Kepler orbit

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.

Planet-stationary orbit

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.

Radiation pressure on circumplanetary dust

The leading uniform stellar radiation pressure is a differential acceleration relative to a planet, of scale

βG M_{⋆} / a_{p}^{2}

. Relative to planetocentric gravity at dust radius

a

, its magnitude is

η_{rad} = β \frac{M _{⋆}}{M _{p}} (\frac{a}{a _{p}})^{2} .

Thus 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.

Shadowing of circumplanetary dust

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.

Circumstellar orbit

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

Binet equation

Binet’s equation rewrites a central-force orbit using

u (θ) = 1/ r

. If

ℓ = m r^{2} \dot{θ}

is angular momentum and the potential is

V (r)

, then

u^{''} + u = \frac{m}{ℓ ^{2} u ^{2}} \frac{d V}{d r}_{r = 1/ u} .

Orbit equation for combined inverse-square and inverse-cube attraction ( $u^{''} + (1 - B / l^{2}) u = A / l^{2}$ )

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

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.

Inferring an inverse-square force from Kepler laws ( $a_{r} = - l^{2} / (A r^{2})$ )

Constant areal velocity gives constant specific angular momentum

l = r^{2} \dot{θ}

, hence zero transverse acceleration and a central force. For a focus-based ellipse

u = 1/ r = (1 + ε cos θ) / A

, one has

u_{θθ} + 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.

Damped Binet equation ( $u^{''} + \frac{k}{h} u^{'} + u = \frac{p}{h ^{2}}$ )

An attractive inverse-square force together with radial drag

- k \overset{r}{˙} / r^{2}

preserves angular momentum

h = r^{2} \dot{θ}

but modifies the Binet equation for

u = 1/ r

by a first derivative. For

k /∣ h ∣ < 2

and

h \neq = 0

, the reciprocal radius is a decaying oscillation about

p / h^{2}

. Forward angular distance

sgn (h) (θ - θ_{0})

makes its damping sign explicit for either orbital orientation.

Circularization under radial inverse-square drag ( $r ⟶ h^{2} / p, \dot{E} = - k \overset{r}{˙}^{2} / r^{2}$ )

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.

Shifted inverse-radius potential

For

V (r) = - \frac{km}{r - r _{0}}, r > r_{0},

Binet’s equation is

u^{''} + u = \frac{K}{( 1 - r _{0} u ) ^{2}}, K = \frac{k m ^{2}}{ℓ ^{2}} .

(2)

Its circular orbits solve

K = u (1 - r_{0} u)^{2}

; the branch

u < 1/ (3 r_{0})

is stable and the branch

u > 1/ (3 r_{0})

is unstable.

Orbital eccentricity

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

Eccentricity vector ( $e$ )

For an attractive Kepler orbit with gravitational parameter

GM

, velocity

v

and specific angular momentum

h = r \times v

, the eccentricity vector is

e = \frac{v \times h}{GM} - r .

Its 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.

Perturbed Kepler-orbit conservation laws

If the relative acceleration is

\ddot{r} = - GM r / r^{3} + f

, with constant masses, then the specific orbital energy

ε

, specific angular momentum

h

and eccentricity vector obey

\overset{ε}{˙} = v \cdot f, \dot{h} = r \times f, GM \dot{e} = f \times h + v \times (r \times f) .

The 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.

Thermal eccentricity distribution ( $n (e) = 2 e$ )

An isotropic population of bound Kepler orbits has eccentricity probability density

n (e) = 2 e

0 \leq e < 1

. It follows from the fixed-energy angular-momentum measure

n (L) d L \propto L d L

and

L = L_{c} 1 - e^{2}

Elliptic orbit

An elliptic orbit is a bound orbit shaped as an ellipse. A Kepler ellipse has negative specific orbital energy and eccentricity

0 \leq e < 1

Hyperbolic Kepler orbit

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

Gravitational scattering angle

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

Non-colliding gravitational velocity-kick bound ( $∣Δ v ∣_{m a x}$ )

For a local two-body hyperbolic encounter with central parameter

μ

and allowed centre separation

r_{m i n} \geq r_{0}

, conservation of energy gives equal asymptotic speeds. The gravitational scattering angle therefore implies

∣Δ v ∣ = \frac{2 v _{\infty}}{1 + r _{m i n} v _{\infty}^{2} / μ} \leq \frac{μ}{r _{0}} = \frac{v _{esc} ( r _{0} )}{2} .

Put

z = v_{\infty} r_{m i n} / μ

and use

2 z / (1 + z^{2}) \leq 1

. The optimizing grazing orbit has

z = 1

, orbital eccentricity two and deflection

π /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.

Gravitational assist

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.

Parabolic Kepler orbit

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

Barker equation

Barker’s equation relates time on a parabolic orbit to the tangent of half the true anomaly through a cubic.

Pericentre-to-crossover flight time ( $n_{p} t_{x} ≃ 2^{7/8} (q / a_{p})^{3/8} /3$ )

A parabolic Kepler orbit obeys

t = 2 q^{3} / (GM) (D + D^{3} /3)

with

D = tan (f /2)

. At the high-eccentricity angular-speed crossover,

q ≪ r_{x}

, so

n_{p} t_{x} ≃ (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

π

of true anomaly while a distant perturber barely moves.

Periapsis

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

Pericentre distance ( $q$ )

Apoapsis

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

Apocentre distance ( $Q$ )

Semi-major axis ( $a$ )

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

Semi-minor axis ( $b$ )

For an ellipse of eccentricity

e

, the semi-minor axis is

b = a 1 - e^{2}

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.

Orbital plane

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.

Osculating orbital element

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.

True anomaly ( $f$ )

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

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

Phase-mixed orbit

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.

Line density on a Kepler orbit ( $λ_{ℓ}$ )

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

Cross-sectional-area current ( $J_{σ}$ )

The geometric cross-section passing a fixed point of a steady orbital wire per unit time is

J_{σ} = λ_{σ} v

, where

λ_{σ} = d σ / d s

. Conservation of fragments requires

J_{σ}

constant and

λ_{σ} \propto 1/ v

. For total area

σ_{tot}

and orbital period

P

J_{σ} = \frac{σ _{tot}}{P} .

With the fractional luminosity of a phase-mixed eccentric wire, this becomes

J_{σ} = 2 f h

. The current describes transported area, not the swept area enclosed by the orbit.

Phase-mixed radial probability of a Kepler orbit ( $p_{r} (r)$ )

For a Kepler orbit with pericentre distance

q

, apocentre distance

Q

, and semi-major axis

a = (q + Q) /2

, a uniform mean anomaly gives

p_{r} (r) = \frac{r}{πa ( r - q ) ( Q - r )}, q < r < Q .

The two radial passages per orbital period are included, and

\int_{q}^{Q} p_{r} (r) d r = 1

Mean anomaly ( $M$ )

Mean anomaly is the uniformly increasing phase

M = n (t - τ)

of a Kepler orbit, measured from periapsis time

τ

Kepler's equation

For an elliptic Kepler orbit, Kepler's equation relates mean anomaly

M

and eccentric anomaly

E

M = E - e sin E .

Eccentric anomaly ( $E$ )

The eccentric anomaly parametrizes a Kepler ellipse through the corresponding point on its auxiliary circle. It gives radius

r = a (1 - e cos E)

R_{z} (Ω) R_{x} (I)

. Its columns are the orbital frame's orthonormal basis expressed in reference coordinates. The unit vector normal is

(sin Ω sin I, - cos Ω sin I, cos I)

; its dot product with another such normal gives the cosine of their mutual inclination.

Mutual inclination ( $I_{m}$ )

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

Mutual inclination of nearly edge-on orbits ( $I_{m}^{2} ≃ (Δ I)^{2} + (ΔΩ)^{2}$ )

For two nearly edge-on orbits with small orbital inclination difference

Δ I

and small longitude of ascending node difference

ΔΩ

, expanding the dot product of their normals gives

I_{m}^{2} = (Δ I)^{2} + (ΔΩ)^{2} + O (δ^{4})

, with angles in radians and all offsets from edge-on of order

δ

. Thus equality of observed sky-plane inclinations alone does not imply aligned orbital planes.

Invariable plane

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.

Inclination to the invariable plane of two orbital planes

For two groups carrying angular momenta

L_{a}, L_{b}

with mutual inclination

Δ I

, resolving total angular momentum parallel and perpendicular to the second group's normal gives its orbital inclination relative to the invariable plane:

I_{b} = atan2 (L_{a} sin Δ I, L_{b} + L_{a} cos Δ I) .

For a positive denominator,

tan I_{b} = sin Δ I / (L_{b} / L_{a} + cos Δ I)

. The larger-angular-momentum group lies closer to the invariable plane.

Edge-on orbit

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

9 0^{\circ}

Orbital-plane orientation degeneracy

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

π

to its nodal longitude.

μ

has

n^{2} a^{3} = μ

Orbital period ( $T$ )

The orbital period is the time required to complete one orbit. It is related to mean motion by

T = 2 π / n

Synodic period ( $T_{syn}$ )

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 π /∣ n_{1} - n_{2} ∣

Specific orbital energy ( $ε$ )

The specific orbital energy is the mechanical energy per unit mass,

ε = \frac{v ^{2}}{2} - \frac{μ}{r} = - \frac{μ}{2 a} .

Its sign distinguishes bound elliptic, marginal parabolic, and unbound hyperbolic Kepler orbits.

Orbit-averaged drag work

A weak perturbing relative acceleration

f

changes specific orbital energy at rate

\overset{ε}{˙} = v \cdot f

. Since

ε = - μ / (2 a)

for a Kepler orbit,

\overset{a}{˙} = \frac{2 a ^{2}}{μ} v \cdot f .

For a circular Kepler orbit under

f = - κ v + g

with constant

g

, the constant-force term averages to zero over an unperturbed orbit. Thus

⟨ \overset{a}{˙} ⟩ = - 2 κa

. This is a leading-order orbital average; a strong constant force can distort the orbit instead.

Vis-viva equation

For a Kepler orbit with semi-major axis

a

, the vis-viva equation is

v^{2} = μ (\frac{2}{r} - \frac{1}{a}) .

For the convention that a hyperbolic Kepler orbit has positive semi-major-axis magnitude

a

, its final term instead has a plus sign.

Specific angular momentum ( $h$ )

The specific angular momentum is angular momentum per unit mass,

h = r \times v

. For a planar Kepler orbit its magnitude is

h = r^{2} \dot{θ} = μ a (1 - e^{2})

High-eccentricity angular-speed crossover ( $r_{x} = h / n_{p}$ )

For a nearly radial Kepler orbit with pericentre

q

, specific angular momentum is

h ≃ 2 GMq

. 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} = h / n_{p} ≃ a_{p} (2 q / 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

π

by approximately

2 q / r_{x}

Periapsis angular speed ( $\dot{θ}_{peri}$ )

The specific angular momentum of an elliptic Kepler orbit is

h = μ a (1 - e^{2})

. At periapsis,

r = a (1 - e)

, so

\dot{θ}_{peri} = \frac{h}{r ^{2}} = n \frac{1 + e}{( 1 - e ) ^{3/2}} .