1 Academic Physics Mechanics Chapter 14 Satellite Motion The Mechanical Universe Kepler's Three Laws (Episode 21) The Kepler Problem (Episode 22) Energy and Eccentricity (Episode 23) Navigating in Space (Episode 24)
14.1: Earth Satellites A satellite is a projectile that falls around a body (star, planet, moon, etc.) rather than into it because of its sideways motion. An object falls about 5 meters each second. The faster its horizontal speed, the further it travels sideways in that time, but it still falls 5 meters. 2
14.1: Earth Satellites The Earth's curvature is about 5 meters for every 8000 meters of horizontal distance. 3 Combining these two facts means than an object thrown horizontally at 8000 m/s would stay the same distance above the Earth's surface. Note that air resistance at this height would cause the object to burn up like a meteor and would slow it enough to cause it crash into the Earth's surface.
14.2: Circular Orbits In circular orbit, the speed of a circling satellite is not changed by gravity. Gravity is neither pulling the satellite forward nor backward, but pulls straight down, perpendicular to the satellite's motion. This means that there is no change in speed, only a change in direction. 4
14.2: Circular Orbits For satellites close to the Earth, the time for a complete orbit (the satellite's period) is about 90 minutes. For satellites parked in orbit about 6.5 Earth radii from Earth's center, the period is 24 hours. These circular orbits are called geostationary orbits. Why? The moon is further away, and has a 27.3-day period. 5
14.2: Orbits of Planets and Satellites A satellite in an orbit that is always the same height above the Earth moves in uniform circular motion, where the centripetal force is provided by gravity. 6 m v 2 r F c =F g =G M E m r 2 - or - a c =a g v 2 r =G M E r 2 These can be for a satellite orbiting any body, by changing the mass M E to the mass of the body being orbited.
14.2: Orbits of Planets and Satellites The speed of a satellite orbiting the Earth is given by 7 v= G M E r Since v is also the circumference of the orbit divided by the period, we can also find that the period of a satellite orbiting the Earth is given by T 2 = 4 π2 G M E r 3
14.2: Orbital Speed and Period The ISS orbits the Earth at an altitude of 342 km. Given that the mass of the Earth is 5.97x10 24 kg and the radius of the Earth is 6.38x10 6 m, What is the ISS's orbital speed? 8 What is the ISS's orbital period? Take a virtual tour
14.2: Orbital Speed and Period The Mars Reconnaissance Orbiter orbits at an altitude of 280 km. Given that the mass of Mars is 6.42x10 23 kg and the radius of Mars is 3.40x10 6 m, What is MRO's orbital speed? 9 What is MRO's orbital period?
14.2: A Satellite's Mass Note that the orbital speed and period of a satellite depend on the mass of the object it is orbiting and on its distance from the center of mass of that body. Neither depend on the mass of the satellite itself. That means that the ISS and space debris at the same distance from the Earth would have the same speed and period. 10
14.2: A Satellite's Mass Are there any factors that limit the mass of a satellite? Yes. Satellites are accelerated to the speeds necessary for them to achieve orbit by large rockets. This acceleration must follow Newton's second law of motion, so the mass of a satellite is limited by our ability to build a rocket that can carry the satellite and itself into orbit. 11
14.3: Western Astronomy Kepler Johannes Kepler, a German mathematician, astronomer, and astrologer, became one of Brahe's assistants when Brahe moved to Prague in 1600. Upon Brahe's death in 1601, Kepler inherited 30 years worth of Brahe's observations. He studied Brahe's data and was convinced that geometry and mathematics could be used to explain the number, distance, and motion of the planets. 12
14.3: Elliptical Orbits A projectile which is launched parallel to the Earth's surface, but whose speed is greater than 13 will overshoot a circular path and trace an oval-shaped path called an ellipse. v= G M E r
14.3: Kepler's First Law of Orbital Motion The paths of the planets are ellipses with the Sun at one focus. 14 The paths of any satellite is an ellipse with the body it orbits at one focus. A circle is a special ellipse where each focus is at the center. Concepts in Motion
14.3: Speed in Elliptical Orbits Satellite speed, which is constant in a circular orbit, varies in an elliptical orbit. Because the path of the satellite is not perpendicular to gravity at every point, gravity slows down the satellite as it moves further away, and speeds up the satellite as it moves closer. The speed lost in receding is regained as it falls back, joining its path with the same speed it had initially, so the ellipse is retraced over and over. 15
14.4: Kepler's Second Law of Orbital Motion 16 An imaginary line from the sun to a planet sweeps out equal areas in equal time intervals. This also implies that the planet moves faster when close to the sun (near perihelion) and slower when further from the sun (near aphelion). Concepts in Motion
14.4: Orbits from an Energy Perspective Circular Orbits Elliptical Orbits 17 At all points in an elliptical orbit, except at perigee and apogee, there is a component of gravitational force that is parallel to the direction of motion. This component, times the distance moved, will be the change in the kinetic energy.
14.4: Orbits from an Energy Perspective 18 When the satellite gains altitude it gains PE and so must lose KE in order to conserve mechanical energy. This means that the higher the altitude, the slower the satellite moves. Note that it moves against the component of gravity parallel to its motion in this case, so negative work is done and the change in kinetic energy is negative. When the satellite loses altitude it moves with the component on gravity parallel to its motion, so positive work is done and the change in kinetic energy is positive.
14.4: Kepler's Third Law of Orbital Motion 19 The squared quantity of the ratio of periods is equal to the cubed quantity of the objects' distances (semi-major axis) from the Sun. (Kepler's Form) ( T A T B)2 =( r A r B)3 The first two laws apply to a planet, moon, or satellite individually, while this law relates the motion of two objects around a single body. Concepts in Motion
14.4: Kepler's Third Law of Orbital Motion Alternatively, this states that the ratio of the square of the period to the cube of the length of the semimajor axis is the same for all satellites about a body. 20
14.4: Solar System Data 21 Name Status Average Radius (m) Mass (kg) Mean Distance From the Sun (m) Orbital Period (Earth Days) r 3 / T 2 Sun Star 6.96x10 8 1.99x10 30 -- -- Mercury Terrestrial Planet 2.44x10 6 3.30x10 23 5.79x10 10 88 Venus Terrestrial Planet 6.05x10 6 4.87x10 24 1.08x10 11 225 Earth Terrestrial Planet 6.38x10 6 5.97x10 24 1.50x10 11 365 Mars Terrestrial Planet 3.40x10 6 6.42x10 23 2.28x10 11 687 Ceres Dwarf Planet 4.87x10 5 9.43x10 20 4.14x10 11 Jupiter Gas Giant Planet 7.15x10 7 1.90x10 27 7.78x10 11 1 681 4 332 Saturn Gas Giant Planet 6.03x10 7 5.69x10 26 1.43x10 12 10 759 Uranus Gas Giant Planet 2.56x10 7 8.68x10 25 2.87x10 12 30 799 Neptune Gas Giant Planet 2.48x10 7 1.02x10 26 4.50x10 12 60 190 Pluto Dwarf Planet 1.20x10 6 1.25x10 22 5.87x10 12 90 613 Haumea Dwarf Planet 5.75x10 5 4.01x10 21 6.45x10 12 103 468 Makemake Dwarf Planet 7.50x10 5 4x10 21 6.85x10 12 113 183 Eris Dwarf Planet 1.30x10 6 1.67x10 22 1.01x10 13 203 600
14.4: Galilean Moons of Jupiter 22 Galileo measured the orbital sizes of Jupiter's moons in 1665 using the diameter of Jupiter as a unit of measure. Using the same units that Galileo used, predict the other three Galilean moons' distances from Jupiter. Moon Galileo's Observation, animated Discovery of the Galilean Moons Animation Period (days) Orbital Radius (Jupiter diameters) Io 1.77 5.9 Europa 3.55 Ganymede 7.16 Callisto 16.7
14.4: Universal Gravitation and Kepler's Third Law 23 Newton stated his law of universal gravitation in terms that applied to the motion of planets around the Sun, which agreed with Kepler's third law and confirmed that Newton's law fit the best observations of the day. Since centripetal acceleration can be written as a c = 4 π 2 r / T 2, we have that F c =F g m planet 4 π 2 r T 2 =G m sun m planet r 2 (Newton's Form) T 2 =( 4 π2 G m sun) r 3
14.4: Universal Gravitation and Kepler's Third Law 24 Note that this is Kepler's third law, (Kepler's Form) (Newton's Form) ( T A T B)2 T 2 =( =( r A r B)3 4 π2 G m sun) r3 but expressed for only one orbiting satellite instead of a pair of satellites around the same body. This formula applies for elliptical orbits as well as circular orbits.
14.4: Planets around Upsilon Andromedae Upsilon Andromedae (υ And) is a binary star located approximately 44 light-years away from Earth in the constellation Andromeda. The primary star (υ And A) is a yellow-white dwarf star that is somewhat younger than the Sun. The second star in the system (υ And B) is a red dwarf located in a wide orbit. Upsilon Andromedae was both the first multipleplanet planetary system to be discovered around a main sequence star, and the first multiple-planet system known in a multiple star system. 25
14.4: Planets around Upsilon Andromedae 26 As of 2008, three confirmed extrasolar planets are known in orbit around the primary star. All three are likely to be Jovian planets that are comparable to Jupiter. υ And b has an average orbital radius of 0.059 AU and a period of 4.6170 days υ And c has an average orbital radius of 0.829 AU and a period of 241.5 days υ And d has an average orbital radius of 2.53 AU and a period of 1284 days. Do these planets obey Kepler's third law? Find the mass of υ And A in units of the Sun's mass.
14.5: Escape Speed A horizontal speed of 8 km/s would ensure an object maintains the same distance it had when fired, neglecting air resistance. However, and object fired up at 8 km/s would still come back down to the Earth. The escape speed, v, from any body is given by v= 2G M d where d is the distance the object starts from the center of the body. 27
14.5: Escape Speed What is the escape velocity from the Earth? The mass of the Earth is 5.98 10 24 kg and the radius of the Earth is 6378 km. What is the escape speed from the Solar System, starting at Earth's orbit? The mass of the Sun is 1.99 10 30 kg and the orbital radius of the Earth is 149.6 10 6 km. The first probe to escape the solar system, Pioneer 10, was launched in 1972 with a speed of 15 km/s. It used the gravity of planets to accelerate it fast enough to escape the solar system in a process called a gravity-assist trajectory. 28
Satellite Motion Review 29 AP B Video 20: Orbits of Planets and Satellites Physics Classroom Tutorial MIT Lecture 14: Orbits 22: Kepler's Laws Hyperphysics Tutorial PhysicsLab Physics Hypertextbook Orbital Mechanics I Miscellaneous Uncertain Principles Measuring Gravity Starts With A Bang Orbiting Earth 101 Orbital Mechanics Bad Astronomy Death from the skies!