Gravity assist (slingshot) — patched conic, v-infinity vectors: Question bank

3.2.24 · D5 · Physics › Orbital Mechanics & Astrodynamics › Gravity assist (slingshot) — patched conic, v-infinity vecto

Shuru karne se pehle, teen words jinhe hum baar baar use karenge — ye sab parent note mein already earn ho chuke hain:

Heliocentric frame — Sun-centered viewpoint, jahan planet khud $V_{p}$ velocity se move kar raha hai.
Planet frame — planet ke saath ride karne wala viewpoint; yahan flyby ek plain hyperbola hai.
== $v_{\infty}$ == — spacecraft ki speed planet ke relative, Sphere of influence (SOI) ki edge par measure ki gayi jahan planet ki pakad effectively chhoot chuki hai.

True or false — justify

True/false — gravity assist craft ko uske planet ke apne frame mein speed up karta hai.

False. Planet frame mein flyby ek symmetric hyperbola hai: speed out equals speed in (

= v_{\infty}

). Wahan sirf

v_{\infty}

ki direction change hoti hai.

True/false — spacecraft energy gain karta hai lekin kuch bhi energy lose nahi karta.

False. Planet exactly utni hi kinetic energy lose karta hai jo craft gain karta hai (dekho Specific orbital energy & angular momentum); planet par effect negligible isliye hai kyunki

m_{sc} ≪ M_{p}

hai, zero nahi.

True/false — SOI mein entry aur exit par

v_{\infty}

ka magnitude same hota hai.

True. Planet frame mein specific energy

ε = v^{2} /2 - μ_{p} / r

conserved hai, aur

ε = v_{\infty}^{2} /2

r \to \infty

, isliye

v_{\infty}^{in} = v_{\infty}^{o u t}

True/false — ek heavier planet hamesha zyada heliocentric speed gain deta hai.

False. Gain

2 v_{\infty}

par capped hai chahe mass kitna bhi ho. Mass sirf tumhe large turn angle

δ

tak pahunchne mein help karta hai; agar

v_{\infty}

tiny hai toh payoff bhi tiny hai.

True/false — SOI ke andar flyby orbit hamesha ek hyperbola hoti hai.

True (ek genuine assist ke liye). Assist ka matlab excess speed hai, isliye

ε > 0

, jo eccentricity

e > 1

deta hai, jo ki ek hyperbola hai. Sirf ek captured craft (

ε < 0

) ellipse hogi — aur woh flyby nahi hai.

True/false — planet ke peeche se guzarna hamesha Sun frame mein tumhe speed up karta hai.

Zyada tar true hai, lekin dhyan se padho. Peeche se guzarna

v_{\infty}

+ V_{p}

ki taraf rotate karta hai, heliocentric speed add karta hai — condition ye hai ki

V_{p}

khud appreciable size ka ho. Agar planet rest par hota (

V_{p} = 0

) toh koi heliocentric gain nahi hota.

True/false — deflection angle

δ

spacecraft ke mass par depend karta hai.

False.

e = 1 + r_{p} v_{\infty}^{2} / μ_{p}

aur

sin (δ /2) = 1/ e

mein sirf planet ka

μ_{p}

, periapsis

r_{p}

, aur

v_{\infty}

hain —

m_{sc}

kabhi nahi. Gravity sabhi masses ko equally accelerate karti hai.

True/false — ek single flyby se

2 v_{\infty}

se zyada gain ho sakta hai.

False. Sirf

v_{\infty}

ki direction change ho sakti hai, isliye maximum possible difference

v_{\infty}^{o u t} - v_{\infty}^{in}

radius

v_{\infty}

ke circle ka diameter hai, yaani

2 v_{\infty}

Spot the error

"Kyunki gravity conservative hai, slingshot craft ki speed change nahi kar sakta — isliye slingshots useless hain."

Flaw ye hai ki sirf ek frame par focus kar rahe ho. Speed planet frame mein unchanged hai, lekin frame transformation

v_{sc} = V_{p} + v_{\infty}

Sun frame mein magnitude change kar deta hai — yahi toh poora point hai.

"Craft dive karte waqt accelerate karta hai, isliye pass ke dौران

v_{\infty}

increase hoti hai."

Periapsis approach karte waqt speed up hota hai phir leave karte waqt slow down — energy conservation speed ko exactly

v_{\infty}

par wapas le aati hai. Jo permanently alter hota hai woh

v_{\infty}

ki direction hai, size nahi.

"Hum poori flyby ko ek heliocentric point par hoti hui treat karte hain, toh hum gravity ko actually ignore kar rahe hain."

Hum gravity ignore nahi kar rahe — hum is fact ka use kar rahe hain ki Sphere of influence (SOI) heliocentric orbit ke comparison mein tiny hai. Uske andar gravity apna poora kaam karte hue

v_{\infty}

ko rotate karti hai; solar scales par position meaningfully shift nahi hoti.

"Craft ko slow down karne ke liye tumhe directly planet mein fly karna hoga."

Nahi — tum planet ke aage se fly karte ho taaki

v_{\infty}

- V_{p}

ki taraf rotate ho, heliocentric speed subtract kare (MESSENGER ne Mercury reach karne ke liye yehi kiya). Koi collision nahi chahiye.

e = 1 + r_{p} v_{\infty}^{2} / μ_{p}

, isliye bada

v_{\infty}

path ko zyada bend karta hai."

Ulta hai. Bada

v_{\infty}

e

badhata hai, aur

sin (δ /2) = 1/ e

bade

e

ke saath shrink karta hai → chhota turn. Fast, far flybys mushkil se bend karte hain; slow, close wale zyada bend karte hain.

"Patched-conic answer exact hai."

Ye ek approximation hai — ye pretend karta hai ki Sun ka pull SOI ke andar vanish ho jata hai aur planet ka bahar vanish ho jata hai, ek sharp boundary par patch kiya gaya. Real trajectories dono ko blend karti hain; patched conics accurate first design pass hain, final truth nahi.

Why questions

Planet ki motion kyun, na ki uski gravity ka kaam, heliocentric boost deta hai?

Gravity planet frame mein zero net work karta hai (speed in = speed out). Extra heliocentric speed poori tarah se planet ki velocity

V_{p}

wapas add karne se aati hai — jaise ek ball ek moving truck se bounce karti hai rather than ek still wall se.

SOI ke andar sahi shape hyperbola (ellipse nahi) kyun hai?

Kyunki craft excess speed ke saath arrive karta hai, uski planet-frame energy

ε = v_{\infty}^{2} /2 > 0

hai. Positive energy ka matlab unbound orbit hai, aur unbound ka matlab eccentricity

e > 1

— ye hai hyperbola ki definition.

Planet ke paas se fly karna velocity ko zyada kyun turn karta hai?

Chhota periapsis

r_{p}

e = 1 + r_{p} v_{\infty}^{2} / μ_{p}

ko 1 ki taraf lower karta hai, aur

sin (δ /2) = 1/ e

badhta hai jab

e \to 1

— isliye deflection

δ

increase hoti hai. Tum gravity well mein deeper dip karte ho aur zyada swing around hote ho.

Jupiter gravity assists ka "king" kyun hai?

Uska enormous

μ_{p}

ratio

r_{p} v_{\infty}^{2} / μ_{p}

ko small banata hai,

e

ko 1 ke paas aur

δ

ko uske maximum ke paas rakhta hai — plus uski solar velocity

V_{p}

real hai, isliye heliocentric payoff large hai. Voyager aur Cassini dono ne faayda uthaya (dekho Voyager & Cassini mission trajectories).

Slingshot alone total gain ko

2 v_{\infty}

se aage kyun nahi push kar sakta?

Operation ek fixed-length vector

v_{\infty}

ka pure rotation hai. Ek rotated vector ka maximum change poora reversal hai, jo circle ke diameter ke barabar difference deta hai,

2 v_{\infty}

Designers slingshots ko periapsis par engine burn ke saath combine kyun karte hain?

Well ke andar deep burn Oberth effect se amplify hoti hai (thrust wahan sabse effective hai jahan speed highest hai), assist se free geometric turn ke upar real propulsive gain stack karta hai.

Heliocentric energy change kyun hoti hai jabki

v_{\infty}

constant hai?

Kyunki

∣ V_{p} + v_{\infty} ∣

unke beech ke angle par depend karta hai.

v_{\infty}

V_{p}

ki taraf rotate karna sum ko lengthen karta hai; usse door karna sum ko shorten karta hai — dekho Two-body problem & vis-viva equation jahan speed orbital energy se map karti hai.

Edge cases

Edge case — agar

v_{\infty} = 0

ho (craft exactly planet ki velocity match kare)?

Rotate karne ke liye kuch nahi hai;

v_{\infty}

zero vector hai, isliye koi heliocentric change possible nahi. Craft simply planet ke saath drift karta rahega — ye capture ki limit hai, assist nahi.

Edge case — agar planet stationary ho (

V_{p} = 0

Frame transformation

v_{sc} = V_{p} + v_{\infty}

ban jaata hai

v_{sc} = v_{\infty}

, jiska magnitude conserved hai. Koi heliocentric speed change nahi — prove karta hai ki planet ki motion, uska mass nahi, jo payoff deti hai.

Edge case —

δ = 180°

ke liye kya chahiye, aur kya ye usually reachable hai?

Poora reversal chahiye

sin (δ /2) = 1

, yaani

e = 1

, yaani

r_{p} v_{\infty}^{2} / μ_{p} \to 0

— ek impossibly close pass ya vanishing

v_{\infty}

. Real periapsis planet ke radius aur atmosphere se limited hai, isliye

δ = 180°

ek idealized cap hai, achievable flyby nahi.

Edge case — agar periapsis

r_{p}

planet ke radius se chhota ho?

Tab "flyby" ek crash hai — hyperbola surface ko intersect karti hai. Physically valid assists require

r_{p}

surface ke upar (aur kisi bhi sensible atmosphere ke upar), jo

r_{p}

par ek hard floor set karta hai aur

δ

par ek ceiling.

Edge case — agar craft ka incoming

v_{\infty}

already

V_{p}

ke parallel ho?

Tab isko rotate karna sirf isse

V_{p}

se door bend kar sakta hai, isliye ye geometry speed lose kar sakti hai lekin zyada add nahi kar sakti — tumhe ek aisa inbound leg chahiye jahan

v_{\infty}

ko reverse karna isse gain ke liye

V_{p}

ki taraf swing kare.

Edge case — agar do flybys chain kiye jayein (gravity-assist tour)?

Har flyby apne gain ko us encounter ke liye

2 v_{\infty}

par cap karta hai, lekin

v_{\infty}

generally har planet par different hoti hai, isliye ek tour large total change accumulate kar sakta hai — exactly aise hi multi-planet tours jaise Cassini ne modest fuel par Saturn reach kiya.

Recall Is page par har trap ki one-line summary

$v_{\infty}$ conserved hai aur sirf rotate hoti hai; heliocentric gain planet ki motion $V_{p}$ se aata hai, $2 v_{\infty}$ par capped hai, aur energy conserved rehti hai kyunki planet quietly bill pay karta hai.