3.2.22
Ek satellite ka orbit ek plane mein hota hai jo Earth ke center se guzarti hai. Yeh change karne ke liye ki woh plane kis taraf jhukti hai (iska inclination ) — tumhe yeh change karne ki zaroorat nahi ki tum kitni tez ja rahe ho — bas tumhe apna velocity vector ek angle Δ i karna hoga. Lekin velocity ek vector hai, aur ek vector ko rotate karna hi velocity mein ek change hai. Us rotation ki cost hai Δ v = 2 v sin ( Δ i /2 ) ek vector ko rotate karna expensive hai. Sirf 60° Δ v = v
Definition Plane change (inclination change) maneuverEk aisa maneuver jo orbital plane ko ek axis ke around rotate karta hai, inclination i Δ i v change kiye bina . Speed ki magnitude pehle aur baad mein same rehti hai; sirf v direction badlti hai.
Kyunki yeh sirf direction change karta hai, ideal plane change ek pure rotation of v hai angle == Δ i ==
Half angle kyun?
Isosceles triangle draw karo (do equal sides v Δ v Δ i v sin ( Δ i /2 ) 2 v sin ( Δ i /2 ) apne aap half-angle build kar leti hai.
Worked example 60° ka shockΔ i = 60° Δ v = 2 v sin ( 30° ) = 2 v ( 0.5 ) = v Yeh step kyun? sin 30° = 0.5 60° Δ v puri orbital speed ke equal hai — LEO mein lagbhag 7.7
Worked example Small angle — ek GPS nudgeInclination ko Δ i = 2° v = 3.9 Δ v = 2 ( 3.9 ) sin ( 1° ) = 7.8 × 0.01745 ≈ 0.136 km/s = 136 m/s Yeh step kyun? sin 1° ≈ 0.01745 tiny 2° 136
Intuition Fuel bachane ki strategyKyunki Δ v ∝ v sabse slow ho — elliptical orbit ke apoapsis par. Ek geostationary transfer wahan apogee burn ke saath plane change combine karta hai, jahan v 1.6 10
Worked example Combined vs separate burnEk speed change aur ek plane change ko angle Δ i Δ v = v 1 2 + v 2 2 − 2 v 1 v 2 cos Δ i Yeh step kyun? Same law-of-cosines triangle, lekin ab do sides ke different magnitudes v 1 = v 2 v 1 = v 2 = v 2 v sin ( Δ i /2 ) kam cost karta hai (triangle inequality).
Δ v = v 2 − v 1 Δ v = 0 Yeh sahi kyun lagta hai: agar speed ki magnitude nahi badlti, toh surely koi fuel nahi chahiye?
Fix: v vector hai. Δ v = ∣ v 2 − v 1 ∣ ∣ v 2 ∣ − ∣ v 1 ∣ vectors subtract karo, phir magnitude lo.
Common mistake "Sine meinΔ i Δ i /2 Yeh sahi kyun lagta hai: tumne Δ i Δ i Fix: Half -angle isosceles-triangle geometry (perpendicular bisector) se aata hai. Sanity check: Δ i = 180° Δ v = 2 v sin ( 90° ) = 2 v v v sin ( 180° ) = 0 0
Common mistake "Plane changes bade satellites ke liye saste hote hain, bas budget kar lo."Yeh sahi kyun lagta hai: rockets plenty of maneuvers karte hain.
Fix: Δ v exponentially map hota hai m 0 / m f = e Δ v / v e 60° Δ v ≈ v or bi t avoid karte hain, ya unhe slow-point burns mein fold karte hain.
Recall Kya tum ise rebuild kar sakte ho?
∣Δ v ∣ = 0 Law of cosines use karke Δ v = 2 v sin ( Δ i /2 )
Orbit mein plane change kahan karna chahiye, aur kyun?
60° Δ v v
Answers: (1) velocity ek vector hai, direction changes mein thrust lagta hai; (2) law of cosines + half-angle identity; (3) apoapsis par jahan v Δ v ∝ v v
Recall Feynman: 12-saal ke bachche ko explain karoSocho tum ek merry-go-round par ho aur ek ball pakde ho, full speed se bhaag rahe ho. Tumhara orbit ek circle hai jo tum trace karte rehte ho. Ab koi kehta hai "same speed se bhago, lekin ek alag direction mein muh karo." Aisa karne ke liye tumhe skid karke turn lena hoga — aur tez jaate hue turn karna ek bada dhakka chahiye. Tum jitni tez ja rahe ho, woh sideways dhakka utna hi bada hona chahiye. Woh dhakka hi Δ v slowly coast kar raha ho.
"Two-Vee-Sine-Half" → 2 v sin ( Δ i /2 ) "Slow mein turn karo, kam kharch karo" — plane sabse slow point (apoapsis) par change karo.
Plane change Δv formula Δ v = 2 v sin ( Δ i /2 )
Δv zero kyun nahi hota jab speed unchanged ho? velocity ek vector hai; uski direction rotate karne mein thrust lagta hai — vectors subtract karo, phir magnitude lo
Formula derive karne ke liye kaun se do tools hain? isosceles velocity triangle par law of cosines + half-angle identity 1 − cos θ = 2 sin 2 ( θ /2 )
60° plane change ke liye Δv (v ke terms mein)? exactly v 2 v sin ( 30° ) = v
180° reversal ke liye Δv? 2 v 2 v sin ( 90° ) = 2 v
Orbit mein plane change karne ki best jagah? apoapsis par, jahan speed v
Combined speed+plane change formula? Δ v = v 1 2 + v 2 2 − 2 v 1 v 2 cos Δ i Burns combine karne se fuel kyun bachta hai? triangle inequality — ek vector step do sequential steps se chhota hota hai
Δv itna bada fuel cost kyun translate karta hai? Tsiolkovsky: mass ratio = e Δ v / v e
only changes direction of
same magnitude before and after
Vector triangle isosceles
Δv2 = 2v2 times 1 minus cos Δi
1 minus cos = 2 sin squared half
Plane changes very expensive
