1. ASTR 1101-001 Spring 2008 Joel E. Tohline, Alumni Professor 247 Nicholson Hall [Slides from Lecture15]

3. Kepler’s Observed Laws of Planetary Motion Kepler’s First Law: –The orbit of a planet about the Sun is an ellipse with the Sun at one focus Kepler’s Second Law: –A line joining a planet and the Sun sweeps out equal areas in equal intervals of time Kepler’s Third Law: –The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit

6. Terminology related to ellipses: Focus (singular) and Foci (plural) Major and Minor axes Semi-major axis (half the major axis) –Average distance between the Sun and planet –In astronomy, usually represented by the letter “a” Eccentricity (e) For a circular orbit, the two foci lie on top of one another at the center of the orbit, e = 0, and “a” is the radius of the circle

7. Planetary Orbits In the solar system, most planets have very nearly circular orbits (that is, “e” is almost zero) Comets, however, often have very eccentric orbits Planeteccentricity Mercury0.206 Venus0.007 Earth0.017 Mars0.093 Jupiter0.048 Saturn0.053 Uranus0.043 Neptune0.010

8. Kepler’s Observed Laws of Planetary Motion Kepler’s First Law: –The orbit of a planet about the Sun is an ellipse with the Sun at one focus Kepler’s Second Law: –A line joining a planet and the Sun sweeps out equal areas in equal intervals of time Kepler’s Third Law: –The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit

10. Terminology related to ellipses (cont.): Perihelion –Point along an orbit when a planet is closest to the Sun Aphelion –Point along an orbit when a planet is farthest from the Sun

11. Kepler’s Observed Laws of Planetary Motion Kepler’s First Law: –The orbit of a planet about the Sun is an ellipse with the Sun at one focus Kepler’s Second Law: –A line joining a planet and the Sun sweeps out equal areas in equal intervals of time Kepler’s Third Law: –The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit

13. Kepler’s Observed Laws of Planetary Motion Kepler’s First Law: –The orbit of a planet about the Sun is an ellipse with the Sun at one focus Kepler’s Second Law: –A line joining a planet and the Sun sweeps out equal areas in equal intervals of time Kepler’s Third Law: –The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit

14. Simplification warning! Kepler’s careful observational work proved that planets orbit the Sun along elliptical paths Frequently, I will discuss planetary orbits as though they are all perfectly circular. Why? –Because the properties of circles are more familiar and easier to deal with than the properties of ellipses –Most planetary orbits are so nearly circular that it is fair to treat them as exact circles when illustrating their behavior The general conclusions I will draw can be generalized to include motion along elliptical orbits – you’ll have to trust me on this!

15. Example: Speed & Velocity associated with Circular Motion We know the size (semimajor axis) of each planet’s orbit, and we know how long it takes each planet to complete an orbit. How fast (at what speed) does each planet move along its orbit? For elliptical orbits, the speed varies along the orbit (as described by Kepler’s Second Law) For circular orbits, however, the speed is constant along the orbit: v = 2  r/P To understand the origin of this formula, consider a related but more familiar situation

16. Example: Speed & Velocity associated with Circular Motion We know the size (semimajor axis) of each planet’s orbit, and we know how long it takes each planet to complete an orbit. How fast (at what speed) does each planet move along its orbit? For elliptical orbits, the speed varies along the orbit (as described by Kepler’s Second Law) For circular orbits, however, the speed is constant along the orbit: v = 2  r/P To understand the origin of this formula, consider a related but more familiar situation

18. Example: Speed & Velocity associated with Circular Motion We know the size (semimajor axis) of each planet’s orbit, and we know how long it takes each planet to complete an orbit. How fast (at what speed) does each planet move along its orbit? For elliptical orbits, the speed varies along the orbit (as described by Kepler’s Second Law) For circular orbits, however, the speed is constant along the orbit: v = 2  r/P To understand the origin of this formula, consider a related but more familiar situation

19. Example: Speed & Velocity associated with Circular Motion We know the size (semimajor axis) of each planet’s orbit, and we know how long it takes each planet to complete an orbit. How fast (at what speed) does each planet move along its orbit? For elliptical orbits, the speed varies along the orbit (as described by Kepler’s Second Law) For circular orbits, however, the speed is constant along the orbit: v = 2  r/P To understand the origin of this formula, consider a related but more familiar situation

20. Example: Speed & Velocity associated with Circular Motion Suppose the time that it takes you to drive a distance d = 100 miles is t = 2 hours. What is your average speed/velocity of travel? Suppose you drive along a road that marks the outer edge of a circular field of sugarcane. If the sugarcane field has a radius r = 1 mile and it takes you t =10 minutes to drive all the way around the field, what was your average speed/velocity of travel? If a planet that moves along a circular orbit whose radius is r = 1 AU takes 1 year to complete an orbit, what is that planet’s average speed/velocity?

21. Example: Speed & Velocity associated with Circular Motion Suppose the time that it takes you to drive a distance d = 100 miles is t = 2 hours. What is your average speed/velocity of travel? Suppose you drive along a road that marks the outer edge of a circular field of sugarcane. If the sugarcane field has a radius r = 1 mile and it takes you t =10 minutes to drive all the way around the field, what was your average speed/velocity of travel? If a planet that moves along a circular orbit whose radius is r = 1 AU takes 1 year to complete an orbit, what is that planet’s average speed/velocity? ANSWER: v = distance/time = 100 miles/2 hrs = 50 mph

22. Example: Speed & Velocity associated with Circular Motion Suppose you drive along a road that marks the outer edge of a circular field of sugarcane. If the sugarcane field has a radius r = 1 mile and it takes you t =10 minutes to drive all the way around the field, what was your average speed/velocity of travel? Suppose the time that it takes you to drive a distance d = 100 miles is t = 2 hours. What is your average speed/velocity of travel? If a planet that moves along a circular orbit whose radius is r = 1 AU takes 1 year to complete an orbit, what is that planet’s average speed/velocity?

23. Example: Speed & Velocity associated with Circular Motion Suppose you drive along a road that marks the outer edge of a circular field of sugarcane. If the sugarcane field has a radius r = 1 mile and it takes you t =10 minutes to drive all the way around the field, what was your average speed/velocity of travel? Suppose the time that it takes you to drive a distance d = 100 miles is t = 2 hours. What is your average speed/velocity of travel? If a planet that moves along a circular orbit whose radius is r = 1 AU takes 1 year to complete an orbit, what is that planet’s average speed/velocity? ANSWER: v = distance/time = 2  (1 mile)/10 minutes = 2  (1 mile)/(1/6) hr = 12  mph = 38 mph

24. Example: Speed & Velocity associated with Circular Motion If a planet that moves along a circular orbit whose radius is r = 1 AU takes 1 year to complete an orbit, what is that planet’s average speed/velocity? Suppose the time that it takes you to drive a distance d = 100 miles is t = 2 hours. What is your average speed/velocity of travel? Suppose you drive along a road that marks the outer edge of a circular field of sugarcane. If the sugarcane field has a radius r = 1 mile and it takes you t =10 minutes to drive all the way around the field, what was your average speed/velocity of travel?

25. Example: Speed & Velocity associated with Circular Motion If a planet that moves along a circular orbit whose radius is r = 1 AU takes 1 year to complete an orbit, what is that planet’s average speed/velocity? Suppose the time that it takes you to drive a distance d = 100 miles is t = 2 hours. What is your average speed/velocity of travel? Suppose you drive along a road that marks the outer edge of a circular field of sugarcane. If the sugarcane field has a radius r = 1 mile and it takes you t =10 minutes to drive all the way around the field, what was your average speed/velocity of travel? ANSWER: v = 2  r/P = 2  (1 AU)/1 yr = 2  (1.5 x 10 11 m)/(3.156 x 10 7 s) = 30,000 m/s = 67,000 mph NOTE: This last step used the knowledge that 1 m/s = 2.2 mph

26. Example: Speed & Velocity associated with Circular Motion We know the size (semimajor axis) of each planet’s orbit, and we know how long it takes each planet to complete an orbit. How fast (at what speed) does each planet move along its orbit? For elliptical orbits, the speed varies along the orbit (as described by Kepler’s Second Law) For circular orbits, however, the speed is constant along the orbit: v = 2  r/P To understand the origin of this formula, consider a related but more familiar situation

27. Orbital Velocities of Planets PlanetP (yr)R (AU)v (km/s) Mercury0.240.3949 Venus0.610.7235 Earth1.00 30 Mars1.881.5224 Jupiter11.865.2013 Saturn29.469.559.7 Uranus84.1019.196.8 Neptune164.8630.075.4

28. Isaac Newton (1642-1727)