00 — Overview
One orbit, six independent descriptions
A two-body elliptical orbit is described by six classical orbital elements. The semi-major axis \(a\) and eccentricity \(e\) determine size and shape. Inclination \(i\), right ascension of the ascending node \(\Omega\), and argument of perigee \(\omega\) orient the ellipse in three-dimensional space. True anomaly \(\nu\) locates the satellite on that ellipse.
The browser lab follows the same convention as the supplied MATLAB app:
\[\mathbf r_{ECI}=R_z(\Omega)R_x(i)R_z(\omega)\mathbf r_{PQW},\qquad r=\frac{a(1-e^2)}{1+e\cos\nu}.\]ECI axis \(I\) points toward the conventional vernal-equinox direction, \(K\) follows Earth's rotation axis, and \(J\) completes the right-handed frame.
The original lecture introduces the vocabulary in the order it appears in the inclined-orbit sketch: orbit and satellite; orbital and equatorial planes; ascending and descending nodes; inclination; the First Point of Aries (spring equinox); right ascension of the ascending node; apogee and perigee; argument of perigee; and true anomaly.
The orbital plane intersects the equatorial plane along the line of nodes. The ascending node is the crossing toward the northern side of the equatorial plane; the descending node is the opposite crossing. Inclination measures the angle between the planes. From the First Point of Aries, \(\Omega\) locates the ascending node; \(\omega\) then measures within the orbital plane to perigee; and \(\nu\) locates the satellite from perigee.
01 — Interactive laboratory
Classical Orbital Elements in 3D
Drag the scene to rotate it, scroll or pinch to zoom, and change one element at a time. Earth remains at the occupied focus rather than at the center of an eccentric ellipse.
No comparison orbits saved.
Drag to orbit camera · scroll or pinch to zoom · right-drag to pan
- Perigee radius
- km
- Apogee radius
- km
- Current radius
- km
- Altitude
- km
- Direction
- ECI position
Read the whole orbit: \(a,e\) define the ellipse; \(i,\Omega,\omega\) orient it; \(\nu\) moves only the satellite.
Semi-major axis \(a\): changes the orbit's scale and energy without rotating its plane.
Eccentricity \(e\): changes shape and focus offset. At \(e=0\), the orbit is circular.
Inclination \(i\): tilts the orbital plane away from the equatorial plane.
RAAN \(\Omega\): rotates the line of nodes around the ECI \(K\)-axis.
Argument of perigee \(\omega\): rotates the ellipse within its fixed orbital plane.
True anomaly \(\nu\): moves the satellite along the fixed osculating ellipse.
Circular-orbit note: perigee is not unique when \(e=0\), so \(\omega\) is geometrically undefined even though the visualization retains a reference direction.
Equatorial-orbit note: the ascending node is not unique when \(i=0^\circ\) or \(180^\circ\), so \(\Omega\) is geometrically undefined.
Original MATLAB App — Classical Orbital Elements GUI (.mlappinstall)
02 — Six elements
Shape, orient, locate
Half the longest diameter of the ellipse. It also sets specific orbital energy.
Zero is circular; increasing \(e\) separates perigee and apogee.
The angle between the orbital and equatorial planes.
The ECI angle from \(I\) to the ascending node.
The in-plane angle from ascending node to perigee.
The in-plane angle from perigee to the satellite.
03 — Discuss and predict
Use the model as an argument
- Predict what will remain unchanged when only \(\Omega\) changes. Test your prediction with a comparison orbit.
- Set \(e=0\). Why can the display still draw a perigee direction even though nature no longer supplies a unique one?
- Compare \(i=60^\circ\), \(90^\circ\), and \(120^\circ\). How does the ECI geometry reveal direction?
- Hold the five orbit-defining elements fixed and vary \(\nu\). Which communication geometry quantities could change?
04 — Summary
A compact mental model
First build the ellipse with \(a\) and \(e\). Place its plane with \(i\) and \(\Omega\). Turn perigee within that plane using \(\omega\). Finally, locate the satellite with \(\nu\). The ECI transform makes those six steps visible in one coordinate frame.