Launch window — phasing with target orbit — Hinglish notes

3.2.39 · Physics › Orbital Mechanics & Astrodynamics

1. Phasing kya hai? (WHAT)

Tum ek lower (ya higher) orbit mein launch hote ho aur ek transfer ellipse par coast karte ho. Transfer time $t_{t}$ ke dauran, target chaltta rehta hai. Tumhe tab fire karna hota hai jab target bilkul itna aage ho ki woh arrival point par coast kare jab tum wahan pahuncho.

2. Required phase angle derive karna (HOW / scratch se derive)

Do co-planar circular orbits set up karo: chaser radius $r_{1}$ , target radius $r_{2}$ . Hum unhe connect karne ke liye Hohmann transfer (half-ellipse) use karte hain.

Step 1 — Transfer time

Transfer ellipse ka semi-major axis hai $a_{t} = \frac{r _{1} + r _{2}}{2}$ Kyun? Perigee $r_{1}$ par hai, apogee $r_{2}$ par; ek ellipse ke liye $r_{p} + r_{a} = 2 a$ .

Transfer exactly half a period hai (perigee → apogee): $t_{t} = \frac{T _{ellipse}}{2} = π \frac{a _{t}^{3}}{μ}$ Kyun? Kepler's third law $T = 2 π a^{3} / μ$ ; half ellipse half period leta hai kyunki major axis ke baare mein symmetry hai.

Step 2 — Transfer ke dauran target kitna move karta hai

Target ki angular speed (mean motion) hai $n_{2} = \frac{μ}{r _{2}^{3}}$ Kyun? Circular orbit ke liye $v = μ / r$ aur $n = v / r = μ / r^{3}$ .

$t_{t}$ ke dauran target sweep karta hai $Δ θ_{2} = n_{2} t_{t} = \frac{μ}{r _{2}^{3}} π \frac{a _{t}^{3}}{μ} = π (\frac{a _{t}}{r _{2}})^{3/2} = π (\frac{r _{1} + r _{2}}{2 r _{2}})^{3/2}$ Kyun yeh step? $μ$ cancel ho jaata hai — physics pure geometry + timing hai.

Step 3 — Rendezvous condition

Chaser Hohmann transfer ke dauran $18 0^{\circ} = π$ fly karta hai (perigee se apogee tak, aadha sky). Target se milne ke liye, target ko usi apogee point par pahunchna hoga. Toh launch ke waqt target ko arrival point se exactly utna aage hona chahiye jo woh abhi bhi travel karega, yaani required lead angle hai:

$ϕ = π - Δ θ_{2} = π [1 - (\frac{r _{1} + r _{2}}{2 r _{2}})^{3/2}]$

Step 4 — Synodic period → window kitni baar repeat hoti hai

Relative angular geometry har synodic period mein repeat hoti hai: $T_{syn} = \frac{2 π}{∣ n _{1} - n _{2} ∣}, n_{i} = \frac{μ}{r _{i}^{3}}$ Kyun? Do bodies relative rate $∣ n_{1} - n_{2} ∣$ par drift karti hain; $2 π$ relative drift se same phasing wapas aati hai.

Toh launch window har $T_{syn}$ mein repeat hoti hai. Ground par, Earth ki rotation add karo taaki tumhara site plane ke neeche se guzre (period ≈ 1 sidereal day) — asli window wahan hai jahan dono conditions ek saath milen.

Figure — Launch window — phasing with target orbit

3. Worked examples

4. Common mistakes (steel-manned)

5. Flashcards

Ek asli launch window define karne wali do conditions kya hain?

Plane match (site orbital plane ke neeche rotate kare) AUR phasing (target correct lead angle par ho rendezvous ke liye).

Hohmann rendezvous mein required phase angle ka formula?

ϕ = π [1 - (\frac{r _{1} + r _{2}}{2 r _{2}})^{3/2}]

Phase-angle formula mein

μ

kyun cancel hota hai?

Kyunki

ϕ

sirf transfer arc times ke ratio par depend karta hai, jo geometric radius ratios mein reduce ho jaata hai;

μ

saare times ko equally scale karta hai.

Hohmann transfer ka transfer time kya hai?

Ellipse period ka half:

t_{t} = π a_{t}^{3} / μ

jahan

a_{t} = (r_{1} + r_{2}) /2

Synodic period define karo aur uska formula do.

Wo time jisme relative phasing geometry repeat hoti hai:

T_{syn} = 2 π /∣ n_{1} - n_{2} ∣

Almost-equal orbits ke synodic periods itne lambe kyun hote hain?

Unke mean motions almost equal hote hain, toh relative drift

∣ n_{1} - n_{2} ∣

tiny hoti hai →

2 π

relative drift mein bahut zyada time lagta hai.

LEO→GEO transfer mein departure par GEO roughly kitna aage hona chahiye?

Lagbhag

10 0^{\circ}

(target slow lambe transfer ke dauran sirf ~

4 4^{\circ}

move karta hai).

Hohmann transfer ke dauran chaser kitna angle sweep karta hai?

Exactly

18 0^{\circ}

(perigee se apogee, ellipse ka half).

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

Socho tum aur tumhara dost do circular tracks par daud rahe ho, tumhara dost outer track par slower ja raha hai. Tum apne dost ko ball throw karna chahte ho, lekin jab tak ball fly kare, tumhara dost move ho chuka hai. Toh tum wahan throw nahi karte jahan woh hai — tum aage throw karte ho, wahan jahan woh hoga. Launch window woh exact moment hai jab tumhara dost sahi distance aage ho taaki tumhara throw (transfer orbit) unse perfectly mile. Aur yeh perfect moment baar baar aata rehta hai — woh "baar baar" synodic period hai.

Connections

Hohmann Transfer Orbit — $a_{t}$ aur transfer time provide karta hai jo yahan use hota hai.
Kepler's Third Law — $T = 2 π a^{3} / μ$ ka source.
Mean Motion and Orbital Period — $n = μ / r^{3}$ define karta hai.
Synodic Period — window kitni baar repeat hoti hai.
Orbital Plane and Inclination — launch window ki geometry side.
Rendezvous and Phasing Orbits — jab single Hohmann kaafi na ho tab fine-tuning.
Lambert's Problem — general (non-Hohmann) transfer timing.