# Revolution of the Spheres

The `revolution' described here is a revolution of ideas. Copernicus started it by recognizing the Earth as one planet among many, but he postulated laws of planetary motion unlike the laws on Earth. Newton completed the revolution by showing that the same laws of motion work everywhere. Between are Kepler, who wrote down the first correct laws of planetary motion, and Galileo, who found evidence supporting Copernicus and studied motion on Earth.

## Topics

• Retrograde Motion
• Kepler's Laws
• Galileo's Discoveries
• Newton's Mechanics
• Universal Gravitation

## Reading

 5.1 A Brief Survey of the Solar System p. 91 5.3 A Heretical Idea: The Sun-Centered Universe p. 95 5.5 Johannes Kepler and His Laws of Orbits p. 98 A Closer Look 5.2: Kepler's Laws p. 98 Figure it Out 5.1: Kepler's Third Law p. 101 5.6 The Demise of the Ptolemaic Model: Galileo Galilei p. 101 5.7 On the Shoulders of Giants: Isaac Newton p. 103 A Closer Look 5.3: Newton's Law of Universal Gravitation     (NOTE: replace the quantity d2 with d 2.) p. 104 6.1c Tides p. 116

### Overview of the Solar System

The Solar System can be divided into two zones. Bodies in the inner zone are composed mostly of rock and metal, while those in the outer zone consist largely of gas and ice.

 Inner Solar System: Mercury, Venus, Earth, Mars and asteroids. Outer Solar System. Left: Jupiter, Saturn, Uranus, Neptune, Pluto, a comet, and the Kuiper belt. Right: The Oort cloud.

## Retrograde Motion

In addition to the Sun and the Moon, ancient astronomers noticed five `Wandering Stars' or planets: Mercury, Venus, Mars, Jupiter, and Saturn.

While the Sun and Moon both appear to circle the Earth -- the Sun once a year, the Moon once a month -- the planets move more erratically. Most of the time they move across the sky in the same West-to-East direction as the Sun and Moon. But from time to time they move in the opposite direction for a while; this is called retrograde motion.

Retrograde motion is hard to explain in any theory which places the Earth at the center. On the other hand, its easily explained as a consequence of our motion if the Earth is recognized as another planet among many circling the Sun.

### The Retrograde Motion of Mars. I

 Mars Retrograde [NASA] Motion of Mars in Fall 2003. Note the retrograde loop.

### The Retrograde Motion of Mars. II

 Mars appears to move backwards whenever the Earth passes on the inside as both planets orbit the Sun. For a month or so before and after the moment of opposition, when Mars is directly opposite the Sun, the planet's motion is retrograde. As this animation shows, Mars does not actually reverse its motion about the Sun; it only seems to do so because we are observing it from a faster-moving planet, the Earth. 2003: Mars Retrograde Animation [NASA]

### The Copernican System

Copernicus assigned each planet, including the Earth, an orbit around the Sun with a known radius and period.

 Planet Radius(AU) Period(yr) Mercury 0.4 0.24 Venus 0.7 0.62 Earth 1.0 1.00 Mars 1.5 1.88 Jupiter 5.2 11.86 Saturn 9.5 29.46

This brings us to Kepler...

## Kepler's Laws

 Law I: The orbit of a planet is an ellipse with the Sun at one focus. The planet is closest to the Sun at perihelion, and furthest at aphelion. At its widest point, the distance across the ellipse is 2a, where a is the semi-major axis. Kepler's laws [Wikipedia]
 Law II: A line drawn from the Sun to a planet sweeps out equal areas in equal times. In other words, (area swept out) ⁄ (time elapsed) is a constant for each orbit. This constant is now called angular momentum, and Law II states that angular momentum is conserved. Kepler's laws [Wikipedia]
 Law III: For all planets, P2 ⁄ a3 = 1 yr2 ⁄ AU3, where P is the period and a is the semi-major axis.

### Using Kepler's Laws. I

As an application of Kepler's laws, we can work out how long a 'Hohmann transfer orbit' takes to get from Mars to Earth. This orbit is an ellipse (Law I) touching the orbit of Mars at aphelion (A) and the orbit of Earth at perihelion (P).

For simplicity, we will assume both planets have circular orbits. These orbits have radii r1 = 1.5 AU (Mars) and r2 = 1 AU (Earth).

 The semi-major axis of the transfer orbit is a = (r1 + r2) ⁄ 2 = (1 AU + 1.5 AU) ⁄ 2 = 1.25 AU Using Law III, the orbital period is given by P2 = (1 yr2 ⁄ AU3) a3 = 3.375 yr2     or     P = 1.84 yr Kepler's Three Laws

The time required to get from A to P is half the orbital period (Law II), or 0.92 yr.

### Using Kepler's Laws. II

Last week we saw that (1) the Moon's orbit is not a circle, and (2) the Moon wobbles side to side as it orbits the Earth. Law II connects these facts.

 The Moon spins smoothly on its axis, taking exactly 27.3 d to make one rotation. Thus in 27.3 d ⁄ 4 = 6.825 d it rotates exactly 90°. The Moon orbits faster when closer to the Earth, and slower when further away. The four wedges outlined here have equal area, so by Law II they show the Moon's position at intervals of 27.3 d ⁄ 4 = 6.825 d -- the time it takes the Moon to rotate 90°.

Since the Moon's speed varies according to Law II, a line of sight from the Earth to the Moon does not turn through 90° in 6.825 d. Thus we get to see a bit more of one side of the Moon as it approaches, and another side as it moves away.

### The Power of Kepler's Laws

Together, Kepler's laws completely describe the motion of a planet. In principle, any question about planetary motion can be answered using these laws.

• Law I gives us the shape of each planet's orbit.

• Law II gives us a planet's speed at each point along its orbit.

• Law III gives us a relationship between sizes and periods of planetary orbits.

These laws also describe the orbits of comets, asteroids, and interplanetary spacecraft. With a minor change to Law III, they can describe orbits of moons and satellites around planets. With a small change to Law I, they can even describe the orbits of double stars.

## Galileo's Discoveries

Galileo used a simple telescope to look at the Sun, Moon, and planets; he also did experiments to understand the laws of motion:

• Astronomical discoveries: Jupiter's Moons, Phases of Venus, Lunar Mountains, Rotation of the Sun.
• Laboratory experiments: law of inertia, law of falling bodies, relative nature of uniform motion.

### Jupiter's Moons

 Looking at Jupiter, Galileo noticed four `stars' which kept close to the planet. Further observation convinced him that these were actually moons of Jupiter itself. This discovery showed that not every object orbited the Earth, and provided a `scale model' of the Solar system. Physics for the Enquiring Mind, Ch. 19, Fig. 7

### Jupiter's Moons

 Looking at Jupiter, Galileo noticed four `stars' which kept close to the planet. Further observation convinced him that these were actually moons of Jupiter itself. This discovery showed that not every object orbited the Earth, and provided a `scale model' of the Solar system. Physics for the Enquiring Mind, Ch. 19, Fig. 7

 Animation of Jupiter's Moons

### Phases of Venus

 Looking at Venus, Galileo saw the planet going through a complete cycle of phases from a slender crescent to full and back again to a crescent. This provided clear proof that Venus circled the Sun. In addition, Galileo noticed that Venus appears much larger at some times than at others. This provided further proof that Venus is not circling the Earth.

### Question 3.1

When would you expect the planet Venus to have the largest diameter as seen from the Earth?

1. When it appears almost full.
2. When it appears gibbous.
3. When it appears about half full.
4. When it appears as a crescent.
5. When it appears almost new.

### Lunar Mountains

Looking at the Moon, Galileo saw a rugged surface with craters and mountains; clearly, not a perfect celestial abode but a landscape with features like those on Earth.

 Physics for the Enquiring Mind, Ch. 19, Fig. 6c By measuring a mountain's shadow, he could estimate its height.

Physics for the Enquiring Mind, Ch. 19, Fig. 6b

### Rotation of the Sun

 Looking at the Sun (which may have eventually blinded him), Galileo saw sunspots. Others had noticed them before, but Galileo studied them systematically and used them to detect the rotation of the Sun.

### Law of Inertia

 A moving body moves in a straight line with a constant speed unless acted on by an outside force. Galileo formulated this law after noticing that a ball rolling down a ramp rolls back up to the same height, and imagining what would happen if the second ramp was replaced with a horizontal track. Physics for the Enquiring Mind, Ch. 19, Fig. 2b

### Laws of Falling Bodies

 I. All bodies fall at the same rate, regardless of their composition or weight. II. The total distance D  a falling body travels is proportional to the square of the time t  it has been falling: D = g t 2 ⁄ 2     where     g = 10 m ⁄ s2 Physics for the Enquiring Mind, Ch. 19, Fig. 1

### Relative Nature of Uniform Motion

Galileo argued that uniform motion is relative. By this he meant that someone moving at a constant rate can only detect their motion by looking at the outside world. For example, if you're in a cabin on a plane which is flying very smoothly you have no way of noticing that you're actually in motion without looking out the window.

## Newton's Mechanics

Newton stated three laws which together provide a complete description of the motion of any object:

• A moving body moves in a straight line with a constant speed unless acted on by an outside force. (Already suggested by Galileo).

• The acceleration (change in velocity) of a body is proportional to the force acting on it and inversely proportional to its mass.

• For every action, there is an equal and opposite reaction (that is, A cannot exert force on B unless B exerts an opposite force on A).

In addition, Newton proposed a law of universal gravitation:

 F = G M1 M2 d 2
where G is a constant of nature

### Orbits. I

Using these laws of motion and gravity, Newton was able to show the relationship between the motion of projectiles (eg, cannon balls) on Earth and the orbits of planets in the Solar System.

 Physics for the Enquiring Mind, Ch. 22, Fig. 29

### Orbits. IIa

Consider what happens to a ball released 200 km above the surface of the Earth. The law of falling bodies tells us that it falls

 18 km after 60 s 72 km after 120 s 162 km after 180 s

It hits the surface after falling for 200 s.

### Orbits. IIb

If we give the ball some sideways velocity, the law of inertia tells that it keeps this sideways velocity as it falls; thus, the distance to the right is proportional to time.

### Orbits. IIc

If we increase the sideways velocity, the ball falls further and further from where it started.

### Orbits. IId

Finally, when the sideways velocity reaches about 8 km/s, something almost magical happens: the surface of the Earth curves away just enough that the ball maintains its starting height of 200 km!

With this sideways velocity, the ball is finally in a circular orbit. It completes one orbit in about 90 minutes. This is a typical period for a satellite in a low orbit about the Earth.

`The secret of flying is to throw yourself at the ground and miss.' -- Douglas Adams

### Orbits. III

Finally, let's see if this orbit and the orbit of the Moon both obey Kepler's third law:

 P a P 2 ⁄ a 3 (s) (km) (s2 ⁄ km3) ball 5.40×103 6.58×103 1.02×10-4 Moon 2.36×106 3.84×105 0.98×10-4

The agreement isn't perfect, but considering the rough approximations we've used it's surprisingly good -- only 4% off. A more accurate calculation would clear up this small discrepancy.

 Joshua E. Barnes (barnes@ifa.hawaii.edu) Last modified: September 7, 2006 http://www.ifa.hawaii.edu/~barnes/ast110_06/rots.html