Fig. 2.1 . The German astronomer Johannes Kepler (1571-1630) published his first two laws of planetary orbital motion in 1609. His first law states that the orbit of a planet about the Sun is an ellipse with the Sun at one focus. The other focus of the ellipse is empty. According to Keplerís second law, the line joining a planet to the Sun sweeps out equal areas in equal times. This is also known as the law of equal areas. It is represented by the equality of the three shaded areas ABS, CDS and EFS. It takes as long to travel from A to B as from C to D and from E to F. A planet moves most rapidly when it is nearest the Sun, at perihelion; a planetís slowest motion occurs when it is farthest from the Sun, at aphelion.
Fig. 2.2 . The orbital periods of the planets in years are plotted against the semi-major axes of their elliptical orbits in astronomical units abbreviated AU, using a logarithmic scale. The straight line that connects the points has a slope of 3/2, thereby verifying Keplerís third law that states that the square of the orbital periods increase with the cubes of the planetary distances. The German astronomer Johannes Kepler (1571-1630) published this third law in 1619. This type of relation applies to any set of bodies in elliptical orbits, including Jupiterís four largest satellites shown in the inset, with a vertical axis in units of days and a horizontal axis that gives the distance from Jupiter in AU units.
Fig. 2.3 . The Moon's gravitational attraction causes two tidal bulges in the Earth's ocean water, one on the closest side to the Moon and one on the farthest side. The Earth's rotation twists the closest bulge ahead of the Earth-Moon line (dashed line), and this produces a lag between the time the Moon is directly overhead and the time of highest tide. The Moon pulls on the nearest tidal bulge, slowing the Earth's rotation down. At the same time, the tidal bulge nearest the Moon produces a force that tends to pull the Moon ahead in its orbit, causing the Moon to spiral slowly outward.
Fig. 2.4 . The height of the tides and the phase of the Moon depend on the relative positions of the Earth, Moon and Sun. When the tide-raising forces of the Sun and Moon are in the same direction, they reinforce each other, making the highest high tides and the lowest low tides. These spring tides (top) occur at either new or full Moon. The range of tides is least when the Moon is at first or third quarter, and the tide raising forces of the Sun and Moon are at right angles to each other. The tidal forces are then in opposition, producing the lowest high tides and the highest low tides, or the neap tides (bottom). In this diagram the height of the tides has been greatly exaggerated in comparison to the size of the Earth.
Fig. 2.5 . A large satellite (top) that moves well within a planetís Roche limit (dashed curve) will be torn apart by the tidal force of the planetís gravity, in an effect first investigated in 1847 by the French mathematician Edouard Roche (1820-1883). The side of the satellite closer to the planet feels a stronger gravitational pull than the side farther away, and this difference works against the self-gravitation that holds the body together. A small solid satellite (bottom) can resist tidal disruption because it has significant internal cohesion in addition to self-gravitation.
Fig. 2.6 . Instead of always tracing out the same ellipse, the orbit of Mercury pivots around the focus occupied by the Sun. The point of closest approach to the Sun, the perihelion, is slowly rotating ahead of the point predicted by Newton's theory of gravitation. This was at first explained by the gravitational tug of an unknown planet called Vulcan that was supposed to revolve about the Sun inside Mercuryís orbit, but we now know that Vulcan does not exist. The German physicist Albert Einstein (1879-1955) explained Mercury's anomalous motion in 1915 by inventing a new theory of gravity in which the Sun's curvature of nearby space makes the planet move in a slowly revolving ellipse.
Fig. 2.7 . A massive object creates a curved indentation upon the ďflatĒ space described by Euclidean geometry, which applies in our everyday life on Earth where we do not directly encounter astronomical amounts of matter. Notice that the amount of space curvature is greatest in the regions near a cosmic object like a star, while further away the effect is lessened.