astr 1101-001 spring 2008 joel e. tohline, alumni professor 247 nicholson hall [slides from...

ASTR 1101-001Spring 2008

Joel E. Tohline, Alumni Professor247 Nicholson Hall

[Slides from Lecture15]

Kepler’s ObservedLaws 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 eccentricityMercury 0.206Venus 0.007Earth 0.017Mars 0.093Jupiter 0.048Saturn 0.053Uranus 0.043Neptune 0.010

Kepler’s ObservedLaws 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 ObservedLaws 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 ObservedLaws 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 = 2r/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 = 2r/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 = 2r/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 = 2r/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 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

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 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

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 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 = 2r/P = 2 (1 AU)/1 yr

= 2 (1.5 x 1011 m)/(3.156 x 107 s) = 30,000 m/s = 67,000 mph

NOTE: This last step used the knowledge that 1 m/s = 2.2 mph

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 = 2r/P

• To understand the origin of this formula, consider a related but more familiar situation

Orbital Velocities of Planets

Planet P (yr) R (AU) v (km/s)Mercury 0.24 0.39 49Venus 0.61 0.72 35Earth 1.00 1.00 30Mars 1.88 1.52 24Jupiter 11.86 5.20 13Saturn 29.46 9.55 9.7Uranus 84.10 19.19 6.8Neptune 164.86 30.07 5.4

Isaac Newton (1642-1727)