Mapping the Path Beneath a Satellite

Satellites circling Earth trace intricate patterns across the planet's surface. The line where the spacecraft appears directly overhead is known as the ground track. Because Earth rotates beneath the orbiting vehicle, this track shifts westward with each successive pass. For many Earth-observation missions, planners desire orbits that revisit the same ground locations on a regular schedule. Repeat ground tracks simplify data comparison, facilitate systematic coverage, and aid in targeting specific regions. The calculator above provides an intuitive way to estimate how many days and orbits elapse before a satellite's ground track repeats, using only the orbital altitude as input. While simplified, the resulting numbers capture the relationship between orbital period and Earth's rotation that governs repeat cycles.

The ground track repeat phenomenon hinges on two key periods: the time it takes for a satellite to complete one orbit and the time it takes for Earth to rotate once relative to the stars. The orbital period for a circular orbit of altitude $h$ is given by $T=2\backslash pi\backslash sqrt\{\backslash frac\{(R\_E+h)^3\}\{\backslash mu\}\}$, where $\mathrm{R\_E}$ is Earth's mean radius (6378 km) and $\mathrm{\backslash mu}$ is the gravitational parameter (3.986×10⁵ km³/s²). The Earth's sidereal rotation period $\mathrm{T\_s}$ is about 86,164 seconds, slightly shorter than the 24-hour solar day because Earth moves along its orbit around the Sun. After each orbit, the ground track shifts westward by $360^\{\backslash circ\}\backslash frac\{T\}\{T\_s\}$. To repeat, this shift must sum to an integer multiple of 360° after some number of orbits.

Mathematically, we seek integers $k$ (number of orbits) and $p$ (number of sidereal days) such that $\mathrm{kT}=\mathrm{pT\_s}$. Solving for $\mathrm{k/p}$ gives $\frac{\mathrm{T\_s}}{T}$, the number of orbits per sidereal day. In general, this ratio is irrational, meaning the ground track never exactly repeats. However, by approximating the ratio with a fraction of relatively small integers, we can find near-repeating patterns that are good enough for practical purposes. The calculator employs a continued-fraction algorithm to approximate $\frac{\mathrm{T\_s}}{T}$ with a fraction whose denominator is less than 400, yielding manageable repeat cycles.

The resulting integers indicate that after $k$ orbits—equivalent to $p$ sidereal days—the satellite's ground track returns close to its starting longitude. For many applications, repeating to within a fraction of a degree suffices. The calculator also reports the orbital period in minutes and the approximate number of orbits per day, helping users contextualize the motion. Because the method assumes a perfectly circular orbit and neglects perturbations such as Earth's oblateness, atmospheric drag, and gravitational influences from other bodies, the repeat times should be viewed as first-order estimates rather than precise mission-planning figures. Nevertheless, the simplified approach reveals the underlying arithmetic that governs these elegant patterns.

Repeat ground tracks prove especially useful in remote-sensing missions. Satellites such as Landsat and Sentinel follow sun-synchronous orbits that maintain a consistent local solar time, allowing the lighting conditions of imagery to remain stable from pass to pass. By selecting altitudes and inclination angles that yield repeat cycles of 16 or 10 days, respectively, these missions systematically revisit every location on Earth, building multi-decade archives of environmental data. The calculator focuses on altitude because for circular orbits the inclination primarily affects coverage latitude rather than the repeat cycle, though sun-synchronous orbits rely on a delicate balance between altitude and inclination to precess at the right rate.

Understanding ground track repetition also assists in collision avoidance and spectrum management for large constellations. When many satellites share the same altitude and inclination, their ground tracks weave together, and designers may purposely offset orbital planes to distribute coverage evenly. Knowing the natural repeat cycle helps engineers plan phasing maneuvers and allocate communication frequencies to minimize interference. The calculator can serve as a starting point for evaluating candidate altitudes before more detailed constellation design tools come into play.

The continued-fraction method used in the script is a time-honored technique for finding rational approximations to real numbers. By iteratively subtracting integer parts and taking reciprocals, the algorithm produces a sequence of fractions known as convergents that successively approximate the original value. Applying this method to the orbits-per-day ratio yields small integer pairs that capture repeating behavior. For example, an altitude around 705 km produces approximately 14 orbits per day; the algorithm may return 233 orbits in 16 days, which corresponds to the operational pattern of the Landsat satellites. Users intrigued by number theory will appreciate how celestial mechanics entwines with rational approximations in this context.

To explore the calculator's behavior, consider a satellite at 500 km altitude. The orbital period is about 94.6 minutes, resulting in roughly 15.2 orbits per sidereal day. The fraction approximation might yield 91 orbits every 6 days, meaning the ground track nearly repeats after this interval. If we increase altitude to 800 km, the period lengthens to about 101 minutes, and the ratio drops to roughly 14.2 orbits per day. The algorithm may produce a 71-orbit, 5-day repeat cycle. These examples demonstrate how modest changes in altitude can significantly alter repeat intervals.

While the calculator assumes a static Earth, the real planet exhibits gravitational harmonics that perturb orbits. The Earth's equatorial bulge causes the orbital plane to precess, gradually shifting the ground track even if the ratio of periods is perfectly rational. Sun-synchronous orbits exploit this effect by choosing an inclination that causes the orbit to precess exactly one degree per day, matching Earth's revolution around the Sun. The simple model here ignores such subtleties but offers a foundation for understanding more sophisticated perturbation theories taught in astrodynamics courses.

Ground track analysis extends beyond Earth. Around other planets or moons, different radii and rotation rates produce distinct repeat patterns. The same equations apply by substituting the appropriate planetary radius and gravitational parameter. For instance, a mapping orbiter around Mars would use the Martian sidereal day of 88,775 seconds and radius of 3396 km in the formulas. The flexibility of the underlying mathematics underscores the universal nature of orbital mechanics.

Mission planners often balance repeat cycles against swath width. A high-resolution imager with a narrow swath may require a short repeat time to ensure global coverage, while a wide-swath sensor can tolerate longer intervals. By coupling this calculator with sensor footprint considerations, designers can gauge how many satellites are required to meet coverage goals. Tables of example altitudes and repeat cycles help illustrate these trade-offs and can guide educational exercises for students learning about satellite orbits.

Beyond technical planning, the geometry of ground tracks captures the imagination. Visualizing the intricate lattices formed by repeating orbits reveals the interplay between celestial motion and Earth's rotation. The classic "figure eight" ground track of a geostationary transfer orbit or the cloverleaf patterns of certain resonant orbits demonstrate the diversity of paths. Engaging with these patterns fosters a deeper appreciation of the dynamical beauty inherent in orbital mechanics.

Finally, the calculator's simplicity invites experimentation. Users can adjust altitude to see how quickly repeat cycles grow with higher orbits or shrink in low Earth orbit. Combining the tool with spreadsheets or programming scripts allows batch analyses of potential mission scenarios. For educators, the calculator provides an interactive demonstration of rational approximation and the relationship between time scales, making abstract astrodynamics concepts tangible for students. By demystifying repeat ground tracks, the tool contributes to more accessible and informed space mission design.

Altitude (km)	Period (min)	Approx. Repeat
500	94.6	91 orbits / 6 days
705	98.9	233 orbits / 16 days
800	101.0	71 orbits / 5 days