# General functions

This package contains some functions that helps in analysis of orbits.

## Angular velocity

The angular velocity of an object in orbit when considering a Keplerian orbit (unperturbed model) is given by:

$$$n = n_0 = \sqrt{ \frac{\mu_0}{a^3} }~,$$$

where $\mu_0$ is the standard gravitational parameter for Earth, and $a$ is the semi-major axis.

If the perturbation terms up to $J_2$ are considered, then the angular velocity is computed by:

$$$n = n_0 + \frac{3}{4} \cdot \frac{R_0^2 \cdot J_2}{a^2\left(1-e^2\right)^2} \cdot n_0 \cdot \left[\sqrt{1-e^2}\cdot(3\cos^2(i)-1) + (5\cos^2(i) - 1) \right]~,$$$

where $e$ is the eccentricity, $i$ is the inclination, and $R_0$ is the Earth equatorial radius.

In this package, the angular velocity [rad/s] can be computed by the following functions:

function angvel(a::Number, e::Number, i::Number, pert::Symbol = :J2)
function angvel(orb::Orbit, pert::Symbol = :J2)

where:

• a is the semi-major axis [m];
• e is the eccentricity;
• i is the inclination [rad];
• pert selects the perturbation terms it should be used, it can be :J0, :J2, or :J4[1]; and
• orb is an instance of Orbit.
julia> angvel(7130982.0, 0.001111, deg2rad(98.405))
0.0010471974485046116

0.0010484431282179

## Time-derivative of the argument of perigee

The time-derivative of the argument of perigee $\dot{\omega}$ when considering perturbation terms up to $J_2$ is:

$$$\dot{\omega} = \frac{3}{4} \cdot \frac{R_0^2 \cdot J_2}{a^2\left(1-e^2\right)^2} \cdot n_0 \cdot (5\cos^2(i) - 1)$$$

where $R_0$ is the Earth equatorial radius, $a$ is the semi-major axis, $e$ is the eccentricity, $i$ is the inclination, and $n_0$ is the unperturbed orbital angular velocity.

In the unperturbed model (Keplerian orbit), the time-derivative of the argument of perigee is always 0.

In this package, the time-derivative of the argument of perigee [rad/s] can be computed by the following functions:

function dargp(a::Number, e::Number, i::Number, pert::Symbol = :J2)
function dargp(orb::Orbit, pert::Symbol = :J2)

where:

• a is the semi-major axis [m];
• e is the eccentricity;
• i is the inclination [rad];
• pert selects the perturbation terms it should be used, it can be :J0, :J2, or :J4[1]; and
• orb is an instance of Orbit.
julia> dargp(7130982, 0.001111, deg2rad(98.405))
-6.082892348533058e-7

-2.433253158726004e-12

0.0

## Time-derivative of the RAAN

The time-derivative of the RAAN (right-ascension of the ascending node) $\dot{\Omega}$ when considering perturbation terms up to $J_2$ is:

$$$\dot{\Omega} = -\frac{3}{2} \cdot \frac{R_0^2 \cdot J_2}{a^2\left(1-e^2\right)^2} \cdot n_0 \cdot \cos(i)$$$

where $R_0$ is the Earth equatorial radius, $a$ is the semi-major axis, $e$ is the eccentricity, $i$ is the inclination, and $n_0$ is the unperturbed orbital angular velocity.

In the unperturbed model (Keplerian orbit), the time-derivative of the RAAN is always 0.

In this package, the time-derivative of the RAAN [rad/s] can be computed by the following functions:

function draan(a::Number, e::Number, i::Number, pert::Symbol = :J2)
function draan(orb::Orbit, pert::Symbol = :J2)

where:

• a is the semi-major axis [m];
• e is the eccentricity;
• i is the inclination [rad];
• pert selects the perturbation terms it should be used, it can be :J0, :J2, or :J4[1]; and
• orb is an instance of Orbit.
julia> draan(7130982, 0.001111, deg2rad(98.405))
1.9909533223838115e-7

0.0

## Period

The orbital period of an object in orbit is given by:

$$$T = \frac{2\pi}{n}$$$

where $n$ is the angular velocity as described in Angular velocity.

In this package, the orbital period [s] can be computed by the following functions:

function period(a::Number, e::Number, i::Number, pert::Symbol = :J2)
function period(orb::Orbit, pert::Symbol = :J2)

where:

• a is the semi-major axis [m];
• e is the eccentricity;
• i is the inclination [rad];
• pert selects the perturbation terms it should be used, it can be :J0, :J2, or :J4[1]; and
• orb is an instance of Orbit.
julia> period(7130982, 0.001111, deg2rad(98.405))/60
100.00000980636328

99.88119746433748
orbit to compute the values. If pert is :J2, then it will be considered the perturbation terms up to $J_2$ to compute the values. Otherwise, if pert is :J4, then it will be considered the perturbation terms $J_2$, $J_4$, and $J_2^2$ to compute the values. If pert is omitted, then it defaults to :J2.
• 1If pert is :J0, then it will be considered a Keplerian, unperturbed