2.1 Orbital Mechanics:
What are orbital mechanics?
"Orbital mechanics, or astrodynamics, is the branch of physics and engineering that calculates the trajectories of artificial satellites and spacecraft, as well as the orbits of natural celestial bodies"
In this part of the article we are going to look in depth at a few different equations and how to derive them those being:
Orbital Elements: Apoapsis, periapsis, semi-major axis, eccentricity
2-Body Problem: Earth-centered gravitational model
Vis-Viva Equation: Accurate orbital velocity calculations
Kepler's Equation: Eccentric anomaly calculations for time predictions
But to be able to understand the derivation/use of these equations first you have to have an understanding of the base of orbital mechanics those being:
Newton's law of universal gravitation$(F = G\frac{M_{1}M_{2}}{r^{2}})$
Specific orbital energy $(\varepsilon = \frac{v^{2}}{2} - \frac{\mu}{r} = - \frac{\mu}{2a})$
Conservation of Angular momentum ${\left({h}={r}\cdot{v}\right)}$
Newton's law of universal gravitation:
$F = G\frac{M_{1}M_{2}}{r^{2}}$
This equation describes the force that one object applies on another, its given in Newtons. Any two objects m1, m2 have a force that's applied one on the other with the larger one attracting the smaller one. G is the gravitational constant which is given as
$G = 6.6743*10^{- 11}\frac{m^{3}}{kgs^{2}}$
To learn more about it: https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
Specific Orbital Energy
$(\varepsilon = \frac{v^{2}}{2} - \frac{\mu}{r} = - \frac{\mu}{2a})$
"Specific orbital energy is the total mechanical energy (sum of kinetic and potential energy) per unit mass of an orbiting body, remaining constant throughout its orbit." This is specifically useful when it comes to elliptical orbits, in elliptical orbits just like any other orbit energy stays constant throughout the orbit but what does that mean when you get further away from the object you're orbiting? Well according to this equation the further we get from the object the slower our speed needs to go to keep our Orbital energy constant. This equation can usually be applied in similar problems to where you would use KE and PE but in orbital settings.
$\mu = GM$
Where G and M are the same as in the force equation
To learn more about it:
https://en.wikipedia.org/wiki/Specific_orbital_energy
Conservation of Angular momentum
Conservation of Angular momentum is particularly important because it relates the speed of a object orbiting to the radius of the orbit, basically it says that the further out you go the lower your velocity is going to be, its given by
$h = r*v\ *sin\theta$
Where theta is the angle between the position vector and the velocity vector, this means that in a circular orbit where they are always perpendicular it will always be $\sin\theta = 1$ This problem comes up in many advanced problems when trying to model an orbit because it gives you a value that's constant and allows you to solve for the others.
To learn more about it: