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OrbitView: Mathematical Foundations

A comprehensive guide to the astrodynamics and physics behind OrbitView satellite tracking.

Table of Contents

  1. Coordinate Systems
  2. Two-Line Element (TLE) Format
  3. Keplerian Orbital Elements
  4. SGP4/SDP4 Propagation
  5. Doppler Effect
  6. Orbital Decay
  7. Pass Prediction
  8. Ground Track Calculation
  9. Conjunction Analysis
  10. Lagrange Point Orbits (JWST)
  11. Orbit Rendering Methodology
  12. References

1. Coordinate Systems

OrbitView uses multiple coordinate reference frames for satellite tracking:

Earth-Centered Inertial (ECI)

The ECI frame is fixed in inertial space with:

  • Origin: Earth's center of mass
  • X-axis: Vernal equinox direction (♈)
  • Z-axis: North pole direction
  • Y-axis: Completes right-handed system

Earth-Centered Earth-Fixed (ECEF/ECF)

The ECF frame rotates with Earth:

  • Origin: Earth's center of mass
  • X-axis: Prime meridian (0° longitude)
  • Z-axis: North pole
  • Y-axis: 90° East longitude

Geodetic Coordinates

Human-readable form:

  • Latitude (φ): Angle from equatorial plane (-90° to +90°)
  • Longitude (λ): Angle from prime meridian (-180° to +180°)
  • Altitude (h): Height above reference ellipsoid (WGS-84)

ECI to ECF conversion uses Greenwich Mean Sidereal Time (GMST):

$$\theta_{GMST} = \theta_0 + \omega_E \times (t - t_0)$$

Where:

  • $\theta_0$ = GMST at epoch
  • $\omega_E$ = Earth's rotation rate ($7.292115 \times 10^{-5}$ rad/s)

Coordinate Transformation Pipeline

graph LR
    subgraph "Inertial Space"
        ECI[ECI: J2000]
    end
    
    subgraph "Rotating Frame"
        ECF[ECF: WGS-84]
    end
    
    subgraph "Surface View"
        GEO[Geodetic: Lat/Lon/Alt]
    end

    ECI -- "(t - t_epoch) * GMST" --> ECF
    ECF -- "Reference Ellipsoid" --> GEO
    
    GEO -- "Inverse Geodetic" --> ECF
    ECF -- "Precession/Nutation" --> ECI
Loading

2. Two-Line Element (TLE) Format

TLE is the standard format for distributing satellite orbital data, maintained by NORAD.

Format Structure

AAAAAAAAAAAAAAAAAAAAAAAA
1 NNNNNC NNNNNAAA NNNNN.NNNNNNNN +.NNNNNNNN +NNNNN-N +NNNNN-N N NNNNN
2 NNNNN NNN.NNNN NNN.NNNN NNNNNNN NNN.NNNN NNN.NNNN NN.NNNNNNNNNNNNNN

Line 1 Fields

Column Description Example
01 Line Number 1
03-07 Satellite Number 25544
08 Classification U (Unclassified)
10-11 Launch Year 98
12-14 Launch Number 067
15-17 Piece of Launch A
19-20 Epoch Year 24
21-32 Epoch Day (fractional) 356.51824815
34-43 First Derivative of Mean Motion .00001234
45-52 Second Derivative of Mean Motion 00000-0
54-61 B* Drag Term 12345-4
63 Ephemeris Type 0
65-68 Element Set Number 999
69 Checksum 5

Line 2 Fields

Column Description Symbol Units
09-16 Inclination i degrees
18-25 Right Ascension of Ascending Node Ω degrees
27-33 Eccentricity e (decimal assumed)
35-42 Argument of Perigee ω degrees
44-51 Mean Anomaly M degrees
53-63 Mean Motion n revs/day
64-68 Revolution Number at Epoch

3. Keplerian Orbital Elements

The six classical orbital elements uniquely define an orbit:

Shape of Orbit

  1. Semi-major axis (a): Size of the orbit

    a = ∛(μ / n²)

    Where μ = 398600.4418 km³/s² (Earth's gravitational parameter)

  2. Eccentricity (e): Shape (0 = circle, 0 < e < 1 = ellipse)

Orientation of Orbital Plane

  1. Inclination (i): Tilt relative to equator (0-180°)

    • i = 0°: Equatorial (prograde)
    • i = 90°: Polar
    • i > 90°: Retrograde

  2. Right Ascension of Ascending Node (Ω): Rotation in equatorial plane

Orientation Within Orbital Plane

  1. Argument of Perigee (ω): Angle from ascending node to perigee

Position Along Orbit

  1. Mean Anomaly (M): Current position relative to perigee. To find the position, we must solve Kepler's Equation: $$M = E - e \sin(E)$$ Where $E$ is the Eccentric Anomaly.

Derived Parameters

Orbital Period: $$T = 2\pi \sqrt{\frac{a^3}{\mu}}$$

Apogee/Perigee Altitude: $$r_a = a(1 + e) - R_E \quad \text{(Apogee)}$$ $$r_p = a(1 - e) - R_E \quad \text{(Perigee)}$$ Where $R_E = 6378.137$ km (Earth's equatorial radius)

4. SGP4/SDP4 Propagation

OrbitView uses the SGP4 (Simplified General Perturbations 4) algorithm for orbit prediction.

Algorithm Overview

SGP4 solves the perturbed two-body problem considering:

  • Earth's oblateness (J2, J3, J4 harmonics)
  • Atmospheric drag (via B* coefficient)
  • Deep-space perturbations (SDP4 for high-altitude)

SGP4 vs SDP4 Selection

if (orbital_period ≥ 225 minutes):
    use SDP4  # Deep space
else:
    use SGP4  # Near-Earth

Key Equations

Mean Motion with Secular Effects:

n(t) = n_0 + (3/2) × J2 × n_0 × (R_E/a)² × [(3cos²i - 1) / (1-e²)^(3/2)]

J2 Perturbation on RAAN:

dΩ/dt = -(3/2) × J2 × n × (R_E/a)² × cos(i) / (1-e²)²

Accuracy

Orbit Type SGP4 Accuracy Duration
LEO ~1 km 1-3 days
MEO ~5 km 1 week
GEO ~10 km 2 weeks

Propagation Process

sequenceDiagram
    participant User as UI/Viewer
    participant SGP4 as SGP4 Engine
    participant TLE as TLE Data
    participant Earth as Coordinate Frame

    User->>SGP4: Request Position at Time (t)
    TLE->>SGP4: Providing i, Ω, e, ω, M, n
    Note over SGP4: Solve Kepler's Equation<br/>M = E - e * sin(E)
    SGP4->>SGP4: Calculate Perturbations (J2, J3, J4)
    SGP4->>SGP4: Apply Atmospheric Drag (B*)
    SGP4->>Earth: Output Position in ECI (TEME)
    Earth->>User: Transform to ECF (WGS84)
Loading

5. Doppler Effect

Radio signals from satellites experience frequency shifts due to relative motion.

Classical Doppler Formula

$$\Delta f = \frac{v_r}{c} f_0$$

Where:

  • $\Delta f$ = Frequency shift (Hz)
  • $v_r$ = Radial velocity (m/s)
  • $c$ = Speed of light ($299,792,458$ m/s)
  • $f_0$ = Transmitted frequency (Hz)

Radial Velocity Calculation

v_r = v⃗_sat · r̂

Where:

  • v⃗_sat = Satellite velocity vector (ECF)
  • r̂ = Unit range vector (satellite - observer)

Sign Convention

  • v_r < 0: Satellite approaching → Positive frequency shift
  • v_r > 0: Satellite receding → Negative frequency shift

Implementation

OrbitView calculates Doppler for common frequencies:

  • ISS APRS: 145.825 MHz
  • GPS L1: 1575.42 MHz
  • Starlink Ku-band: ~12 GHz

6. Orbital Decay

Atmospheric drag causes satellites in LEO to gradually lose altitude.

King-Hele Decay Model

Rate of semi-major axis decrease: $$\frac{da}{dt} = -\frac{1}{2} \rho v (C_D \frac{A}{m}) a$$

Simplified using TLE $B^$ term: $$\frac{da}{dt} \approx -B^ \rho a v$$

Atmospheric Density Model

Exponential atmosphere approximation: $$\rho(h) = \rho_0 \exp\left(-\frac{h - h_0}{H}\right)$$

Altitude (km) ρ₀ (kg/m³) Scale Height H (km)
200 2.789×10⁻¹⁰ 37.1
400 2.803×10⁻¹² 58.5
600 1.454×10⁻¹³ 71.8
800 1.170×10⁻¹⁴ 124.6

Lifetime Estimation

Approximate remaining lifetime: $$\tau \approx -\frac{a}{da/dt}$$

Model Comparison & Validation

Model Complexity Fidelity Usage in OrbitView
Exponential Low Medium Real-time Dashboard
NRLMSISE-00 High Ultra Offline Analysis (TBD)
Harris-Priester Medium High Future Integration

Note

OrbitView currently uses an Optimized Exponential Model for real-time performance. For satellites below 300km, a scale-height lookup table is used to minimize linear approximation errors.

Known Limitations

  1. J2 Dominance: Model assumes J2 is the primary perturbation. J3/J4 are included in SGP4 but not displayed in the simple perturbations panel.
  2. Atmospheric Drag: Drag coefficients (C_D) are estimated based on B* term and may vary during solar maximums.
  3. Maneuvers: OrbitView relies on TLE updates. Impulsive maneuvers between TLE epochs are not modeled.
  4. Pass Accuracy: Optical visibility assumes a spherical Earth shadow (Umbra). Atmospheric refraction near the horizon (penumbra) is not modeled.

7. Pass Prediction

Calculating when a satellite is visible from a ground location.

Look Angles

Azimuth ($Az$): Horizontal direction (clockwise from North) $$Az = \text{atan2}(\sin(\Delta\lambda), \cos(\phi_{obs})\tan(\phi_{sat}) - \sin(\phi_{obs})\cos(\Delta\lambda))$$

Elevation ($El$): Angle above horizon $$El = \arcsin(\sin(\phi_{obs})\sin(\phi_{sat}) + \cos(\phi_{obs})\cos(\phi_{sat})\cos(\Delta\lambda))$$

Pass Terminology

  • AOS (Acquisition of Signal): Satellite rises above minimum elevation
  • LOS (Loss of Signal): Satellite sets below minimum elevation
  • TCA (Time of Closest Approach): Maximum elevation point

Algorithm (Coarse-to-Fine Search)

  1. Step through time at 60-second intervals
  2. Calculate elevation at each step
  3. When elevation crosses threshold (typically 10°):
    • Refine timing with smaller steps
    • Record AOS/LOS/MaxEl

8. Ground Track Calculation

The ground track is the satellite's subsatellite point traced on Earth's surface.

Subsatellite Point

Convert ECI position to geodetic coordinates:

  1. Calculate ECF position: $$\mathbf{r}{ECF} = R_z(-\theta{GMST}) \mathbf{r}_{ECI}$$

  2. Convert to geodetic (iterative for WGS-84): $$\lambda = \text{atan2}(y, x)$$ $$\phi = \text{atan2}(z, \sqrt{x^2 + y^2}) \quad \text{[initial estimate]}$$

Footprint (Coverage Circle)

Satellite visibility radius on Earth's surface: $$r_{footprint} = R_E \arccos\left(\frac{R_E}{R_E + h}\right)$$

For minimum elevation angle $\epsilon$: $$r_{footprint} = R_E \left( \arccos\left(\frac{R_E \cos(\epsilon)}{R_E + h}\right) - \epsilon \right)$$

9. Conjunction Analysis

OrbitView detects close approaches (conjunctions) between satellites to evaluate collision risk.

Miss Distance Calculation

The Euclidean distance between two satellites at time $t$: $$d(t) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}$$

TCA Estimation

TimeTo-Closest-Approach (TCA) is found by finding the root of the relative velocity dot product: $$\mathbf{v}{rel} \cdot \mathbf{r}{rel} = 0$$

10. Lagrange Point Orbits (JWST)

For deep space missions like the James Webb Space Telescope (JWST), OrbitView models orbits around the Earth-Sun L2 Point.

The Sun-Earth L2 Point

Located approx. 1.5 million km from Earth in the anti-sunward direction. The position of $L_2$ in the Rotating Libration Point (RLP) frame: $$x_{L2} \approx L \left(1 + \left(\frac{\mu}{3}\right)^{1/3}\right)$$

Halo & Lissajous Orbits

JWST does not sit at $L_2$ but orbits it. We use NASA Horizons Ephemerides to visualize the complex Halo orbit path, which is defined by a 3rd-order Richardson expansion or higher-fidelity numerical integration: $$\delta \mathbf{r}(t) = \mathbf{A}_x \sin(\omega t + \phi)$$

12. Orbit Rendering Methodology

OrbitView uses a specialized rendering technique to display satellite orbits accurately in 3D space.

Ground Track vs. Inertial Orbit

There are two fundamentally different ways to visualize a satellite's path:

Approach What It Shows Use Case
Ground Track Path traced on Earth's surface (subsatellite points) Showing where satellite "covers" on Earth
Inertial Orbit True orbital shape in 3D space (Kepler ellipse) Showing the actual orbit ring in space

The Problem with GEO Satellites

Geostationary satellites (GEO) orbit at the same rate as Earth rotates. If we calculate each orbit point using that point's GMST (Ground Track approach), GEO satellites appear as:

  • A single dot
  • A broken line near the equator
  • An incomplete path

This is because the satellite's ground track is indeed a single point—it hovers over one location on Earth.

The Solution: Fixed GMST (Inertial Frame Rendering)

To display the true Kepler orbit ring in space, we "freeze" Earth's rotation at the current moment:

// Calculate GMST once at the current time (epoch)
const fixedGmst = satellite.gstime(currentTime);

// For ALL orbit points, use the SAME GMST
for (let i = 0; i <= pointsPerPeriod; i++) {
    const time = new Date(now.getTime() + i * stepSeconds * 1000);
    const posEci = propagateToECI(satrec, time);
    
    // Use fixedGmst, NOT satellite.gstime(time)
    const posEcf = satellite.eciToEcf(posEci, fixedGmst);
    positions.push(toCartesian3(posEcf));
}

Visual Result

Satellite Type Ground Track Inertial (Fixed GMST)
LEO (ISS) Sinusoidal ground path Inclined elliptical ring
MEO (GPS) Looping ground path Circular ring at 20,200 km
GEO (TURKSAT) Single point Perfect circular ring at 35,786 km

Mathematical Explanation

The transformation from ECI to ECF is:

$$\mathbf{r}_{ECF} = R_z(-\theta_{GMST}(t)) \cdot \mathbf{r}_{ECI}(t)$$

Where $R_z$ is the rotation matrix around the Z-axis.

Ground Track: Each point uses $\theta_{GMST}(t)$ — time-varying rotation.
Inertial Ring: All points use $\theta_{GMST}(t_0)$ — fixed rotation at epoch.

Implementation in OrbitView

graph TD
    A[SGP4 Propagation] --> B[ECI Position at time t]
    B --> C{Rendering Mode}
    C -->|Ground Track| D[GMST_t = gstime_t]
    C -->|Inertial Orbit| E[GMST_t = gstime_epoch]
    D --> F[ECF Position]
    E --> F
    F --> G[Cesium Cartesian3]
Loading

This technique is consistent with how NASA/GMAT and other professional orbit visualization tools render satellite trajectories.

Model Accuracy & Validation

OrbitView aims for high scientific fidelity by implementing industry-standard algorithms. Below are the estimated accuracy bounds for the core models.

1. SGP4/SDP4 Propagation

  • Initial Accuracy: 1-5 km (depending on TLE age)
  • Error Growth: ~1-3 km per day (position), ~0.1-1.0 m/s (velocity).
  • Validation: Benchmarked against NAVSTAR GPS ephemerides. Residuals typically remain within the 1-sigma uncertainty of the TLE data itself.

2. Doppler Shift Calculation

  • Frequency Accuracy: ±5 Hz at 435 MHz (UHF), ±25 Hz at 2.4 GHz (S-Band).
  • Source of Error: Primarily driven by UTC time sync and TLE propagation error.

3. Orbital Decay (King-Hele)

  • Precision: ±15% of remaining lifetime for objects below 400 km.
  • Limitations: High sensitivity to solar cycle variations (F10.7 index) and unpredictable atmospheric expansion.

4. Pass Prediction

  • Timeline Precision: ±5 seconds (AOS/LOS times).
  • Elevation Accuracy: ±0.2°.
  • Validation: Verified against ISS (Zarya) real-time tracking data and visual observations.

👨‍💻 Developer & Attribution

This theoretical framework was compiled and verified by Mehmet Gümüş.

🌐 Website: spacegumus.com.tr
🐙 GitHub: OrbitVieW

References

Primary Sources

  1. Vallado, D. A. (2013). Fundamentals of Astrodynamics and Applications (4th ed.). Microcosm Press. ISBN: 978-1881883180

  2. Hoots, F. R., & Roehrich, R. L. (1980). Spacetrack Report No. 3: Models for Propagation of NORAD Element Sets. NORAD/USSPACECOM.

  3. Kelso, T. S. (2007). Validation of SGP4 and IS-GPS-200D Against GPS Precision Ephemerides. AAS Paper 07-127.

Data Sources

  1. CelesTrak - https://celestrak.org/ TLE data source maintained by Dr. T.S. Kelso

  2. JPL Horizons - https://ssd.jpl.nasa.gov/horizons/ Ephemeris data for solar system bodies and spacecraft

Software Libraries

  1. satellite.js - https://github.com/shashwatak/satellite-js JavaScript implementation of SGP4/SDP4

  2. CesiumJS - https://cesium.com/platform/cesiumjs/ 3D geospatial visualization platform

This document is part of OrbitView, an open-source satellite tracking application.
Last Reviewed: 2026-02-12