Question bank — Sphere of influence — radius derivation

False. SOI (Laplace) do viewpoints mein ratio perturbation/main force ko balance karta hai, jisse $2/5$ power milti hai; raw force-equality $1/2$ power deti hai aur ek chhota radius (Earth: ~260,000 km vs SOI ka ~924,000 km). :::

False. Earth ke paas Sun abhi bhi craft ko roughly utna hi khichta hai jitna wo Earth ko khichta hai. Jo negligible hai woh sirf us pull ka tidal difference hai us chhoti si doori $r$ ke across — woh hissa jo actually ek planet-centred orbit ko perturb karta hai. :::

False (approximately spherical). Real balance surface Sun ki taraf aur usse door bulge karti hai kyunki tidal term direction par depend karta hai; "sphere" ek convenient idealisation hai, isliye hum $r_{SOI}\approx$ likhte hain na ki $=$. :::

True. $r_{SOI}\propto (m/M{)}^{2/5}$, isliye planet mass $m$ badhane se radius badhta hai — bada boss, bada bubble. :::

False. $2/5$ power ki wajah se, $m$ ko double karne par radius sirf $2^{2/5}\approx 1.32$ se multiply hoti hai — 100% nahi balki sirf 32% ka increase. :::

True. $r_{SOI}\propto R$ (pehli power tak), isliye bade $R$ par outer planet ka proportionally bada bubble hota hai — Jupiter ka SOI bahut bada hai partly isliye kyunki $R$ large hai. :::

True. Isko rakhne se $r_{SOI}$ lagbhag 13% shrink ho jaati; isko drop karne se value thodi inflate ho jaati hai, jo standard convention hai. :::

False. Har force term per unit mass (acceleration) likhi gayi thi, isliye craft ki khud ki mass har jagah cancel ho jaati hai — bubble sirf planet–Sun pair ki property hai. :::

True. Tidal perturbation $\frac{d}{dR}(1/R^2)\propto 1/R^3$ se aayi; $R$ ki woh extra power hi exactly hai jo ek  ​ balance ko $2/5$ Laplace result mein badal deti hai. Dekho Tidal Forces. :::

Dimensionally broken. Sirf dimensionless ratio $m/M$ ko $2/5$ tak raise kiya jaata hai; length $R$ pehli power par rehni chahiye taaki answer ke units metres hoon: $r_{SOI}=R(m/M{)}^{2/5}$. :::

Galat pairing. SOI us pair ke liye define hoti hai jiske beech aap patch kar rahe ho. Moon ke Earth ke relative SOI ke liye, $m=$ Moon aur $M=$ Earth hoga — local primary, Sun nahi. :::

Galat. Craft aur planet dono Sun ki taraf saath-saath girte hain, isliye common pull relative orbit par kuch nahi karta. Sirf $r$ ke across difference, yaani tidal term $2GMr/R^3$, use perturb karta hai. :::

 ​ ek force balance ki pehchaan hai, Laplace SOI ki nahi. Ek sahi SOI derivation mein $2/5$ power aani chahiye; $1/2$ power ka matlab hai tumne raw forces barabar ki. :::

Nahi. SOI ke andar ratio $(P/F{)}_A$, $(P/F{)}_B$ se chhota hota hai, yaani planet-centred picture zyada saaf wali hoti hai — switch is baare mein hai ki kaun sa two-body approximation kam galat hai, na ki kaun si raw force badi hai. :::

Nahi. Sun-centred view mein planet perturber hai, aur planet craft par directly act karta hai — uska pura $Gm/r^2$ pull, tidal difference nahi. Sirf Sun (shared distant body) tidal term ke roop mein enter karta hai. :::

Perturbation tab matter karta hai jab main force ke relative dekha jaaye jo orbit ko baandhe rakhti hai. Ek tightly-bound orbit ke around badi disturbance bhi harmless ho sakti hai; ek weakly-bound orbit ke around chhoti disturbance fatal ho sakti hai — isliye meaningful quantity hai perturbation ÷ main. :::

Pura three-body problem ka koi clean closed-form solution nahi hai. SOI humein do exact two-body orbits ko stitch karne deta hai — Patched Conic Approximation — craft boundary cross karte waqt central body swap karte hue. :::

Kyunki planet frame mein Sun ka disturbing influence sirf ek gentle tidal effect hai ($\propto r/R^3$), planet crude force-equality point se kahin zyada door tak "in charge" rehta hai — criterion zyada forgiving hai. :::

Ye encode karta hai ki Sun ek direct force ki bajaye ek derivative (tidal) ke roop mein enter karta hai. Exponent ek fingerprint hai: $2/5$ dekhna batata hai ki tidal reasoning sahi se use ki gayi. :::

Boundary fuzzy aur non-spherical hai, aur model mein central body ka switch instantaneous hai lekin reality mein gradual hai. Ek Gravity Assist / Flyby ke dauran patched-conic seam ek chhoti si error introduce karta hai jise real missions tracking se correct karte hain. :::

$r_{SOI}=R(m/M{)}^{2/5}\to 0$. Ek massless body ka koi bubble nahi hota — dominate karne ke liye kuch hai hi nahi, isliye uska "planet-centred" region ek point tak shrink ho jaata hai. :::

Derivation ne $m\ll M$ aur $r\ll R$ assume kiya tha; woh approximations toot jaati hain. Comparable masses ke liye tumhe pura Restricted Three-Body Problem aur Hill Sphere chahiye, jo Lagrange-point geometry ko sahi se handle karta hai. :::

Nahi, lekin dono cousins hain. Hill Sphere rotating frame mein gravity aur centrifugal effects ko balance karke aata hai aur $(m/3M{)}^{1/3}$ scale karta hai; Laplace SOI $(m/M{)}^{2/5}$ perturbation-ratio criterion use karta hai. Alag exponents, alag sizes. :::

Koi bhi behtar nahi hai — yahi definition hai: dono approximations waahan exactly utni hi imperfect hain. Practice mein convention se ek side choose karte ho aur accept karte ho ki seam largest patched-conic error ka region hai. :::

Real tidal term directional hota hai, isliye balance surface planet–Sun line ke along stretch hoti hai aur perpendicular compress hoti hai. Single number $r_{SOI}$ ek averaged, idealised radius hai — reality ek lobed surface hai. :::