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. (cite Wikipedia). 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 #:
| Newton's Cannon |
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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 2nd law, Newton’s Universal Gravitation, and Uniform Circular Motion (Jin).
Combining the equations, we get the expression for the required speed for an object to orbit at the desired radius.
Note that the mass
is the mass of the object that is the center of the orbit. Such as the sun in the case of the earth-sun system. And the massis the mass of the oject in orbit. And of course the constant
is 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 (Taylor):
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 (Taylor figure 8.4):
| Effective Potential Energy |
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When the total 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 (Taylor).
Orbital Parameters
There are other 6 parameters that are used to describe an orbit (Jin):
- 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 |
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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 perihelion
for an object of mass
orbiting a much larger object of mass
(Taylor):
Whereis a constant and
is the angular momentum of the object in orbit. The value of the eccentricity (ϵ) will determine the orbit’s shape (Taylor).
- Circular Orbits (ε = 0)
- Elliptical Orbits (0 < ε < 1)
- Unbounded Orbits (ε ≥ 1)
| Common Shapes of Orbits |
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An elliptical orbit can be described by the equation for an ellipse in cartesian form:
Where
| Geometry of an Elliptical Orbit |
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Kepler's Laws
Kepler’s first law states that the orbit of a planet is an ellipse, with the sun at the focus (Taylor).
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) (Taylor).
The square of the orbital period is proportional to the cube of the semi-major axis (Taylor).