Interactive 3D visualization of the classical orbital elements of a binary star system in accordance with the Keplerian convention.
Scroll down to see the 3D model ↓In astrometry of visual binary stars there are used six orbital elements, called the Keplerian orbital elements, to describe the 3D orbit of a secondary star's relative motion to its primary star.
The longitude of the ascending node Ω is the angle between the reference direction (on this site its to the north, +x-axis) and the ascending node N, which is the point alon the orbit where the secondary star M₂ crosses the plane of the sky (xy-plane) moving away from the observer (towards +z).
The inclination i is the angle between the orbital plane and the plane of the sky (xy-plane). It is shown on this website at the ascending node N, where the secondary star M₂ crosses the sky plane from the observer's side (towards -z) to the far side (towards +z). It is one of the most important elements, as it determines how the orbit is tilted with respect to us and if we can see the stars eclipsing each other (i close to 90°) or if we see the orbit more face-on (i close to 0° or 180°).
The argument of periastron ω is the angle in the orbital plane from the ascending node N line to the Periastron Π, which is the point of the orbit where the secondary star M₂ is closest to the primary star M₁. It is measured in the direction of the orbital motion. This means that it describes the orbital orientation within the orbital plane, while Ω describes the orientation of the orbital plane in space. Often the argument of periastron ω and the longitude of the ascending node Ω are combined ϖ =Ω + ω.
The eccentricity e is a parameter that tells us the amount the ellipse is squashed. It ranges for a stable orbit between 0 and 1, where 0 means a perfect circle and values close to 1 mean a very elongated ellipse. It is defined through:
The Period P is not part of the orginial six keplerian elements, but it is important for calculating the position of the secondary star at a given time. It is the time that it takes for the secondary star M₂ to complete one full orbit around the primary star M₁. It is not shown in the 3D model, but it is important for calculating the position of the secondary star at a given time. For the position of the star we use a parameter (which is not a parameter of the Keplerian elements itself, but is derived from them) called the true anomaly θ. The true anomaly θ is the position angle of the secondary star in its orbit at a given time. It is measured in the direction of motion from the periastron Π. It is 0° at periastron Π and 180° at apoastron A. In the 3D model you can press play to see how the true anomaly changes as the secondary star moves along its orbit.
The semi-major axis a is the distance from the center of the ellipse to the farthest point on the ellipse along the longest axis.
The time of periastron is the last Keplerian orbital element, which is not shown in the 3D model, but it is important for calculating the position of the secondary star at a given time. It works as a time reference point for the true anomaly θ, where one can define an orbital phase Φ that ranges from 0 to 1 (start to end of one full orbit) and is calculated as:
*The model uses a right-handed coordinate system with the observer looking along the +z-axis. The sky plane is the xy-plane, with north in the +x direction and east in the +y direction. That can feel counterintuitive at first, but it follows the standard astronomical convention.
| Axis | Direction | Description |
|---|---|---|
| +x | North | Reference direction; Ω is measured from here |
| −x | South | Opposite direction to +x |
| −y | West | Negative y-direction after sky reflection |
| +y | East | East lies in the positive y-direction |
| +z | Line of Sight | Away from the observer |
| −z | To Observer | To the observer |