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

3.2.4 · D5 · Physics › Orbital Mechanics & Astrodynamics › Orbit shape from eccentricity — circle (e=0), ellipse (0 - e

Yeh parent topic ke liye ek question bank hai jo conceptual traps pe focus karta hai. Yahan koi number-crunching nahi hai — har item yeh probe karta hai ki kya tum actually samajhte ho kyun ek number, eccentricity $e$ , "wapas aata hai" aur "hamesha ke liye chala gaya" ke beech fark karta hai. Har answer tab hi reveal karo jab tum apni reasoning khud commit kar lo.

Is page par use hone wale symbols (pehle vocabulary build karo)

Kisi bhi trap se pehle, yahan har woh symbol hai jo tumhe milega — har ek simple shabdon mein aur neeche diye figure se linked. Koi bhi cheez yahan appear hone se pehle use nahi ki jaayegi.

Definition Symbols ki cast

$r$ — central mass se orbiting body tak ki distance. Figure mein dot (mass) se chhote circle (body) tak ke arrow ki length dekho.
$θ$ (theta) — polar angle, central mass par measure kiya gaya, closest-approach direction se lekar body abhi jahan hai tak. Yeh figure mein "kitna ghuma" wala angle hai.
$e$ — eccentricity, ek pure number (koi units nahi) jo batata hai curve kitna squashed hai. $e = 0$ matlab round, bada $e$ matlab zyada stretched.
$c$ — focal distance: ek focus ellipse ke center se kitna door baithta hai (dono foci ke beech ke gap ka aadha). Figure mein chhote segment ke roop mein mark kiya gaya.
$a$ — semi-major axis: ellipse ke sabse lambe diameter ka aadha (uska "size").
$p$ — semi-latus rectum: $r$ ki woh value jab $θ = 9 0^{\circ}$ ho (focus se seedha upar). Yeh orbit ka "width scale" hai. Figure mein vertical bar ke roop mein drawn hai.
$μ$ (mu) — gravitational parameter $μ = GM$ ( $G$ = Newton's constant, $M$ = central mass). Yeh "pull kitna strong hai" ko ek jagah bundle karta hai.
$h$ — specific angular momentum $h = r^{2} \dot{θ}$ , yaani angular momentum per unit mass. Yeh orbit ki "sideways sweep" measure karta hai aur constant rehta hai.
$ε$ (epsilon) — specific orbital energy $ε = E / m$ , yaani total energy per unit mass. Iska sign decide karta hai bound hai ya unbound.

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

Do master formulas jo inhe tie karte hain (parent note se):

$r (θ) = \frac{p}{1 + e c o s θ}, p = \frac{h ^{2}}{μ}, e = 1 + \frac{2 ε h ^{2}}{μ ^{2}} .$

Focus-form equation kahan se aati hai (taaki neeche kuch bhi black box na lage)

Intuition Solution "constant + cosine" kyun hai, aur woh conic kyun HAI

Binet equation $\frac{d ^{2} u}{d θ ^{2}} + u = \frac{μ}{h ^{2}}$ (jahan $u = 1/ r$ ) bilkul usi shape ki equation hai jaise koi mass spring par ho aur usse ek steady force push kare. Aisi equation ka solution hamesha ek steady part aur ek wobble part hota hai: $u (θ) = steady (particular) \frac{μ}{h ^{2}} + wobble (homogeneous) A cos θ .$ Steady part ko factor out karo aur $A = \frac{μ}{h ^{2}} e$ rename karo (yahi $e$ ko wobble ki relative size ke roop mein define karta hai): $u (θ) = \frac{μ}{h ^{2}} (1 + e cos θ) .$ Ab $u = 1/ r$ ko $r$ mein flip karo: $r = \frac{1}{u} = \frac{h ^{2} / μ}{1 + e c o s θ} = \frac{p}{1 + e c o s θ}, p = \frac{h ^{2}}{μ} .$ Yeh exact form — " $r$ = (constant) over (1 + e·cos of the angle from the focus)" — conic ki textbook focus-directrix definition hai. Isliye orbit guaranteed kisi ek conic mein se hogi: algebra kuch aur produce hi nahi kar sakta.

True ya false — justify karo

Har item ek statement hai. True/false decide karo, phir mechanism do — sirf verdict nahi.

A larger semi-major axis

a

always means a larger eccentricity

e

False.

a

ek size hai aur

e

ek shape ratio (

e = c / a

). Ek tiny near-circular orbit aur ek huge near-circular orbit dono

e \approx 0

share kar sakte hain; ek chhoti stretched orbit ka

e

ek badi round orbit se bada ho sakta hai.

An orbit with

e = 0.999

is a different kind of curve from one with

e = 1

True.

e < 1

ek closed ellipse hai (body wapas aati hai);

e = 1

ek open parabola hai (

a \to \infty

, kabhi wapas nahi aati).

e = 1

cross karna ek bound→unbound phase change hai, sirf "aur patla" nahi.

Two orbits with the same

e

must have the same shape but not the same size.

True. Same

e

matlab geometrically similar conics (ek doosre ka scaled copy);

a

(ya

p

) absolute size alag se set karta hai.

The central mass sits at the center of the orbital ellipse.

False. Yeh ek focus par baithta hai. Sirf special case

e = 0

(circle) mein dono foci center par merge hote hain.

For any conic orbit,

r

reaches infinity somewhere.

False.

e < 1

ke liye denominator

1 + e cos θ

kabhi zero nahi hota (kyunki

- 1/ e < - 1

unreachable hai), isliye

r

har jagah finite rehta hai — ek bound loop. Sirf

e \geq 1

par

r \to \infty

possible hai.

A parabolic comet returns to the Sun after an extremely long period.

False.

e = 1

matlab

ε = 0

: yeh zero speed se infinity tak pahunchta hai aur kabhi wapas nahi aata. Koi period nahi hai — orbit open hai.

Increasing the launch speed at a fixed perihelion always raises

e

True, lekin sirf tangential boost itna clean hai. Perihelion par tangential (sideways) burn speed badhata hai bina pull-line change kiye, isliye

ε

aur

h

dono raise hote hain;

e = 1 + 2 ε h^{2} / μ^{2}

mein plug karne par

e

upar jaata hai. Radial (along-

r

) burn

ε

raise karta hai lekin

h

alag tarah change karta hai aur perihelion ki jagah bhi shift ho jaati hai — isliye "fixed perihelion par" ab valid nahi rehta. Clean monotone rise tangential case assume karta hai.

A hyperbolic orbit exists for all angles

θ

from

0

36 0^{\circ}

False. Sirf jab

cos θ > - 1/ e

ho tab

r

positive aur physical hai. Asymptote angles ke baad formula

r < 0

deta hai, jo body kabhi occupy nahi karti — yeh sirf ek open arc par exist karti hai.

Error dhundho

Har line mein ek flawed claim ya reasoning step hai. Use dhundho aur correct karo.

"Eccentricity equals the distance between the two foci."

Error:

e

woh distance

a

se divide kiya hua hai — ek dimensionless ratio

e = c / a

(jahan

c

= focus-to-focus gap ka aadha), koi length nahi. Dono foci separation aur orbit size dono double karne par

e

unchanged rehta hai.

"Since a circle has no focus offset, the conic formula

r = p / (1 + e cos θ)

can't describe it."

Error:

e = 0

set karo aur formula

r = p

deta hai (ek constant, kyunki

p = h^{2} / μ

orbit se fixed hai), exactly ek circle. Circle smooth

e = 0

limit hai, koi exception nahi.

"At

e = 1

the ellipse is just maximally stretched, so its semi-major axis is smallest."

Error:

e \to 1

par semi-major axis

a \to \infty

hota hai, smallest nahi. Orbit khulti hai, sihriti nahi.

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

and

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

for an ellipse."

Error: Yeh swap hain. Closest approach mein minus use hota hai:

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

; farthest mein plus:

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

, kyunki

1 - e < 1 + e

"For a hyperbola the mass sits outside the curve, so there's no focus."

Error: Hyperbola ka bhi ek focus hota hai (upar overlay figure dekho), aur attracting mass us branch ke focus par baithta hai jis par body travel karti hai. Formula

r = p / (1 + e cos θ)

charon conics ke liye ek focus-origin equation hai — same origin, sirf

e

badalta hai.

"Because seasons happen, Earth's orbit must be strongly eccentric."

Error: Earth ka

e \approx 0.0167

nearly circular hai (

r_{m a x} / r_{m i n} \approx 1.034

). Seasons mainly axial tilt se aate hain, tiny distance change se nahi.

"A body with total energy

E > 0

can still be captured into a closed orbit."

Error:

E > 0

force karta hai

e > 1

(hyperbola), jo unbound hai. Closed orbit ke liye

E < 0

chahiye. (Capture ke liye energy loss chahiye hogi, jaise drag ya third body.)

Why questions

Mechanism answer karo, label nahi.

Why does the sign of the energy, not its magnitude, decide bound vs unbound?

Kyunki

e = 1 + 2 ε h^{2} / μ^{2}

: crossover

e = 1

exactly

ε = 0

par hota hai. Negative

ε

→

e < 1

(potential well mein trapped); positive →

e > 1

(infinity par leftover kinetic energy).

Why is

e = 1

called the "marginal" or "knife-edge" escape orbit?

Yeh exact boundary hai jahan body zero speed se infinity tak pahunchti hai (

ε = 0

). Thodi kam energy aur woh wapas girti hai; thodi zyada aur woh extra speed ke saath escape karti hai.

Why does the orbit equation automatically come out as a conic and not some other curve?

Binet equation ek simple-harmonic-plus-forcing equation hai, jiska solution "constant +

A cos θ

" hota hai.

A

\frac{μ}{h ^{2}} e

rename karke aur

u = 1/ r

invert karne par exactly

r = p / (1 + e cos θ)

milta hai — conic ka focus-form (upar work out kiya gaya hai).

Why does

r \to \infty

require

e \geq 1

r

diverge hota hai jab

1 + e cos θ = 0

, yaani

cos θ = - 1/ e

. Cosine sirf

[- 1, 1]

mein kisi number ke barabar ho sakta hai, isliye

- 1/ e

tabhi reachable hai jab

1/ e \leq 1

, yaani

e \geq 1

Why does angular momentum set the semi-latus rectum

p

but not directly the eccentricity?

p = h^{2} / μ

sirf

h

par depend karta hai (orbit ki "width"), jabki

e

dono energy aur

h

mix karta hai. Tum

h

fixed rakhte hue energy change karke

e

badal sakte ho.

Why does a comet on a parabola visit the Sun exactly once?

Parabolas open curves hain — perihelion ke baad comet single arm ke saath infinity ki taraf jaata hai aur kabhi re-close nahi hota, isliye koi return pass nahi hota.

Why can a hyperbolic flyby deflect an object without capturing it?

e > 1

ke saath object ka

E > 0

hai; gravity uski path bend karti hai (deflection) lekin surplus kinetic energy remove nahi kar sakti, isliye woh perihelion se guzar kar outgoing asymptote ke saath escape karta hai.

Edge cases

Boundary aur degenerate scenarios — yahan intuition usually toot jaati hai.

What happens to the two foci as

e \to 0

Woh ek single point par center mein merge ho jaate hain (focal distance

c = e a \to 0

), aur ellipse constant

r = p

ke saath circle ban jaata hai.

What happens to the semi-major axis

a

e \to 1

from below?

a \to \infty

. Ellipse ka door wala end infinity ki taraf bhaag jaata hai, aur closed loop ek parabola mein khul jaata hai.

e

allowed to be negative?

Nahi.

e \geq 0

by definition; yeh roundness se deviation ka magnitude hai. "Negative

e

" sirf perihelion direction relabel karta hai (ek

18 0^{\circ}

rotation of

θ

What is

p

physically, and does it exist for all four shapes?

p

semi-latus rectum hai,

θ = 9 0^{\circ}

par

r

ki value. Yeh circle, ellipse, parabola, aur hyperbola sab ke liye positive aur well-defined hai, kyunki

p = h^{2} / μ

ko sirf nonzero

h

chahiye.

What if the angular momentum

h = 0

(a purely radial drop)?

Tab

p = h^{2} / μ = 0

aur conic ek straight line mein degenerate ho jaata hai focus se guzarta hua — ek radial fall/escape, koi curved orbit nahi.

For a hyperbola, what does a "negative

r

" at forbidden angles actually mean?

Polar form mein, angle

θ

par

r < 0

matlab "opposite direction mein

∣ r ∣

door, yaani

θ + 18 0^{\circ}

par" — woh points hyperbola ki doosri branch ke hain. Physical body sirf near-focus branch par hai, isliye woh kabhi un angles par nahi hoti; woh asymptotes ke baad wale space ko mark karte hain.

At the exact asymptote angle of a hyperbola, what is happening physically?

Denominator

1 + e cos θ \to 0^{+}

, isliye

r \to \infty

: body infinitely door hai, straight-line asymptote ke saath move kar rahi hai — yeh infinity par uski incoming ya outgoing direction hai.

e \to \infty

, what does the orbit approach?

cos θ = - 1/ e \to 0

se asymptote angle

9 0^{\circ}

ki taraf jaata hai, aur orbit nearly straight-line flyby ki taraf flatten hoti hai — almost undeflected pass.

Recall Har trap ka ek-line summary

$e$ ek shape ratio hai, mass ek focus par baithta hai, $ε$ ka sign (uska size nahi) bound vs unbound decide karta hai, aur $e = 1$ ek genuine open/closed phase change hai — koi bahut patla ellipse nahi.

Connections

Conic Sections — jahan se "same $e$ = similar curve" aata hai
Specific Orbital Energy — $ε$ sign jo har bound/unbound trap ke peeche hai
Angular Momentum in Orbits — kyun $p = h^{2} / μ$ aur $h = 0$ degenerate case
Vis-viva Equation — kisi bhi $r$ par speed, energy bookkeeping
Escape Velocity — $e = 1$ , $ε = 0$ knife-edge
Kepler's First Law — ellipse, Sun at a focus ( $0 \leq e < 1$ )