Home About Data Launches Tools Product Updates Live Solar System

Two Line Element (TLE) Parser and Orbital State Vector Calculator

Free online TLE parser for satellite tracking. Convert NORAD Two-Line Element Sets to classical orbital elements (COEs) and Earth-Centered Inertial (ECI) state vectors. Calculate apogee, perigee, orbital period, and more with our orbital mechanics calculator.

Classical Orbital Elements

State Vector (ECI)

Position (km): x = y = z = Velocity (km/s): vx = vy = vz =

Understanding TLEs, Orbital Elements, and State Vectors

What is a Two-Line Element Set (TLE)?

A Two-Line Element Set (TLE) is a standardized data format used to convey orbital elements of Earth-orbiting satellites. Developed by NORAD (North American Aerospace Defense Command), TLEs enable accurate tracking and prediction of satellite positions. Each TLE consists of three lines: a title line (optional) and two data lines containing orbital parameters and identification information.

TLEs are updated regularly to account for orbital perturbations caused by atmospheric drag, solar radiation pressure, and gravitational anomalies. They use the SGP4/SDP4 propagation models for position prediction.

TLE Format Structure

Satellite Name (Optional Line 0)
ISS (ZARYA)
Line 1
1 25544U 98067A 08264.51782528 .00002182 00000-0 42534-4 0 9993
Line 2
2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.72125391563537
Field Descriptions:
  • Satellite Number: NORAD catalog number (5 digits)
  • International Designator: Launch year, launch number, piece of launch
  • Epoch: Year and day of year with fractional portion
  • Mean Motion Derivatives: Orbital decay parameters
  • Drag Term: Atmospheric drag / radiation pressure coefficient
  • Inclination: Orbital tilt relative to equator (degrees)
  • Right Ascension (RAAN): Angle of ascending node (degrees)
  • Eccentricity: Orbital shape (decimal point assumed)
  • Argument of Perigee: Angle to closest approach (degrees)
  • Mean Anomaly: Position along orbit (degrees)
  • Mean Motion: Orbits per day
  • Revolution Number: Total orbits completed at epoch

Classical Orbital Elements (COEs)

Classical Orbital Elements are six parameters that uniquely define an orbit in space:

  • Semi-major axis (a): Half the longest diameter of the elliptical orbit, measured in kilometers. Determines the orbital size and period.
  • Eccentricity (e): Describes the shape of the orbit (0 = circular, 0 < e < 1 = elliptical, e = 1 = parabolic, e > 1 = hyperbolic).
  • Inclination (i): The tilt of the orbital plane relative to Earth's equatorial plane, measured in degrees (0° = equatorial, 90° = polar).
  • Right Ascension of Ascending Node (RAAN/Ω): The angle from the vernal equinox to the point where the orbit crosses the equatorial plane moving northward.
  • Argument of Perigee (ω): The angle from the ascending node to the perigee (closest approach point) within the orbital plane.
  • True Anomaly (ν): The angle from perigee to the satellite's current position, measured along the orbital path.

State Vector (Position and Velocity)

A State Vector represents a satellite's position and velocity at a specific moment in time using Cartesian coordinates in an Earth-Centered Inertial (ECI) reference frame. It consists of six components:

  • Position Vector (r): Three Cartesian coordinates (x, y, z) in kilometers representing the satellite's position.
  • Velocity Vector (v): Three Cartesian components (vx, vy, vz) in km/s representing the satellite's velocity.

The ECI frame is non-rotating and commonly uses the J2000 epoch as its reference. This representation is essential for orbit propagation and spacecraft navigation.

Converting Classical Orbital Elements to State Vectors

The conversion from orbital elements to state vectors involves several steps:

Step 1: Calculate Position and Velocity in Orbital Frame

First, compute the position magnitude and orbital frame coordinates:

Semi-latus rectum: p = a(1 - e²)

Radius magnitude: r = p / (1 + e cos ν)

Position in orbital plane:

xorb = r cos ν

yorb = r sin ν

zorb = 0

Velocity in orbital plane:

μ = 398,600.4418 km³/s² (Earth's gravitational parameter)

vx,orb = -√(μ/p) sin ν

vy,orb = √(μ/p) (e + cos ν)

vz,orb = 0

Step 2: Construct Rotation Matrix

Build the rotation matrix using the three Euler angles (Ω, i, ω):

Rotation Matrix R:

R11 = cos Ω cos ω - sin Ω sin ω cos i

R12 = -cos Ω sin ω - sin Ω cos ω cos i

R13 = sin Ω sin i

R21 = sin Ω cos ω + cos Ω sin ω cos i

R22 = -sin Ω sin ω + cos Ω cos ω cos i

R23 = -cos Ω sin i

R31 = sin ω sin i

R32 = cos ω sin i

R33 = cos i

Step 3: Transform to ECI Frame

Apply the rotation matrix to convert from orbital frame to ECI coordinates:

Position Vector:

x = R11xorb + R12yorb

y = R21xorb + R22yorb

z = R31xorb + R32yorb

Velocity Vector:

vx = R11vx,orb + R12vy,orb

vy = R21vx,orb + R22vy,orb

vz = R31vx,orb + R32vy,orb

Deriving Orbital Elements from TLEs

This tool extracts orbital elements from TLE data using the following relationships:

Mean Motion to Semi-major Axis:

n = mean motion (revolutions per day)

nrad = n × 2π / 86400 (convert to rad/s)

a = (μ / nrad²)^(1/3)

Mean Anomaly to True Anomaly:

This requires solving Kepler's equation iteratively:

E - e sin E = M (where E is eccentric anomaly, M is mean anomaly)

Then convert to true anomaly:

tan(ν/2) = √[(1+e)/(1-e)] tan(E/2)

Additional Orbital Parameters

Orbital Period: T = 2π√(a³/μ)

Apogee: rmax = a(1 + e)

Perigee: rmin = a(1 - e)

Specific Angular Momentum: h = √(μp)

Orbital Energy: ε = -μ/(2a)