Lagrange points L1–L5 — derivation, stability

3.2.30 · Physics › Orbital Mechanics & Astrodynamics

Lagrange points KYUN exist karte hain?

Setup: Restricted Three-Body Problem (R3BP)

Mass ratio define karo: $μ = \frac{M _{2}}{M _{1} + M _{2}}, 1 - μ = \frac{M _{1}}{M _{1} + M _{2}} .$

Aisi units use karo jahan total mass $= 1$ , separation $D = 1$ , aur $G (M_{1} + M_{2}) = 1$ . Tab Kepler's third law angular velocity deta hai:

$M_{1}$ ko $x = - μ$ par aur $M_{2}$ ko $x = + (1 - μ)$ par x-axis par rakho; origin barycenter hai (isliye $M_{1} x_{1} + M_{2} x_{2} = 0$ : check karo $(1 - μ) (- μ) + μ (1 - μ) = 0$ ✓).

Effective potential (force balance derive karna)

Unit test mass ke liye position $(x, y)$ par:

$M_{1}$ se doori: $r_{1} = (x + μ)^{2} + y^{2}$
$M_{2}$ se doori: $r_{2} = (x - (1 - μ))^{2} + y^{2}$

Jacobi constant $C_{J} = - 2Ω - v^{2}$ conserved hota hai; iske level curves (zero-velocity curves) batate hain ki given energy wala spacecraft kahan ja sakta hai aur kahan nahi.

Figure — Lagrange points L1–L5 — derivation, stability

Collinear points L1, L2, L3 (axis par, $y = 0$ )

X-axis par $y = 0$ , $\partial Ω/ \partial x = 0$ set karo:

Ye ek quintic hai — numerically solve hoti hai. Iske 3 real roots hain:

L1: dono bodies ke beech mein. Yahan $M_{2}$ ka outward pull, $M_{1}$ ke pull se subtract karta hai aur centrifugal baaki balance karta hai.
L2: $M_{2}$ se aage (smaller body ke bahar). Dono gravities inward pull karti hain, centrifugal bahar pull karti hai.
L3: $M_{1}$ se aage (opposite side), almost $M_{2}$ ke opposite.

Triangular points L4, L5 (off-axis)

Real example: Jupiter ke Trojan asteroids uske L4 aur L5 par cluster karte hain. Earth ka ek known Trojan (2010 TK7) L4 par hai.

Stability — crucial difference

Check: Sun–Jupiter mein $M_{1} / M_{2} \approx 1047 ≫ 24.96$ ⇒ Trojans stable hain. ✓ Earth–Moon: $81 ≫ 24.96$ ⇒ Moon ke L4/L5 (weakly) stable hain (Kordylewski dust clouds).

Recall Feynman: 12-saal ke bachche ko explain karo

Socho do bacche spin kar rahe hain hath pakad ke aur unke beech ek chhoti ball strings par swing ho rahi hai. Kuch magic spots hain jahan ball wahan latkti rehti hai, unke saath spinning karti hai, drift nahi karti. Teeno spots aisa hain jaise ek ball ko pahaadi ki top par balance karna — zehra sa push aur woh roll kar jaayegi (isliye un spots par space telescopes ko rukne ke liye chhote rocket puffs lagaane padte hain). Do aur spots chhote bachche ke aage aur peeche hain, ek perfect triangle banate hue. Wo ek whirlpool jaisi hain: chahe marble drift bhi kare, spinning motion (Coriolis) use wapas ghuma deti hai — isliye asteroids billions of years tak wahan khushi se reh sakte hain.

Common mistakes

Forecast-then-verify

Flashcards

Kitne Lagrange points hote hain aur unki shapes/axes kya hain?

5 total: L1,L2,L3 collinear dono bodies ke beech wali line par; L4,L5 equilateral triangles ke apexes par (

\pm 60°

Rotating frame mein Lagrange point par kaunsi forces balance hoti hain?

M_{1}

ki gravity +

M_{2}

ki gravity + centrifugal force sum to zero hoti hain (Coriolis rest par body ke liye zero hai).

R3BP ka effective potential likhो.

Ω = - \frac{1 - μ}{r _{1}} - \frac{μ}{r _{2}} - \frac{1}{2} (x^{2} + y^{2})

jahan

μ = M_{2} / (M_{1} + M_{2})

Coriolis, Lagrange points ki location ko affect KYUN nahi karti?

- 2 ω \times v

hai, jo stationary particle ke liye vanish ho jaati hai; ye sirf stability (dynamic) analysis mein enter karti hai.

Sun–Earth ke liye L1 aur L2 kahan hain, aur roughly kitni door?

L1 Sun aur Earth ke beech, L2 Earth se aage; dono Earth se ~1.5 million km door

= D (μ /3)^{1/3}

L4/L5 ke coordinates?

x = \frac{1}{2} - μ

y = \pm 3 /2

; dono masses ke saath equilateral triangle (

r_{1} = r_{2} = D

Triangular points ke liye stability criterion?

Stable iff

M_{1} / M_{2} > \frac{25 + 621}{2} \approx 24.96

, yaani

μ < 0.0385

Kya L4/L5 potential minima hain?

Nahi — ye potential maxima hain; stability Coriolis force ki wajah se hai, potential shape ki wajah se nahi.

Kaunsi real objects Lagrange points par hain?

JWST/Gaia Sun–Earth L2 par; SOHO L1 par; Jupiter ke Trojan asteroids L4/L5 par.

Hill/L1 distance formula

D (μ /3)^{1/3}

mein 3 ka factor KYUN hai?

3 = tidal gravity gradient (2) + centrifugal gradient (1) jo small body ki local gravity balance karte hain.

Connections

Restricted Three-Body Problem
Effective Potential & Jacobi Constant
Coriolis and Centrifugal Forces (Rotating Frames)
Hill Sphere
Trojan Asteroids
Kepler's Third Law
Halo Orbits & Station-Keeping (JWST, SOHO)
Roche Limit (contrast: tidal forces)