It has a semi-major axis, a, of approximately 149,598,260 km and an eccentricity of 0.0174.
The specific mechanical energy, is constant, so potential energy and kinetic energy are traded off according to the relationshipĪn example of an elliptical orbit is the Earth’s orbit around the sun. The focus of the conic section is the center of the central body. Conic sections are the only possible paths for an orbiting object governed by the ideal two body equation of motion. Let us review what we know about orbits so far before taking a look at special types of orbits. For a detailed derivation of the period equation, see Bate. The time for a satellite to go once around its orbit is called the period. Since ellipses are closed curves, an object in an ellipse repeats its path over and over. P = semi-latus rectum = the magnitude of the position vectors at = 90 degrees and 270 degrees Īt exactly apogee and perigee on an ellipse, the position and velocity vectors will be perpendicular so the velocity vector is parallel to the local horizon, hence = 0. When travelling from apogee to perigee, the velocity vector will always be below the local horizon (losing altitude) so <0 for. ThusĮccentricity defines the shape of the conic section. R a = Radius of apoapsis (farthest point) = Radius of apogee when the satellite is around the earth = a(1+e)Ģc is the distance between the foci = R a – R pĮ = eccentricity, the ratio of the distance between the foci (2c) to the length of the ellipse (2a), or. R p = Radius of periapsis (closest point) = Radius of perigee when the satellite is around the earth = a(1-e) = satellite velocity vector, tangent to the orbital pathį and F’ = primary (occupied) and vacant (unoccupied) foci of ellipse R is the radius from the focus of the ellipse (center of the earth) to the satellite, or, the magnitude of the vector = satellite position vector, measured from the center of the Earth Let us take a closer look at the geometry of an orbital ellipse that we considered in Chapter 1 and describe each of its parameters more precisely. The velocity in elliptical orbits is always less than that needed to escape from the central object’s influence. Not only do the planets and minor planets have elliptical orbits, most comets and binary stars also do. These are the most common orbits because one object is ‘captured’ and orbits another larger object. Kepler’s First Law said the orbits of the planets are ellipses. The fuel that the rocket saves and then be used to accelerate it horizontally in order to attain high speed and more easily enter into orbit. This technique allows the rocket to use Earth’s gravity, rather than its own fuel, to change its direction. This technique of optimizing a trajectory of a rocket so that it attains the desired path is called a gravity turn. However, you will notice that it begins to tilt horizontally and gradually increases this tilt until it achieves orbit around the earth. Try the simulator yourself and see what happens! (Schroeder, 2020).Ī rocket launches vertically to get as quickly as possible out of the earth’s thick atmosphere and minimize drag. Each time the cannonball goes a little farther before hitting the earth. In this simulation, the cannon was fired at 3000 m/s, 4000 m/s, 5000 m/s and 6000 m/s. As an example, try the Newton’s Cannon simulator below. The horizontal and vertical motions are independent, hence, the faster you fire the cannonball, the farther it will go. The vertical motion of the projective, or cannonball, is influenced by the gravitational acceleration, g, pulling downward.
The horizontal motion of the projectile is the result of the tendency of any object in motion to remain in motion at a constant velocity. We can think of the cannonball as a projectile, an object that, when it is set in motion, continues in its path by its own inertia and is influenced only by the downward force of gravity. We will then review the elliptical orbit and its parameters and then extend these results to consider the other conic sections.įirst, how does a satellite get into orbit? Imagine you are at the top of a mountain and and start firing a cannon from it. We will start this chapter with a discussion of how a satellite gets into orbit and relate it to the conic sections. However, in general the solution can be any of the four conic sections: circles, ellipses, parabolas and hyperbolas. In Chapter 1, the Two Body Equation of Motion was developed and we discussed how the elliptical orbit was one possible solution.
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