Groundtrack analysis — swath, revisit — Physics notes

WHAT is a groundtrack?

WHY it looks like a sine wave on a flat map: the orbit plane is tilted by the inclination $i$ to the equator. As the satellite moves, its latitude oscillates between $+i$ and $-i$ (or $\pm(180°-i)$ for retrograde). Plotting $\phi$ vs $\lambda$ on a rectangular (Mercator-ish) map traces a sinusoid.

HOW the westward drift arises (derive it)

Start from first principles. In one orbital period $T$ , the Earth rotates by an angle equal to (Earth's rotation rate)×(one period):

$\Delta\lambda_{\text{node}} = -\,\omega_\oplus \, T$

where $\omega_\oplus = \dfrac{2\pi}{86164\,\text{s}}$ (rotation rate w.r.t. stars, the sidereal day, not the 86400 s solar day — WHY: the groundtrack cares about Earth's spin in the same inertial frame the orbit lives in).

The minus sign: Earth spins eastward, so the next equator crossing appears farther west. This westward shift per orbit is the nodal spacing:

$\boxed{\;S = \omega_\oplus\, T = \frac{2\pi}{86164}\; T \quad[\text{rad, west per orbit}]\;}$

In degrees: $S = 360° \cdot \dfrac{T}{86164\,\text{s}}$ .

WHAT is swath?

Formula Swath from field-of-view half-angle

For a sensor looking straight down with half field-of-view $\eta$ , the Earth-central half-angle $\lambda_s$ it subtends satisfies (from the triangle Earth-center → satellite → ground edge, using the sine rule): $\frac{\sin\eta}{R_E} = \frac{\sin(\eta+\lambda_s)}{R_E + h}$ so the half-swath ground angle is $\lambda_s = \arcsin\!\left(\frac{(R_E+h)\sin\eta}{R_E}\right) - \eta$ and the swath width is the arc length $\boxed{\,W = 2 R_E\, \lambda_s\,}$ WHY the sine rule: the two rays (to satellite, to ground edge) and the surface form a triangle; the angle at the ground edge is $180°-(\eta+\lambda_s)$ , and $\sin(180°-x)=\sin x$ .

Worked example Swath of a nadir imager

$R_E = 6371$ km, $h = 700$ km, $\eta = 15°$ . Step 1: $\sin(\eta+\lambda_s) = \frac{R_E+h}{R_E}\sin\eta = \frac{7071}{6371}\sin15° = 1.110\times0.2588 = 0.2872$ . Why? Rearranging the sine rule to isolate the far angle. Step 2: $\eta+\lambda_s = \arcsin(0.2872)=16.69°\Rightarrow \lambda_s = 1.69°$ . Step 3: $W = 2(6371)(1.69°\times\tfrac{\pi}{180}) = 2\times6371\times0.0295 = 376$ km. Why the $\pi/180$ ? Arc length needs radians: $\text{arc}=R\theta_{\text{rad}}$ .

WHAT is revisit time?

WHY it depends on swath: if adjacent groundtracks are spaced $S$ apart at the equator but the swath covers width $W$ , then whether a target is re-seen every orbit or only after the pattern "closes" depends on whether $W$ ≥ the gap between neighboring tracks.

Formula Repeat groundtrack condition

The track exactly repeats after $N$ orbits in $D$ days when $\boxed{\,N\,T = D\,T_{\text{Earth}}\quad\Rightarrow\quad \frac{N}{D} = \frac{T_{\text{Earth}}}{T} = \text{(orbits per day)}\,}$ $N,D$ integers with no common factor. WHY: after $D$ sidereal days the Earth is back to the same orientation and after $N$ orbits the satellite is back to the same spot in its orbit — both conditions met simultaneously ⇒ track overlays itself.

The fundamental interval between neighboring tracks at the equator: $\delta = \frac{360°}{N}$ and the worst-case revisit ≈ $D$ days (full cycle), while best-case revisit can be one orbit if swaths overlap.

Figure — Groundtrack analysis — swath, revisit

Forecast-then-Verify

Recall Forecast before you read the answer

Q: If you increase orbital altitude, does the westward nodal spacing $S$ increase or decrease? Forecast, then verify: Higher altitude ⇒ longer $T$ (Kepler) ⇒ more Earth rotation per orbit ⇒ larger $S$ ⇒ fewer, more widely spaced tracks per day. ✔

Common Mistakes

Mnemonic

Feynman: explain to a 12-year-old

Recall Explain it simply

Imagine you're on a merry-go-round (Earth spinning) and a friend is running in a big circle above you (the satellite). Every time your friend comes around, YOU'VE turned a bit, so they zoom over a different part of the merry-go-round. The swath is how far your friend can see out of the corner of their eye. Revisit is how many spins until they finally fly right back over your exact seat again. If they see far enough, they cover everyone quickly; if they see only a narrow strip, it takes many spins!

Flashcards

What is a subsatellite point?

The point on Earth's surface directly below the satellite (on the line from Earth's center to the satellite).

Why does a groundtrack drift westward each orbit?

Because Earth rotates eastward under the fixed orbit plane; each equator crossing appears farther west by

S=\omega_\oplus T

Which day length is used for groundtrack calculations and why?

The sidereal day (86164 s), because the orbit and Earth's spin are compared in the same inertial (star-fixed) frame.

Formula for nodal (westward) spacing per orbit?

S = 360°\times T/86164\text{ s}

What sets the maximum latitude of a groundtrack?

The inclination:

\phi_{\max}=i

(or

180°-i

i>90°

), independent of altitude.

Repeat-groundtrack condition?

N T = D\,T_{\text{Earth}}

, i.e. orbits/day

=N/D

with

N,D

coprime integers.

Equatorial track spacing after a full repeat cycle?

\delta = 360°/N

Half-swath ground angle formula?

\lambda_s=\arcsin\!\big(\frac{(R_E+h)\sin\eta}{R_E}\big)-\eta

, swath

W=2R_E\lambda_s

Condition for gap-free global coverage?

Swath width

\ge

track spacing:

W \ge \delta

Does higher altitude increase or decrease nodal spacing?

Increase — longer period means more Earth rotation per orbit.

Connections

Orbital Period & Kepler's Third Law — gives $T$ that feeds $S$ .
Nodal Regression & J2 Perturbation — extra westward node drift affects repeat cycles.
Sun-Synchronous Orbits — combine $J_2$ regression with repeat groundtracks.
Inclination & Orbital Elements — sets $\phi_{\max}$ .
Remote Sensing Sensor Geometry — field of view $\eta$ that determines swath.
Earth Rotation & Sidereal Time — source of the 86164 s.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, groundtrack ka funda simple hai: satellite ek fixed orbit plane mein ghoomta hai (space ke stars ke respect mein fixed), lekin Earth neeche apne axis pe ghoom rahi hai. Isliye har baar jab satellite equator cross karta hai, Earth thodi ghoom chuki hoti hai, aur next crossing thoda west ki taraf shift ho jaata hai. Is shift ko hum nodal spacing $S = 360° \times T/86164$ kehte hain. Yaad rakho — yahan sidereal day (86164 s) use karna hai, solar day (86400 s) nahi, kyunki hum orbit aur Earth-spin dono ko same inertial frame mein compare kar rahe hain.

Latitude ka maximum? Wo sirf inclination $i$ decide karta hai, altitude ka isse koi lena-dena nahi. Agar $i=55°$ hai to satellite kabhi bhi $55°$ latitude se pole ki taraf nahi ja sakta — pure geometry hai, orbit plane ka tilt.

Swath matlab sensor ek pass mein zameen pe kitni chaudi patti dekh paata hai. Ye sensor ke field-of-view angle $\eta$ aur altitude pe depend karta hai, sine rule se nikaalte hain. Revisit matlab kitne time baad satellite wapas usi jagah pe aata hai. Ye orbital period se decide NAHI hota (galti mat karna!) — ye repeat cycle $N/D$ (orbits per day) aur swath width pe depend karta hai. Agar swath itni chaudi hai ki neighbouring tracks ka gap $\delta=360°/N$ cover ho jaaye, to poori Earth bina gap ke cover ho jaati hai aur revisit fast ho jaata hai. Yehi reason hai ki remote-sensing satellites (Landsat jaise) ko carefully design karte hain — jaise $N=233$ orbits in $D=16$ days.

Groundtrack analysis — swath, revisit

WHAT is a groundtrack?

HOW the westward drift arises (derive it)

WHAT is swath?

WHAT is revisit time?

Forecast-then-Verify

Common Mistakes

Mnemonic

Feynman: explain to a 12-year-old

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Orbital Mechanics & Astrodynamics

Connections