Hohmann transfer — derivation, minimum energy transfer

3.2.19 · Physics › Orbital Mechanics & Astrodynamics

Hohmann transfer KYA hota hai?

Tangent burns KYUN? Kyunki ek burn speed ko tab sabse efficiently change karta hai jab thrust velocity ke parallel ho. Tangent points par transfer ellipse aur circle ek hi velocity direction share karte hain, isliye burn ko sirf $v$ ki magnitude change karni hoti hai — koi bhi sideways thrust waste nahi hota.

Woh physics jisse aap sab kuch banate hain

Hume two-body problem ( $μ = GM$ ) ke baare mein sirf teen first-principles facts chahiye:

1. Vis-viva equation (orbits ke liye energy conservation): $v^{2} = μ (\frac{2}{r} - \frac{1}{a})$

2. Circular speed (special case, $r = a$ ): $v^{2} = μ (\frac{2}{r} - \frac{1}{r}) = \frac{μ}{r} \Rightarrow v_{circ} = \frac{μ}{r}$

3. Transfer ellipse ka semi-major axis. Iska sabse lamba diameter inner se outer orbit tak phela hua hai: $2 a_{t} = r_{1} + r_{2} \Rightarrow a_{t} = \frac{r _{1} + r _{2}}{2}$ Kyun? Periapsis $= r_{1}$ , apoapsis $= r_{2}$ , aur definition se $r_{p} + r_{a} = 2 a$ .

Do burns KAISE nikaalte hain (derivation)

Figure — Hohmann transfer — derivation, minimum energy transfer

Step 1 — Inner circle par speed

$v_{c 1} = \frac{μ}{r _{1}}$ Yeh step kyun? Yeh woh speed hai jo throttle touch karne se pehle aapke paas actually hai.

Step 2 — Transfer ellipse ke periapsis par zaroori speed

Vis-viva use karo $r = r_{1}$ par $a = a_{t}$ ke saath: $v_{p} = μ (\frac{2}{r _{1}} - \frac{1}{a _{t}}) = μ (\frac{2}{r _{1}} - \frac{2}{r _{1} + r _{2}})$ Yeh step kyun? Periapsis par aap ellipse par planet ke sabse paas hote hain, isliye circle se tez move karna zaroori hai taaki $r_{2}$ tak swing karne ki energy ho.

Step 3 — Pehla burn

$Δ v_{1} = v_{p} - v_{c 1} = \frac{μ}{r _{1}} (\frac{2 r _{2}}{r _{1} + r _{2}} - 1)$ Yeh step kyun? Dono velocities tangent hain (same direction), isliye $Δ v$ sirf magnitudes ka fark hai.

Step 4 — Transfer ellipse ke apoapsis par speed

Vis-viva $r = r_{2}$ par: $v_{a} = μ (\frac{2}{r _{2}} - \frac{2}{r _{1} + r _{2}})$ Yeh outer circular speed se slow hai — aap $r_{2}$ par pahunchte waqt "steam khatam ho chuka" hota hai.

Step 5 — Circularize karne ke liye doosra burn

$v_{c 2} = \frac{μ}{r _{2}}$ $Δ v_{2} = v_{c 2} - v_{a} = \frac{μ}{r _{2}} (1 - \frac{2 r _{1}}{r _{1} + r _{2}})$ Yeh step kyun? Bade circle ko hold karne ke liye speed badhani padti hai.

Total cost

$Δ v_{total} = Δ v_{1} + Δ v_{2}$

Yeh minimum energy KYUN hai? (Pehle forecast, phir verify)

Oberth insight (Kyun tangential + low-altitude burns jeetते hain): kinetic energy $\frac{1}{2} v^{2}$ hai. Ek fixed $Δ v$ wahan add kiya jaaye jahan aap already fast hain (gravity well mein deep, periapsis par) toh zyada orbital energy milti hai: $d (\frac{1}{2} v^{2}) = v d v$ $v$ ke saath badhta hai. Isliye pehla burn $r_{1}$ par place kiya jaata hai.

Worked Examples

Common Mistakes

Recall

Recall 12-saal ke bachche ko explain karo (Feynman)

Socho tum ek merry-go-round par ho aur baahr wali badi merry-go-round par jump karna chahte ho. Tum seedha baahr nahi step kar sakte — tum ud jaoge. Toh tum ek zor ka push dete ho ek bade loopy path par swing karne ke liye jo baahri ring tak just pahunchti hai, aur jab tum wahan pahunchte ho toh tum wahan rehne ke liye bahut slow ho, isliye tum ek aur push dete ho uski speed pakadne ke liye. Do pushes, sabse sasta ride. Woh looping path ek squished circle (ellipse) hai, aur yeh orbits change karne ka aaram-talab, fuel-saving tarika hai.

Recall Active recall — answers cover karo

Do burns kyun, ek kyun nahi? → Kyunki ek burn sirf circle ko ellipse mein stretch karta hai; nayi radius par re-circularize karne ke liye doosra zaroori hai.
Burns kahan apply hote hain? → Transfer ellipse ke periapsis ( $r_{1}$ ) aur apoapsis ( $r_{2}$ ) par, tangentially.
$a_{t}$ kya hai? → $(r_{1} + r_{2}) /2$ .
Bi-elliptic Hohmann ko kab beat karta hai? → Jab $r_{2} / r_{1} ≳ 11.94$ .

Flashcards

Hohmann transfer kya hota hai?

Ek two-impulse, minimum-energy maneuver jo do coplanar circular orbits ke beech ek ellipse ke through hota hai jo dono ko tangent ho (periapsis

r_{1}

par, apoapsis

r_{2}

par).

Vis-viva equation?

v^{2} = μ (2/ r - 1/ a)

ε = \frac{1}{2} v^{2} - μ / r = - μ /2 a

se derive hoti hai.

Transfer ellipse ka semi-major axis?

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

Pehle burn

Δ v_{1}

ka formula?

μ / r_{1} (2 r_{2} / (r_{1} + r_{2}) - 1)

Doosre burn

Δ v_{2}

ka formula?

μ / r_{2} (1 - 2 r_{1} / (r_{1} + r_{2}))

Tangential burns kyun?

Velocities collinear hoti hain, isliye

Δ v

ek pure magnitude change hai — turning par koi thrust waste nahi.

Transfer time?

Ellipse period ka aadha:

t = π a_{t}^{3} / μ

Raise karte waqt apoapsis par circularize karne ke liye speed up karo ya slow down?

Speed up (prograde), kyunki

v_{a} < v_{c 2}

Bi-elliptic Hohmann ko kab beat karta hai?

Radius ratios

r_{2} / r_{1} ≳ 11.94

ke liye.

Bada burn periapsis par kyon sabse best jagah hai (Oberth)?

Energy gain

= v d v

; jahan

v

bada ho wahan

Δ v

add karne se har unit fuel par zyada orbital energy milti hai.

Connections

Vis-viva equation
Kepler's Third Law
Orbital energy & semi-major axis
Oberth effect
Bi-elliptic transfer
Delta-v budget
Two-body problem