PTYS/ASTR 206Kepler’s Laws / Gravity 1/18/07 Planetary Orbits: Kepler’s Laws

PTYS/ASTR 206 Kepler’s Laws / Gravity1/18/07

Planetary Orbits: Kepler’s Laws


Announcements

• The correct link for the course webpage

• The first homework due Jan 25 (available for download on website).

• Starry Night Backyard software program – blue CD that comes with textbook

• Preceptors– Please fill out a preceptor application.– Please also give me your schedule on the back of the form

• Mission update volunteers?

• LPL Public Lecture Series next Tuesday

http://www.lpl.arizona.edu/undergrad/classes/spring2007/Giacalone_206-2




Today’s Topics

• Finish discussion on Copernican revolution– A long-standing Earth-centered view of the solar

system is turned on its head with the invention of the telescope

– Phases of Venus

• The modern view of the solar system and the epiphany that gravity is important!– Kepler’s revision to the system devised by Copernicus

• Kepler’s Laws– Galileo’s experiments on gravity– Newton’s law of universal gravitation


The Ptolemaic System


Nicholas Copernicus

• 1473-1543, Polish

• Re-proposed heliocentric theory

• Put the Sun at the center, but still believed the orbits of the planets were circles + epicycles

• He felt that this was a more natural explanation of the solar system


Tycho Brahe

• 1546-1601, Danish

• Nose cut off in duel by a fellow student

• Observed a supernova, and periodic comets– Proof that the stars and planets are not

constant, as the ancient astronomers believed

• Best pre-telescope observer– Given island of Hven to build an

observatory– Best, most observations– Still no parallax


Galileo Galilei

• 1564-1642, Italian

• Used telescope– Jupiter’s moons– Lunar mountains, sunspots– Phases of Venus

• Experiments on motion– Including attempts to measure

the speed of light

– Famous gravity experiment using the leaning tower of Pisa


• Galileo also discovered four moons, now called the Galilean satellites, orbiting Jupiter

• The positions of these moons change rapidly and can be seen during the course of one evening– Especially Io and

Europa, the two closest to Jupiter


Jupiter and 3 of the Galilean satelites (only Callisto is missing from this amateur photograph)

This is what Jupiter looks like through a standard amateur telescope. Galileo probably saw it similarly


Phases of Venus

Galileo’s observations of Venus’s gibbous (“full-ish”) phase definitively ruled out Ptolemy’s geocentric model !


The phases of Venus in the

Ptolemaic Model

only a crescent Venus is possible


The Phases of Venus in Heliocentric Model – including the gibbous phase of

Venus, as observed


Conclusion

• The 1000-year-old paradigm of a Earth-centered solar system, was proved incorrect as the result of simple observations made by a new technology -- the telescope.

• New technologies often lead to re-writing of textbooks.

• Current “new technologies” include– Space age– Super computers


Johannes Kepler proposed elliptical pathsfor the planets about the Sun

• Using data collected by Tycho Brahe, Kepler deduced three laws of planetary motion:

1. the orbits are ellipses2. a planet’s speed

varies as it moves around its elliptical orbit

3. the orbital period of a planet is related to the size of its orbit


Kepler’s First Law

• The planets move about the sun in elliptical orbits with the Sun at one foci of the ellipse

Perihelion Aphelion


Semi-major axis

The Semi-major axis is ½ the “long” width of the ellipse

Planetary fact sheets usually quote the semi-major axis of the orbit


Ellipses

• Eccentricity – a measure of how oblong an ellipse is.


• Here are two elliptical orbits with the Sun at one focus. The orbits of most of the planets are nearly circles. But other objects, especially comets, have orbits with high eccentricities.


Kepler’s Second Law

• The Planet sweeps out equal areas of the elliptical orbit, in equal time intervals

• The planet moves fastest when it is closest to the Sun and slowest when it is farthest away


Kepler’s Third Law

The square of a planet’s period (P) is proportional to the cube of its semi-major axis (a).

or

The farther a planet is from the sun, the longer it takes to go around the sun

P = the period (in years)a = semi-major axis (in AU)



A warning about this formula !

It only applies to planetary motion about the Sun! Also, the units must be P in years, and a in AU. Otherwise you will NOT get the correct result

P = the period (in years)a = semi-major axis (in AU)


Acceleration Due to Gravity

• Galileo discovered that the higher an object is dropped, the greater its speed when it reaches the ground

• All falling objects near the surface of the Earth have the same acceleration (9.8 m/s2)

• The acceleration of gravity on the surface of other solar-system bodies depends on their mass and radius– Mars and the Moon have a smaller

acceleration of gravity– Saturn is about the same as Earth– Jupiter is more than Earth


Objects fall at the same rate – independent of mass !

• Galileo’s famous experiment– Showed that objects of different

masses dropped from the same height arrive to Earth at the SAME time

– Disproved Aristotle’s “theory”– Did he really do it ?

• The experiment was also performed by an astronaut on the moon?


Astronaut Alan Bean

Did Galileo’s experiment on the Moon


Isaac Newton (1642 – 1727)

• Developed the Laws of Motion

• Discovered the law of gravity

• Used physics to derive Kepler’s 3rd Law– (see page 81 and box 4-4

of the textbook)


Newton’s Law of Universal Gravitation

• Gravitation force between two bodies

• G = Gravitational Constant = 6.67x10-11 N • m2 / kg2

• m1 and m2 = the masses of the two objects

• r = the distance between the two (more precisely, the distance between the two centers)


Galilean Satellites and Kepler’s Laws

• Newton derived Kepler’s third law using physics and his universal law of gravitation. His form of Kepler’s 3rd law for the orbits of the planets about the Sun is:


Today’s in-class activity• You may work together in small (2-3 people) groups

– You must hand in your own work

• Please remain in your seats until we collect all of the finished activities

• After the activity is turned in, we will toss a coin to decide if it will be graded– Heads – it is graded– Tails – it will not be graded

• If time permits, we will go over the activity afterwards


Reading for next class

• Chapter 4– Box 4-2 (p. 67)– Section 4-6, 4-7, 4-8 (pp. 76-85)

• Chapter 2– Section 2-3, 2-4, 2-5 (pp. 22-31)

Documents

PTYS/ASTR 206Kepler’s Laws / Gravity 1/18/07 Planetary Orbits: Kepler’s Laws