Foundations — Groundtrack analysis — swath, revisit

3.2.38 · D1 · Physics › Orbital Mechanics & Astrodynamics › Groundtrack analysis — swath, revisit

Parent note Groundtrack analysis — swath, revisit padhne se pehle, tumhe har woh symbol already obvious lagni chahiye jo woh use karta hai. Yeh page unhe ek-ek karke, bilkul scratch se build karta hai.

0. Woh picture jisme poora topic rehta hai

Neeche sab kuch ek hi drawing par ek label hai: ek spinning sphere (Earth), ek tilted ring (orbit), aur surface par ek dot (subsatellite point). Is picture ko apne dimaag mein rakho.

• Grey sphere Earth hai.
• Yellow ring orbit hai — ek circle jiska plane tilted hai.
• Surface par pink dot woh jagah hai jahan satellite ki shadow parti hai (subsatellite point).
• Equator par blue arrow Earth ki spin direction hai (eastward).

1. Sphere par angle measure karna: latitude $\varphi$ aur longitude $\lambda$

Kisi bhi orbit se pehle, humein Earth par ek jagah ka naam rakhne ka tarika chahiye.

Picture: Earth ke centre par khade ho aur kisi city ki taraf point karo. Do angles us direction ko pin down karte hain — equator se kitna upar ($\varphi$), aur kitna ghuma ($\lambda$). Yahi exactly globe ki grid lines hain.

Topic ko yeh kyun chahiye: groundtrack ek positions ki list hai moving dot ki. Sphere par ek position precisely ek $(\varphi ,\lambda )$ pair hai. Jab parent kehta hai "$\varphi$ vs $\lambda$ plot karo", woh us dot ka path draw kar raha hai.

2. Inclination $i$ — orbit ring kitni tilted hai

Picture (upar ki figure): yellow orbit ring lo aur use tip karo. Flat equator se woh jitna angle kholta hai woh $i$ hai. Jitna zyada tip karo, satellite poles ke utna zyada paas fly karta hai.

Topic ko yeh kyun chahiye: dot kabhi bhi jis highest latitude tak pahunchta hai woh sirf is tilt se set hota hai. Ring dekho: uska sabse upar wala point exactly equator se $i$ latitude par baithta hai. Isliye parent kehta hai ${\varphi }_{\max }=i$.

3. Prograde vs retrograde — tilt ke effect ka sign

Yeh yahan kyun matter karta hai: "ring kis taraf tilted hai" ($i$) aur "tum us par kis taraf run karte ho" ko alag nahi kar sakte. Dono milke decide karte hain ki ${\varphi }_{\max }=i$ hai ya $180°-i$. Special retrograde orbits jo fixed Sun angle maintain karte hain woh Sun-Synchronous Orbits hain — parent ka "Landsat-like" example ek hai.

4. Ek baar ghoomne mein kitna time: orbital period $T$

Picture: satellite ek bead hai jo yellow ring ke around ek baar poori tarah slide karta hai. $T$ ek lap ke liye stopwatch reading hai.

Topic ko yeh kyun chahiye: Earth ek lap ke dauran spin karti rehti hai. Toh $T$ woh clock hai jo batata hai "do equator crossings ke beech Earth ko rotate karne ke liye kitna time milta hai?" — westward drift ka seed. $T$ height par kaise depend karta hai yeh Orbital Period & Kepler's Third Law se aata hai: zyada upar orbit → bada loop → lamba $T$.

5. Earth kitni tezi se spin karti hai: rotation rate ${\omega }_{\oplus }$ aur sidereal day

Symbol $\oplus$ (circle with cross) Earth ke liye standard astronomy shorthand hai. Toh ${\omega }_{\oplus }$ = "Earth's spin rate".

$86164$ s kyun, $86400$ s nahi? Do "days" hote hain:

• Solar day (86400 s, woh 24 h jisme hum jeete hain) woh time hai jab Sun wapas overhead aata hai. Lekin Earth ne us time mein apni yearly orbit mein aage bhi move kiya hai, isliye Sun ki taraf face karne ke liye use thoda extra spin karna padta hai.
• Sidereal day (86164 s) woh time hai jab Earth distant stars ke relative exactly $360°$ spin karti hai — ek sachchi rotation, koi extra nahi.

Satellite ki orbit stars ke relative fixed hai, isliye orbit aur spin ko fairly compare karne ke liye hum star-fixed (sidereal) rotation use karte hain — woh $86164$ s wala din. Yahi parent ki sabse common warning hai. Full detail: Earth Rotation & Sidereal Time.

6. Spin aur period ko saath mein rakhna: nodal spacing $S$

Yeh exact product kyun: ek lap mein stopwatch $T$ seconds read karta hai; $T$ seconds mein Earth (rate $\times$ time) $={\omega }_{\oplus }T$ turn karti hai. Crossing utna zyada west mein appear hoti hai kyunki ground uske neeche se east mein rotate ho gaya. Yeh sirf "distance = speed × time" hai jo angle ke kapde pehne hue hai.

Worked check (LEO): $T\approx 5760$ s ⇒ $S=360°\times 5760/86164=24.07°$. Lagbhag $360/24\approx 15$ crossings din ko tile karte hain.

7. Angle se ground distance: radians aur arc length

Hum baar baar kilometres on the ground chahte hain, sirf angles nahi. Yeh woh ek tool hai jo convert karta hai.

Radians aur kuch kyun nahi: arc-length rule $R\theta$ tabhi sahi hai jab $\theta$ radians mein ho — yahi literally radian ki definition hai (woh angle jiska arc radius ke barabar ho). Degrees daalne se nonsense milta hai. Isliye parent mein har "angle → km" step pehle ek $\pi /180$ slip karta hai.

Symbol $R_E$ Earth ki radius hai ($\approx 6371$ km); $h$ surface se satellite ki height hai. Toh satellite Earth ke centre se $R_E+h$ distance par hai.

8. Sideways dekhna: swath aur uska half-angle $\eta$

Parent ${\lambda }_s$ ko triangle (Earth-centre → satellite → swath edge) par sine rule se nikalti hai. Hum use yahan re-derive nahi karte — Remote Sensing Sensor Geometry karta hai — lekin tumhe teen symbols $\eta$, ${\lambda }_s$, $W$ pehchanne chahiye aur yeh ki final ground width ek arc-length hai: $W=2R_E{\lambda }_s$ (§7 use karke).

9. Wapas aana: repeat cycle $N/D$, spacing $\delta$, aur revisit

Kyun: $D$ poore sidereal days baad Earth wapas same orientation par aa jaati hai, aur $N$ poore orbits baad satellite wapas apni ring ke same spot par aa jaata hai. Jab dono ek saath hote hain, track khud ke upar overlay ho jaata hai aur pattern repeat hota hai. Beech mein, $N$ equally spaced strips equator ke $360°$ around fill karte hain, toh har strip $\delta =360°/N$ apart hoti hai.

Orbit plane ka slow drift jo in numbers ko nice integers par land karta hai woh $J_2$ effect hai Nodal Regression & J2 Perturbation mein.

Worked check: $N=233,D=16$ ⇒ orbits/day $=14.5625$, aur $\delta =360°/233=1.545°$, jo equator par $6371\times 1.545\times \pi /180\approx 172$ km hai.

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