Chapter 4

Gravitation and the Waltz of the

Planets

Guiding Questions1. How did ancient astronomers explain the motions of the planets?2. Why did Copernicus think that the Earth and the other planets revolved

around the Sun?3. What did Galileo see in his telescope that confirmed that planets orbit the

Sun?4. How did Tycho Brahe attempt to test the ideas of Copernicus?5. What paths do the planets follow as they move around the Sun?6. What fundamental laws of nature explain the motions of objects on Earth

as well as the motions of the planets?7. Why don’t the planets fall into the Sun?

8. What keeps the same face of the Moon always pointed toward the Earth?

Ancients knew that five “wandering stars” seemed to slowly move among the constellations.

These wandering stars, commonly known as “planetes,” typically move from west to east, except during brief periods where they move backwards or retrograde. The early Greek model of a celestial sphere

did not adequately account for these retrograde loops.

Ancient astronomers invented geocentric models to explain complex planetary motions

Claudius Ptolemy devised the longest used geocentric model to explain retrograde loops by

putting planets on epicycles and deferents.

• Copernicus imagined a universe where the Sun was at the center instead of Earth.

• He suggested that Earth’s motion around the Sun provided a more “natural” explanation for retrograde loops as Earth passed the other planets.

Nicolaus Copernicus devised the first comprehensive heliocentric (Sun-centered) model

Heliocentric planetary positions are described relative to Earth

Opposition

Inferior conjunction

Superior conjunction

Greatest eastern elongation (appears east of the Sun in the sky)

Greatest western elongation (appears west of the Sun in the sky)

In this heliocentric model, the planets just “appear” to move backwards as the faster moving Earth “laps” the more distant

planet once each year when it is at opposition.

Galileo’s discoveries of Jupiter’s moons with his telescope showed that Earth was not the center of all orbits strongly supported a heliocentric model

even though Copernicus’ model was no more accurate than Ptolemy’s.

Galileo’s discoveries of Venus’ phases with his telescope showed that Venus must orbit the Su,.

strongly supporting a heliocentric model even though Copernicus’ model was no more accurate than Ptolemy’s.

Venus is clearly smallest when it is at superior conjunction and largest when it is close to inferior conjunction.

Venus can go through phases only if it orbits the Sun.

Tycho Brahe measured positions precisely,before telescopes were available.

Brahe constructed enormous instruments to meticulously record the precise positions of the planets in the sky to an accuracy never previously obtained.

Tycho Brahe’s astronomical observations disproved ancient ideas about the heavens.

Using PARALLAX, Brahe was able to demonstrate that the comet of 1577 was beyond the Moon’s orbit and that the supernova of 1572 was in “the distant realm of the stars.”

Kepler’s First Law of Planetary Motion

• The orbit of a planet about the Sun is an ellipse with the Sun at one focus.

Johannes Kepler proposed elliptical paths for the planets about the Sun.

Elliptical Eccentricity (e): a number ranging between zero (for a flat line) and one (for a perfectly round circle).

Kepler’s elliptical paths for planets’ orbits

Kepler’s Second Law of Planetary Motion

• A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

Kepler’s Third Law of Planetary Motion• The square of the sidereal period of a planet is

directly proportional to the cube of the semi-major axis of the orbit.

p2 = a3

Period p is in years (p=1 for Earth)

distance a from Sun is in AU (a=1 for Earth)

Kepler’s laws explain how the universe works, but they do not explain “why.”

Newton’s explanation shows why Kepler’s laws work.

Applying Kepler’s 3d law:

A satellite is placed in a circular orbit around the Sun, orbiting the Sun once every 10 months. How far is the satellite from the Sun?

23 2 10

a = p = _______12

a ______

Sidereal and Synodic periods:

A satellite is placed in a circular orbit around the Sun, orbiting the Sun once every 10 months. How often does the satellite pass between the Earth and the Sun?

1 1 1

'

1 1 1

1 1 110 1

121

________________

________________

sidereal period Earth s sidereal year synodic period

P E S

S

SS

1. A body remains at rest, or moves in a straight line at a constant speed, unless acted upon by a net outside force.

2. F = m a (the force on an object is directly proportional to its mass and acceleration).

3. Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body.

Isaac Newton formulated three laws that describe fundamental properties of

physical reality.

Newton’s Law of Universal GravitationTwo bodies attract each other with a force that is proportional to the mass of each body and inversely proportional to the square of the distance between them.

Where G = 6.67 X 10-11 N m2/kg2 and r is the distance between the objects with masses m1 and m2.

Newton’s description of gravity accounts for Kepler’s laws and explains the

motions of the planets.

1 22

G m mF =

r

The law of universal gravitation accounts for planets not falling into the Sun nor the Moon crashing into the Earth:

Paths A, B, and C do not have enough horizontal velocity to “escape” Earth’s surface whereas Paths D, E, and F do.

Path E is where the horizontal velocity is exactly what is needed so its orbit matches the circular curve of the Earth.

Mathematically speaking, Newton discovered that orbiting bodies may follow any one of a family of curves

called “conic sections.”

Newton’s laws also explain tidal forces which can deform planets, reshape galaxies.

Newton’s laws also explain tidal forces which can deform planets,

reshape galaxies.

Newton’s laws also explain tidal forces which can deform planets, reshape galaxies.

Gravitational forces from the Sun in addition to the Moon can create abnormally high tides,

called spring tides.

Gravitational forces from the Sun can

also diminish each other’s effects. When

the Sun and Moon are at right angles,

they create abnormally low

tides, called neap tides.

We can use Newton’s gravity to approximate the size of a black hole!

21

2____________

Gravitational energy kinetic energy

GmMmv

rSolve for r

Not even light can escape (v=c) if it is closer than r to a black hole. This is the Schwarzschild radius:

R=_____________________

Guiding Questions1. How did ancient astronomers explain the motions of the planets?2. Why did Copernicus think that the Earth and the other planets of around

the Sun?3. What did Galileo see in his telescope that confirmed that planets orbit the

Sun?4. How did Tycho Brahe attempt to test the ideas of Copernicus?5. What paths do the planets follows as they move around the Sun?6. What fundamental laws of nature explain the motions of objects on Earth

as well as the motions of the planets?7. Why don’t the planets fall into the Sun?

8. What keeps the same face of the Moon always pointed toward the Earth?