Orbital Mechanics Equation

Orbital Mechanics Equations

Mathematical formulations describing the motion of celestial bodies under gravitational influence, including two-body and N-body problems.

Kepler's Laws of Planetary Motion - Empirical orbital laws
Observable Universe - Cosmic scales
Game Theory - N-body problem complexity

Two-Body and N-Body Dynamics

The fundamental equation describing the motion of two celestial bodies is given by Newton's Law of Universal Gravitation:

F = \frac{G \times m_{1} \times m_{2}}{r^{2}}

where $F$ is the force of gravity between the two objects, $G$ is the gravitational constant ( $6.67430 \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}$ ), $m_{1}$ and $m_{2}$ are the masses of the objects, and $r$ is the distance between their centers.

Kepler's Equation

Kepler's equation relates the mean anomaly $M$ , the eccentric anomaly $E$ , and the eccentricity $e$ of an orbit:

M = E - e \sin E

Orbital Period

The orbital period $P$ of an object can be calculated using the following equation:

P = 2 π \sqrt{\frac{a^{3}}{μ}}

where $a$ is the semi-major axis of the orbit and $μ$ is the standard gravitational parameter ( $μ = G \times M$ , where $M$ is the mass of the central body).

Vis-Viva Equation

The vis-viva equation relates the current distance $r$ from the central body to the specific orbital energy $h$ :

v^{2} = μ (\frac{2}{r} - \frac{1}{a})

where $v$ is the relative velocity between the two objects, and $a$ is the semi-major axis.

Specific Mechanical Energy

The specific mechanical energy $E$ of an object in orbit is given by:

E = \frac{v^{2}}{2} - \frac{μ}{r}

where $v$ is the velocity of the object, $r$ is its distance from the central body, and $μ$ is the standard gravitational parameter.

Escape Velocity

The escape velocity $v_{e}$ from a gravitational body is calculated as:

v_{e} = \sqrt{2 μ (\frac{1}{r})}

where $r$ is the distance from the center of the body, and $μ$ is the standard gravitational parameter.

Orbital Velocity

For a circular orbit, the orbital velocity $v$ is given by:

v = \sqrt{μ (\frac{1}{r})}

where $r$ is the radius of the circular orbit, and $μ$ is the standard gravitational parameter.

Angular Velocity

The angular velocity $ω$ of an object in orbit is related to the period $P$ by:

ω = \frac{2 π}{P}

Universal Variable Formulation

The universal variable formulation is used to address the two-body problem for various orbit types. It introduces the universal variable $u$ :

r = a (1 - e \cos u)

\dot{r} = - a e \sin u \dot{u}

θ - θ_{0} = u + e \sin u

where $r$ is the distance, $a$ is the semi-major axis, $e$ is the eccentricity, $θ$ is the true anomaly, and $\dot{u}$ is the derivative of $u$ with respect to time.

Two-Body Equation of Motion

The fundamental equation describing the motion of two celestial bodies is given by Newton's law of universal gravitation:

F = \frac{G \times m_{1} \times m_{2}}{r^{2}}

Three-Body and N-Body Problems

Three-Body Problem

The three-body problem does not have a general closed-form solution. However, we can describe it using numerical methods and approximations:

Euler's Three-Body Problem:
This formulation involves solving for the motions of three bodies interacting gravitationally. It is a specific instance of the three-body problem with particular initial conditions.

Restricted Three-Body Problem:
In this formulation, one body is assumed to have negligible mass compared to the other two, effectively making it a two-body problem with an additional perturbing force. The equations of motion for the restricted three-body problem are:

{\ddot{r}}_{1} = - \frac{G M_{2}}{r_{2}^{3}} r_{2} - \frac{G M_{3}}{r_{3}^{3}} r_{3}

{\ddot{r}}_{2} = - \frac{G M_{1}}{r_{1}^{3}} r_{1} - \frac{G M_{3}}{r_{3}^{3}} r_{32}

where $G$ is the gravitational constant, $M_{i}$ and $r_{i}$ are the masses and position vectors of the bodies, and $r_{i j} = | r_{i} - r_{j} |$ .

N-Body Problem

The N-body problem involves determining the motions and interactions of $N$ celestial bodies under their mutual gravitational attractions. There is no general closed-form solution for $N > 2$ . Numerical methods are employed to approximate solutions:

Gravitational N-Body Simulation:
This involves discretizing time, calculating accelerations due to pairwise gravitational interactions, and updating positions and velocities iteratively:

a_{i} = \sum_{j = 1, j \neq i}^{N} \frac{G M_{j} (r_{j} - r_{i})}{| r_{j} - r_{i} |^{3}}

r_{i} (t + Δ t) = r_{i} (t) + v_{i} (t) Δ t + \frac{1}{2} a_{i} (t) Δ t^{2}

v_{i} (t + Δ t) = v_{i} (t) + \frac{1}{2} (a_{i} (t) + a_{i} (t + Δ t)) Δ t

where $r_{i}$ , $v_{i}$ , and $a_{i}$ are the position, velocity, and acceleration of the $i$ -th body, $M_{i}$ is its mass, and $Δ t$ is the time step.

Perturbation Theory:
Perturbation theory treats the N-body problem as a two-body problem with additional forces causing deviations from the unperturbed two-body trajectory:

a_{i} = - \frac{G M_{0}}{r_{i}^{3}} r_{i} + \sum_{j = 1, j \neq i}^{N} \frac{G M_{j} (r_{j} - r_{i})}{| r_{j} - r_{i} |^{3}}

where $M_{0}$ is the mass of the central body.

Next Action

Add examples of orbital transfers (Hohmann, bi-elliptic)