Specific angular momentum h = √(GMp): Question bank

3.2.12 · D5 · Physics › Orbital Mechanics & Astrodynamics › Specific angular momentum h = √(GMp)

Why questions

Kyun hum angular momentum ko mass se divide karte hain

h

define karne ke liye?

Two-body problem mein orbiting mass

\ddot{r} = - \frac{GM}{r ^{2}} \overset{r}{^}

se cancel ho jaata hai, toh dynamics ko uski parwah nahi — natural conserved quantities bhi mass-free honi chahiye, taaki ek formula kisi bhi orbiter ke liye kaam kare.

Kyun

v \times v = 0

conservation proof mein use hota hai?

Koi bhi vector khud ke parallel hota hai, aur parallel vectors ka cross product zero hota hai. Yeh product rule ka ek term khatam kar deta hai toh sirf

r \times a

bachta hai — jo phir bhi vanish hota hai kyunki

a ∥ r

Kyun semi-latus rectum, periapsis distance ki jagah,

h = GM p

mein appear karta hai?

p

seedha orbit equation se nikalta hai as

p = h^{2} / GM

, ise

h

ka natural geometric partner banata hai. Periapsis

r_{p} = p / (1 + e)

mein

e

mix ho jaata hai, toh spin ko handle karne ka yeh kam clean tarika hai.

Kyun flight-path angle

ϕ

general relation

h = r v sin ϕ

mein enter karta hai?

Sirf

v

ka woh component jo

r

ke perpendicular hai, angular momentum produce karta hai;

sin ϕ

exactly woh perpendicular component extract karta hai. Dekho Flight-path angle ki

ϕ

orbit ke around kaise vary karta hai.

Kyun circular orbit sabse simple

h = GM r

deta hai?

e = 0

ke saath har point ek "peri = apo"-jaisa point hai:

v ⊥ r

hamesha aur

r

kabhi nahi badlata, toh

p = a = r

aur formula ek single radius mein collapse ho jaata hai.

Kyun hum

p = h^{2} / GM

sirf do likhe hue equations compare karke read off kar sakte hain?

Dono

r = \frac{h ^{2} / GM}{1 + e c o s θ}

(derived) aur

r = \frac{p}{1 + e c o s θ}

(definition) sab

θ

ke liye hold karte hain; do fractions jo sab

θ

ke liye equal hain unke numerators equal hone chahiye, toh

p = h^{2} / GM

Kyun

h

yeh nahi bata sakta ki orbit bound hai ya unbound?

h

sirf half-width

p

fix karta hai; ek parabola aur ek slim ellipse same

p

share kar sakte hain. Tumhe energy ka sign (ya

e

via Eccentricity vector) chahiye bound vs unbound decide karne ke liye.

Edge cases

Edge: Purely radial fall ke liye

h

kya hai (seedha star ki taraf, koi sideways motion nahi)?

v ∥ r

toh

r \times v = 0

h = 0

. Zero-

h

"orbit" centre se guzarna wali ek degenerate line hai, area-sweeping ellipse nahi.

Edge: Kya

h

zero ho sakta hai kisi orbit ke liye jisme nonzero speed ho?

Haan — agar

v

exactly

r

ke saath (radially) ho, speed badi hai lekin

sin ϕ = 0

h = 0

bana deta hai. Nonzero speed ≠ nonzero spin.

Edge:

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

mein kya hota hai jab

e \to 1

(parabola)?

a \to \infty

jabki

a (1 - e^{2})

finite rehta hai, toh

p

ek well-defined nonzero number rehta hai chahe

a

blow up kare — isliye

p

a

nahi, parabolic orbits ke paas safe parameter hai.

Edge: Hyperbolic orbit (

e > 1

) ke liye, kya

h

abhi bhi real aur conserved hai?

Haan.

p = h^{2} / GM > 0

e

ki parwah kiye bina, toh

h = GM p

real hai aur central-force conservation abhi bhi

d h / d t = 0

deta hai; sirf

a

negative ho jaata hai.

Edge: Agar do orbits ka same instant par same

r

aur

v

magnitude ho lekin alag

ϕ

ho, toh kya woh

h

share karte hain?

Nahi.

h = r v sin ϕ

arrows ke beech ke angle par depend karta hai, toh alag flight-path angles alag

h

dete hain — same speeds kaafi nahi hain.

Edge: Jab

e \to 0

se upar se aata hai, toh kya periapsis-based formula

h = r_{p} v_{p}

break ho jaata hai?

Nahi — yeh valid rehta hai aur smoothly

h = r v_{c i r c}

ban jaata hai, kyunki ek circle ko ek aisi ellipse maana ja sakta hai jiska "periapsis" har point hai. Formula gracefully degenerate hota hai.

Edge: Theek us instant par jab planet

θ = 9 0^{\circ}

cross karta hai (jahan

r = p

), kya

h

ke baare mein kuch special hai?

h

ke baare mein kuch bhi special nahi — yeh har jagah constant hai. Jo special hai woh geometry hai: wahan

r

semi-latus rectum ke barabar hota hai, jo

p

ko physically interpret karne ka tarika hai, spin mein koi change nahi.

Recall Quick self-audit

Agar tumne har "why" aur har "edge" sahi kiya, toh tum samajhte ho ki $h$ (a) mass-free hai, (b) constant hai, (c) plane ke perpendicular hai, aur (d) $p$ ke through shape se juda hai lekin energy se nahi. Koi bhi miss exactly woh pillar point karta hai jise dobara padhna hai.

Connections

Parent — full derivation & examples
Flight-path angle — $h = r v sin ϕ$ mein $ϕ$ .
Kepler's Second Law — $d A / d t = h /2$ wala trap.
Specific orbital energy — kyun $h$ akela orbit fix nahi kar sakta.
Eccentricity vector — missing $e$ .
Vis-viva equation — speed aur radius ko energy se pair karta hai.

Question bank — Specific angular momentum h = √(GMp)

True ya False — justify karo

Error dhundho

Why questions

Edge cases

Connections