Groundtrack analysis — swath, revisit — Hinglish notes

3.2.38 · Physics › Orbital Mechanics & Astrodynamics

WHAT is a groundtrack?

WHY yeh flat map par sine wave jaisi lagti hai: orbit plane equator se inclination $i$ ke angle par tili hoti hai. Jaise-jaise satellite move karta hai, uski latitude $+ i$ aur $- i$ ke beech oscillate karti hai (ya retrograde ke liye $\pm (180° - i)$ ). Rectangular (Mercator-ish) map par $ϕ$ ko $λ$ ke against plot karna ek sinusoid trace karta hai.

HOW the westward drift arises (derive it)

Pehle principles se shuru karte hain. Ek orbital period $T$ mein, Earth itne angle se rotate karti hai jitna (Earth ki rotation rate)×(ek period) ho:

$Δ λ_{node} = - ω_{\oplus} T$

jahan $ω_{\oplus} = \frac{2 π}{86164 s}$ (rotation rate stars ke respect mein, yani sidereal day, na ki 86400 s ka solar day — WHY: groundtrack ko Earth ke spin ki zaroorat hai usi inertial frame mein jisme orbit hai).

Minus sign: Earth eastward spin karti hai, isliye agla equator crossing farther west appear karta hai. Har orbit mein yeh westward shift nodal spacing hai:

$S = ω_{\oplus} T = \frac{2 π}{86164} T [rad, west per orbit]$

Degrees mein: $S = 360° \cdot \frac{T}{86164 s}$ .

WHAT is swath?

WHAT is revisit time?

WHY yeh swath par depend karta hai: agar adjacent groundtracks equator par $S$ apart hain lekin swath width $W$ cover karti hai, toh koi target har orbit mein dobara dikhe ya sirf pattern "close" hone ke baad, yeh depend karta hai ki $W$ ≥ neighboring tracks ke beech ka gap hai ya nahi.

Figure — Groundtrack analysis — swath, revisit

Forecast-then-Verify

Recall Jawab padhne se pehle forecast karo

Q: Agar aap orbital altitude badhate hain, toh westward nodal spacing $S$ badhti hai ya ghatti hai? Forecast, then verify: Zyada altitude ⇒ lamba $T$ (Kepler) ⇒ har orbit mein zyada Earth rotation ⇒ bada $S$ ⇒ kam, zyada widely spaced tracks per day. ✔

Common Mistakes

Mnemonic

Feynman: explain to a 12-year-old

Recall Simply samjhao

Socho aap ek merry-go-round par ho (Earth ghoom rahi hai) aur ek dost aapke upar ek bade circle mein daud raha hai (satellite). Har baar jab dost aata hai, TUM thoda ghoom chuke hote ho, isliye woh merry-go-round ke ek alag hisse ke upar se guzarta hai. Swath yeh hai ki aapka dost apni aankhon ke kone se kitna dur dekh sakta hai. Revisit yeh hai ki kitne chakkar ke baad woh finally seedha aapki exact seat ke upar wapas aata hai. Agar woh kaafi dur dekh sake, toh sab logo ko jaldi cover kar leta hai; agar sirf ek narrow strip dekhta hai, toh bahut saare chakkar lagte hain!

Flashcards

What is a subsatellite point?

Woh point jo Earth ki surface par seedha satellite ke neeche hota hai (Earth ke center se satellite tak ki line par).

Why does a groundtrack drift westward each orbit?

Kyunki Earth fixed orbit plane ke neeche eastward rotate karti hai; har equator crossing

S = ω_{\oplus} T

se farther west appear karti hai.

Which day length is used for groundtrack calculations and why?

Sidereal day (86164 s), kyunki orbit aur Earth ki spin ko usi inertial (star-fixed) frame mein compare kiya jaata hai.

Formula for nodal (westward) spacing per orbit?

S = 360° \times T /86164 s

What sets the maximum latitude of a groundtrack?

Inclination:

ϕ_{m a x} = i

(ya

180° - i

agar

i > 90°

), altitude se independent.

Repeat-groundtrack condition?

N T = D T_{Earth}

, yaani orbits/day

= N / D

jahan

N, D

coprime integers hain.

Equatorial track spacing after a full repeat cycle?

δ = 360°/ N

Half-swath ground angle formula?

λ_{s} = arcsin (\frac{( R _{E} + h ) s i n η}{R _{E}}) - η

, swath

W = 2 R_{E} λ_{s}

Condition for gap-free global coverage?

Swath width

\geq

track spacing:

W \geq δ

Does higher altitude increase or decrease nodal spacing?

Increase — lamba period matlab har orbit mein zyada Earth rotation.

Connections

Orbital Period & Kepler's Third Law — $T$ deta hai jo $S$ mein feed hota hai.
Nodal Regression & J2 Perturbation — extra westward node drift repeat cycles ko affect karti hai.
Sun-Synchronous Orbits — $J_{2}$ regression ko repeat groundtracks ke saath combine karta hai.
Inclination & Orbital Elements — $ϕ_{m a x}$ set karta hai.
Remote Sensing Sensor Geometry — field of view $η$ jo swath determine karta hai.
Earth Rotation & Sidereal Time — 86164 s ka source.