Orbit shape from eccentricity — circle (e=0), ellipse (0 - e - 1), parabola (e=1), hyperbola (e - 1)

3.2.4 · Physics › Orbital Mechanics & Astrodynamics

Ek akela number — eccentricity $e$ — decide karta hai ki koi body hamesha loop karti rahegi (bound) ya infinity tak escape kar jaayegi (unbound). Yeh note derive karta hai KYU, conic equation se.

The big picture

Orbit equation derive karna (first principles)

KAISE. Newton ki gravity radial acceleration $- μ / r^{2}$ deti hai jahan $μ = GM$ . Substitution $u = 1/ r$ use karo aur specific angular momentum $h = r^{2} \dot{θ}$ (angular momentum per unit mass; total hai $L = m h = m r^{2} \dot{θ}$ ). Tab Binet equation yeh aati hai:

$\frac{d ^{2} u}{d θ ^{2}} + u = \frac{μ}{h ^{2}}$

Solution: $u = \frac{μ}{h ^{2}} (1 + e cos θ)$ , jahan $e$ integration constant hai (initial conditions se set hota hai). $u = 1/ r$ invert karo:

$r (θ) = \frac{p}{1 + e cos θ}, p = \frac{h ^{2}}{μ}$

Yeh conic ka polar equation hai jisme focus origin par hai (central mass ek focus par hoti hai). $p$ semi-latus rectum hai ( $θ = 9 0^{\circ}$ par $r$ ).

Chaar shapes

Figure — Orbit shape from eccentricity — circle (e=0), ellipse (0 - e - 1), parabola (e=1), hyperbola (e - 1)

$e$ ko energy se link karna

KAISE (eccentricity–energy relation). Vis-viva aur conservation laws se yeh derive hota hai (total energy $E$ , total angular momentum $L$ ):

$e = 1 + \frac{2 E L ^{2}}{μ ^{2} m ^{3}}$

Equivalently, specific quantities mein (per unit mass): $ε = E / m$ likho specific energy ke liye aur $h = L / m$ specific angular momentum ke liye, toh relation simplify hoti hai

$e = 1 + \frac{2 ε h ^{2}}{μ ^{2}} .$

$ε$ ka sign dekho:

$ε < 0 \Rightarrow e < 1$ (ellipse/circle),
$ε = 0 \Rightarrow e = 1$ (parabola),
$ε > 0 \Rightarrow e > 1$ (hyperbola). ✓ table se match karta hai.

Ek ellipse ke liye, $p = a (1 - e^{2})$ toh perihelion aur aphelion hain: $r_{m i n} = a (1 - e), r_{m a x} = a (1 + e)$

Common mistakes

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

Imagine karo ek ball kisi planet ke around throw karna. Dheeray throw karo → woh ek squashed circle (ellipse) mein loop karke wapas aati hai. Exactly sahi speed se throw karo → woh door fly kar jaati hai aur barely kabhi wapas nahi aati, infinitely door jaake zero par slow ho jaati hai (parabola). Aur tez throw karo → woh past zoom karke hamesha ke liye chalti jaati hai speed bacha ke (hyperbola). Number $e$ ek "shape score" hai: $0$ = perfect circle, almost-1 = stretched loop, $1$ = escape edge, 1 se zyada = hamesha ke liye gone.

Active recall

Origin par focus ke saath orbit ka polar equation kya hai?

r = \frac{p}{1 + e cos θ}

jahan

p = h^{2} / μ

Kaun si eccentricity circle deti hai?

e = 0

(dono foci center par merge ho jaate hain,

r

constant).

Ellipse ke liye

e

ki range?

0 < e < 1

(bound,

E < 0

Marginal escape orbit ki shape aur uska

e

kya hai?

Parabola,

e = 1

, total energy

E = 0

e > 1

ka physically kya matlab hai?

Hyperbola, unbound, positive total energy — leftover speed ke saath escape.

e \geq 1

ke liye

r \to \infty

kyun hota hai lekin

e < 1

ke liye nahi?

Denominator

1 + e cos θ = 0

cos θ = - 1/ e

chahiye; yeh sirf tab reachable hai jab

e \geq 1

Specific aur total angular momentum mein farq?

h = r^{2} \dot{θ}

per unit mass hai; total hai

L = mh = m r^{2} \dot{θ}

e

aur specific energy/ang. momentum ka relation?

e = 1 + 2 ε h^{2} / μ^{2}

jahan

ε = E / m, h = L / m

;

ε

ka sign shape fix karta hai.

Ellipse ka perihelion aur aphelion?

r_{m i n} = a (1 - e)

r_{m a x} = a (1 + e)

e = 2

wali hyperbola ka asymptote angle?

cos θ = - 1/ e = - 0.5 \Rightarrow θ = 12 0^{\circ}

Eccentricity ek length hai ya ratio?

Ek dimensionless ratio

e = c / a

(focal distance over semi-major axis).

Connections

Conic Sections — chaar shapes ki geometric origin
Vis-viva Equation — kisi bhi $r$ par speed deta hai, energy se jodta hai
Specific Orbital Energy — sign bound vs unbound determine karta hai
Angular Momentum in Orbits — semi-latus rectum $p$ set karta hai
Kepler's First Law — focus par Sun ke saath ellipse ( $0 \leq e < 1$ )
Escape Velocity — $e = 1$ boundary condition