Methodology

OrbitLink is a software-only platform for a communication engineering graduation project. It uses publicly available orbital data and mathematical models, with no hardware. It sets out to:

1. Track satellites around the Earth with live orbital data.
2. Show satellite positions and orbital trajectories on a 3D globe and a map.
3. Predict satellite passes over chosen ground station locations.
4. Simulate ground stations with their altitude and elevation mask.
5. Calculate satellite-to-ground link budgets.
6. Analyse the Doppler frequency shift over real passes.
7. Simulate digital modulation over a noisy channel.
8. Generate engineering reports.

A single web application holds the interface, the calculation engines and a small server layer. The server downloads the CelesTrak catalog, stores it in SQLite, and serves element sets, ground stations and saved analyses as JSON. The engines run in the browser on those element sets.

CelesTrak publishes general perturbations (GP) data, mean orbital elements fitted for the SGP4 model, in the CCSDS Orbit Mean-Elements Message (OMM) format. The legacy two-line element format cannot carry catalog numbers above five digits, which ran out on 11 July 2026, so OrbitLink reads OMM keywords exclusively and supports six-digit and longer numbers.

CelesTrak updates its data every two hours and allows one download per update for each server address. OrbitLink therefore downloads only the active group (about 17,000 satellites), refreshes at most every six hours, never contacts CelesTrak twice within two hours, and never retries a refused request. Groups such as Starlink, GNSS or weather are worked out from names and orbits, because downloading overlapping groups is discouraged.

Each record keeps the epoch, mean motion, eccentricity, inclination, right ascension of the ascending node, argument of perigee, mean anomaly, the B* drag term and the derived period, semi-major axis, apogee and perigee.

Positions and velocities come from SGP4 (near-Earth) and SDP4 (deep space, periods over 225 minutes), as implemented by satellite.js, which follows the reference code of Vallado et al. (2006) also used by Skyfield. SGP4 returns the state in the True Equator Mean Equinox (TEME) frame. It is rotated to the Earth-fixed frame with Greenwich Mean Sidereal Time, converted to WGS-84 geodetic coordinates, and projected into the station's topocentric frame for look angles.

Inertial (TEME) to Earth-fixed

$$\vec r_{ECEF} = R_3(\theta_{GMST})\,\vec r_{TEME},\quad R_3(\theta) = \begin{pmatrix}\cos\theta & \sin\theta & 0\\ -\sin\theta & \cos\theta & 0\\ 0&0&1\end{pmatrix}$$

Earth-fixed velocity

$$\vec v_{ECEF} = R_3(\theta_{GMST})\,\vec v_{TEME} - \vec\omega_\oplus \times \vec r_{ECEF}$$

With the Earth rotation rate 7.2921150e-5 rad/s.

Topocentric south-east-zenith frame

$$\begin{pmatrix}S\\E\\Z\end{pmatrix} = \begin{pmatrix}\sin\varphi\cos\lambda & \sin\varphi\sin\lambda & -\cos\varphi\\ -\sin\lambda & \cos\lambda & 0\\ \cos\varphi\cos\lambda & \cos\varphi\sin\lambda & \sin\varphi\end{pmatrix}\vec\rho,\quad \vec\rho = \vec r_{sat} - \vec r_{station}$$

Look angles

$$Az = \operatorname{atan2}(E, -S),\qquad El = \arcsin\!\left(\dfrac{Z}{|\vec\rho|}\right)$$

Derived orbit

$$a = \left(\dfrac{\mu}{n^2}\right)^{1/3},\quad h_a = a(1+e) - R_e,\quad h_p = a(1-e) - R_e$$

With mu = 398600.4418 km^3/s^2 and Re = 6378.137 km (WGS-84).

SGP4 position errors are about one kilometre at the element epoch and grow by roughly one to three kilometres per day for low orbits, which is why the element age is shown with every satellite.

The pass finder scans the time window at about 1/180 of an orbit (20 s to 5 min). Where the elevation crosses the mask it bisects to 0.1 s to find AOS and LOS, then a golden-section search finds the time of closest approach (TCA) and the maximum elevation. Satellites that never drop below the mask, as geostationary satellites in view, are reported as always visible.

Visibility condition

$$El(t) \ge \varepsilon_{min}$$

Coverage half-angle of the footprint

$$\lambda_0 = \arccos\!\left(\dfrac{R_e}{R_e + h}\cos\varepsilon\right) - \varepsilon$$

Naked-eye visibility

$$\text{satellite sunlit} \ \wedge\ El_\odot(\text{station}) < -6^\circ$$

Civil twilight or darker at the station while sunlight still reaches the satellite.

The link budget follows the standard decibel chain from transmitter to receiver. Distance comes from a typed value, from the slant range formula for a given altitude and elevation, or from a predicted pass, where the margin is evaluated along the whole pass.

Free-space path loss

$$L_{fs} = 20\log_{10}\!\left(\dfrac{4\pi d}{\lambda}\right) = 32.45 + 20\log_{10} f_{MHz} + 20\log_{10} d_{km}$$

EIRP and received power

$$EIRP = P_t + G_t - L_t,\qquad P_r = EIRP - L_{fs} - L_a + G_r - L_r$$

Antenna gain

$$G = \eta\left(\dfrac{\pi D}{\lambda}\right)^2,\qquad \theta_{3dB} \approx 70\,\dfrac{\lambda}{D}\ \text{deg}$$

Noise

$$T_{sys} = T_a + 290\,(F - 1),\qquad N = kT_{sys}B$$

Carrier to noise

$$\dfrac{C}{N_0} = EIRP - L + \dfrac{G}{T} + 228.6,\qquad \dfrac{E_b}{N_0} = \dfrac{C}{N_0} - 10\log_{10}R_b$$

Margin and capacity

$$M = \dfrac{E_b}{N_0} - \left(\dfrac{E_b}{N_0}\right)_{req},\qquad C = B\log_2(1 + SNR)$$

The required Eb/N0 is the value at which the chosen uncoded modulation reaches the target bit error rate, found by inverting the theoretical curve. It can be entered by hand to account for coding gain or implementation losses.

The range rate comes from the relative position and the Earth-fixed velocity of the satellite. The first-order shift is used throughout; the relativistic correction is about f (v/c)^2 / 2, roughly 0.1 Hz at UHF, and is shown for comparison in the Doppler lab.

Range rate

$$\dot\rho = \dfrac{\vec\rho\cdot\vec v_{ECEF}}{|\vec\rho|}$$

Doppler shift (first order)

$$\Delta f = -\dfrac{\dot\rho}{c}\,f_t$$

Uplink pre-compensation

$$f_{tx} = f_0 - \Delta f(f_0)$$

The signal lab uses normalised time (one bit lasts Tb = 1) and unit average bit energy. Waveforms are sampled, noise is added sample by sample, and correlator receivers make coherent decisions. For the bit error rate curves the same detection is simulated in signal space, which is exactly what a matched filter outputs, so millions of bits run in seconds. A seeded random generator makes every run repeatable.

Signals over one bit (Eb = 1, Tb = 1)

$$\text{BPSK: } \pm\sqrt2\cos(2\pi f_c t),\quad \text{ASK: } \{0, 2\cos(2\pi f_c t)\},\quad \text{FSK: } \sqrt2\cos(2\pi f_{0,1} t)$$

QPSK symbol (two bits, Gray coded)

$$s(t) = I\cos(2\pi f_c t) - Q\sin(2\pi f_c t),\quad I, Q \in \{\pm 1\}$$

Sampled AWGN

$$n_k \sim \mathcal N\!\left(0, \dfrac{N_0}{2\,\Delta t}\right)$$

Theoretical bit error rates

$$P_{b,BPSK} = P_{b,QPSK} = Q\!\left(\sqrt{2E_b/N_0}\right),\qquad P_{b,FSK} = P_{b,ASK} = Q\!\left(\sqrt{E_b/N_0}\right)$$

The Gaussian tail is $Q(x) = \tfrac12\,\mathrm{erfc}(x/\sqrt2)$, with erfc evaluated by a Chebyshev fit accurate to 1.2e-7. Every engine is checked against references on the Validation page.

• Accuracy is limited by SGP4 and the age of the element set, not by the numerics; elements older than a week are flagged.
• Atmospheric refraction is not modelled, so rise and set times near 0° elevation can differ from optical observations by tens of seconds.
• The Earth's shadow uses the solar ephemeris of satellite.js with a spherical Earth; penumbra is reported but not used for brightness.
• The link budget assumes free-space propagation plus the losses entered; rain and atmospheric losses do not change with elevation automatically.
• Modulation results assume an ideal coherent receiver with perfect carrier and timing recovery and no channel coding.
• Satellite groups are derived from names and orbits and can misplace a few objects.

1. D. A. Vallado, P. Crawford, R. Hujsak and T. S. Kelso, "Revisiting Spacetrack Report #3", AIAA/AAS Astrodynamics Specialist Conference, AIAA 2006-6753, 2006.
2. F. R. Hoots and R. L. Roehrich, "Spacetrack Report No. 3: Models for Propagation of NORAD Element Sets", 1980.
3. T. S. Kelso, "A New Way to Obtain GP Data", CelesTrak documentation (GP data formats and usage policy).
4. CCSDS 502.0-B-3, Orbit Data Messages (Orbit Mean-Elements Message, OMM).
5. National Imagery and Mapping Agency, "World Geodetic System 1984", TR8350.2.
6. ITU-R Recommendation P.525, Calculation of free-space attenuation.
7. G. Maral, M. Bousquet and Z. Sun, Satellite Communications Systems, 6th ed., Wiley.
8. J. G. Proakis and M. Salehi, Digital Communications, 5th ed., McGraw-Hill.
9. B. Sklar, Digital Communications: Fundamentals and Applications, 2nd ed., Prentice Hall.