Satellite in Orbit —  Study Notes

Section 1

1. Foundations: The Satellite's Orbital World

1.1 Why Orbital Mechanics for ECE Students

A satellite engineer does not need to become an astrodynamicist, but must be fluent in the language of orbits: the satellite's trajectory determines its coverage (who it can see and when), its link geometry (range, Doppler, contact duration), its environment (radiation belts, drag, eclipse), and its propulsion budget (station-keeping, disposal). Radar & satellite engineering in particular relies on orbit knowledge for target tracking, space-surveillance radars, and ground-station scheduling.

1.2 Reference Frames

1.3 Fundamental Constants

The sidereal day — not the solar day — sets the GEO altitude: the satellite must complete one orbit in the time Earth takes to rotate once relative to the stars.

1.4 The Two-Body Assumption

All elementary satellite work begins with two point masses: satellite + Earth, with gravity the only force. The assumptions are: spherical Earth (mass distribution is spherically symmetric), no atmosphere, no third bodies (Sun, Moon), no solar radiation pressure, no thrust. Each assumption is relaxed later as a perturbation (Section 5). Two-body orbits are conic sections: ellipses (closed orbits, E < 0), parabolas (E = 0, escape), hyperbolas (E > 0, flyby).

Section 2

2. Kepler's Laws and the Two-Body Solution

2.1 Kepler's Three Laws (Statement for Satellites)

1. Ellipses: every satellite travels on an ellipse with Earth's centre at one focus (circle: special case e = 0).
2. Equal areas: the radius vector sweeps equal areas in equal times — the satellite moves fastest at perigee, slowest at apogee.
3. Harmonic law: T² = (4π²/μ)·a³ — the orbital period depends only on the semi-major axis.

2.2 The Equation of Motion and Its First Integrals

Newton's law of gravitation gives the relative equation of motion:

2.3 Vis-Viva and Velocity Relations

2.4 Time on Orbit: Kepler's Equation

Position as a function of time requires solving Kepler's equation (E = eccentric anomaly, M = mean anomaly, M = n(t − t_p), n = 2π/T):

For small e, M ≈ ν − 2e sin ν (series expansion). Practical note: mean motion n = √(μ/a³) gives the satellite's "average" angular rate; the true angular rate varies by ±2e rad/percent.

Section 3

3. The Six Keplerian Orbital Elements

An orbit needs six numbers to be fully specified: three say where the orbital plane and ellipse sit, three say where the satellite is in the ellipse.

3.1 Special Cases

• Equatorial (i = 0): Ω undefined (no node); GEO comsats live here.
• Circular (e = 0): ω and ν merge into one angle (argument of latitude u); element sets use (u) instead.
• Circular equatorial: only (a, mean longitude) remain meaningful.

Section 4

4. Orbit Selection: Matching Orbit to Mission

4.1 The Mission-Driven Trade

Orbit selection trades coverage, latency, link budget, launch cost, radiation environment, and propulsion burden. The dominant question: what must the satellite see, how often, and from how far?

4.2 The Standard Menu

4.3 Quantitative Comparison Anchors

• Free-space path loss at GEO vs 600 km LEO: ΔL = 20log(35,786/600) ≈ 35.5 dB — the fundamental comms penalty of GEO, offset by 24/7 coverage.
• Round-trip delay: GEO ~250–280 ms (two hops ~500+ ms); LEO ~2–10 ms — decisive for telephony vs broadband.
• Coverage per satellite: GEO instantaneous view of ~42% of Earth's surface (elevation ≥ 0°); a 600 km LEO sat sees ~2.6% — hence constellation scaling.
• Radiation: LEO below ~1,000 km skirts the inner Van Allen belt (South Atlantic Anomaly excepted); GEO/MEO cross the belts during transfer but operate above them — still high total-dose environments.

4.4 Launch Site Constraint

Launch sites sit at fixed latitudes: Cape Canaveral 28.5°N, Kourou 5.2°N, Baikonur 45.9°N, Satish Dhawan 13.7°N. A direct launch cannot exceed the site latitude without a dogleg (plane-change) manoeuvre, which is extremely expensive in Δv. Hence SSO missions launch near-dawn from high-latitude sites, GEO missions launch eastward from near-equatorial sites (Kourou saves ~0.46 km/s vs Baikonur), and ISS orbit (51.6°) reflects a compromise of international partner launch capabilities.

Section 5

5. Orbital Perturbations

Real orbits deviate from Keplerian ellipses. The dominant perturbations and their practical signatures:

5.1 Earth Oblateness (J2) — the Dominant Term

Earth's equatorial bulge (J2 ≈ 1.0826×10⁻³) produces torques that change the orbital plane and perigee direction. First-order secular rates:

Consequences: the critical inclination (63.4°) is chosen for Molniya/Tundra so the apogee stays fixed over the northern hemisphere; Sun-synchronous orbits exploit nodal regression — setting Ω̇ = +0.9856°/day (Earth's motion around the Sun) requires i ≈ 97–99° depending on altitude.

5.2 Atmospheric Drag (LEO)

• Drag is strongest at perigee, lowest perigee altitude, and high solar activity (thermospheric heating expands the atmosphere).
• Rule of thumb: at 400 km a CubeSat (B ≈ 0.01–0.02 m²/kg) decays in months–years; at 600 km, decades. VLEO (< 450 km) sats need frequent reboost or drag compensation (e.g., GOCE's ion thruster).
• Effect: semi-major axis and period shrink (orbital energy loss), eccentricity decreases (circularizes), inclination nearly unchanged; epoch/element predictions degrade rapidly without density modeling (e.g., NRLMSISE-00, JB2008, HASDM).

5.3 Third-Body Gravity (Sun and Moon)

The Sun and Moon torque the orbital plane, most strongly at high altitude where Earth's gravity gradient weakens. GEO satellites experience inclination drift ~0.85–0.95°/year (combined luni-solar), forcing N–S station-keeping of ~40–55 m/s per year. Luni-solar effects on MEO navigation constellations (GPS) similarly drive inclination/RAAN station-keeping (~1–2 m/s/year).

5.4 Solar Radiation Pressure (SRP)

SRP perturbs eccentricity and longitude (east–west drift for GEO: ~1–2 m/s/yr budget) and is the dominant force for high area-to-mass objects (debris, balloons). It is effectively non-conservative over an orbit (eclipse cycles modulate it), causing slow eccentricity growth.

5.5 Other Perturbations

• Earth albedo and IR: small, warms/perturbs drag-like for VLEO.
• Tides: solid-Earth and ocean tides shift the geopotential field at the metre level over a day — relevant to precise orbit determination (altimetry at cm level).
• Relativity: GPS clocks run ~38 µs/day faster (gravitational blueshift −6 µs, velocity −7 µs, net +38 µs incl. Sagnac) — must be corrected in the system design; perigee precession for very eccentric orbits (Mercury-like effect, ~0.01°/century for LEO — negligible for ops).
• Maneuvers, venting, momentum dumping: small Δv's must be modeled in OD; unmodeled thruster firings are the largest source of prediction error in practice.

Section 6

6. Orbit Transfer and Launch Windows

6.1 The Hohmann Transfer

The minimum-Δv two-impulse transfer between coplanar circular orbits is the Hohmann ellipse: transfer perigee at r₁, apogee at r₂, a_t = (r₁ + r₂)/2.

6.2 Launch Windows and Phase Angle

• To a plane: the launch site passes through the target orbital plane once per sidereal day — the daily window is minutes wide (or seconds for rendezvous).
• To a satellite (rendezvous/phasing): the target's in-plane phase angle determines window; phasing orbits adjust the chaser's period to match.
• Sun-synchronous: the ascending-node local time is fixed by orbit; launch time sets the initial node and β angle seasonality.
• GEO: window set by longitude drift rate needed to reach slot; typically < 1 h/day.

6.3 Electric Propulsion Orbit Raising

Spiral raising with Isp 1,500–3,000 s takes months (e.g., all-electric GEO satellites: 4–6 months from GTO) but cuts propellant by ~40% vs chemical GTO→GEO. The transfer uses continuous thrust arcs around the orbit, optimizing the argument of perigee advance (thrust near perigee for raising, near apogee for plane change).

Section 7

7. Ground Track, Coverage, and Revisit

7.1 Ground Track Geometry

The sub-satellite point (SSP) traces a sinusoidal-like curve on the rotating Earth. Key parameters:

• Maximum latitude reached = i (for direct orbits) — a satellite never sees beyond its inclination latitude.
• Node spacing: Earth rotates 360° per sidereal day (Ω_E = 0.2507°/min); the equator-crossing longitude shifts west by 360°·(T_sidereal/T_orbit) each orbit (for prograde orbits, roughly).
• Inclination vs. latitude coverage: GEO sees to latitude ≈ i = 0°±81° at 5° min elevation; LEO inclination sets the latitude band of coverage.

7.2 Horizon and Instantaneous Coverage

7.3 Revisit and Contact Duration (LEO)

• 600 km, 5° min elevation: pass ≈ 8–9 min; horizon-to-horizon ≈ 15 min.
• Passes per day from one station: ~2–5 depending on inclination vs. station latitude; coverage improves with station latitude up to i.
• Revisit (global, nadir-pointing swath): repeating ground-track orbits (resonant T/n_orbits vs. sidereal day) give exact repeat cycles — e.g., Sentinel-2: 10-day repeat, 786 km.

Section 8

8. Eclipse Seasons and Sun Angle

8.1 Eclipse Geometry

A satellite enters Earth's shadow when the Sun–Earth–satellite geometry passes behind the planet. The governing angle is the Sun's declination δ and the satellite's orbit geometry:

• LEO: eclipses occur every orbit when the orbit plane is oriented away from the Sun — typically up to ~35–36 min per ~93 min orbit (summer/winter symmetric), twice per year with midnight–noon orbits; total eclipse minutes grow near equinox.
• GEO: eclipses only near equinoxes (Sun near equatorial plane), up to 72 min per day for ~45 days around each equinox (spring and autumn seasons), none at solstice.
• Eclipse season definition: the period when |declination of the Sun relative to orbit plane| < arcsin(R_e/(R_e+h)) — the same geometric condition for both LEO and GEO.

8.2 Beta Angle

β drives both eclipse duration (small |β| → long eclipses) and thermal flux (β near 90° = continuous sunlight on the orbit plane, benign; β near 0 = deep cold-soak cycles). Mission thermal and power designs must cover the worst β of the mission.

Section 9

9. Constellations and Formation Flying

9.1 Why Constellations

Single satellites give intermittent coverage; constellations trade number of satellites for continuous or near-continuous global (or regional) coverage. The design variables: altitude, inclination, number of planes, sats per plane, phasing between planes.

9.2 Walker Constellations

The standard symmetric notation: i: T/P/F — inclination i, T total satellites, P planes, F phasing factor between planes.

• Starlink (v1): 53° inclination shells at 550 km; ~1,600 sats per shell for single-fold coverage.
• OneWeb: 87.4° (near-polar), 1,200 km — continuous polar + global coverage with fewer planes due to high latitude convergence.
• Iridium: 86.4° LEO (780 km), 66 sats in 6 planes — pole-to-pole with 100% Earth coverage including oceans/poles (aviation/maritime market).
• GPS: 24 sats in 6 planes at 55° inclination, 4 sats/plane — 4+ visible everywhere at all times (6+ in practice with 31 active).

9.3 Formation Flying and Relative Motion

For close formations (clusters, SAR interferometry, rendezvous), linearize about the reference orbit: Hill–Clohessy–Wiltshire equations:

• Along-track/cross-track: closed relative orbits (ellipse, no drift) exist with matching energy (same period) — e.g., satellite trailing 120° in a GPS plane.
• Rendezvous: station-keeping with periodic Δv (~1–2 m/s per day for ISS proximity ops), R-bar/V-bar approaches.
• Tandem interferometry: TerraSAR-X/TanDEM-X flew 200–500 m apart for single-pass DEM generation — differential drag, not thrusters, fine-tuned the separation.

Section 10

10. Orbit Determination and Tracking

10.1 The OD Problem

Given noisy measurements (range, Doppler, angles, GNSS) over an arc, estimate the epoch state (r, v) and force-model parameters (drag coefficient, SRP area-to-mass, maneuver Δv's). Methods:

• Batch least squares (weighted LSQ over an arc — e.g., Goddard Trajectory Determination System).
• Sequential filters: Extended Kalman Filter (EKF), Unscented KF, square-root information filters — standard for onboard and ground real-time.
• Consider-covariance: account for uncertain parameters (density, clock) without estimating them.

10.2 Measurement Types and Accuracies

10.3 Tracking Coverage

A radar or TTC network must geometrically see the satellite to measure it. Prediction requirements: catalog maintenance (cataloging debris > 10 cm in LEO requires multiple dedicated radars, e.g., Space Fence S-band at Kwajalein), conjunction screening (satellite operators receive conjunction data messages; screening against the public TLE catalog with 7-day prediction is standard practice). TLE + SGP4 (simplified perturbation model) is the de-facto interchange format — accuracies ~1–10 km at epoch, degrading ~1–2 km/day for LEO without updates.

Section 11

11. Doppler, Range, and the Communication Link Geometry

11.1 Doppler for LEO

• 600 km LEO at VHF (145 MHz): max |Doppler| ≈ ±1.5 kHz one-way at horizon approach/recession (≈ ±3.3 kHz for a two-way radar link, doubled).
• At Ku band (12 GHz) the same geometry gives ≈ ±270 kHz — receiver AFC loops and Doppler-compensated demodulators are mandatory.
• LEO Doppler S-curve over a pass: near-zero at rise/set, maximum rate of change (Doppler "jerk") near closest approach — drives synthesizer step size in burst modems.

11.2 Delay and Handover

• LEO one-way delay: 2–10 ms; GEO: ~120 ms one-way.
• LEO broadband constellations need inter-satellite links (ISLs) or dense gateways; cell handover every ~30–90 s per satellite at user level.
• Link budget closure must use time-varying slant range ρ(t): EIRP/L(ρ) swings several dB across a pass at Ku band — margins must be allocated at minimum elevation.

11.3 Space Surveillance Radar View

For radar & satellite engineering, the satellite-in-orbit material closes the loop: a space-surveillance radar applies the two-way radar equation to a target whose RCS is aspect-dependent (flat panels, solar arrays) and whose trajectory is perturbed. Track custody (maintaining association) requires prediction accuracies better than the radar beamwidth; catalog updates refresh state vectors multiple times per day for LEO.

Section 12

12. Station-Keeping, End-of-Life, and Debris

12.1 GEO Station-Keeping Boxes

• Typical allocation: ±0.05–0.1° in longitude/latitude (ITU coordination protects assignments); the satellite is kept inside its box with weekly thruster firings (N–S) and fortnightly (E–W), or continuous low-thrust electric SK.
• Annual Δv budget: N–S ~40–55 m/s (luni-solar inclination drift), E–W ~1–2 m/s (SRP + tesseral resonance), total ~45–60 m/s/yr — dominating 15-year propellant mass (~600+ m/s lifetime for chemical; ~2% mass fraction electric).
• Longitude perturbations: geopotential tesseral resonances (Earth's gravity field triaxiality) create stable points at 75.1°E and 252.9°E (sats drift toward them) and unstable points at 164°E and 346°E — dead sats migrate to the stable longitudes over years.

12.2 LEO Maintenance

• Drag makeup: VLEO (~300–400 km): meters/second per day — constellation sats carry electric propulsion for weekly reboost.
• Phasing & slot keeping: constellation sats maintain assigned slots within ±few km along-track (relative station-keeping with crosslink ranging).
• Collision avoidance: with ~30,000+ tracked objects (TLE catalog), operators screen and maneuver (typical threshold: probability of collision > 1e-4); a maneuver costs ~0.1–1 m/s and reserves propellant for ~2–3 avoidance burns per year per LEO sat (rising with constellation density).

12.3 End-of-Life Disposal

The 25-year rule is driving design: sats at 600–1,200 km now carry propulsion sized for deorbit; natural decay from 550 km takes ~5–10 years depending on solar cycle.

Section 13

13. Worked Problems

Section 14

14. Formula Sheet

Section 15

15. Glossary of Key Terms

References

1. D. A. Vallado, Fundamentals of Astrodynamics and Applications, 4th ed., Microcosm, 2013.
2. B. Wie, Space Vehicle Dynamics and Control, 2nd ed., AIAA, 2008.
3. W. J. Larson & J. R. Wertz (eds.), Space Mission Analysis and Design, 3rd ed., Microcosm, 1999.
4. M. H. Kaplan, Modern Spacecraft Dynamics and Control, Wiley, 1976.
5. O. Montenbruck & E. Gill, Satellite Orbits: Models, Methods and Applications, Springer, 2000.
6. J. R. Wertz (ed.), Spacecraft Attitude Determination and Control, Springer, 1978.