#### Transcript Lecture15 - LSU Physics & Astronomy

ASTR 1101-001 Spring 2008 Joel E. Tohline, Alumni Professor 247 Nicholson Hall [Slides from Lecture15] 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 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 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 Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune eccentricity 0.206 0.007 0.017 0.093 0.048 0.053 0.043 0.010 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 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 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 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 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! 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 = 2pr/P • To understand the origin of this formula, consider a related but more familiar situation 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 = 2pr/P • To understand the origin of this formula, consider a related but more familiar situation 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 = 2pr/P • To understand the origin of this formula, consider a related but more familiar situation 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 = 2pr/P • To understand the origin of this formula, consider a related but more familiar situation 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? 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 has a radius = 1 mile ANSWER: v =field distance/time = 100rmiles/2 hrs and = 50 itmph 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? 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? 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 ANSWER: v = distance/time = 2pof(1travel? mile)/10 minutes your average speed/velocity • If a planet that alonghra =circular = 2pmoves (1 mile)/(1/6) 12p mphorbit = 38 mph whose radius is r = 1 AU takes 1 year to complete an orbit, what is that planet’s average speed/velocity? 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? 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 ANSWER: v = 2pr/P = 2p (1 AU)/1ofyrtravel? • Suppose you drive a road that the 7 s) = 2p (1.5 along x 1011 m)/(3.156 x 10marks outer edge =of30,000 a circular field of sugarcane. If the m/s = 67,000 mph sugarcane field has a radius r = 1 mile and it takes you t =10 minutes to drive all the way around field, was your average NOTE: Thisthe last step usedwhat the knowledge that 1 m/s = 2.2 mph speed/velocity of travel? 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 = 2pr/P • To understand the origin of this formula, consider a related but more familiar situation Orbital Velocities of Planets Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune P (yr) 0.24 0.61 1.00 1.88 11.86 29.46 84.10 164.86 R (AU) 0.39 0.72 1.00 1.52 5.20 9.55 19.19 30.07 v (km/s) 49 35 30 24 13 9.7 6.8 5.4 Isaac Newton (1642-1727)