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The Motions of Celestial Objects

The Sun is an average star of small significance in the overall scheme of the universe but is of central importance to our immediate neighborhood of space, the solar system. It sends life-supporting energy to the Earth, and its gravitational force dictates the motions of nine planets and numerous smaller objects such as asteroids and comets. While only average compared to many stars, the Sun is more than 100 times the size of the Earth and many times larger than Jupiter, the largest of all our solar system’s planets.

The Sun’s family of objects move around it in elliptical orbits of varying shape and size. In order of increasing distance from the Sun the planets are Mercury, Venus, the Earth, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto. The four inner planets together with Pluto are often referred to as terrestrial planets because of their similarity to the Earth, and the four remaining planets are frequently called Jovian planets because in many ways they resemble Jupiter. From the closest planet, Mercury, to the most distant planet, Pluto, the solar system extends some 5,914 million kilometers. At even larger distances, the remote realm of the comets defines the outer reaches of the solar system.

All of the planets revolve about the Sun in the same direction as the Earth and in very nearly the same plane. The Earth’s orbital plane is known as the ecliptic plane and the time required to orbit the Sun is called the sidereal period of revolution. For the Earth this period is one year but each planet has a different period of revolution. Mercury takes only 88 days while distant Pluto requires 248.5 years to make its journey around the Sun!

As planets revolve around the Sun they also rotate, and the distinction between these terms should be noted. Revolution always refers to the motion of one object about another object whereas rotation relates to the motion of a single object about an imaginary axis passing through it. The Earth’s sidereal period of rotation is what is commonly called one day. The Earth rotates 365.25 times as it revolves once around the Sun. Some planets have periods of rotation that are longer then the Earth’s and some have rotational periods that are shorter. Strangely enough, the largest planet, Jupiter, rotates once in a little less than 10 hours while little Mercury takes more than 58 days.

The acceptance of this Sun-centered picture, called the heliocentric theory, is the result of centuries of scientific thought and observations. One of the first advocates of the heliocentric theory was the Greek astronomer Aristarchus, who lived about 300 BC. However, it was not until centuries later that the Polish astronomer Copernicus revived and revised the ancient idea of a Sun-centered system. His heliocentric theory was published in a book called De Revolutionibus (Concerning the Revolutions) in 1543 AD, the year of his death.

It was more than 60 years later that the first accurate description of planetary motion was made by the German astronomer Johannes Kepler. Between 1609 and 1618 Kepler, using the observations of the Danish astronomer Tycho Brahe, discovered three laws that describe planetary motion. The first two laws were published in the book Commentaries on the Motions of Mars in 1609; the third appeared in a 1618 book called The Harmony of the Worlds.

FIRST LAW: Each planet moves about the Sun in an elliptical orbit with the Sun at one focus.

SECOND LAW: The line connecting the Sun and a planet sweeps out equal areas in equal intervals of time.

THIRD LAW: The semi-major axis cubed is equal to the period of revolution squared.

Kepler’s laws are a significant departure from earlier heliocentric theories that had planets orbiting the Sun in circular orbits. Kepler discarded the circles and replaced them with ellipses. Whereas all circles have the same shape but can differ in size, ellipses can differ in both shape and size. The size of a circle is given by its radius. The size of an ellipse is specified by its semi-major axis, which is one-half of the major axis (see Figure 1.1). The semi-major axis is usually designated by "a".

The shape of an ellipse is specified by the eccentricity, which can be defined as the ratio, cf/a, where cf is the distance between the center of the ellipse and the focus. All circles have eccentricities equal to 0. For ellipses, eccentricities are always greater than 0 but less than 1. If the eccentricity is equal to 0 the ellipse is, in fact, a circle, but as the eccentricity approaches 1 the ellipse becomes more and more elongated in appearance.

Kepler discovered that planet orbits have eccentricities very close to but not exactly equal to 0. That is, the orbits are very nearly circular. Were it not for the accurate observations of Tycho Brahe, Kepler would not have been able to discover the planets’ small deviations from circular motion.

Because a planet moves around the Sun in an elliptical orbit with the Sun at one focus, its distance from the Sun continually changes. When a planet is closest to the Sun it is said to be at perihelion and when it is farthest from the Sun, at aphelion. On the average, however, a planet’s distance from the Sun is equal to the semi-major axis of its elliptical orbit. For the Earth this distance is approximately 93 million miles.

Kepler’s second law says that if the time required for a planet to move from T1 to T2 in Figure A is the same as the time from T4 to T3, then the areas swept out must be equal; area 1 must equal area 2. This law requires that as a planet’s distance from the Sun changes, so must its orbital speed. A planet moves fastest when it is closest to the Sun at the perihelion point and slowest when it is farthest from the Sun at the aphelion point.

Kepler’s third law states that the square of a planet’s sidereal period of revolution, P, is equal to the cube of its semi-major axis, that is,

In the above equation the sidereal period, P, must be expressed in years. The semi-major axis, a, must be expressed in terms of a unit of distance called the astronomical unit (AU). One astronomical unit is defined as the Earth’s average distance from the Sun. Thus, it is the semi-major axis of the Earth and is equivalent to 93 million miles. In terms of this unit Mercury is 0.39, the Earth is 1.0, Jupiter is 5.2, and Pluto is 39.5 astronomical units from the Sun. The third law indicates that the periods of revolution increase with increasing distance from the Sun. Mercury’s sidereal period is 0.24 years (88 days), while Pluto’s is 248.5 years.

 Newton’s Laws

Kepler’s laws describe how planets move but do not explain why they move in the way they do. Neither Kepler nor his friend Galileo, who in 1609 first used the telescope for astronomical observation, could answer the question why. The answer was found by the Englishman Isaac Newton, who during 1666 and 1667 discovered the force called gravity. Newton explained planetary motion in terms of the gravitational forces acting between the Sun and planets and demonstrated that Kepler’s laws are a natural consequence of gravity.

Newton’s universal law of gravity asserts that the gravitational force attracting any two objects varies directly as the product of their masses and inversely as the square of the distance between them. The law is written as

F = GmM / D²

where D is the distance between the masses m and M, and G is a number called the universal gravitational constant.

The force of gravity rapidly decreases as the distance between the two masses increases. For example, Pluto at 40 astronomical units from the Sun, experiences a gravitational force that is 1,600 times smaller than the force at the Earth’s distance. Yet, an object would have to be infinitely far from the Sun to feel no pull from the Sun’s gravity! In fact, every object in the universe attracts every other object according to Newton’s law. The Sun, the solar system, our galaxy, and the universe are all held together by the glue of gravity.

In the same year that he discovered the law of gravity Newton proposed three general laws of motion.

FIRST LAW: An object at rest remains at rest and an object moving in a straight line at a constant speed continues to do so unless acted upon by an unbalanced force.

SECOND LAW: Force (F) is equal to mass (M) times acceleration (A). That is,

F = M x A.

THIRD LAW: For every force there is an equal but opposite force.

These three laws together with the law of gravity can be used to accurately predict the motions of objects in the universe.

Newton’s insight was the culmination of centuries of speculation, thought, and observation. But he himself pointed out, "if I have seen farther than others it is because I have stood on the shoulders of giants." By this he meant that he could not have discovered his laws had it not been for the work done by such individuals as Copernicus, Brahe, Kepler, and Galileo.

The heliocentric theory is not only correct, it is also easy to visualize. Everyone can imagine standing outside the solar system watching the planets as they move around the Sun. Although this theory was first proposed as early as 300 BC, most astronomers up until the time of Copernicus believed in another theory known as the geocentric theory. This theory held that the Earth was the center of the solar system and that it neither rotated on an axis nor revolved about the Sun. In this theory everything moved about a central, stationary Earth. The daily east to west motion of celestial objects was thought, for example, to be caused by an actual rotation of the sky.

When we observe celestial objects from the Earth, motions seem more complex than they really are because they are a combination of the object’s motion and the Earth’s rotation and revolution. In reality it is the west to east rotation of the Earth that makes the sky appear to move from east to west. The apparent daily motion is called diurnal motion. Diurnal motion is comparable to the apparent westward motion of the landscape when you are riding in a car going east.

Similarly, the Earth’s west to east revolution around the Sun makes the Sun appear to move eastward relative to the stars. Each day the Sun moves about one degree eastward and every month it enters a new constellation. In one year the Sun makes a complete circuit through the 12 constellations known as the zodiacal constellations. The Sun’s apparent yearly path is called the ecliptic, and the narrow band of sky on either side of the ecliptic is known as the Zodiac.

Because planets orbit the Sun in the same direction and in about the same plane as the Earth, they also seem to move eastward through the zodiacal constellations and they never get very far from the ecliptic. The planets, however, do not circle the ecliptic in one year. The time it takes a planet to move around the ecliptic depends on the planet’s period of revolution and hence on its distance from the Sun. The combined motions of a planet and the Earth sometimes make a planet reverse its eastward motion and seem to move westward. This "backing up" is called retrograde motion. For planets farther from the Sun than the Earth, retrograde motion occurs when the faster moving Earth overtakes and passes the slower moving planet. You may have observed a similar phenomenon if you have ever sped by a slower moving car and looked back at the car you passed in your rear view mirror!

The Moon’s monthly revolution around the Earth causes it to appear to move eastward through the Zodiac. Since the Moon revolves around the Earth in about one month, it also circles the ecliptic every month. The Moon’s motion is very rapid compared to that of the Sun and the planets. Every hour the Moon moves its own apparent size, 1/2 degree, to the east so each day it travels some 12 degrees. This motion is so large that it is easily observed with the naked eye. As it moves through the zodiacal constellations, we see the Moon going through phases.

Viewed from the Earth the motions of celestial objects seem exceedingly complex and it is easy to understand why it required centuries to sort out and explain these motions.