehte hain, ==radius $r$ par circular speed==.

$v$ (bas "vee") = spacecraft ki speed: woh kitne metres per second cover karta hai, direction ignore karke. Hum hamesha speeds ko positive numbers maanenge.

Picture mein do circular orbits dikhti hain: ek chhoti fast wali (lamba velocity arrow) aur ek badi slow wali (chhota velocity arrow).

4. Ellipses, apsides, aur semi-major axis $a$

Circle ek special shape hai; zyada general orbit ek ellipse hai — ek squashed circle, jaise ek oval racetrack. Transfer orbit hamesha ek ellipse ka piece hoti hai.

Periapsis = ellipse ka planet ke sabse paas wala point. Wahan tum fastest move karte ho.
Apoapsis = sabse door wala point of the ellipse. Wahan tum slowest move karte ho.
Yeh donon milke apsides hain (apsis ka plural).

$a$ = semi-major axis: ellipse ka sabse lamba diameter ka aadha. Yeh ek akela number hai jo batata hai ki ellipse "kitna bada" hai.

Picture: ek ellipse jiske ek focus par planet hai, periapsis aur apoapsis marked hain, aur semi-major axis $a$ long axis ki half-length ke roop mein draw ki gayi hai.

5. Vis-viva equation — master speed formula

Ab sabse important tool. Poori story ke liye Vis-viva equation aur Semi-major axis and orbital energy dekho; yahan woh hai jo tumhe chahiye.

Recall Check: circular speed vis-viva se nikalti hai

Circle par, $r$ kabhi nahi badalta, isliye $a = r$ . Substitute karo: $v_{c}^{2} = μ (\frac{2}{r} - \frac{1}{r}) = \frac{μ}{r} \Rightarrow v_{c} = \frac{μ}{r}$ Yahan se $v_{c} (r) = μ / r$ aata hai — yeh koi alag law nahi hai.

6. Burn aur $Δ v$

Ek rocket burn engine fire karta hai speed change karne ke liye. Kyunki tum kisi orbit par kisi point par ho, wahan apni speed change karna turant change karta hai ki tum kis orbit par ho (naya $a$ ).

$Δ$ (Greek "delta") ka matlab hai "mein change." Toh $Δ v$ = burn produce karta hai speed mein change.

Poore maneuver ki total cost har burn ke $∣Δ v ∣$ ka sum hai. Hohmann do add karta hai; bi-elliptic teen add karta hai. Wahi sum exactly woh hai jo parent note compare karta hai.

7. Radius ratio $R$

$R = r_{2} / r_{1}$ = outer radius ka inner radius se ratio: "target orbit, start se kitne times badi hai?"

8. Bi-parabolic limit ( $r_{b} \to \infty$ )

Detour point $r_{b}$ ko aur aur door push kiya ja sakta hai. Jaise $r_{b} \to \infty$ ("infinity ki taraf jaata hai," yani bina bound ke badhta hai), door wale ellipses ellipses nahi rehte aur parabolas ban jaate hain — barely-bound escape paths. Yeh ideal Bi-parabolic transfer hai, woh theoretical best case hai jise bi-elliptic approach kar sakta hai lekin kabhi reach nahi kar sakta (isme infinite time lagta). Famous thresholds is limit mein compute hote hain.

Yeh topic ko kaise feed karte hain

Related tools jo tum baad mein miloge: Oberth effect (deep burns efficient kyun hote hain), Plane change maneuvers (direction reshape karna), aur Transfer time vs delta-v tradeoffs (hidden time cost).

Equipment checklist

Khud ko test karo — har line answer chhupaati hai.

μ

ka matlab kya hai aur ise bundle kyun karein?

μ = GM

;

G

aur

M

hamesha multiply hokar aate hain, isliye ek symbol cleaner hai.

Kya

r

surface se measure hota hai ya centre se?

Centre of the central body se.

Badi orbit mein spacecraft slow kyun move karta hai?

Weak far-out gravity sirf ek gently-moving craft ko circle mein bend kar sakti hai.

Circular speed formula kya hai aur yeh kahan se aata hai?

v_{c} = μ / r

, vis-viva se jab

a = r

ho.

Periapsis aur apoapsis kya hain?

Closest point (fastest) aur farthest point (slowest) of an ellipse.

Do apsidal radii se

a

kaise nikaalte hain?

a = (r_{peri} + r_{apo}) /2

, unka average.

Vis-viva equation batao.

v^{2} = μ (\frac{2}{r} - \frac{1}{a})

Vis-viva tumhe kya karne deta hai?

v, r, a

mein se koi bhi do do, teesra nikaal lo — burn ke baad

a

change hone par nayi speed padhlo.

Δ v

kya hai aur hum uska magnitude kyun lete hain?

Speed mein change; fuel cost sirf size par depend karti hai, sign par nahi.

Kya slow down karna free hai?

Nahi — braking ab bhi engine fire karta hai aur

∣Δ v ∣

cost karta hai.

R

kya hai aur analysis ise kyun use karti hai?

R = r_{2} / r_{1}

; sirf proportion matter karta hai, isliye results unit-free numbers hain.

Jab

r_{b} \to \infty

ho toh kya hota hai?

Transfer bi-parabolic ban jaata hai — ideal limit, infinite time leta hai.