3.4.24 · D1Aerocapture — using atmosphere to decelerate into orbit
Ek spacecraft jo kisi planet ke paas aata hai usmein bahut zyada energy hoti hai trap hone ke liye, isliye woh seedha nikal jaata — jab tak koi us extra energy ko chura na le. Aerocapture planet ki apni hawa ko yeh kaam karne deta hai ek hi dip mein, ek "fly-by-forever" path ko "loop-around-forever" orbit mein badal deta hai.
Neeche jo bhi hai woh ek tool hai jo tumhe chahiye parent note Aerocapture padhne ke liye. Hum har symbol ko zero se build karte hain — koi prior notation assumed nahi. Upar se neeche padho; har piece agla use karta hai.
Definition Ek consistent unit system (pehle yeh padho)Physics mein units mix hoti hain, isliye hum ek rule fix karte hain aur uspe tikke rehte hain : orbit maths ke liye hum kilometres aur seconds mein kaam karte hain (toh r a v v ∞ μ km 3 / s 2 ε km 2 / s 2 Drag maths ke liye, atmospheric data metres aur kilograms mein quote hoti hai (density ρ kg/m 3 A m 2 m ek ko convert karo taaki dono match karein (1 km = 1000 m
Kisi bhi formula se pehle, scene picture karo. Ek round planet still baitha hai. Ek tiny spacecraft uske paas move kar raha hai. Woh ek quantity jo hum baar baar measure karte hain woh hai craft planet ke centre se kitni door hai .
r r planet ke centre se spacecraft tak, kilometres (km) mein measured.
Picture: upar figure mein yellow line ki length, planet ke core ke dot se chhote craft tak.Kyun zaroori hai: gravity zyada kheenchti hai jab tum paas hote ho aur kam jab door hote ho. Har gravity formula ko jaanna chahiye "kitna paas?" — woh exactly r Dhyan raho: r centre se measure hota hai, surface se nahi. Altitude h
R R surface tak ki doori hai (km) — ball kitni badi hai.
Picture: upar figure mein blue disk ki radius, white centre-dot se edge tak.Kyun zaroori hai: yeh "centre se doori" r h r = R h r = R + h h = r − R
v v
Picture: figure mein blue arrow ki length — lamba arrow matlab tez.Kyun zaroori hai: tez craft mein bahut motion-energy hoti hai; use slow karna aerocapture ka poora game hai.
Tumhare paas ek coin hai. Uski money-value hai. Ek moving, zyada upar spacecraft mein energy-value hoti hai jo do parts se banti hai: move karne se energy aur gravity mein zyada upar hone se energy. Aerocapture poori tarah is energy ke ghattne ki kahani hai.
Definition Kinetic energy per unit mass,2 1 v 2 har kilogram mein energy track karo. Motion ke liye yeh 2 1 v 2
Picture: blue speed-arrow ko squared socho — speed double karo aur yeh energy chaar guna ho jaati hai (v 2 Kyun per-mass: craft ki actual mass har orbit question se cancel ho jaati hai, isliye physicists use drop karte hain aur "per kilogram" quantities track karte hain. Maths saaf rehta hai.
v 2 v aur lambi doori dono energy add karte hain, aur dono speed ke saath badhte hain — toh energy speed times speed ki tarah badhti hai, yaani v 2
μ μ = GM G M km 3 / s 2 μ ≈ 4.28 × 1 0 4 km 3 / s 2
Picture: ek "gravity dial" — heavier planet ka bada μ Bundle kyun karein: orbit problems mein G M
Common mistake "Slower hamesha matlab captured."Bilkul sahi nahi — capture combination v 2 /2 − μ / r v r − μ / r ε
Definition Hyperbola vs. ellipse
Hyperbola: ek open path jo planet ke paas se ek baar swing karta hai aur forever ud jaata hai. Yeh "escape / not captured" shape hai, ε > 0 Hyperbolic Trajectories & Hyperbolic Excess Velocity .Ellipse: ek closed oval loop jo baar baar wapas aata hai — ek bound orbit . ε < 0 Picture: neeche figure — red hyperbola bend karta hai aur chala jaata hai; green ellipse loop mein close ho jaata hai.
a a long axis ki half length hai — ek single number jo kehta hai "orbit kitna bada hai."
Picture: upar figure mein green loop ki half width.Kyun zaroori hai: aerocapture ek specific target orbit aim karta hai, yaani specific a ε o u t
Ab humein woh tool chahiye jo energy hataata hai: drag .
ρ ρ kg/m 3 ρ ρ
Picture: neeche figure mein dots-per-box — zameen ke paas dense, oopar sparse.
Intuition Exponential kyun?e fraction hatao aur pressure usi fraction se drop karta hai — "har step mein constant proportion se shrink" wala rule. Jo bhi har equal step mein fixed proportion se shrink karta hai woh exponential hota hai, aur e ρ = ρ 0 e − h / H
m C D A F D = 2 1 ρ v 2 C D A
m A m 2 C D C D C D Force: F D = 2 1 ρ v 2 C D A ρ squared (v 2 C D A Ballistic Coefficient .
Δ Δ Δ v = v in − v o u t Δ ε
v e v e v e v e ≈ 3
Picture: craft ko aage push karte waqt peeche blast ho rahi flame-jet ki speed.
Δ v a er o Δ v a er o = v in − v o u t free mein maarta hai. Ek rocket normally fuel se speed change khareedta hai rocket equation ke zariye: exhaust speed v e Δ m / m = 1 − e − Δ v / v e
sign of epsilon decides capture
ballistic coefficient beta
Right side cover karo aur aloud jawab do; check karne ke liye reveal karo.
r Planet ke centre se craft tak seedhi-line doori (km), surface se nahi.
r R h h = r − R
ε Motion-energy 2 1 v 2 − μ / r
ε ε < 0 ε > 0
μ μ = GM G M
ε = − μ / ( 2 a ) Kyunki ε a − μ / ( 2 a )
a D = ρ v 2 / ( 2 β ) a D = F D / m β = m / ( C D A )
Drag v 2 Tez jaane par tum zyada hawa sweep karte ho aur har bit ko zyada momentum dete ho — v
β Heaviness-per-face m / ( C D A ) β β
v e Exhaust speed — rocket apna propellant kitni tez bahar fenkta hai; bada v e
Atmosphere altitude mein exponential kyun hai? Density har scale height H proportion se shrink hoti hai, jo exactly ρ 0 e − h / H
Δ v a er o v in − v o u t , woh braking jo hawa free deti hai —
rocket equation se bachaya hua fuel.
