Hohmann Δv calculation — both maneuvers — Hinglish notes

3.2.20 · Physics › Orbital Mechanics & Astrodynamics

KYA calculate ho raha hai?

Hamare paas teen orbits hain:

Orbit 1: circular, radius $r_{1}$ (starting).
Transfer: ellipse jiska periapsis $r_{1}$ hai, apoapsis $r_{2}$ .
Orbit 2: circular, radius $r_{2}$ (target), jahan $r_{2} > r_{1}$ .

KAISE: Har speed ko first principles se derive karo

Step 1 — Circular orbit speed (Newton se)

Kyun? Hume do circles ki speeds chahiye taaki transfer ellipse se compare kar sakein.

Circle of radius $r$ ke liye gravity = centripetal force set karo: $\frac{GM m}{r ^{2}} = \frac{m v ^{2}}{r}$

Ye step kyun? Circle par net inward force gravity hi hai, aur required inward force $m v^{2} / r$ hai. $m$ cancel karo, $r$ se multiply karo: $v_{circ}^{2} = \frac{GM}{r} = \frac{μ}{r} \Rightarrow v_{circ} = \frac{μ}{r}$ jahan $μ = GM$ standard gravitational parameter hai.

Step 2 — Transfer ellipse par speed (vis-viva)

Kyun? Burns ek circular speed ko usi radius par ek elliptical speed se compare karte hain, isliye hume ellipse par speed chahiye.

Ellipse par energy conservation se shuru karo. Specific orbital energy: $ε = \frac{v ^{2}}{2} - \frac{μ}{r} = - \frac{μ}{2 a}$

$ε = - μ /2 a$ kyun hai? Kisi bhi Keplerian orbit ki total energy sirf semi-major axis $a$ par depend karti hai (ek standard Kepler result). $v$ ke liye rearrange karne par vis-viva equation milta hai: $v^{2} = μ (\frac{2}{r} - \frac{1}{a})$

Ye step kyun? Ye ek formula size $a$ ke orbit ke kisi bhi point $r$ par speed deta hai. Circles ( $a = r$ ) special case ke roop mein nikalte hain: $v^{2} = μ (2/ r - 1/ r) = μ / r$ . ✔ (Step 1 ke saath consistency check).

Step 3 — Transfer ellipse ka semi-major axis

Kyun? Vis-viva ko $a$ chahiye. Ellipse $r_{1}$ ko periapsis par aur $r_{2}$ ko apoapsis par touch karti hai. Major axis periapsis-se-apoapsis tak phailta hai: $2 a_{t} = r_{1} + r_{2} \Rightarrow a_{t} = \frac{r _{1} + r _{2}}{2}$ Ye step kyun? Major axis = periapsis distance + apoapsis distance (dono planet ke center par focus se measure), ellipse ke paar sum kiya gaya.

Step 4 — Char key speeds

Location	Formula	Meaning
Circle 1	$v_{c 1} = μ / r_{1}$	burn 1 se pehle speed
Transfer periapsis	$v_{p} = μ (\frac{2}{r _{1}} - \frac{1}{a _{t}})$	burn 1 ke baad speed
Transfer apoapsis	$v_{a} = μ (\frac{2}{r _{2}} - \frac{1}{a _{t}})$	burn 2 se pehle speed
Circle 2	$v_{c 2} = μ / r_{2}$	burn 2 ke baad speed

Step 5 — Do Δv's

Burn 1 ( $r_{1}$ par): circle se transfer periapsis tak speed badhaao. Kyunki $a_{t} > r_{1}$ , ellipse yahan faster hai: $Δ v_{1} = v_{p} - v_{c 1} = \frac{μ}{r _{1}} (\frac{2 r _{2}}{r _{1} + r _{2}} - 1)$

Burn 2 ( $r_{2}$ par): transfer apoapsis se (faster) outer circle tak speed badhaao: $Δ v_{2} = v_{c 2} - v_{a} = \frac{μ}{r _{2}} (1 - \frac{2 r _{1}}{r _{1} + r _{2}})$

Total: $Δ v_{tot} = Δ v_{1} + Δ v_{2}$

Figure — Hohmann Δv calculation — both maneuvers

Worked Example 1 — LEO se GEO (Earth)

Given $μ_{\oplus} = 3.986 \times 1 0^{5} km^{3} / s^{2}$ , $r_{1} = 6678$ km (300 km LEO), $r_{2} = 42164$ km (GEO).

$a_{t} = \frac{6678 + 42164}{2} = 24421$ km. Kyun? periapsis+apoapsis ka midpoint.
$v_{c 1} = 398600/6678 = 7.726$ km/s. Kyun? $r_{1}$ par circular speed.
$v_{p} = 398600 (2/6678 - 1/24421) = 10.15$ km/s. Kyun? periapsis par vis-viva.
$Δ v_{1} = 10.15 - 7.726 = 2.42$ km/s.
$v_{c 2} = 398600/42164 = 3.075$ km/s.
$v_{a} = 398600 (2/42164 - 1/24421) = 1.61$ km/s.
$Δ v_{2} = 3.075 - 1.61 = 1.47$ km/s.
Total $Δ v \approx 3.89$ km/s. Sum kyun? do independent impulsive burns.

Worked Example 2 — Kaun sa burn bada hai?

LEO→GEO ke liye, $Δ v_{1} > Δ v_{2}$ . Ye surprising kyun lagta hai? Kyunki outer orbit slower hai, students expect karte hain ki outer burn chota hoga... aur hai bhi, lekin note karo ki pehla burn zyaadatar energy lifting karta hai: use itni energy inject karni padti hai ki door ke apoapsis tak pahunche, jahan speeds largest aur gravity strongest hoti hai. Lesson: gravity well mein deep burn wala expensive hota hai (isliye Oberth effect low burns ko energy gain ke liye efficient banata hai).

Common mistake Steel-manned errors

Mistake A: "Bas do circular speeds subtract karo, $v_{c 1} - v_{c 2}$ ." Kyun sahi lagta hai: Tum do circles ke beech move kar rahe ho, toh cost clearly circular speeds ka difference hona chahiye. Kyun galat hai: Tum actually dono circles par continuously ride nahi karte — beech mein ek ellipse ride karte ho. Sahi comparisons har radius par circle↔ellipse hain. Yahan $v_{c 1} - v_{c 2} = 4.65$ km/s use karna bakwaas hoga (reasoning ka galat sign aur galat number). Fix: hamesha vis-viva ke zariye transfer speeds $v_{p}, v_{a}$ insert karo.

Mistake B: Sign errors — $Δ v_{2} = v_{a} - v_{c 2}$ likhna. Kyun sahi lagta hai: Tum apoapsis par "arrive" kar rahe ho, toh natural lagta hai ki transfer se circle subtract karo. Kyun galat hai: Apoapsis par ellipse target circle se slower hai, isliye tum speed add karte ho: $Δ v_{2} = v_{c 2} - v_{a} > 0$ . Fix: physically socho — Hohmann mein hamesha prograde burn hota hai, dono Δv's positive speed increases hain (orbit raise karne ke liye).

Mistake C: $μ = GM$ bhool jaana, sirf $G$ plug karna. Fix: central body ka $μ$ directly use karo; units km³/s² km aur km/s ke saath pair karte hain.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho tum ek merry-go-round (inner orbit) par ho aur door bahar ek bade, slower merry-go-round par hop karna chahte ho. Seedha across jump nahi kar sakte. Isliye tum ek bada dhakka dete ho taaki ek curved path (oval) ke along bahar udo, door ke edge tak coast karo, aur phir ek doosra dhakka do taaki bade ride ki speed match karo aur us par settle ho jao. Do dhakke, bas itna. Pehla dhakka (neeche jahan cheezein fast spin karti hain) mushkil wala hota hai.

Active Recall

Ek classic Hohmann transfer kin do orbits ko connect karta hai?

Do coplanar, circular orbits.

Exactly do burns kyun?

Ek periapsis par apoapsis ko target radius tak raise karne ke liye, ek apoapsis par circularize karne ke liye.

Vis-viva equation batao.

v^{2} = μ (\frac{2}{r} - \frac{1}{a})

Transfer ellipse ka semi-major axis?

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

Pehle Δv ka formula (orbit raise karna)?

Δ v_{1} = μ / r_{1} (2 r_{2} / (r_{1} + r_{2}) - 1)

Doosre Δv ka formula?

Δ v_{2} = μ / r_{2} (1 - 2 r_{1} / (r_{1} + r_{2}))

Δ v_{2} = v_{c 2} - v_{a}

kyun hai aur

v_{a} - v_{c 2}

kyun nahi?

Apoapsis par ellipse outer circle se slower hai, isliye speed add karni padti hai (prograde burn), jo positive Δv deta hai.

LEO→GEO ke liye kaun sa burn bada hai aur kyun?

Pehla (periapsis) burn, kyunki wo gravity well mein deep hoke far apoapsis tak pahunchne ki energy inject karta hai.

Circular orbit speed kis cheez se derive hoti hai?

Gravity = centripetal force set karke:

GM m / r^{2} = m v^{2} / r \Rightarrow v = μ / r

r_{2} \to \infty

par

Δ v_{1}

kya hota hai?

v_{c 1} (2 - 1)

— periapsis speed escape/parabolic speed increment ke paas pahunchti hai.

Connections

Vis-viva equation — yahan har speed ke peeche ka engine.
Circular orbital velocity — special case $a = r$ .
Specific orbital energy — $ε = - μ /2 a$ ka source.
Oberth effect — deep burns efficient kyun hote hain (Ex. 2 explain karta hai).
Bi-elliptic transfer — Hohmann ko tab beat karta hai jab $r_{2} / r_{1} ≳ 11.94$ .
Standard gravitational parameter — poore mein use hone wala $μ = GM$ .
Impulsive maneuver approximation — kyun hum burns ko instant Δv treat karte hain.