Orbital element

An orbital element is one of a set of parameters specifying an orbit at a chosen epoch. For a Kepler orbit, the semi-major axis and eccentricity determine its size and shape, while angular elements determine its orientation and the body's position.

The argument of periapsis is the angle in the orbital plane from the ascending node to periapsis, measured in the direction of motion. It differs from the longitude of periapsis $\varpi =\Omega +\omega$. For an inclined orbit this sum is a conventional longitude, not generally the azimuth of the projected periapsis. The argument is undefined for a circular orbit and the node-based description is singular when the orbit and reference plane coincide.

True longitude is the conventional orbital angle formed from the longitude of ascending node, argument of periapsis and true anomaly. For a coplanar Kepler orbit it is the actual polar longitude of the position vector. For an inclined orbital plane the sum of angles is generally not the projected azimuth. The mean longitude uses mean anomaly $M$ in place of $f$, so ${\ell }_{\mathrm{true}}-\lambda =f-M$. For small orbital eccentricity, Kepler's equation gives $E=M+e\sin M+O(e^2)$, while the eccentric-to-true anomaly relation gives $f=E+e\sin E+O(e^2)$; hence $f-M=2e\sin M+O(e^2)$. This periodic difference matters when separating a slow resonant argument from a rapidly varying geometric angle.

The plane containing a central-force Kepler orbit, perpendicular to its conserved specific angular momentum. Its orientation can be specified by an orbital inclination and an orbital node. Perturbing forces can change that orientation.

An orbital element of the Kepler orbit having the same instantaneous position and velocity as a perturbed trajectory. Its elements vary as the perturbing force acts. Their interpretation depends on the central gravitational parameter used, for example whether static radiation pressure is included in the unperturbed attraction.

A circular distant barycentric Kepler orbit around a circular star–planet binary need not have constant star-centred osculating orbital elements. Let $\mu$ be the planet's fraction of the binary mass, $\alpha =a_{\mathrm{pl}}/a\ll 1$, and $\epsilon =\mu /\sqrt{\alpha }\ll 1$. Subtracting the stellar reflex velocity produces radial and tangential perturbations $(-v_0\epsilon \sin \Delta ,v_0\epsilon \cos \Delta )$, where $\Delta$ is the planet's longitude minus the particle's longitude. The vis-viva equation gives the displayed semi-major axis variation. The eccentricity vector components follow from $e_r=r{v}_{\theta }^2/(G{M}_{\star })-1$ and $e_{\theta }=-r{v}_r{v}_{\theta }/(G{M}_{\star })$. The errors are $O(\mu +{\epsilon }^2)$; the spatial translation itself is only $O(\mu \alpha )$. These are changes of instantaneous orbital description, even when the assumed barycentric trajectory is fixed.

The true anomaly is the angle at the focus from periapsis to the current position of an orbiting body, measured in the direction of motion.

Orbital phase specifies where a body lies within its orbital cycle relative to a chosen reference event.

A phase-mixed orbit has a probability distribution uniform in its canonical angle variable. For a Kepler ellipse, this means uniform mean anomaly and therefore probability proportional to time spent in each orbital segment.

Uniform mean anomaly implies a line probability inversely proportional to the orbital speed. For small orbital eccentricity and true anomaly $f$, it is proportional to $1-e\cos f$.

The geometric cross-section passing a fixed point of a steady orbital wire per unit time is $J_{\sigma }={\lambda }_{\sigma }v$, where ${\lambda }_{\sigma }=d\sigma /ds$. Conservation of fragments requires $J_{\sigma }$ constant and ${\lambda }_{\sigma }\propto 1/v$. For total area ${\sigma }_{\mathrm{tot}}$ and orbital period $P$,With the fractional luminosity of a phase-mixed eccentric wire, this becomes $J_{\sigma }=2fh$. The current describes transported area, not the swept area enclosed by the orbit.

For a Kepler orbit with pericentre distance $q$, apocentre distance $Q$, and semi-major axis $a=(q+Q)/2$, a uniform mean anomaly givesThe two radial passages per orbital period are included, and ${\int }_q^Q{p}_r(r)dr=1$.

Mean anomaly is the uniformly increasing phase $M=n(t-\tau )$ of a Kepler orbit, measured from periapsis time $\tau$.

For an elliptic Kepler orbit, Kepler's equation relates mean anomaly $M$ and eccentric anomaly $E$ by

The eccentric anomaly parametrizes a Kepler ellipse through the corresponding point on its auxiliary circle. It gives radius $r=a(1-e\cos E)$.

An orbital node is an intersection of an orbital plane with its reference plane.

The ascending node is the orbital node where a body crosses from the negative to the positive side of the reference plane.

The longitude of ascending node is the reference-plane angle from a fixed direction to the ascending node.

Orbital inclination is the angle between an orbital plane and a reference plane.

In a right-handed reference frame, the rotation matrix from an orbital plane frame whose first axis points along the ascending node is $R_z(\Omega )R_x(I)$. Its columns are the orbital frame's orthonormal basis expressed in reference coordinates. The unit vector normal is $(\sin \Omega \sin I,-\cos \Omega \sin I,\cos I)$; its dot product with another such normal gives the cosine of their mutual inclination.

The mutual inclination is the angle between two orbital planes, equivalently the angle between consistently oriented orbital normal vectors.

For two nearly edge-on orbits with small orbital inclination difference $\Delta I$ and small longitude of ascending node difference $\Delta \Omega$, expanding the dot product of their normals gives $I_m^2=(\Delta I{)}^2+(\Delta \Omega {)}^2+O({\delta }^4)$, with angles in radians and all offsets from edge-on of order $\delta$. Thus equality of observed sky-plane inclinations alone does not imply aligned orbital planes.

The invariable plane is perpendicular to a system's total angular momentum. For an isolated planetary system, it is fixed while individual orbital planes precess.

For two groups carrying angular momenta $L_a,L_b$ with mutual inclination $\Delta I$, resolving total angular momentum parallel and perpendicular to the second group's normal gives its orbital inclination relative to the invariable plane:For a positive denominator, $\tan I_b=\sin \Delta I/(L_b/L_a+\cos \Delta I)$. The larger-angular-momentum group lies closer to the invariable plane.

An orbit is edge-on to an observer when its orbital plane contains the line of sight, so its inclination to the plane of the sky is $90^{\circ }$.

Projected observations that do not identify which node is ascending may determine an orbital plane only up to reversal of its normal or addition of $\pi$ to its nodal longitude.

The longitude of periapsis gives the angular direction of periapsis in the reference plane.

Mean longitude is an angle that advances uniformly at the mean motion in an unperturbed Kepler orbit.

Mean motion is the average orbital angular frequency. A Kepler orbit about gravitational parameter $\mu$ has $n^2{a}^3=\mu$.

The orbital period is the time required to complete one orbit. It is related to mean motion by $T=2\pi /n$.

The synodic period is the interval between successive repetitions of the relative orbital configuration of two bodies. For coplanar prograde mean motions $n_1,n_2$, it is $2\pi /|n_1-n_2|$.