Note: check attached files for visuals of specific concepts.
Mankind’s fascination with the stars is seemingly perennial, a relationship which would form the basis of early human culture, religion, and navigation. Although these influences remain pertinent, humanity’s connection with the night sky has evolved over time and given way to a new age of exploration and intense scrutiny of the cosmos. With the advent of rocketry came the necessity for precision spacecraft travel and thus humanity’s minds began working on solutions.
Conic sections are the shapes that result from the cross section of a cone at various angles and areas. One of the most well known conic sections is the circle. Although technically an ellipse, the circle is unique because the radius is always uniform no matter what point on the circle one joins the center to. There are three conic sections, the ellipse, the parabola, and the hyperbola, respectively. A conic section is assigned a number called eccentricity; when eccentricity is equal to zero, the conic section can be identified as a circle. Consider the case of an ellipse, the eccentricity is always greater than zero and less than one, essentially, the eccentricity is a ratio to help us understand how stretched or warped this conic section is compared to a circle (although this definition is not very technical). Equidistant from the center of the ellipse and along the major axis lies two points called foci. The parabola is a trajectory which can be a ballistic; however, more commonly, it is an escape trajectory.
Now we must extend discussion to Kepler’s laws, which lay out the basis of planetary motion. Kepler’s first law states that the orbit of the planets take an elliptical shape, with the planet orbiting around a celestial body which exists at one of the two foci on the ellipse’s major axis. This law suggests that the orbiting body experience fluctuating altitude and thus has an extreme far point, known as apoapsis, and an extreme low point, known as a periapsis. However, for an orbit with an eccentricity of zero (circular), there are no extreme points. Continuing, Kepler’s second law states, “the line joining the sun and the planet sweeps out equal areas in equal times.” This second law has important implications. Consider an object orbiting at apoapsis, because it is farther away, it will end up covering a larger area at any given instant. Now consider an object orbiting at periapsis, because it is closer to the celestial body, at any given instant it will cover an area less than that of an object that is orbiting farther away. Thus, because of the second law, we are forced to understand that at apoapsis, the object has less velocity than an object at periapsis. Side note: this is an interesting idea which relates back to the idea of conservation energy because as the object coasts away from the celestial body it loses kinetic energy and gains gravitational potential energy and vise versa.
Kepler, now confident in his ability to comb through data published his third law: “The squares of the periods of the planets are proportional to the cubes of their mean distances from the sun”. The wording makes this law needlessly confusing, but essentially this law relates the orbital period or amount of time it takes for one full orbit to the semi-major axis, which is the major axis of the ellipse divided by 2. It is unfair for us to not mention Isaac Newton, a man who mainly dabbled in occult studies like chronology, alchemy, and biblical interpretation, yet on his off days published the greatest scientific findings about physics and mathematics ever. Newton’s law of universal gravitation states that the farther you are from an object the less gravitational force you experience, furthermore the larger the mass, the more force due to gravitation you will experience. This relates back to the idea of the energy exchange in the second law, and also helps to provide a much needed element which was unaccounted for in Kepler’s third law. Because all objects have mass, they exert gravitational forces, thus, two objects orbiting each other that are massive objects have less than negligible gravitational forces which implies that objects in a system will orbit around the center of mass. This is true for all groups of celestial objects, including our own star system.
The most frequently utilized orbital maneuver is referred to as the Hohmann transfer orbit. When in a circular parking orbit, one will burn in the direction of the prograde vector until the orbit becomes elliptical. At this point, one cruises until reaching the apoapsis of the new elliptical orbit and burns in the direction of the prograde vector once more until the orbit is of zero eccentricity. The change in velocity needed to perform a maneuver such as this can be calculated using the vis-viva equation which is derived from Newton’s law of universal gravitation and kepler’s third law. The Hohmann transfer has a variety of applications, especially in regards to travelling to other celestial bodies; however, it is more nuanced as you must include phase angles and other complex concepts.
Baker, Robert and Maud Makemson. An Introduction to Astrodynamics 2nd ed., New York, Academic Press, 1967.
Braeunig, Robert. Rocket and Space Technology. http://www.braeunig.us/space/index_top.htm
Lodgson, Tom. Orbital Mechanics: Theory and Applications. New York, Wiley, 1997.
Elert, Glenn. The Physics Hypertextbook. https://physics.info/
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