Chapter 2: Orbits and Launching Methods
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1 9/20/ Chapter 2: Orbits and Launching Methods Prepared by Dr. Mohammed Taha El Astal EELE 6335 Telecom. System Part I: Satellite Communic ations Winter Content Kepler s First, Second, and Third Law Definitions of Terms for Earth-Orbiting Satellites Orbital Elements Apogee and Perigee Heights Orbit Perturbations Inclined Orbits 1
2 9/20/ 2.1 Introduction Satellite/Spacecraft orbiting the earth follow the same laws that govern the motion of the planets around to sun. Johannes Kepler ( ) was able to derive empirically three laws describing planetary motion. Later, Isaac Newton derived Kepler's laws from his own laws of mechanics and developed the theory of gravitation. Kepler s laws can be applied almost for any two bodies in space interact through gravitation. interact through gravitation? Variables : mass1 mass 2 Distance r (Less massive) : Secondary or Satellite (More massive) : Primary 2
3 9/20/ 2.2 Kepler s First Law The path followed by a satellite around the primary will be an ellipse The center of mass of the two-body system (barycenter) is always centered on one of the foci. In our case (sat.+earth), the barycenter coincides with the center of the earth The earth is always at one of the foci. The eccentricity e if given by e = a2 b 2 Elliptical orbit: 0 < e < 1. e= 0, the orbit becomes circular. Refer to App. B for details of the e and ellipse. The e & a are two of the orbital parameters. a Question: From the original definition of e, derive the most common formula of e : a2 b 2 a x a 2 + y b area = π a b perimeter 2π a2 + b 2 2 = 1 2 gets a more elongated the tangent line has equal angles with the two lines going to each focus f : linear eccentricity e is the ratio of distance between two focus to the length of major axis=2f/2a=f/a 3
4 9/20/ 2.3 Kepler s Second Law For equal time intervals, a satellite will sweep out equal areas in its orbital plane, focused at the barycenter because of the equal area law, it follows that the velocity at S2 is less than that at S1. Mean that the satellite takes longer to travel a given distance when it is farther away from earth. They used this property to increase the length of time a satellite can be seen from particular geographic regions of the earth. 4
5 9/20/ 2.4 Kepler s Third Law The square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies: P 2 /a 3 is constant value Orbital period: P = 2π n (n in rad/sec), where, n is the mean motion of the satellite in rad/sec Mean motion: it is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. Mean distance the arithmetic mean of the greatest and least distances of a satellite from the earth. Mean distance = semimojor axis (a) Third Law become as: a 3 = μ n 2, where, μ is the earth s geocentric gravitational constant = m 3 /s 2 N.B. This equation applies only to the ideal situation : a perfectly spherical earth of uniform mass, with no perturbing forces acting, such as atmospheric drag. (see Sec. 2.8) (3 rd law: another perspective) In a way, you can deduce directly that P 2 /a 3 is constant value for any planetary (satellite/spacecraft) motion, this mean? a++ P++ large ellipse/orbit results longer time to complete a period (in other words, slower motion) a-- P-- Smaller ellipse/orbit results shorter time to complete a period (in other words, faster motion) 5
6 9/20/ 2.5 Definition of Terms of Earth-Orbiting Satellites: Subsatellite path: this is the path traced out on the earth s surface directly below the satellite. Apogee: the point farthest from earth. Apogee height (ha) Perigee: The point of closest approach to earth. Perigee height (hp). Line of apsides: The line joining the perigee and apogee through the center of the earth. Ascending node: The point where the orbit crosses the equatorial plane going from south to north. Descending node: The point where the orbit crosses the equatorial plane going from north to south. Line of nodes: The line joining the ascending and descending nodes through the center of the earth. 6
7 9/20/ Inclination (i): the angle between the orbital plane and the earth s equatorial plane. It is measured at the ascending node from the equator to the orbit, going from east to north. i relate to subsatellite path?? It will be seen that the greatest latitude, north or south, reached by the subsatellite path is equal to the inclination. Prograde orbit: an orbit in which the satellite moves in the same direction as the earth s rotation. (also known as a direct orbit) i of a prograde orbit lies between Most satellites are launched in a prograde orbit, why?? because the earth s rotational velocity provides part of the orbital velocity with a consequent saving in launch energy. Retrograde orbit: An orbit in which the satellite moves in a direction counter to the earth s rotation. i always lies between 90 and
8 9/20/ Up : mean prograde orbit Down : mean retrograde Exercise : do it for Earth, Uranus, and Venus Argument of perigee (ω): the angle from ascending node to perigee, measured in the orbital plane at the earth s center, in the direction of satellite motion. Mean anomaly (M): gives an average value of the angular position of the satellite with reference to the perigee. For a circular orbit, M gives the angular position For elliptical orbit, the position is much more difficult to calculate, and M is used just as an intermediate step. True anomaly: is the angle from perigee to the satellite position, measured at the earth s center. This gives the true angular position of the satellite in the orbit as a function of time. 8
9 9/20/ 2.6 Orbital Elements: Earth-orbiting artificial satellites are defined by six orbital elements referred to as the keplerian element set. (a & e ) give the shape of the ellipse (v or M) gives the position of the satellite in its orbit at a reference time known as the epoch (I & Ω) relate the orbital plane s position to the earth. ω gives the rotation of the orbit s perigee point relative to the orbit s line of nodes in the earth s equatorial plane. 9
10 9/20/ Because of : 1. the equatorial bulge causes slow variations in w and Ω 2. and other perturbing forces may alter the orbital elements slightly, the values are specified for the reference time or epoch (called as two-line elements (TLE)) A two-line element set (TLE) is a data format encoding a list of orbital elements of an Earth-orbiting object for a given point in time (the epoch). Appendix C lists the two-line elements provided to users by NASA (see Celestrak site). Using suitable prediction formula, the state (position and velocity) at any point in the past or future can be estimated to some accuracy. An example TLE for the International Space Station: The meaning of this data is as follows:(see attached cases study file) Ref: Note: the semimajor axis is not specified, but this can be calculated from the data given. 10
11 9/20/ 2.7 Apogee and Perigee Heights Although not specified as orbital elements, they are often required. As shown earlier : r a = a 1 + e r p = a 1 e To find h a and h p, the radius of the earth must be subtracted from the radii lengths Next time 2.8 Orbit Perturbations 2.9 Inclined Orbits 11
12 9/20/ Dr. Mohammed Taha El Astal 20/9/ 12
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