Foundations — Launch window — phasing with target orbit

3.2.39 · D1 · Physics › Orbital Mechanics & Astrodynamics › Launch window — phasing with target orbit

Pehle tum parent note Launch Window — Phasing with Target Orbit padh sako, uske liye tumhe har symbol dekhke ek picture dikhni chahiye. Yeh page unhe sab zero se build karta hai. Upar se neeche padho — har idea agle ke liye ek brick hai.

1. Do circles aur ek central body

Sab kuch ek central body ke aas-paas hota hai — hamare liye, Earth. Earth ke liye ek dot banao. Ab uske around do circles banao: ek chhota aur ek bada. Ek circle par rehne wala satellite forever Earth se same distance par rehta hai, kyunki "circular orbit" ka yahi matlab hai.

Figure 1 neeche do circles, Earth-dot, aur do radius-spokes dikhata hai.

Humhe do chahiye:

• $r_1$ = chaser (interceptor) ka radius — woh rocket jo tum launch karte ho. Usually inner circle.
• $r_2$ = target ka radius — woh cheez jis tak tum pahunchna chahte ho.

2. Angle — circle ke around position measure karna

Satellite ki distance $r$ kabhi nahi badlti, toh hum kaise batayein ki woh apne circle par kahan hai? Ek angle se.

Radians kyun, degrees nahi? Kyunki radian mein ek poora circle exactly $2\pi$ hai — wahi $\pi$ jo har orbit formula mein aata hai — toh algebra clean rehta hai. (Agar degrees pasand hain: $\pi$ rad $=180^{\circ }$, $2\pi$ rad $=360^{\circ }$.)

3. Phase angle $\varphi$ — lead

Ab dono satellites ko ek hi instant mein apne circles par rakho. Dono spokes banao. Do spokes ke beech ka angle, Earth par measure kiya gaya, is poore topic ka star hai.

Figure 2 neeche woh wedge hai — chaser spoke, uske aage target spoke, aur unke beech lead angle $\varphi$.

Yeh jawaab ek line mein hai, is page par aage define kiye pieces se bana. Chaser transfer ellipse ka exactly aadha chakkar ($\pi$) lagaata hai; usi flight time $t_t$ mein target angle $n_2{t}_t$ sweep karta hai. Unke milne ke liye, departure par target ka lead wahi hona chahiye jo chaser use gain karta hai — isliye:

Parent mein use hone wale Greek letters:

4. $\mu$ — planet kitna strong pull karta hai

Hum $\mu$ use karte hain ($G$ aur $M$ alag-alag nahi) kyunki yahi exact combination har orbit equation mein appear hota hai — ise directly measure karna $M$ measure karne se easier aur zyada precise hai.

5. Speed, period, aur mean motion

Satellite apne circle ke around kisi speed $v$ par daudta hai, aur ek lap period $T$ mein complete karta hai. Lekin jab hum angles ki baat karte hain, sabse handy quantity yeh hai ki spoke kitni fast sweep karta hai — mean motion $n$.

Figure 3 neeche mean motion ko radius ke against plot karta hai, taaki tum dekh sako ki outer orbits crawl karti hain.

Ise "mean" motion kyun kehte hain? Circle ke liye speed bilkul steady hai, toh $n$ exact hai. "Mean" (average) word isliye hai kyunki stretched ellipses mein real angular speed vary karti hai, aur $n$ uska average hai — ek detail jo tum Mean Motion and Orbital Period mein milte ho.

Ab, circular speed $v=\sqrt{\mu /r}$ kyun hai? Kyunki circle do forces ka balance hai:

Isse, mean motion ek step mein follow hota hai ($n=v/r$):

6. Semi-major axis $a$ aur transfer ellipse

Inner circle se outer circle par saste mein jaane ke liye hum ek ellipse ka aadha fly karte hain — ek Hohmann Transfer Orbit. Ellipse ek squashed circle hai; iska ek longest diameter hota hai, aur us longest diameter ka aadha semi-major axis $a$ hai.

Figure 4 neeche transfer ellipse draw karta hai jo dono circles ko touch karta hai, uski long axis marked ke saath.

7. Kepler's Third Law — size ko time mein badalna

Hum jaante hain ki transfer us ellipse ka aadha lap leta hai. "Aadha lap" ko seconds mein badalne ke liye humhe ellipse ka period chahiye, aur woh Kepler's Third Law se aata hai.

Hohmann transfer time is ka aadha hai, $a=a_t$ use karke:

8. Angles aur time ko saath laana — $\Delta {\theta }_2$ aur synodic period

Upar ke pieces se bilkul bane do final combinations.

Tumhari flight ke dauran target kitna sweep karta hai. Target ka spoke rate $n_2$ par time $t_t$ ke liye turn karta hai, toh woh jo angle cover karta hai woh rate × time hai: $\Delta {\theta }_2=n_2{t}_t.$ Yeh "jab main pahunchunga tab woh kahan hoga" wala number hai. Isse $\pi$ (chaser ka half-lap) se subtract karo aur tumhe required lead milta hai — exactly Section 3 ka $\varphi =\pi -n_2{t}_t$ formula.

Perfect moment kitni baar wapas aata hai. Do spokes alag-alag rates $n_1$ aur $n_2$ par turn karte hain, toh unke beech wedge slowly khuld aur band hota hai. Yeh kisi bhi given value par wapas aata hai jab relative spoke poora $2\pi$ sweep kar le. Woh waiting time Synodic Period hai.

Prerequisite map

Upar ki har foundation is map ka ek arrow feed karti hai; do outputs — phase angle $\varphi$ aur synodic period $T_{\text{syn}}$ — exactly woh do sawaal hain jo parent topic answer karta hai.

Equipment checklist

Khud test karo. Right side cover karo; agar tum har line answer kar sako, tum parent note ke liye ready ho.