Gravitation 4 PDF - Kepler's Laws & Satellites

5.3-6 Planets and Satellites: Kepler's Laws Learning Objectives 13.17 Identify Kepler's three 13.20 For an elliptical orbit, laws. apply the relationships between the semimajo...

5.3-6 Planets and Satellites: Kepler's Laws Learning Objectives 13.17 Identify Kepler's three 13.20 For an elliptical orbit, laws. apply the relationships between the semimajor axis, 13.18 Identify which of Kepler's the eccentricity, the laws is equivalent to the law perihelion, and the aphelion. of conservation of angular momentum. 13.21 For an orbiting natural or artificial satellite, apply 13.19 On a sketch of an Kepler's relationship between elliptical orbit, identify the the orbital period and radius semimajor axis, the and the mass of the eccentricity, the perihelion, astronomical body being the aphelion, and the focal orbited. points. © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-6 Planets and Satellites: Kepler's Laws The motion of planets in the solar system was a puzzle for astronomers, especially curious motions such as in Figure 13-11 Johannes Kepler (1571-1630) derived laws of motion using Tycho Brahe's (1546-1601) measurements Figure Figure © 2014 John Wiley & Sons, Inc. All rights reserved. 13-12 5.3-6 Planets and Satellites: Kepler's Laws The orbit is defined by its semimajor axis a and its eccentricity e An eccentricity of zero corresponds to a circle Eccentricity of Earth's orbit is 0.0167 Equivalent to the law of conservation of angular momentum © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-6 Planets and Satellites: Kepler's Laws Figure 13-13 The law of periods can be written mathematically as: Eq. (13-34) Holds for elliptical orbits if we replace r with a © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-6 Planets and Satellites: Kepler's Laws Table 13-3 Answer: (a) satellite 2 (b) satellite 1 © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-7 Satellites: Orbits and Energy Learning Objectives 13.22 For a satellite in a 13.23 For a satellite in an circular orbit around an elliptical orbit, calculate the astronomical body, calculate total energy. the gravitational potential energy, the kinetic energy, and the total energy. © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-7 Satellites: Orbits and Energy Relating the centripetal acceleration of a satellite to the gravitational force, we can rewrite as energies: Eq. (13-38) Meaning that: Eq. (13-39) Therefore the total mechanical energy is: Eq. (13-40) © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-7 Satellites: Orbits and Energy Total energy E is the negative of the kinetic energy For an ellipse, we substitute a for r Therefore the energy of an orbit depends only on its semimajor axis, not its eccentricity All orbits in Figure 13-15 have the same energy Figure 13-16 Figure 13-15 © 2014 John Wiley & Sons, Inc. All rights reserved. 5.3-7 Satellites: Orbits and Energy Answer: (a) orbit 1, since the energy has decreased (b) the semimajor axis has decreased, so the period decreases © 2014 John Wiley & Sons, Inc. All rights reserved.