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Case Study: The Lost Signal

Problem Statement

A nominal GPS satellite (Slot A1) has been launched into a Medium Earth Orbit (MEO). Your mission is to plan ground station contacts for The Aerospace Corporation (El Segundo, CA) over the next 72 hours.

You need to answer the following questions: 1. When can we talk to it? Identify all contact windows where the satellite is at least 5 degrees above the horizon. 2. When is the signal strongest? Find the time and value of the maximum elevation angle. 3. How far off-axis are we? At the moment of maximum elevation, assuming the satellite is pointing directly at the center of the Earth (Nadir), what is the angle between the satellite's boresight and the ground station?

Simulation Parameters

  • Orbit: GPS Nominal
    • Semi-Major Axis: 26,559.7 km
    • Inclination: 55.0 deg
    • Eccentricity: 0.0 (Circular)
    • Start Time: GPS Week 2294, 0.0 s
  • Ground Station:
    • Latitude: 33.9167° N
    • Longitude: 118.4167° W

Mathematical Background

To solve this, we rely on coordinate frames and transformations provided by gps_frames.

1. Orbit Propagation (Kepler -> ECI)

We start with Keplerian Orbital Elements (\(a, e, i, \Omega, \omega, M\)). While the physics of propagation are standard, we wrap the result in a gps_frames.position.Position object. This ties the coordinates to a specific frame ("ECI") and time (GPSTime).

from gps_frames.position import Position
from gps_frames.rotations import Rotation

# ... calculate r_pqw ...
orbit_rotation = Rotation(dcm=dcm_combined)
r_eci_coords = orbit_rotation.rotate(r_pqw)

# Wrap in a Position object
sat_pos_eci = Position(r_eci_coords, current_time, "ECI")

2. Time Control (GPSTime Arithmetic)

The gps_time.GPSTime objects support direct addition of seconds (as float or int), making simulation loops extremely clean:

# Direct arithmetic with seconds
current_time = base_time + t_sec

3. Automatic Frame Transformations

Once the satellite is in a Position object, we can transform it to the Earth-Fixed (ECEF) frame without manually calculating Sidereal Time. The library handles the time-dependent rotation internally through the get_position method.

# The library handles the ECI -> ECEF rotation for current_time
sat_pos_ecef = sat_pos_eci.get_position("ECEF")

4. Topocentric Horizon Frame (ENU Basis)

To compute Azimuth and Elevation, we project the satellite position into the station's local East-North-Up (ENU) frame using the gps_frames.basis.Basis object.

from gps_frames.basis import Basis, coordinates_in_basis

# Define the local ENU basis at the station
enu_basis = Basis(station_ecef, u_east, u_north, u_up)

# Project the satellite position into this basis directly!
rho_enu = coordinates_in_basis(sat_pos_ecef, enu_basis)

Note: Azimuth is calculated using arctan2(East, North), resulting in a range of -180° to +180°, which avoids wrapping discontinuities on plots.

Solution Script

The refactored solution is implemented in examples/case_study_satellite_tracking.py. It demonstrates how to leverage high-level objects to minimize manual coordinate bookkeeping.

Running the Code

Ensure you have the optional dependencies installed: 

pip install .[examples]
python examples/case_study_satellite_tracking.py

Expected Output

The script will output contact times and angles. Example snippet: 

================================================================
            SATELLITE TRACKING CASE STUDY (OBJECT-ORIENTED)     
================================================================

Simulating 72 hours...

Simulation Results:
Total passes observed: 6
Max Elevation: 68.87 deg at T+8.17h
Off-Boresight Angle at Peak: 13.6053 deg

Contact Windows (>5 deg):
  Pass 01:  5.83h to 10.52h (Duration: 281.0 min)
  Pass 02: 17.80h to 22.48h (Duration: 281.0 min)
  Pass 03: 29.77h to 34.45h (Duration: 281.0 min)
  Pass 04: 41.73h to 46.42h (Duration: 281.0 min)
  Pass 05: 53.70h to 58.38h (Duration: 281.0 min)
  Pass 06: 65.67h to 70.35h (Duration: 281.0 min)

visualization plots will also be generated showing the Azimuth and Elevation profiles for all visible passes.