Kepler orbit

An inverse-square central force produces conic-section trajectories with the force centre at a focus.

A weak tangential force can move a satellite through a slowly varying sequence of nearly circular Kepler orbits. For constant satellite mass, $E=-\mu m/(2r)$ and $v_c=\sqrt{\mu /r}$ give $\dot{E}=F{v}_c$, hence $F/m=\sqrt{\mu }\dot{r}/(2r^{3/2})$. Integrating the thrust magnitude for a monotone outward transfer proves the displayed formula. The integrated thrust per mass is not the vector difference of the endpoint velocities; gravity acts throughout the transfer. The approximation requires slow orbital evolution and negligible eccentricity.

An unbound Kepler orbit under attractive inverse-square law gravity is a hyperbola with the attracting center at a focus of a conic. Writing $\mu =GM$ and $h$ for angular momentum per unit mass, its eccentricity satisfies $e^2=1+2\epsilon h^2/{\mu }^2$, where $\epsilon >0$ is the specific orbital energy. A nonzero incoming speed $v$ at infinity gives $\epsilon =v^2/2$.

For a positive-mass object in the inverse-square attractive field with potential zero at infinity, energy conservation gives $v^2(r)=2E/m+2GM/r$. Along the outgoing unbound branch, speed decreases toward the displayed limiting value. It is positive for a hyperbola and zero for a parabola. This is an infimum approached at infinite separation, not a finite-radius stopping point or the local escape velocity.

At a genuine pericentre, the radial velocity vanishes and the radial acceleration is nonnegative. The radial equation therefore requires $v_p^2\ge GM/r_p$. In terms of the displayed factor k, a circle has k=1, a noncircular ellipse has $1<k<\sqrt{2}$, a parabola has $k=\sqrt{2}$ and a hyperbola has $k>\sqrt{2}$. If $0<k<1$ at a tangential turning point, that point is the apocentre, despite the negative orbital energy. Bound-energy classification alone does not establish which turning point is being specified.

In axes directed along the periapsis and transverse to it, a Kepler orbit with gravitational parameter ${\mu }_g$ and specific angular momentum $h$ has the displayed velocity. The velocity tip traces a circle of radius ${\mu }_g/h$ centred at $(0,e{\mu }_g/h)$. To derive it, differentiate $r=h^2/[{\mu }_g(1+e\cos f)]$, use $r^2\dot{f}=h$, and resolve $\dot{r}\hat{\mathbf{r}}+r\dot{f}\hat{\mathbf{\theta }}$. A rotation of the spatial axes rotates the entire hodograph by the same angle.

For unit mass with inverse-square potential $-Q/r$, $Q>0$, and positive conserved angular momentum $h$, a bounded orbit has $-Q^2/(2h^2)\le E<0$. Its periapsis distance and apoapsis distance are roots of $|E|r^2-Qr+h^2/2=0$, so their sum is $Q/|E|$ and product $h^2/(2|E|)$. Speed is largest at the periapsis and smallest at the apoapsis, because $v^2=2(E+Q/r)$ decreases with $r$. Radial velocity is zero there, so the two speeds are $h/r_{\min }$ and $h/r_{\max }$. Their sum is $h(r_{\min }+r_{\max })/(r_{\min }{r}_{\max })=2Q/h$, independent of $E$, including the circular limit.

For an instantaneous in-plane change of velocity of magnitude $\gamma v_K$ on a circular orbit, with angle $\theta$ measured from the forward tangential direction, the new specific orbital energy gives $a^{\prime }/a=(1-2\gamma \cos \theta -{\gamma }^2{)}^{-1}$. The specific angular momentum gives $e^{\prime 2}=1-(1-2\gamma \cos \theta -{\gamma }^2)(1+\gamma \cos \theta {)}^2$. The signed semi-major axis becomes negative for an unbound hyperbolic Kepler orbit.

Uniform planar kick angle $\theta$ gives $c=\cos \theta$ with probability density $1/(\pi \sqrt{1-c^2})$. This maps onto a one-dimensional curve in orbital eccentricity–semi-major axis space. The two radial-kick signs share that curve but have opposite tangential components of the eccentricity vector. The resulting density is enhanced near tangential-kick endpoints.

For unit mass in $\Phi =-GM/r$, conservation of angular momentum splits the velocity norm into ${\dot{r}}^2+L^2/r^2$. Conservation of energy then gives $\frac{1}{2}{\dot{r}}^2+L^2/(2r^2)=E+GM/r$. It separates radial dynamics from the reconstruction of the orbital angle.

In the centre-of-mass frame of two Newtonian point masses, total orbital energy is reduced mass times specific orbital energy: $E=\mu [|\dot{\mathbf{r}}{|}^2/2-GM/r]$. The factor $\mu$ multiplies both kinetic and potential contributions, while $GM$ is the relative gravitational parameter.

For an attractive inverse-square force, $r=p/(1+e\cos f)$, where $f$ is true anomaly, $e$ is orbital eccentricity, and $p=h^2/\mu$ is the semilatus rectum. The specific angular momentum relation $\dot{f}=h/r^2$ converts this geometric equation into an angular evolution law.

An orbit about a planet. Stellar tides, radiation pressure and other perturbations can limit its stability; a sufficiently small semi-major axis compared with the Hill sphere permits a local approximately planet-centred Kepler orbit.

Orbit-maintenance maneuvers counteracting perturbations that would move a spacecraft away from its desired orbital parameters.

An elliptical transfer trajectory whose apogee is near the radius of a geostationary orbit. Apogee maneuvers can circularize the trajectory and reduce inclination.

An Earth-centred orbit at relatively low altitude, commonly within roughly 2000 km of the surface. It supports frequent orbital revolutions and relatively fine spatial sampling for remote sensing.

Height above a stated planetary reference surface. For a spherical body of radius $R_p$ and a circular orbit of central radius $r$, it is $r-R_p$. Confusing altitude with central radius changes both the orbital period calculation and surface viewing geometry.

An orbit passing near both geographic poles of its central body. Planetary rotation brings different longitudes beneath successive passes, useful for global remote sensing; one satellite still samples each location intermittently.

An orbit whose nodal precession tracks the central planet's annual motion, keeping approximately the same local solar time at corresponding crossings. Near-Earth implementations are nearly polar and exploit the planet's nonspherical gravitational field. Fixed local-time sampling supports comparison between passes while limiting observations of the diurnal cycle.

A circumplanetary orbit whose orbital period equals the central planet's spin period. Its semi-major axis follows from Kepler's third law. Matching periods alone does not fix the satellite above one ground location; that additionally requires a planet-stationary orbit.

An Earth orbit with period equal to one sidereal day. A geostationary orbit additionally requires circular, prograde, equatorial geometry.

A prograde, equatorial circular orbit with orbital period equal to the planet's spin period. In the ideal model its satellite remains above one equatorial ground location. Nonzero orbital inclination or orbital eccentricity destroys exact stationarity even when the periods agree.

The leading uniform stellar radiation pressure is a differential acceleration relative to a planet, of scale $\beta G{M}_{\star }/a_p^2$. Relative to planetocentric gravity at dust radius $a$, its magnitude isThus lying inside a Hill sphere alone does not guarantee small radiative perturbations. Approximately circular tightly bound dust requires this ratio sufficiently small, as well as weak tides; the uniform radiative force can otherwise excite orbital eccentricity or unbind grains.

A planet can block stellar radiation during part of a circumplanetary orbit. Then the full-orbit constant-flux averages for radiation pressure and Poynting–Robertson drag require modification. The detailed shadow duty cycle depends on the orbital inclination, stellar angular extent and dust radius.

An orbit about a star. Dust on such an orbit can experience radiation pressure and Poynting–Robertson drag, in addition to gravitational perturbations.

Binet’s equation rewrites a central-force orbit using $u(\theta )=1/r$. If $\ell =m{r}^2\dot{\theta }$ is angular momentum and the potential is $V(r)$, then

For an attractive central force of magnitude $c/r^5$, a circle of radius $R$ through the force center has polar equation $r=2R\cos \theta$. With specific angular momentum $h$, substitution into the Binet equation gives $c=8m{h}^2{R}^2$. Consequently $R$ is inversely proportional to the magnitude of the angular momentum, and the angular-time integral gives $T=2\pi R^2/|h|\propto R^3$. This is the finite travel time along the punctured circle from one approach to the singular center to the next; the force law alone does not specify continuation through the collision.

For radial acceleration $-A/r^2-B/r^3$, conserved specific angular momentum $l$ and $u=1/r$ reduce the Binet equation to the displayed linear equation. If $l^2>B$, its angular oscillation frequency is $\sqrt{1-B/l^2}$; attraction proportional to $r^{-3}$ changes the apsidal angle. A zero of $u$ ends the physical positive-radius branch and corresponds to escape, not passage to a negative radius. A spatial self-intersection need not be a periodic return because velocity may differ.

For $u=u_c[1+e\cos \nu (\psi -{\psi }_0)]$ with $\nu >0$ and $e\ge 1$, the positive-radius scattering branch has $|\nu (\psi -{\psi }_0)|<\arccos (-1/e)\le \pi$. Thus the principal complex argument in the displayed expression is continuous throughout that branch, unlike at an apocentre of a bound orbit. Its derivative is $\nu$, so the angle modulo $2\pi$, or the associated unit apsidal vector in the orbital plane, is a stationary integral of motion fixing the periapsis direction. Together with energy and three components of specific angular momentum, it gives five regular scattering labels even for irrational $\nu$. Periodicity is therefore necessary for five orbit labels in recurrent bound motion, not in general for escaping trajectories.

In the gravitational potential $-\mu (1/r+a/r^2)$, a regular noncircular bound orbit requires $L^2>2\mu a$ and has $u=\mu (L^2-2\mu a{)}^{-1}[1+e\cos \nu (\psi -{\psi }_0)]$ with $0<e<1$. Each radial period advances the angle by $2\pi /\nu$. The full position and velocity return after finitely many radial periods precisely when $\nu$ is rational. Irrational $\nu$ gives a dense rosette on the two-frequency invariant torus. Rationality permits a fifth orbit label fixing the apsidal phase on that resonant angular momentum family, but does not give global superintegrability for all nearby angular momenta. When $a=0$, every regular bound Kepler orbit is closed and the Laplace-Runge-Lenz vector supplies the globally defined additional integral.

For radial acceleration $-\mu /r^2+C/r^3$, the Binet equation gives $u^{\prime \prime }+(1+C/h^2)u=\mu /h^2$, where $h$ is specific angular momentum. A regular bound orbit has $h^2+C>0$, $a=-\mu /(2\mathcal{E})$ and $e^2=1+2\mathcal{E}(h^2+C)/{\mu }^2$. Its radius is $a(1-e^2)/[1+e\cos (\alpha \theta )]$, with $\alpha =\sqrt{1+C/h^2}$. The apsidal precession per radial period is $2\pi ({\alpha }^{-1}-1)$, which reduces to the displayed expression for a small perturbation. Negative $C$ produces prograde precession; positive $C$ produces retrograde precession. The dimensionless coefficient is $C/(\mu a)$, since $C$ has units of length to the fourth power divided by time squared.

For radial acceleration $-GM(1+\gamma /r)/r^2$, conserved specific angular momentum $h$ gives an angular radial frequency $\alpha =\sqrt{1-GM\gamma /h^2}$ through the Binet equation. Successive pericentres differ by $2\pi /\alpha$, so a small perturbation produces apsidal precession $\Delta \varpi \simeq \pi GM\gamma /h^2$. Writing the unperturbed semilatus rectum as $p=h^2/(GM)$ gives the displayed inverse relation. For a nearly circular orbit of period $P$, use $p\simeq [GM(P/(2\pi ){)}^2{]}^{1/3}$. Positive $\gamma$ means prograde precession; $\gamma$ is a length. A measured rate per century must first be converted to an angle per orbit.

Constant areal velocity gives constant specific angular momentum $l=r^2\dot{\theta }$, hence zero transverse acceleration and a central force. For a focus-based ellipse $u=1/r=(1+\epsilon \cos \theta )/A$, one has $u_{\theta \theta }+u=1/A$. The Binet equation then gives the displayed radial acceleration, proving an attractive inverse-square law. An orbit's shape alone does not specify its acceleration without a time law.

An attractive inverse-square force together with radial drag $-k\dot{r}/r^2$ preserves angular momentum $h=r^2\dot{\theta }$ but modifies the Binet equation for $u=1/r$ by a first derivative. For $k/|h|<2$ and $h\ne 0$, the reciprocal radius is a decaying oscillation about $p/h^2$. Forward angular distance $\mathrm{sgn}(h)(\theta -{\theta }_0)$ makes its damping sign explicit for either orbital orientation.

A bounded trajectory in the damped Binet equation with nonzero angular momentum asymptotically becomes circular. Energy loss is nonnegative in magnitude and vanishes only for zero radial velocity. The centrifugal effective potential energy excludes collision; bounded radius forces indefinitely increasing forward angular distance, so the reciprocal-radius oscillation decays. Escaping trajectories are excluded by the boundedness hypothesis.

ForBinet’s equation isIts circular orbits solve $K=u(1-r_{0}u{)}^2$; the branch $u<1/(3r_0)$ is stable and the branch $u>1/(3r_0)$ is unstable.

Orbital eccentricity distinguishes circular, elliptic, parabolic, and hyperbolic conics.

Launching at the correct circular orbital speed but at in-plane angle $\alpha$ from the tangent leaves specific orbital energy $-GM/(2R)$, hence semi-major axis $R$. The specific angular momentum becomes $\sqrt{GMR}\cos \alpha$, so $j^2=GMa(1-e^2)$ gives $e=|\sin \alpha |$. The actual periapsis and apoapsis are $R(1\mp e)$, equally far from the intended circular orbit. The maximum geometric radial separation is therefore $Re$. This is not an equal-time separation between two moving satellites.

An orbit with nonzero orbital eccentricity. A bound Kepler orbit with $0<e<1$ is a noncircular ellipse; $e=1$ and $e>1$ describe the parabolic and hyperbolic unbound cases. A semi-major axis fixed by Kepler's third law need not equal the time average of the orbital radius.

For an attractive Kepler orbit with gravitational parameter $GM$, velocity $\mathbf{v}$ and specific angular momentum $\mathbf{h}=\mathbf{r}\times \mathbf{v}$, the eccentricity vector isIts magnitude is the orbital eccentricity, and it points toward periapsis for a noncircular orbit. It is a normalized attractive Laplace-Runge-Lenz vector; it remains constant under the pure inverse-square force.

If the relative acceleration is $\ddot{\mathbf{r}}=-GM\mathbf{r}/r^3+\mathbf{f}$, with constant masses, then the specific orbital energy $\epsilon$, specific angular momentum $\mathbf{h}$ and eccentricity vector obeyThe inverse-square-force terms cancel in each derivative. These identities express perturbing work and torque, and track the instantaneous change of the ellipse's shape and orientation.

An isotropic population of bound Kepler orbits has eccentricity probability density $n(e)=2e$ on $0\le e<1$. It follows from the fixed-energy angular-momentum measure $n(L)dL\propto LdL$ and $L=L_c\sqrt{1-e^2}$.

For bound Kepler orbits, the Delaunay variables obey $0<G<L$, $-G<H<G$, and three periodic angles. A canonical transformation preserves phase space volume. At fixed $L$, integrating $H$ supplies the measure $2GdG$. Since $G=L\sqrt{1-e^2}$, the fraction with orbital eccentricity less than $e$ is $L^{-2}{\int }_{L\sqrt{1-e^2}}^{L}2GdG=e^2$. Thus the probability density function is $2e$, independently of any normalizable energy-only phase-space distribution function. A Boltzmann distribution is unnecessary. Even the time-dependent density $F(E(L))[1+\eta \cos (l-n(L)t)]$, $|\eta |<1$, has this same marginal because integrating the mean anomaly removes its oscillatory part; an eccentricity marginal alone does not imply equilibrium.

For bound Kepler orbits with gravitational parameter $\mu$, integrating the Delaunay variables over $0<G<L$, $-G<H<G$ and their three full angle periods gives the displayed population. A globally defined thermal eccentricity distribution requires this integral to converge, or an explicit finite action range. Merely having finite $F(E)$ at every energy does not suffice: $F=1$ yields an infinite integral at large $L$. At each fixed energy, or on every finite normalized action band, the cumulative eccentricity fraction remains $e^2$.

An elliptic orbit is a bound orbit shaped as an ellipse. A Kepler ellipse has negative specific orbital energy and eccentricity $0\le e<1$.

A positive-energy inverse-square orbit is a hyperbola with two scattering asymptotes.

The hyperbolic excess speed is the limiting relative speed far from an attracting body on an unbound hyperbolic Kepler orbit. Its specific orbital energy is $\epsilon =v_{\infty }^2/2>0$. At radius $r$ from a body of gravitational parameter $\mu$, conservation of energy gives $v^2=v_{\infty }^2+2\mu /r$. The outgoing velocity direction as well as its magnitude matters when this orbit is matched to another attracting body's frame in a patched conic approximation.

The gravitational scattering angle is the change between incoming and outgoing asymptotic velocity directions.

For a local two-body hyperbolic encounter with central parameter $\mu$ and allowed centre separation $r_{\min }\ge r_0$, conservation of energy gives equal asymptotic speeds. The gravitational scattering angle therefore impliesPut $z=v_{\infty }\sqrt{r_{\min }/\mu }$ and use $2z/(1+z^2)\le 1$. The optimizing grazing orbit has $z=1$, orbital eccentricity two and deflection $\pi /3$. For finite spherical bodies $r_0$ is the sum of their radii; excluding actual contact makes the upper bound a supremum. Without a finite exclusion radius the ideal point-mass problem has no finite bound.

In a gravitational assist, a small body's speed relative to a moving massive body is deflected with nearly unchanged magnitude. Transforming back to the inertial frame can transfer orbital energy between the bodies.

A zero-energy inverse-square orbit is parabolic and has eccentricity one.

For a parabolic trajectory with periapsis radius $r_p$, put $D=\tan (f/2)$, where $f$ is true anomaly. The Barker equation is $t-t_p=\sqrt{2r_p^3/(GM)}(D+D^3/3)$. Since $r=r_p(1+D^2)$, it is equivalent to the radius-time relation for a parabolic Kepler orbit. The sign of $D$ distinguishes approach from recession; $t_p$ is the periapsis time.

On a nonradial zero-energy Kepler orbit, conserved specific angular momentum $h$ gives periapsis $r_p=h^2/(2GM)$ and ${\dot{r}}^2=2GM(r-r_p)/r^2$. Integrating with $s=\sqrt{r-r_p}$ yields $t-t_p=\pm \sqrt{2/(GM)}(r+2r_p)\sqrt{r-r_p}/3$. Squaring gives the displayed version of the Barker equation, valid on both incoming and outgoing branches. The time origin $t_p$ is passage through periapsis, not an arbitrary point on the orbit.

A parabolic Kepler orbit obeys $t=\sqrt{2q^3/(GM)}(D+D^3/3)$ with $D=\tan (f/2)$. At the high-eccentricity angular-speed crossover, $q\ll r_x$, so $n_p{t}_x\simeq (\sqrt{2}/3)(r_x/a_p{)}^{3/2}=2^{7/8}(q/a_p{)}^{3/8}/3$. The small time fraction explains why an eccentric orbit turns through almost $\pi$ of true anomaly while a distant perturber barely moves.

The periapsis is the point of an orbit nearest its force centre.

The periapsis of an Earth-centred orbit. For an elliptic orbit its distance from the Earth's centre is $a(1-e)$. Both this distance and the apogee distance are measured from the force centre, rather than as heights above Earth's surface.

The apoapsis is the point of an orbit farthest from its force centre.

The apoapsis of an Earth-centred orbit. For an elliptic orbit its distance from the Earth's centre is $a(1+e)$, in terms of semimajor axis and orbital eccentricity.

The semi-major axis is one half of an ellipse's longest diameter. For a Kepler ellipse with apsidal distances $r_1,r_2$, it is $a=(r_1+r_2)/2$.

This is one of the semi-major and semi-minor axes; the two-axis geometry is the broader concept.

For an ellipse of eccentricity $e$, the semi-minor axis is $b=a\sqrt{1-e^2}$.

This is one of the semi-major and semi-minor axes; the two-axis geometry is the broader concept.

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.

Half the full range of an orbit-induced line-of-sight velocity variation, after accounting for a constant systemic offset.

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 $T=2\pi /n$.

For a circular orbit, the angular rate about the central body equals $2\pi$ divided by its orbital period. Matching it to planetary spin is necessary for a planet-stationary orbit, which additionally requires a prograde equatorial circular geometry.

For a spherical body of radius $R$, mass $M$ and mean density $\bar{\rho }=3M/(4\pi R^3)$, a low circular orbit has period $P\simeq 2\pi \sqrt{R^3/(GM)}$, by equating the gravitational acceleration from Newton's law of universal gravitation to the centripetal acceleration. Substitution proves the displayed relation. Thus similar low-orbit periods indicate similar mean densities, even for bodies with very different masses and radii. At altitude $h$, the right side is multiplied by $(1+h/R{)}^3$; the simple density inference requires $h\ll R$ and neglects rotation, nonsphericity and other gravitating bodies.

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 $2\pi /|n_1-n_2|$.

The specific orbital energy is the mechanical energy per unit mass,Its sign distinguishes bound elliptic, marginal parabolic, and unbound hyperbolic Kepler orbits.

A weak perturbing relative acceleration $\mathbf{f}$ changes specific orbital energy at rate $\dot{\epsilon }=\mathbf{v}\cdot \mathbf{f}$. Since $\epsilon =-\mu /(2a)$ for a Kepler orbit,For a circular Kepler orbit under $\mathbf{f}=-\kappa \mathbf{v}+\mathbf{g}$ with constant $\mathbf{g}$, the constant-force term averages to zero over an unperturbed orbit. Thus $⟨\dot{a}⟩=-2\kappa a$. This is a leading-order orbital average; a strong constant force can distort the orbit instead.

For a Kepler orbit with semi-major axis $a$, the vis-viva equation isFor the convention that a hyperbolic Kepler orbit has positive semi-major-axis magnitude $a$, its final term instead has a plus sign.

The specific angular momentum is angular momentum per unit mass, $\mathbf{h}=\mathbf{r}\times \mathbf{v}$. For a planar Kepler orbit its magnitude is $h=r^2\dot{\theta }=\sqrt{\mu a(1-e^2)}$.

For a nearly radial Kepler orbit with pericentre $q$, specific angular momentum is $h\simeq \sqrt{2GMq}$. Relative to a circular reference orbit of radius $a_p$ and mean motion $n_p$, the instantaneous azimuthal rate changes from faster to slower at $r_x=\sqrt{h/n_p}\simeq a_p(2q/a_p{)}^{1/4}$. In a rotating reference frame the polar angle has a turning point at this radius. The true anomaly there differs from $\pi$ by approximately $2\sqrt{q/r_x}$.