# Planetary Mechanics:

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1 Planetary Mechanics:

2 Satellites A satellite is an object or a body that revolves around another body due to the gravitational attraction to the greater mass. Ex: The planets are natural satellites of the Sun and moons are natural satellites of the planets themselves.

3 Satellites Artificial satellites, conversely, are human-made objects that orbit Earth or other bodies in the solar system. Ex: CSA s RADARSAT-1 and RADARSAT-2 and the International Space Station (ISS) are examples of artificial satellites.

4 Satellites Another well-known example of artificial satellites is the network of 24 satellites that make up the Global Positioning System (GPS). The data from 1 satellite will show that the object is somewhere along the circumference of the circle.

5 Satellites Two satellites consulted simultaneously will refine the location to one of two intersection spots.

6 Satellites With three satellites, the intersection of the three circles will give the location of the object to within 15 m of its actual position.

7 Satellites in Circular Orbits When Newton developed the idea of universal gravitation, he also theorized that the same force that pulls objects to Earth also keeps the Moon in its orbit. But of course the Moon does not hit the Earth s surface! The Moon orbits Earth at a distance from Earth s centre - called the orbital radius. The motion of the Moon depends upon the centripetal force due to Earth s gravity and the Moon s orbital velocity.

8 Moon Orbiting Earth The Moon s orbit, similar to the orbits of the planets around the Sun, is actually elliptical. The orbits are approximated as circular orbits.

9 Analyzing Satellites in Circular Orbits For the motion of a satellite experiencing uniform circular motion in a gravitational field: From Newton s Law of Universal Gravitation for an object in Earth s gravitational field: g = Gm E r 2 From centripetal acceleration: a c = v2 r Since the gravitational force provides the centripetal force for a satellite, m, in orbit: F c = F g ma c = mg v 2 = Gm E r r 2 v = Gm E r This eq n gives the speed of an orbiting satellite/body within Earth s gravitational field.

10 Analyzing Satellites in Circular Orbits To calculate the orbital speed around any large body of mass m: v = Gm r where v is the orbital speed of the satellite (m/s) G is the universal gravitational constant (6.67 x N m 2 /kg 2 ) m is the central object s mass about which the satellite orbits (kg) r is the orbital radius (m) ORBITAL SPEED = speed needed by a satellite to remain in orbit

11 Orbits: Example Problem 1 Determine the speeds of the 2 nd and 3 rd planets from the Sun. The 2 nd planet has an orbital radius of 1.08 x m while the 3 rd has an orbital radius of 1.49 x m. The mass of the Sun is 1.99 x kg. v v = Gm S r V = 3.51 x 10 4 m/s v E = Gm S r E = 2.98 x 10 4 m/s As Earth is further from the Sun it travels more slowly than Venus.

12 Orbits: Example Problem 2 The International Space Station (ISS) orbits Earth at an altitude of about 350 km above Earth s surface. Determine: A) The speed needed for the ISS to maintain its orbit. B) The orbital period of the ISS in hours and minutes. C) How many times in a 24 hour day would astronauts aboard the ISS see the sun rise and set? Given: m E = 5.98 x kg r E = x 10 6 m h ISS = 350 km = 3.5 x 10 5 m Note: r ISS = r E + h ISS = x 10 6 m

13 Orbits: Example Problem 2 Cont d A) v = Gm E = x 10 3 m/s The ISS requires a r ISS constant speed of 7.7 x 10 3 m/s to maintain its orbit. B) The distance travelled in 1 revolution is 2πr. T = 2πr = s v = min = h The ISS goes around the entire Earth in 1.5h!!! c) Astronauts aboard the ISS would see the sun rise and set apprx. 16 times a day! (every 45 min.)

14 Orbits: Example Problem 3 What is the difference between a geosynchronous orbit and a geostationary orbit? A geosynchronous orbit is a satellite with an orbital speed that matches Earth s period of rotation; it takes exactly 1 day to travel around the Earth. To an observer on Earth, the satellite will appear to travel through the same point in the sky every 24 h. A geostationary orbit is a special type of geosynchronous orbit in which the satellite orbits directly over the equator. To an observer on Earth, the satellite would appear to remain fixed in the same point in the sky at all times.

15 More Info on Orbits To put you into a real spin.. Try pg. 303 #6,7,9,12 Check out: Train Like an Astronaut: esamultimedia.esa.int/docs/.../en/primedukit_ch 4_en.pdf Physicsclassroom.com: Planetary and Satellite Motion

16 Early Astronomy Early Philosophers, Scientists, and Mathematicians (Aristotle, Plato, Ptolemy) believed in the geocentric view of the universe; Geo meaning Earth + centric meaning centre Scientists tried to explain the motion of the stars and planets

17 A Scientific Revolution Nicolas Copernicus ( ) proposed the heliocentric view of the solar system in which planets revolved in circles around the Sun; helios meaning Sun He also deduced that planets closer to the Sun have a higher speed than those farther away; supported by the orbital speed equation v = Gm r His work was supported and verified by Galileo for which Galileo was persecuted by the Catholic Church

18 Renaissance Astronomers Tycho Brahe ( ) carried out naked-eye observations using large instruments (quadrants) to accumulate the most complete and accurate observations over 20 years to support the heliocentric view Tycho hired a brilliant young mathematician, Johannes Kepler ( ), to assist in the analysis of the data

19 Johannes Kepler Kepler s objective was to find an orbital shape for the motions of the planets that best fit Tycho s data Worked mainly with the orbit of Mars which had the most complete records The only shape that fit ALL of the data was the ellipse He formulated three laws to explain the true orbits of planets

20 Kepler s First Law of Planetary Motion Law of Ellipses: Each planet moves around the Sun in an elliptical orbit with the Sun at one focus of the ellipse. Note: The orbits still very much resemble circles; distance from Earth to Sun varies by only apprx. 3% annually.

21 Kepler s Second Law of Planetary Motion Law of Equal Areas: The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time. Kepler determined that Mars sped up as it approached the Sun and slowed down as it moved away

22 Kepler s Third Law of Planetary Motion Law of Harmonies: The cube of the average radius r of a planet s orbit is directly proportional to the square of the period T of the planet s orbit. r 3 T 2 r 3 = C s T 2 C s = r3 T 2 where C s = Kepler s constant or the constant of proportionality for the Sun measured as 3.35 x m 3 /s 2

23 Kepler s Laws: Example Problem 1 The average radius of orbit of Earth around the Sun is x 10 8 km. The period of revolution is days. Determine: A) The constant C s to four sig. digs. B) An asteroid has a period of revolution around the Sun of 8.1 x 10 7 s. What is the avg. radius of its orbit?

24 Kepler s Laws: Example Problem 1 Cont d A) r E = x 10 8 km = x m T E = days = x 10 7 s C s = r3 T 2 = x m x 10 7 s 2 = x m 3 /s 2 The Sun s constant is x m 3 /s 2. B) T = 8.1 x 10 7 s 3 r = C s T 2 r = x m 3 /s x 10 7 s 2 3 r = 2.8 x m is the avg. radius of the asteroid s orbit.

25 Kepler s Laws: HW Problems The equation for Kepler s 3 rd Law can be obtained from a relationship between Newton s Law of Universal Gravitation and the uniform circular motion of a planet around the Sun. From 1 st principles derive an equation for the Sun s constant that is dependent only on the mass of the Sun. What does orbital eccentricity mean? And just for fun.how was the mass of the Earth originally determined? How was Earth s radius calculated?

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