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)