Forward: I am not writing this guide because I am an expert, quite the contrary. I am writing this guide because I am learning myself. This is made with more AI than I am used to using. However, the information is accurate, and if it helps you, too, good.
There are three primary methods for tracking known space objects. The most commonly used method is the Two-Line Element Set (TLE), which provides orbital parameters in a concise format, enabling orbit prediction. An Element Set (Elset) is similar to a TLE and also facilitates trajectory determination. Lastly, state vectors, which include an object’s position and velocity components at a given time, are crucial for calculating current states and planning orbital maneuvers. Ephemeris data is an array of state vectors used for accurate positioning. Orbital Determination (OD) combines all of the above data along with sensor data, error checking, and more. There is an example of an OD at the bottom of the document.
A TLE (Two-line element set) consists of two lines of text following a title line that often includes the satellite’s name. TLEs were designed in the early 60’s and represent the most information one could fit on an IBM punch card. Each line of the TLE provides specific orbital elements as follows:
ISS Zarya:
1 25544U 98067A 23001.50000000 .00016717 00000+0 10270-3 0 9000
2 25544 51.6456 357.2560 0004907 276.4237 63.8597 15.49544077 91916
Line 1:
- Satellite Catalog Number: A unique identifier for the object Resident Space Object (RSO) assigned by Norad
- International Designator: Launch year, launch number of that year, and a piece code denoting the piece of the launch (e.g., the primary satellite or a piece of debris).
- Epoch Time: The Julian Date and fractional portion of the day for which the elements are valid.
- First Derivative of the Mean Motion: Reflects the rate of change of the satellite’s orbital period, mainly due to atmospheric drag. Often referred to as ṅ
- Second Derivative of the Mean Motion: Usually, a small number that indicates the rate of change of the first derivative over time. Noted as a n with two dots n¨
where Δn˙is the change in the first derivative of the mean motion, and Δt is the time over which this change occurs.
Positive n¨: Indicates that the rate at which the satellite’s orbital speed is decreasing is itself increasing. This typically happens when atmospheric drag or other decelerative forces are strengthening.
Negative n¨: Suggests that the rate at which the satellite’s orbital speed is decreasing is reducing, possibly indicating a lessening of atmospheric drag or moving to a slightly higher and less dense part of the atmosphere.
- Drag Term (B coefficient): The Ballistic Coefficient is a crucial parameter used in orbital mechanics to describe the effects of atmospheric drag on a satellite or any space object in Low-Earth Orbit (LEO).
- Ephemeris Type: Typically set to a default value (usually 0).refers to the specific model used to interpret the TLE data for orbital prediction and satellite tracking.
- Element Set Number: a sequential identifier that indicates the version of the TLE data for a specific satellite. This number helps users identify the most recent update or version of the orbital data provided for any given satellite.
- Checksum: A modulo 10 sum used for error checking, calculated from all preceding digits in the line. The checksum method used in TLEs provides only a basic level of error detection and is not foolproof. It can catch simple errors involving alterations of a single digit or the addition or removal of digits.
Line 2:
- Satellite Catalog Number: Repeated for consistency.
- Inclination (degrees): The orbit plane’s angle relative to the Earth’s equatorial plane. The inclination is measured at the orbit’s ascending node, which is where the satellite crosses the equatorial plane from south to north.
- Right Ascension of Ascending Node (RAAN): The angle from a fixed direction in space (the vernal equinox) to the point where the satellite crosses the equatorial plane going northward.
- Eccentricity: A decimal point followed by a 7-digit number describing the orbit’s shape. It is a dimensionless number that quantifies how much an orbit deviates from being circular. Eccentricity (eee) ranges from 0 to 1 for elliptical orbits, equals 1 for a parabolic trajectory, and is greater than 1 for hyperbolic trajectories.
e=0: A perfectly circular orbit.
0<e<1: An elliptical orbit, with the level of ellipticity increasing as eee approaches 1.
e=1 A parabolic escape orbit is the energy needed for an object to escape the gravitational pull of another object without returning.
e>1 A hyperbolic orbit, indicating that the object is on an escape trajectory and will not return.
Eccentricity can be calculated from the orbital state vectors (position and velocity) at any point in the orbit. The formula for eccentricity eee in vector form is:
v is the velocity vector of the orbiting body,
r is the position vector from the central body to the orbiting body,
h is the specific angular momentum vector(h⃗=r⃗×v⃗)
μ is the standard gravitational parameter of the central body. - Argument of Perigee (ω): The angle from the ascending node to the orbit’s point of closest approach to the Earth (perigee). Along with the inclination, the right ascension of the ascending node (RAAN), eccentricity, and mean anomaly, the argument of perigee helps fully describe the shape and orientation of an orbit around Earth or any other celestial body.
After calculating the eccentricity vector (e) as above, compute the node vector which points towards the ascending node. N=k×h
K is the unit vector in the direction of the Earth’s north pole.
The argument of perigee is then the angle between N and e, which can be calculated using the dot product:
and adjusted for the correct quadrant with the cross-product.
- Mean Anomaly (degrees): orbital element used to describe the position of a satellite or any celestial body along its orbit at a specific time, relative to perigee (the closest point to the central body in an elliptical orbit). It is one of the six orbital elements (Keplerian elements) used to define the motion of a body in space.
M=M0+n×(t−t0)
M is the mean anomaly at time t.
M0 is the mean anomaly at the epoch t0.
n is the mean motion.
t is the current time.
t0 is the epoch time.
The mean anomaly is crucial for determining the body’s position along its orbit at a given time. It is essential for satellite tracking, mission planning, and navigation. The mean anomaly helps define the position within the orbit, particularly in calculating the true anomaly and the distance from the central body using Kepler’s equation. - Mean Motion (revolutions per day): How many times the satellite orbits the Earth in one day.
- Revolution Number at Epoch: How many times has the satellite orbited the Earth since launch, up to the epoch
- Checksum: For error checking, as in Line 1.
A new version of the TLE called TLE-XP has more data elements, including an Improved Atmospheric Drag Model, Solar Radiation Pressure (SRP), Higher-Fidelity Gravity Models, Empirical Corrections, and Numerical Integration Techniques. These new factors will be integrated into the taplab’s ways and means within the next few months.
“Elset,” or element set, is a data format used to describe the orbit of an object in space, typically a satellite. These sets of orbital elements are most commonly provided in a format known as the Two-Line Element set (TLE), widely used for satellite tracking and space domain awareness. The TLE format is compact and contains all the necessary information to describe a satellite’s orbit and predict its future position. TLEs are specific to Earth’s orbit, where Elsets can be used for other orbital bodies in Astrophysics.
In addition to the standard TLE functions, Elsets can include classical orbital elements like semi-major axis, eccentricity, inclination, longitude of ascending node, argument of perigee, and true anomaly.
Name: ISS (ZARYA)
Catalog Number: 25544
Epoch (UTC): 2023-001.50000000
Semi-major Axis (a): 6796.20 km
Eccentricity (e): 0.0004907
Inclination (i): 51.6456 degrees
Longitude of Ascending Node (Ω): 357.2560 degrees
Argument of Perigee (ω): 276.4237 degrees
True Anomaly (ν): 63.91 degrees
Mean Motion: 15.49544077 rev/day
Drag Term (Bstar): 10270-3
Orbital Period: Approximately 92.65 minutes
A state vector in orbital mechanics represents the position and velocity of an object in space at a specific time. This vector is crucial for understanding where the object is and how it moves through its orbit. There are 6 specific orbital elements. Three for the position coordinates (x, y, z) and three for the velocity components (velocity in the x-direction, y-direction, and z-direction). State Vectors use the center of a planetary body, which is the Earth’s center in most cases. They can be used directly in trajectory analysis and orbital maneuver calculations without an orbital model to interpret them.
ISS Zarya:
Position Vector (x, y, z)
- x: -5831.67 km
- y: 1989.36 km
- z: -2863.99 km
Velocity Vector (v_x, v_y, v_z)
- v_x: -3.92 km/s
- v_y: -4.20 km/s
- v_z: 5.06 km/s
The Math for calculating a State Vector is rather complex.
Step 1: Calculate the Position and Velocity in the Perifocal Coordinate System
- Calculate the distance (r) from the central body (Earth) using the formula:
Where:
a is the semi-major axis,
e is the eccentricity,
ν is the true anomaly.
- Determine position coordinates (x, y) in the perifocal plane: r⋅cos(ν) yp=r⋅sin(ν)y_p = r \cdot \sin(\nu)yp=r⋅sin (v)
zp is zero because the perifocal plane is two-dimensional. - Compute the velocity components (v_x, v_y) using the vis-viva equation:
(transverse component) is zero, mirroring the two-dimensional nature of the plane.
Step 2: Transform to Geocentric Equatorial Coordinate System
- Apply rotations for the three orbital angles (argument of perigee ω omegaω, inclination i, and right ascension of the ascending node Ω) using rotation matrices:
Rotate around the z-axis by −ω
Rotate around the x-axis by −i
Rotate around the z-axis by −Ω
Combine the rotation matrices and apply them to the position and velocity vectors calculated in the perifocal plane:
This process converts the position and velocity from the orbit plane (perifocal) into a three-dimensional Earth-centered inertial frame, providing the comprehensive state vector for tracking the satellite’s current position and velocity relative to Earth.
Time measurement:
J2000 refers to a specific epoch used in astronomy and astrodynamics as a standard reference time. It marks the Julian Date of January 1, 2000, at 12:00 Terrestrial Time (TT), which is equivalent to January 1, 2000, 11:58:55.816 UTC. This epoch serves as a baseline for many calculations involving the positions and motions of celestial bodies, including stars, planets, and artificial satellites.
Terrestrial Dynamical Time (TDT) or Terrestrial Time (TT) is a uniform, dynamical time scale used primarily in high-precision astronomical and orbital calculations. For Earth satellites, TT is preferred because it avoids the irregularities inherent in time scales tied to the actual rotation of the Earth, such as UT1, and aligns directly with the ephemerides used in celestial mechanics.
Barycentric Dynamical Time (TDB) is a uniform time scale referenced to the solar system’s center of mass. It is essential for high-precision astronomy, mainly when dealing with the dynamics of bodies in the solar system.
TEME (True Equator, Mean Equinox) is a coordinate system used primarily to represent artificial Earth satellites’ orbital positions and velocities. It is widely used in conjunction with the Two-Line Element set (TLE) data for satellite tracking.
More Definitions
Six orbital elements The six orbital elements defined as Kepler’s laws of planetary motion are a set of parameters that fully characterize the size, shape, orientation, and position of an orbiting body in three-dimensional space. They are used to describe satellite or planetary orbits. 1. Length of the semi-major axis – half the diameter of the orbit
2. Eccentricity – shape of orbit
3. Inclination – degree of orbit
4. Right Ascension of the Ascending Node
5. Arguement of Perigee
6. True Anomaly
Stepped output The processing, updating, or alerting in discrete increments is very relevant to how data is handled in this field. Each step represents a snapshot or a decision point in the continuous monitoring and management 4
Ephrem list is a detailed compilation of ephemeris data for one or more celestial bodies over a set period, state vectors. Ephemeris data, as noted earlier, includes precise information about the positions and velocities of objects in space at various times.
State Estimation is the art and science of continuously updating our understanding of an object’s position and movement by merging noisy observations with a mathematical model of its dynamics.
Linearization is a powerful method to simplify the analysis and computation of complex, nonlinear systems by approximating them with linear models around a chosen operating point in small regions.
Two-Body Problem deals with predicting the motion of two masses under their mutual gravitational influence. With Newton’s laws, conservation of angular momentum and energy, and Kepler’s laws, the problem can be reduced to a simpler one-body equation that describes the relative motion. This fundamental problem illuminates the dynamics of simple orbital systems and serves as a stepping stone for analyzing more complex gravitational interactions in astronomy and space missions.
Time of periapsis passage is a critical parameter in orbit determination that marks the exact moment an orbiting object reaches its closest point to the central body. When combined with other orbital elements, this timing information allows for precise tracking and prediction of the object’s future positions, enabling effective mission planning and operations.
Heliocentric ecliptic – Origin at the center of the sun.
A geocentric equatorial reference frame is a coordinate system centered on the Earth, with the Earth’s equator as its base plane. It is extensively used for satellite tracking and astronomical observations because it aligns well with the Earth’s natural orientation and simplifies the description of orbital elements.
Geopotential model is a mathematical representation of Earth’s gravitational potential. The potential energy per unit mass due to Earth’s gravity in a form that accurately captures the complexities of Earth’s shape and mass distribution. Because Earth is not a perfect sphere but an oblate and irregular body with variations in density and topography
Declination is analogous to latitude on Earth. It measures the angular distance of an object north or south of the celestial equator, with the origin at the Earth’s center or point on the surface.
Two-body Orbital Dynamics provides an analytical description of how two objects move under mutual gravitational attraction. The resulting equations and conservation laws describe the orbit’s shape, orientation, and period, typically resulting in conic sections (ellipses for bound orbits, hyperbolas or parabolas for unbound motions).
Newton’s law of universal gravitation describes the force of attraction between two masses as proportional to the product of their masses and inversely proportional to the square of the distance between them.
Atmospheric Resistance is the force that opposes an object’s motion as it travels through the Earth’s atmosphere. It depends on air density, the object’s velocity, cross-sectional area, and drag coefficient. This resistance is central to many practical applications.
Orbital Precession describes the gradual rotation of the orbit of a celestial body. In other words, even though a satellite or planet may follow an elliptical path, the orientation of that ellipse can slowly rotate over time within its orbital plane.
Nutation refers to the small, periodic oscillations in the orientation of a celestial body’s rotational axis. On Earth, nutation is superimposed on the much slower precession of the equinoxes and is primarily driven by the gravitational interactions with the Moon and Sun.
Rotation is a key property of planets, stars, and galaxies in celestial mechanics. It influences everything from day-night cycles and weather on a planet to the observed phenomena in stars and galaxies.
Orbital Accuracy is a critical measure of how well a computed orbital model represents the actual path of a satellite or celestial body. It depends on high-quality observational data, precise dynamic models, robust numerical methods, and careful error management.
Force model Accuracy measures how well the mathematical models of all forces acting on an object represent the true accelerations experienced.
Statistical Orbit Determination is a sophisticated process that merges observational data with dynamical models through statistical estimation. Accounting for noise and uncertainties inherent in both measurements and modeling not only yields the best estimate of an orbit but also quantifies its accuracy.
The polar motion component quantifies the displacement of Earth’s instantaneous rotational pole along the axis of a terrestrial reference frame. It is one of the key Earth Orientation Parameters used to correct and transform positional data between the Earth-fixed and space-fixed coordinate systems.
Optical Sensors:
Photodiodes and phototransistors are basic optical sensors that detect light intensity and are commonly used for presence detection or brightness control. Charge-coupled devices (CCDs) are employed in cameras and telescopes for high-quality imaging, capturing light, and converting it into digital data. They are especially valued in astronomy and satellite tracking due to their sensitivity and accuracy. Complementary Metal-Oxide-Semiconductor (CMOS) sensors, similar to CCDs but generally more cost-effective and power-efficient, are prevalent in digital cameras, smartphones, and complex imaging systems in satellites. Spectrometers, which analyze the spectrum of light emitted, absorbed, or scattered by objects, are crucial in scientific research for determining the properties of stars, planets, and other celestial bodies.
Optical Sensor Data Variables
- Timestamp: The exact date and time the observation was recorded, usually in UTC.
- Right Ascension (RA): Celestial longitude is used to pinpoint an object’s location in the sky, typically given in hours, minutes, and seconds.
- Declination (Dec): Celestial latitude describes the vertical angle of an object in the sky, typically given in degrees, arcminutes, and arcseconds.
- Brightness (Magnitude): An object’s apparent brightness can provide insights into its size and surface properties.
- Exposure Time: The duration for which the sensor’s detector was exposed to light, affecting the image’s brightness and clarity.
- Image Data: This is the actual pixel data from the captured image, which may require processing to enhance visibility and extract meaningful information.
- Pixel Scale: The angular size represented by each pixel, usually expressed in arcseconds per pixel, crucial for precise positional measurements.
- Filter Type: The specific wavelength or band of light used during observation, such as visible, infrared, or ultraviolet, which can affect what features are visible.
- Air Mass is the path length that light from the observed object travels through Earth’s atmosphere; it affects observations and is important for calibrations.
- Seeing Conditions: This is a measure of the blurring and twinkling of astronomical objects caused by the atmosphere; it impacts the quality of the data.
- Sky Background: The level of ambient light in the background, including light pollution or natural sky glow, can affect the contrast of the object.
- Astrometric Calibration: Corrections are applied to the raw data to account for instrumental and atmospheric distortions, improving the positional accuracy.
- Photometric Calibration is adjustments made to measure the brightness of objects accurately, accounting for sensor sensitivity and atmospheric conditions.
- CCD Temperature: The charge-coupled device (CCD) temperature during the observation can affect the noise level and quality of the image.
Radar Sensors Parabolic, Phaseed Array, Dipole, Yagi-Uda Antennas using Signal Operations. (SIGINT)
- Timestamp: The precise time when the observation was made.
- Range: The distance to the object from the radar station.
- Azimuth: Horizontal angle relative to a reference direction (typically true north).
- Elevation: Vertical angle above the horizon.
- Doppler Shift: Change in frequency of the radar signal due to the relative motion of the target.
- Radar Cross Section (RCS): Measure of how detectable an object is with radar.
- Signal-to-Noise Ratio (SNR): The strength of the radar return signal relative to background noise.
- Polarization: The orientation of the radar wave’s electric field.
- Velocity: Calculated from Doppler Shift, indicating how fast the object moves towards or away from the radar.
Key Variables in Radio Signal Tracking
- Frequency: The specific frequency or range of frequencies over which the signals are transmitted.
- Bandwidth: The width of the frequency band used by the signal.
- Modulation Type: The method used to encode information in the signal, such as frequency modulation (FM), amplitude modulation (AM), or phase modulation (PM).
- Signal Strength: The power level of the received signal.
- Polarization: The orientation of the electromagnetic wave’s oscillations.
- Time Stamp: Exact time of signal detection.
- Direction (Azimuth and Elevation): The angle from which the signal is received is useful for determining its origin.
Satellite Laser Ranging (SLR) is a powerful and precise technique that uses laser pulses to determine the distance to satellites outfitted with retroreflectors. By emitting laser pulses and timing their round-trip travel, SLR provides accurate measurements essential for satellite orbit determination, Earth science research, and geodesy. Understanding the variables involved in SLR is crucial for interpreting the measurements and ensuring high precision. Below is a comprehensive overview of these variables, categorized for clarity:
Laser Pulse Transmission Time The exact time when a laser pulse is sent from the ground station.
Round-Trip Time{total} The total time taken for the laser pulse to travel to the satellite and return.
Speed of Light (ccc) A constant (≈299,792,458 m/s) is used to calculate distance from time.
Ground Station Coordinates The precise Earth-centered location of the SLR ground station.
Satellite Coordinates The satellite’s position at the time of laser pulse reflection.
Elevation Angle (θ)The angle between the local horizontal and the line of sight to the satellite.
Azimuth Angle (ϕ) The compass direction from the ground station to the satellite.
Atmospheric Delay Corrections Adjustments for time delays caused by the atmosphere.
Retroreflector Characteristics Properties of the satellite’s reflectors that affect signal return.
Laser Pulse Characteristics Features of the emitted laser pulses, such as frequency and width.
Signal Return Rate The frequency at which reflected laser pulses are successfully detected.
Timing Accuracy The precision of the timing systems used for measuring pulse times.
Geometric Range (RRR) The straight-line distance between the ground station and the satellite.
Photon Detection Efficiency The effectiveness of detectors in capturing returned laser photons.
Reference Frames and Coordinate Systems The spatial frameworks used to define ground and satellite positions.
Instrument Calibration Parameters Constants and settings used to ensure measurement accuracy.
Satellite Orbital Elements Parameters that define the satellite’s orbit, refined using SLR data.
Data Processing Algorithms Methods used to analyze raw SLR data and compute accurate ranges.
Time Standards Coordinated time references (e.g., TT, TDB) used for synchronization.
Two-Way Range (TWR) is a highly accurate method for determining the distance between a ground station and a spacecraft. By sending a signal to the satellite and measuring the time it takes for the signal to be returned, operators can compute the range using the speed of light.
Doppler The Doppler effect is the apparent change in the frequency of a wave when the source and observer are in relative motion. If the source moves toward the observer, the wave is compressed (higher frequency, or “blue shift”); if the source moves away, the wave is stretched (lower frequency, or “red shift”). By measuring the Doppler shift (the difference between the observed frequency and the transmitted frequency), engineers can determine the relative radial velocity of the spacecraft: