Convert between Classical Orbital Elements (COEs) and
Cartesian position–velocity state vectors \( (\mathbf{r}, \mathbf{v}) \).
This tool links orbital geometry to the dynamical state representation used in astrodynamics
modeling, trajectory analysis, orbit propagation, and spacecraft simulation.
What this tool computes
A Keplerian orbit can be described in two complementary ways. One is the
geometric orbital-element form:
\( a, e, i, \Omega, \omega, \nu \).
The other is the state-vector form:
position \( \mathbf{r} \) and velocity \( \mathbf{v} \) in an inertial frame.
This converter provides both directions:
COE → State Vector: converts orbital elements into ECI position and velocity
State Vector → COE: reconstructs semi-major axis, eccentricity, orientation, and true anomaly
Displays the perifocal frame and ECI frame relationship
Shows the full transformation matrix
Plots the orbit in 3D space around the selected central body
Interactive converter
Interpretation
Use either direction of the converter to move between orbital geometry and the inertial state used in astrodynamics simulation.
Results
Position Vector \( \mathbf{r} \)
[ — , — , — ] km
Magnitude: — km
Velocity Vector \( \mathbf{v} \)
[ — , — , — ] km/s
Magnitude: — km/s
Recovered / Input Semi-major Axis
— km
Orbit size parameter
Eccentricity
—
Shape of the conic section
Inclination
— deg
Tilt of the orbital plane
RAAN / Argument of Periapsis / True Anomaly
— / — / —
Orientation and position angles
Specific Angular Momentum \( |\mathbf{h}| \)
— km²/s
Conserved in two-body motion
Specific Orbital Energy \( \epsilon \)
— km²/s²
Negative for bound elliptical motion
Perigee Radius \( r_p \)
— km
Closest orbital distance
Apogee Radius \( r_a \)
— km
Farthest orbital distance
3D orbit visualization
The orbit is plotted in an Earth-centered inertial-style frame. The current spacecraft location is shown on the orbit.
Core physics
In the perifocal frame, the orbit is expressed naturally in terms of the true anomaly \( \nu \).
The semilatus rectum is
These vectors are then rotated into the inertial frame using the classical
\( R_3(\Omega)\,R_1(i)\,R_3(\omega) \) transformation.
Transformation matrix
The perifocal-to-ECI direction cosine matrix is shown below for the current orbit:
1
2
3
Row 1
—
—
—
Row 2
—
—
—
Row 3
—
—
—
This matrix maps \( \mathbf{r}_{pf} \) and \( \mathbf{v}_{pf} \) into the inertial frame:
\( \mathbf{r}_{ECI} = \mathbf{Q}_{pf\rightarrow ECI}\mathbf{r}_{pf} \),
\( \mathbf{v}_{ECI} = \mathbf{Q}_{pf\rightarrow ECI}\mathbf{v}_{pf} \).
Assumptions and limitations
This converter assumes:
Ideal two-body motion
Point-mass central gravity field
No atmospheric drag, J2, SRP, or third-body perturbations
Classical orbital elements remain well-defined away from singular cases
Special caution is needed for:
Circular orbits \( e \approx 0 \), where periapsis direction becomes undefined
Equatorial orbits \( i \approx 0^\circ \), where RAAN becomes undefined
Circular equatorial orbits, where multiple classical angles lose uniqueness
Why this tool matters
This is one of the most important core tools in orbital mechanics because it connects
orbital geometry with the dynamical state used by propagators,
guidance laws, estimation systems, and mission-analysis codes.
In practice, engineers constantly move between:
element sets for interpretation and orbit design
state vectors for propagation, simulation, and control
frame transformations for real mission analysis workflows
So this tool is not just educational. It is foundational.
Example validation case
A standard test case is:
\( a = 7000\ \text{km},\ e = 0.10,\ i = 45^\circ,\ \Omega = 30^\circ,\ \omega = 40^\circ,\ \nu = 60^\circ \).