Orbital Dynamics

The base solution of the two body problem are the Keppler's Laws.

All the equations can be found in the formulary:

The 6 orbital elements

2 orbit parameters:

$a$ Semi-major axis
$e$ eccentricity
alternative:
$b$ Semi-minor axis
$p$ Semi-parameter
$r_{p}$ Periapsis
$r_{a}$ Apoapsis

3 orientation parameters

$Ω$ Longitude of the ascending node
$i$ Inclination
$ω$ Argument of periapsis

1 position parameter

$ϑ$ True Anomaly
for circular orbits: longitude of the satellite from ascending node
for equatorial orbits: basic angle from vernal equinox

From $a$ and $e$ to $r_{p}$ and $r_{a}$

\begin{aligned} a & = \frac{r_{a} + r_{p}}{2} \\ e & = \frac{r_{a} - r_{p}}{r_{a} + r_{p}} \\ r_{p} & = a (1 - e) \\ r_{a} & = a (1 + e) \end{aligned}

True vs Mean vs Eccentric anomaly

Definition-of-true-v-eccentric-E-and-mean-M-anomaly.png

True Anomaly (v): The effective angle that is formed by the Planet-Focus Point-Periapsis.
Eccentric Anomaly (E): The Angle between the Periapsis-Center-Projection from the Planet on the surrounding Circle.
Mean Anomaly (M): The angle between Periapsis-Center-the average point after this time from the periapsis.

Orbit Terminology

Orbit classification by altitude:

LEO (low-Earth orbit) < 1′000 km
GEO (geosynchronous orbit) = 35′856 km
MEO (medium-Earth orbit)
HEO (highly elliptical orbit)
Super-Synchronous
Lunar and Lagrange point orbit ≈ 350 ′000 km
Interplanetary orbit
Interstelar orbit

Orbit classification by inclination

prograde orbit < 90°
retrograde orbit > 90°
polar orbit = 90°
equatorial orbit = 0°

Orbit classification by function

mission orbit
transfer orbit
parking orbit
graveyard orbit
phasing orbit

Ballistic trajectory

elliptic orbit intersecting with Earth's surface

Orbit Perturbations

Small perturbations change the perfect Kepler systems, so the orbital elements are no longer constant in time anymore.

Perturbations can come from:

asymmetry of the gravitational field, mainly Earth's oblateness
friction in the upper atmosphere
gravitational attraction of the Moon or/and the Sun
other forces (radiation pressure, electromagnetic forces, general relativity) mostly negligible

The perturbations are added as a small force linearly to the equation of motion and then decomposed in the rotating frame of reference. This can then be expanded into the first derivatives in time for the orbital elements, the Lagrange planetary equations.

\frac{d a}{d t} = 2 \sqrt{\frac{a^{3}}{μ (1 - e^{2})}} [f_{ξ} e \sin ϑ + f_{η} (1 + e \cos ϑ)]

\frac{d e}{d t} = \sqrt{\frac{a (1 - e^{2})}{μ}} [f_{ξ} \sin ϑ + f_{η} (\cos ϑ + \cos E)]

\frac{d i}{d t} = \sqrt{\frac{a (1 - e^{2})}{μ}} \frac{1}{1 + e \cos ϑ} f_{ζ} \cos (ϑ + ω)

\frac{d Ω}{d t} = \sqrt{\frac{a (1 - e^{2})}{μ}} \frac{1}{1 + e \cos ϑ} f_{ζ} \frac{\sin (ϑ + ω)}{\sin i}

\frac{d ω}{d t} = \sqrt{\frac{a (1 - e^{2})}{μ}} \frac{1}{e} [- f_{ξ} \cos ϑ + f_{η} \frac{(2 + e \cos ϑ) \sin ϑ}{1 + e \cos ϑ}] - \frac{d Ω}{d t} \cos i

Asymmetry of the Gravitational Field

The gravitational potential can be represented by a spherical harmonic expansion:

U = - \frac{μ}{r} [1 - \sum_{n = 2}^{\infty} {(\frac{a_{E}}{r})}^{n} J_{n} P_{n} (\sin φ) + \sum_{n = 2}^{\infty} \sum_{m = 1}^{n} {(\frac{a_{E}}{r})}^{n} (C_{n m} \cos m λ + S_{n m} \sin m λ) P_{m n} (\sin φ)]

The dominant perturbing term is the oblateness of earth:

U_{2} = \frac{μ}{r} {(\frac{a_{E}}{r})}^{2} J_{2} \frac{1 - 3 \sin^{2} φ}{2}

$U_{2}$ produces a torque that which rotates the orbit in the equatorial plane, leading to a nodal regression, which is zero if the inclination is 90°:

\frac{d \bar{Ω}}{d t} = - \frac{3}{2} {(\frac{μ a_{E}^{4}}{a^{7}})}^{1 / 2} \frac{J_{2} \cos i}{(1 - e^{2})^{2}}

\frac{d \bar{Ω}}{d t} = - 9.964 {(\frac{a_{E}}{a})}^{7 / 2} \frac{\cos i}{(1 - e^{2})^{2}} °/24 h for Earth

$U_{2}$ also produces a pull that moves the orbit towards the equatorial plane, leading to a apsidal precession with is zero at an inclination of 63.435°:

\frac{d \bar{ω}}{d t} = \frac{3}{4} {(\frac{μ a_{E}^{4}}{a^{7}})}^{1 / 2} \frac{J_{2}}{(1 - e^{2})^{2}} (4 - 5 \sin^{2} i)

\frac{d \bar{ω}}{d t} = 4.982 {(\frac{a_{E}}{a})}^{7 / 2} \frac{4 - 5 \sin^{2} i}{(1 - e^{2})^{2}} °/24 h for Earth

$U$ : Gravitational potential
$U_{2}$ : Dominant perturbing potential due to Earth's oblateness (J2 effect)
$μ$ : Standard gravitational parameter ( $G M$ )
$r$ : Distance from the center of the Earth
$a_{E}$ : Earth's equatorial radius
$a$ : Semi-major axis of the orbit
$e$ : Orbital eccentricity
$i$ : Orbital inclination
$φ$ : Geocentric latitude
$λ$ : Geocentric longitude
$J_{n}$ : Zonal harmonics (unnormalized)
- $J_{2}$ represents the oblateness (Earth: $J_{2} = 1.08263 \cdot 10^{- 3}$ )
$C_{n m}, S_{n m}$ : Tesseral and sectorial harmonic coefficients
$P_{n}, P_{m n}$ : Legendre polynomials and associated Legendre functions
$\frac{d \bar{Ω}}{d t}$ : Rate of nodal regression (precession of the ascending node)
$\frac{d \bar{ω}}{d t}$ : Rate of apsidal precession (rotation of the argument of perigee)

Atmospheric Drag

The drag induces a radius decrease for initially circular orbits.

Orbital Maneuvers

These are moved to Propulsed Dynamics