Orbital Mechanics

Table of Contents

What is Orbital Mechanics?

Orbital mechanics is the study of applying celestial mechanics and ballistics to the motion of rockets and other spacecraft. It is essential for the planning and design of space missions [1]. Knowledge of fundamental calculus and dynamics is important to understand the concepts of orbital mechanics. But more advanced calculus and geometry will also be required when diving into more advanced topics within orbital mechanics.

How does an orbit work?

A simple way of describing how an orbit works is a concept called Newton’s Cannon. Imagine firing a canon pointed tangent to the earth. Also imagine that there is no atmosphere, so the canon will be travelling through a vacuum and the only force being exerted is that of earth's gravity. At a slow speed, the cannon ball will just fall right back to earth. But if the speed is fast enough. The cannonball will continue to fall to the earth but will never reach the ground and will continue to ‘orbit’ the earth. This phenomenon can be seen in figure to the right.

Newton's Cannon [2]

Kepler Orbits

In this article, we will be focusing on Keplerian orbits. Which make certain assumptions about the conditions of the motion between two bodies in space. The assumptions are as follows [3]:

The orbit forms a two-dimensional plane in three-dimensional space.
Consider only point-like gravitational attraction of two bodies (this is known as the two-body problem).
Neglect non-spherical central bodies.
Orbit is in a pure vacuum
Neglect relativistic effects.

In addition to these assumptions we will also assume that the mass of the object in orbit is much smaller than the object in which is it orbiting. In all equations, the symbolwill represent the mass of the object in orbit and the symbolwill represent the mass of the object in which massorbits.

Law of Gravitation and Centripetal Motion

The simplest orbital scenario to consider is the two-body problem. Like the earth orbiting the sun without the presence of any other planets. Where the only force acting on the earth is the gravitational force exerted by the sun. This makes the analysis a lot simpler without the need for numerical methods. A very simple circular orbit can be derived using Newton’s Law 2^nd law, Newton’s Universal Gravitation, and Uniform Circular Motion [4].

Combining the equations, we get the expression for the required speed for an object to orbit at the desired radius.

Note that the constantis Newton's gravitational constant.

Conservation of Energy

Consider the energy of a comet orbiting the earth (where the mass of the comet is significantly smaller than the earth’s. The effective potential energy of the system is [5]:

The nature of the orbit will be determined by what the effective potential energy is. A graph of the energy with respect to the radius can be seen below:

Effective Potential Energy [5]

When the effective potential energy of the system is greater than 0. The orbit will be unbounded, such like a hyperbolic or a parabolic orbit. But an orbit will be bounded if the energy is below 0 but greater than the minimum. As seen in the graph, there will be two radii in which the comet will oscillate between. Which will resemble an elliptical orbit. But if the energy is equal to the minimum. The comet will remain at that radius and will be in a circular orbit [5].

Orbital Parameters

There are other 6 parameters that are used to describe an orbit [4]:

Semi-Major Axis: represents half of the major axis and the mean distance from its primary.
Eccentricity: the distance between the foci divided by the major axis and is between 0 and 1. The eccentricity defines the shapes of the orbit.
Inclination: the angular distance between the orbit plane and the equator of it’s primary.
Argument of Periapsis: the angular distance between the ascending node and the point of periapsis. Periapsis is the point of the orbit closest to the primary. The furthest point to the primary is the Apoapsis.
Time of Periapsis Passage: the time to move through its point of periapsis.
Longitude of Ascending Node: nodes are points where the orbit crosses a plane. The ascending node is when the orbit crosses the plane going from south to north. The longitude of the ascending node is the node’s celestial longitude.

Orbital Elements [3]

Equation of an Orbit and Types of Orbits

The derivation of an equation for an orbit is complex and requires substitutions and the Lagrange equations. But the result of the radiusof the orbit as a function of the radial position with respect to periapsis(commonly called perihelion for orbits around the sun) (Taylor) [5]:

Whereis a constant andis the angular momentum of the object in orbit. The value of the eccentricity (ϵ) will determine the orbit’s shape [5].

Circular Orbits (ε = 0)
Elliptical Orbits (0 < ε < 1)
Unbounded Orbits (ε ≥ 1)

Common Shapes of Orbits [5]

An elliptical orbit can be described by the equation for an ellipse in cartesian form:

Where

Geometry of an Elliptical Orbit [5]

Kepler's Laws

Kepler’s first law states that the orbit of a planet is an ellipse, with the sun at the focus [5].

As a planet moves around the sun, the line from the sun to the planet sweeps out an equal are in equal time (constant sweeping of area) [5].

The square of the orbital period is proportional to the cube of the semi-major axis [5].

References

"Orbital Mechanics," Wikipedia, [Online]. Available: https://en.wikipedia.org/wiki/Orbital_mechanics. [Accessed 12 August 2021].
"Newton's Cannon," [Online]. Available: https://physics.weber.edu/schroeder/software/NewtonsCannon.html. [Accessed 12 August 2021].
"Kepler Orbit," Wikipedia, [Online]. Available: https://en.wikipedia.org/wiki/Kepler_orbit#Keplerian_elements. [Accessed 12 August 2021].
R. A. Braeunig, "Orbital Mechanics," 2013. [Online]. Available: http://www.braeunig.us/space/index.htm. [Accessed 12 August 2021].
J. R. Taylor, Classical Mechanics, Boulder: University Science Books, 2002.