Orbit equation r = p - (1 + e·cos θ) — derivation from equations of motion: Question bank

3.2.3 · D5 · Physics › Orbital Mechanics & Astrodynamics › Orbit equation r = p - (1 + e·cos θ) — derivation from equat

Traps dhundne se pehle, woh picture dekho jis par har question secretly depend karta hai. Is ek diagram mein page par use hone wale har symbol ka naam hai — ise apne mind mein pin kar lo.

Figure — Orbit equation r = p - (1 + e·cos θ) — derivation from equations of motion

Definition Symbols ki cast (sab pehle se earn kiye hue)

$r$ = focus $F$ se tumhari doori (jahan Sun baitha hai), oval ke middle se nahi. Figure mein blue segment dekho.
$\overset{r}{^}$ = radial unit vector: lambaai $1$ ka ek arrow jo focus se seedha planet ki taraf point karta hai. Yeh bas tumhari current jagah par "bahar ki" direction ka naam deta hai. Gravity $- \overset{r}{^}$ ke along pull karti hai (seedha andar ki taraf).
$θ$ = true anomaly — perihelion (sabse kareeb wala point) se sweep kiya gaya angle, focus $F$ par measure kiya gaya, $0 \leq θ < 2 π$ tak chalta hai (ek poora loop). Figure mein yellow angle.
$\dot{θ}$ = angle kitni tezi se sweep hota hai (praati second $θ$ kitna badta hai); kisi bhi symbol par ek dot matlab "time mein uski change ki rate".
$p$ = semi-latus rectum: $r$ ki woh special value jab tum bilkul "sideways" ho ( $θ = 9 0^{\circ}$ ). Pink segment.
$a$ = semi-major axis: ellipse ka aadha lamba diameter, center $C$ se measure kiya gaya.
$b$ = semi-minor axis: aadha chota diameter, bhi center $C$ se.
$e$ = eccentricity: ek pure number ( $e \geq 0$ ) jo shape set karta hai (circle→ellipse→parabola→hyperbola).

True or false — justify karo

Har answer True ya False se shuru hota hai aur phir kyun — sirf verdict dene se kuch nahi milega.

TF1 —

r = p / (1 + e cos θ)

mein, angle

θ

ellipse ke center se measure kiya jaata hai.

False.

θ

true anomaly hai, perihelion se measure kiya gaya focus (Sun) se dekha gaya, geometric center se nahi; center par measure kiya gaya angle eccentric anomaly hai, jo bilkul alag angle hai.

TF2 —

e = 0

set karne par equation

r = p

ban jaati hai, yaani ek constant, yaani ek circle.

True.

e = 0

ke saath

cos θ

term gayab ho jaata hai, isliye sabhi

θ

ke liye

r = p

— ek fixed radius precisely ek circle hai, aur yahan

p = a

bhi hota hai.

TF3 — Ek bound ellipse ke liye,

r

sabse bada hota hai jab

cos θ

sabse bada hota hai.

False.

r

ek fraction hai jiske neeche

1 + e cos θ

hai;

r

sabse bada hota hai jab denominator sabse chhota hota hai, yaani jab

cos θ = - 1

hota hai

θ = π

par (aphelion).

TF4 — Semi-latus rectum

p

har orbit ke liye semi-major axis

a

ke barabar hota hai.

False. Generally

p = a (1 - e^{2})

; yeh tabhi barabar hote hain jab

e = 0

(circle). Kisi bhi real ellipse ke liye

p < a

TF5 — Specific angular momentum

h

conserved hai kyunki gravity planet par koi kaam nahi karti.

False.

h

conserved hai kyunki gravity ek central force hai (hamesha radial direction

\overset{r}{^}

ke along), focus ke baare mein zero torque deti hai — yeh direction ke baare mein ek statement hai, kaam ke baare mein nahi.

TF6 — Ek parabolic orbit (

e = 1

) ka ek finite aphelion distance hota hai.

False.

e = 1

par denominator

1 + cos θ \to 0

jab

θ \to π

, isliye

r \to \infty

; body escape kar jaati hai aur kabhi nahi lauti, isliye koi aphelion nahi hai.

TF7 — Same

p

lekin alag

e

wale do orbits

θ = 9 0^{\circ}

par ek hi point se guzarte hain.

True.

θ = 9 0^{\circ}

par,

cos θ = 0

isliye

r = p

regardless of

e

; same

p

share karne wale sabhi conics semi-latus-rectum point par saath guzarte hain.

TF8 —

p

fixed rakhte hue

e

badhane se perihelion focus ke kareeb aata hai.

True.

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

e

badhne par ghatta hai, isliye fixed

p

par ek zyada eccentric orbit Sun ke kareeb dive karta hai closest approach par.

TF9 — Orbit equation is assumption par derive ki gayi thi ki dono masses equal hain.

False. Yeh assume karta hai ki ek mass dominate karti hai isliye

μ = G (M + m) \approx GM

; yeh kisi bhi two-body system ke relative coordinate ke liye bhi

μ = G (M + m)

ke saath exactly hold karta hai.

Error dhundho

Har line mein ek plausible-lagta-hai lekin galat statement hai. Reveal mein flaw ka naam bataya gaya hai.

SE1 — "

cos

har

36 0^{\circ}

par repeat karta hai, isliye orbit equation

θ

aur

θ + 18 0^{\circ}

ke liye do alag radii deta hai, to orbit closed nahi hai."

Error yeh hai:

cos

ka repeat karna precisely wahi hai jo orbit ko har revolution par retrace karaata hai; alag

θ

ke liye alag

r

oval shape hai, closure mein break nahi. Ek bound orbit precisely isliye close hoti hai kyunki

r (θ)

0 \leq θ < 2 π

mein

θ

mein periodic hai.

SE2 — "Perihelion woh jagah hai jahan planet sabse dheere chalti hai, kyunki woh sharp corner le rahi hai."

Error yeh hai: perihelion woh jagah hai jahan planet sabse tezi se chalti hai.

h = r^{2} \dot{θ} = const

se, chhota

r

bada

\dot{θ}

force karta hai; corner sharp isliye hai kyunki planet usme se whip kar rahi hai, crawl nahi kar rahi.

SE3 — "

μ \approx GM

ki wajah se, ek bhaari orbiting satellite same launch conditions ke liye wider orbit follow karta hai."

Error yeh hai: kyunki

μ = G (M + m) \approx GM

sirf central body par depend karta hai,

F = m a

m

se divide karne par satellite ki apni mass hat jaati hai. Same launch conditions ⇒ choti mass chahe jo bhi ho, identical path.

SE4 — "

1 + e cos θ

ki jagah

r = p / (1 - e cos θ)

likhna bilkul alag orbit describe karta hai."

Error yeh hai: yeh same orbit hai, bas perihelion ko

θ = 0

ki jagah

θ = π

par relabel kiya gaya hai. Sign sirf angle origin rotate karta hai; geometric shape untouched rehti hai.

SE5 — "

p = h^{2} / μ

, isliye ek faster-spinning orbit (bada

h

) hamesha ek chhota orbit hota hai."

Error yeh hai: bada

h

p

ko badhata hai (yeh numerator mein hai), orbit ko bada banata hai, chota nahi. Zyada angular momentum body ko wider fling karta hai.

SE6 — "

u = 1/ r

substitution sirf algebraic bookkeeping hai jiska koi physical reason nahi."

Error yeh hai: yeh isliye choose kiya gaya hai kyunki

u = 1/ r

θ

ke against plot karne par orbit ka curve ek plain cosine wave ban jaata hai —

1/ r^{2}

force

μ u^{2}

ban jaata hai aur ODE linear oscillator

u^{''} + u = μ / h^{2}

ban jaata hai, jiska solution

u = const + cos

hai. Poori derivation us tool choice par hinge karti hai.

SE7 — "

θ = 9 0^{\circ}

par planet ellipse ke minor axis ke end par hota hai."

Error yeh hai:

θ = 9 0^{\circ}

par hum

r = p

par hain focus se measure karke, jo latus rectum ka end hai, minor-axis tip nahi. Woh tip center se measure ki jaati hai aur wahan uski distance semi-minor axis

b

ke barabar hoti hai — ek alag point aur alag length.

Why questions

WQ1 — Derivation time derivatives se

θ

derivatives par kyun switch karti hai?

Kyunki hum shape

r (θ)

chahte hain, timing

r (t)

nahi, aur

θ

monotonically

0 \leq θ < 2 π

par badhta hai, isliye yeh ek clean, single-valued independent variable hai — jabki

r

time mein aage-peeche jaata hai.

WQ2 — Oscillator equation mein driving term

μ / h^{2}

ek constant kyun hai?

Dono

μ

(central-body property) aur

h

(conserved specific angular momentum) orbit set hone ke baad fix hote hain, isliye right-hand side

θ

ke saath kabhi vary nahi karta, ek simple constant-forced harmonic oscillator deta hai.

WQ3 — Conserved angular momentum immediately ek flat (planar) orbit kyun imply karta hai?

h = r \times v

dono position aur velocity ke perpendicular hai; agar

h

kabhi direction nahi badalta, to position aur velocity hamesha ke liye us single plane mein confined rehte hain jo

h

ke perpendicular hai.

WQ4 — Eccentricity

e

ek free constant ke roop mein kyun aata hai, physics se fixed hone ki bajay?

Jab hum oscillator

u^{''} + u = μ / h^{2}

solve karte hain, wave part ek amplitude constant

A

carry karta hai (cosine ki height, initial conditions se set hoti hai); ise

e = A h^{2} / μ

ke roop mein bundle karne se

e

free ho jaata hai. Equations koi bhi shape permit karti hain, aur yeh woh hai jahan tum launch karte ho aur kitna tez ki circle, ellipse, parabola ya hyperbola select hoti hai.

WQ5 —

θ = 9 0^{\circ}

par exactly

r = p

kyun hota hai regardless of

e

Kyunki

cos 9 0^{\circ} = 0

poore

e

-dependent term ko kill kar deta hai,

r = p /1 = p

chhodta hai; isliye

p

literally kisi bhi conic ke liye perihelion se "sideways" ki doori hai.

WQ6 — Same formula ek captured planet aur ek escaping comet dono describe kyun kar sakta hai?

Denominator mein

1 - e

ka sign behaviour badalta hai:

e < 1

ke liye denominator sabhi

θ

ke liye positive rehta hai (closed ellipse), jabki

e \geq 1

ke liye yeh zero tak pahunch sakta hai,

r \to \infty

bhejta hai (open escape) — ek equation, sabhi conics.

Edge cases

EC1 — Formula

e \to 1^{-}

par aphelion distance

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

ke liye kya predict karta hai?

Yeh blow up ho jaata hai:

r_{m a x} \to \infty

. Ellipse bina bound ke stretch karta hai aur exactly

e = 1

par, ek open parabola ban jaata hai jiska koi far turning point nahi hota.

EC2 —

e > 1

ke liye us angle par

r

ka kya hota hai jahan

cos θ = - 1/ e

Denominator

1 + e cos θ = 0

ho jaata hai, isliye

r \to \infty

: yeh do angles hyperbola ke asymptote directions hain, jin ke baad formula unphysical negative

r

dega (woh branch travel nahi ki jaati).

EC3 — Kya

θ = π

(aphelion) hyperbolic orbit ke liye meaningful hai?

Nahi.

e > 1

ke liye,

1 + e cos π = 1 - e < 0

negative

r

deta hai, jo unphysical hai; body

θ = π

tak kabhi nahi pahunchti kyunki woh pehle hi ek asymptote ke along escape kar chuki hoti hai.

EC4 — Agar exactly

h = 0

ho to orbit kaisi hogi?

Ek degenerate radial "orbit" — zero specific angular momentum ka matlab

\dot{θ} = 0

hai, isliye body seedhe andar (ya seedhe bahar) ek line ke along girta hai; polar formula collapse ho jaata hai kyunki

p = h^{2} / μ = 0

EC5 — Circle limit

e = 0

mein perihelion kahan hai?

Har jagah aur kahi nahi —

r = p

constant ke saath koi unique closest point nahi hai, isliye

θ = 0

ki choice arbitrary ho jaati hai; perihelion sirf tabhi well-defined hai jab

e > 0

ho.

EC6 — Ek bound ellipse ke liye, kya denominator

1 + e cos θ

kabhi zero tak pahunch sakta hai?

Nahi. Kyunki

0 \leq e < 1

, sabse chhoti value

1 - e > 0

hai

θ = π

par; denominator strictly positive rehta hai, isliye

r

hamesha finite hota hai — precisely isliye orbit closed hoti hai.

Active recall

Recall One-line reflexes (answers cover karo!)

$r$ kis point se measure kiya jaata hai? → focus (Sun) se, center se nahi.
Teen anomalies mein se orbit equation mein kaun sa aata hai? → true anomaly $θ$ (focus par measure kiya gaya).
Largest $r$ ellipse ke liye kis $θ$ par hota hai? → $θ = π$ (aphelion), jahan denominator sabse chhota hota hai.
Kya choti orbiting mass path shape ko affect karti hai? → Nahi; $μ \approx GM$ sirf central body par depend karta hai.
$r \to \infty$ kya possible banata hai? → denominator $1 + e cos θ \to 0$ , jiske liye $e \geq 1$ chahiye.

Yeh bhi dekho: Kepler's Laws, Conic Sections, Specific Angular Momentum h, Eccentricity and Orbital Energy, Central Force Motion, Vis-viva Equation.