Sphere of influence — radius derivation — Hinglish notes

3.2.25 · Physics › Orbital Mechanics & Astrodynamics

Setting up the two viewpoints

Maano:

$M$ = Sun ki mass, $m$ = planet ki mass, jahan $m ≪ M$ .
$R$ = planet ↔ Sun ke beech ki doori.
$r$ = spacecraft ↔ planet ke beech ki doori (chhoti, $r ≪ R$ ).

Spacecraft ki motion describe karne ke do tarike hain:

Viewpoint A — Planet-centred. Main force = planet ka craft par pull. Perturbation = Sun ka differential (tidal) pull.

Viewpoint B — Sun-centred. Main force = Sun ka craft par pull. Perturbation = planet ka pull.

HOW: har ratio banana

Viewpoint A (planet central hai)

Main force (planet ka craft par), per unit mass: $F_{A} = \frac{G m}{r ^{2}}$

Perturbation = Sun se tidal term. Sun ka acceleration ek body par jo $R$ doori par hai woh $\frac{GM}{R ^{2}}$ hai. Iska variation ek chhote displacement $r$ par: $P_{A} \approx \frac{d}{d R} (\frac{GM}{R ^{2}}) r = \frac{2 GM}{R ^{3}} r$

Viewpoint B (Sun central hai)

Main force (Sun ka craft par), per unit mass. Kyunki $r ≪ R$ hai, craft Sun se $\approx R$ doori par hai: $F_{B} = \frac{GM}{R ^{2}}$

Perturbation = planet ka craft par direct pull: $P_{B} = \frac{G m}{r ^{2}}$

The Laplace boundary condition

Laplace ka criterion: SOI ka edge wahan hai jahan donon viewpoints equally (im)perfect hain: $(\frac{P}{F})_{A} = (\frac{P}{F})_{B}$

Substitute karo: $\frac{2 M}{m} \cdot \frac{r ^{3}}{R ^{3}} = \frac{m}{M} \cdot \frac{R ^{2}}{r ^{2}}$

$r$ ke liye solve karo. Dono sides multiply karo $r$ ki powers gather karne ke liye: $2 \frac{M}{m} r^{3} \cdot r^{2} = \frac{m}{M} R^{2} \cdot R^{3}$ $2 \frac{M}{m} r^{5} = \frac{m}{M} R^{5}$ $r^{5} = \frac{1}{2} (\frac{m}{M})^{2} R^{5}$ $r = R (\frac{m}{M})^{2/5} (\frac{1}{2})^{1/5}$

Factor $(1/2)^{1/5} \approx 0.87$ 1 ke kareeb hai aur conventionally standard formula ke liye drop kar diya jaata hai:

Figure — Sphere of influence — radius derivation

Worked examples

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho ek bachcha ek bade parade balloon (Sun) ke saath chal raha hai aur ek chhota pet (planet) pakde hua hai, aur ek makhi (spacecraft) pet ke paas bhanbhana rahi hai. Pet se door, makhi poori parade (Sun) ki parwah karti hai. Pet ke bahut paas, makhi pet ke chakkar lagaati hai aur parade ko barely notice karti hai — use sirf parade ke dono sirron ko thoda alag khichte feel hota hai (yahi tidal nudge hai). Sphere of Influence woh invisible ball hai pet ke aas-paas jahan makhi "main pet ko follow karti hoon" se "main parade ko follow karti hoon" switch karti hai. Kyunki sirron-ko-kheechna ek halka effect hai, yeh ball surprisingly badi hoti hai — badi us jagah se jahan pull sirf half-and-half hota hai.

Active recall

Laplace SOI define karne ke liye kaunsi quantity balance ki jaati hai?

Ratio (perturbing force / main force) jo planet-centred aur Sun-centred dono viewpoints ke liye compute ki jaati hai — dono ko ek doosre ke equal set karo. :::

SOI radius formula batao.

r_{S O I} \approx R (m / M)^{2/5}

, jahan

R

primary separation hai,

m

chhota body,

M

bada. :::

Planet-centred view mein Sun ka effect "tidal" term kyun hai?

Craft aur planet dono Sun ki taraf girte hain; sirf doori

r

par Sun ke pull ka antar orbit ko perturb karta hai, jo deta hai

\propto r d / d R (1/ R^{2}) = 2 r / R^{3}

. :::

Kaunsa exponent batata hai ki yeh SOI hai ya force-balance?

SOI ⇒ 2/5 power; equal-force point ⇒ 1/2 power (ek

m / M

). :::

Earth ka SOI radius roughly kitna hai?

~9.2×10^5 km (≈ 924,000 km, ~0.006 AU). :::

SOI bada hai ya chhota, jahan Sun aur planet ki forces equal hain us point se?

Bada (Earth ke liye 924,000 km vs ~260,000 km). :::

Dropped factor

(1/2)^{1/5}

kahan se aata hai?

Tidal derivative

2 GM / R^{3}

mein factor 2 se; yeh ≈0.87 hai aur conventionally chhod diya jaata hai. :::

Connections

Patched Conic Approximation — SOI woh boundary hai jahan aap conics switch karte ho.
Two-Body Problem — SOI ke har side ko pure two-body orbit maana jaata hai.
Tidal Forces — planet-centred view mein perturbation ek tidal term hai.
Hill Sphere — ek related (rotating-frame) stability radius, $\sim R (m /3 M)^{1/3}$ ; exponents confuse mat karo.
Restricted Three-Body Problem — exact context jise SOI approximate karta hai.
Gravity Assist / Flyby — trajectories SOI mein enter/exit karke plan ki jaati hain.