Lecture Notes Elliptical OrbitsIn the previous lecture, we discussed the basics of circular orbits. Mastering even circular orbits provides quite a bit of intuitive behavior about the motion of spacecraft about planets We learnedprovides quite a bit of intuitive behavior about the motion of spacecraft about planets. We learned that orbiting spacecraft speed up as they get closer to their planets, how to classify earth orbits based on their altitudes, and touched upon basic trade-offs of these different orbits.In this lecture, our analysis becomes more general and elegant. We’ll deal with the general case of elliptical orbits using Kepler’s 3 laws.Page 1Lecture Notes Elliptical OrbitsKepler’s laws consist of 3 rules that two-body orbits must satisfy.The first law states that the orbit of a smaller body (spacecraft) about a larger body (planet or sun) is always an ellipse, with the center of mass of the larger body as one of the two foci. Above is the mathematical description of an elliptical path (use lower formula). Recall from geometry that the e-parameter is eccentricity, a measure of how oblong the ellipse is. Always between the values of 0 and 1, inclusive, eccentricity of 0 implies a perfect circle (like the number “0”) and an eccentricity of 1 implies an ellipse that has been stretched to zerothickness along one directioneccentricity of 1 implies an ellipse that has been stretched to zero-thickness along one direction (like the number “1”).Kepler’s second law states that the orbit of the smaller body sweeps out equal areas in equal time. What does this mean? A space craft in orbit around a planet, regardless of that orbit’s shape, will speed up when it is closer to the center of the planet and will slow down when it is further from the t Th t th ti l d fi iti i l t i t f l t thicenter. The swept area mathematical definition is an elegant, precise way to formulate this principle; we see this at work trivially in the case of circular orbits where the “pie-slice area” swept out was constant at all points of the orbit.Page 2Lecture Notes Elliptical OrbitsKepler’s third law allows us to compute the period of a satellite in elliptical orbit given its geometrical propertiesgeometrical properties.Consider the geometry of an ellipse as described by its semi-major axis (a) and its semi-minor axis (b). The greater the eccentricity, the large (a) becomes relative to (b). In fact, there is a simple relationship (given above) between all 3 of these quantities.The planet must rest on one of the foci of this ellipse. The perigee (on earth; periapsis in the general case) is the distance between the origin and the spacecraft at the closest distance to the planet. This distance occurs along the major axis and is basically (1-e)a. The apogee (on earth; apoapsis in the general case) is the distance between the origin and the spacecraft at the furthest distance to the planet. This distance also occurs along the major axis on the opposite end of the orbit, for a total distance (1+e)a. Note that the average of apogee and perigee will always produce th ijithe semi-major axis a.At apogee and perigee, the orbiting spacecraft has no radial velocity component – all velocity is tangential. Also note that, although the speed of the spacecraft changes during its orbit, there is a temporal symmetry about the major axis to its travels. A journey from perigee to apogee, and vice versa, takes exactly one half of a period T.Page 3Lecture Notes Elliptical OrbitsIt is an unavoidable fact of life in a fallen world that much of learning in a profession is couched in specific jargon the language of which often obscures what is actually very simple concepts Inspecific jargon, the language of which often obscures what is actually very simple concepts. In some fields, practitioners take advantage of this jargon and use it as a sort of blockade to any novice from entering a profession that is otherwise straightforward and even a little lite on concepts (fill in your own joke here).Normally we engineers and the more practical scientists try to avoid this, but in the case of apsis and periapsis it would appear that we have tried to make this as complicated as possible Vastlyand periapsis, it would appear that we have tried to make this as complicated as possible. Vastly different terms are applied to this very simple concept depending on what the spacecraft is orbiting. In fact, it gets so ridiculous, that one usually must delve into the legal profession or the field of biology to find this level of nomenclature chicanery.The terms apsis and periapsis are general; apogee and perigee are applied when the orbit is around th h li d ih li li d h th bit i d th d f th Obearth; aphelion and perihelion are applied when the orbit is around the sun; and so forth. Observe the enormous variety in the table above. As soon as scientists discover something new to orbit around, they feel compelled to invent new terminology to re-describe an ellipse.Particularly humorous is the apparent multiplicity of orbital terms for black holes. We’ve never sent a satellite into orbit about a black hole and likely have no intentions of doing so for the next f ill i d t th t l d i t th ti t i l i f hPage 4few millennia, and yet there appears to already exist three competing terminologies for such an orbit.Lecture Notes Elliptical OrbitsAn initial rocket stage launches satellites into LEO. From there, depending on how high the final orbit is a transfer orbit or trajectory is required to move the satellite into its final orbit Aorbit is, a transfer orbit or trajectory is required to move the satellite into its final orbit. A Hohmann transfer orbit is one such example, commonly used because of its low-energy requirements and short transit time.The Hohmann transfer orbit requires a short, energetic burst of thrust from the spacecraft in LEO. By adding kinetic energy at what is nearly a single point in the orbit (Point A), the spacecraft has been transferred to an elliptical orbit, swinging round the planet to a much higher altitude at apogee (Point B). Another burst of thrust at the apogee of the elliptical orbit transfers theapogee (Point B). Another burst of thrust at the apogee of the elliptical orbit transfers the spacecraft into a higher-energy orbit. If the amount of thrust is carefully calculated, this orbit can be “circularized” and the final MEO, GEO, or HEO achievied.The Hohmann