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Johannes Kepler (1571–1630) was one of the most important characters in the story of unraveling how the Solar System works. The German‐born mathematician was appointed assistant to Tycho Brahe (1546–1601), the most famous observer of the day. Granted access to Brahe's catalog of positional data, Kepler was given the task of explaining the orbit of Mars. After four years of calculations, Kepler finally realized in 1605 that the orbits of the planets were not perfect circles, but elongated circles known as ellipses.

Whereas a circle has one central point, an ellipse has two key interior points called foci (singular: focus). The sum of the distances from the foci to any point on the ellipse is a constant. For Solar System objects, the Sun always lies at one focus.

In order to draw an ellipse, place two drawing pins some distance apart and loop a piece of string around them. Place a pencil inside the string, draw the string tight and move the pencil around the pins. Now move one of the pins and repeat the process. Note how the shape of the ellipse has changed.

The amount of “stretching” or “flattening” of the ellipse is termed its eccentricity. All ellipses have eccentricities lying between zero and one. A circle may be regarded as an ellipse with zero eccentricity. As the ellipse becomes more stretched, its eccentricity approaches one.

Figure 1.13 A circle has an eccentricity of zero. As the ellipse becomes more stretched (i.e. the foci move further apart) the eccentricity approaches one. Half of the major axis is termed the semi‐major axis. The average distance of a planet from the Sun as it follows its elliptical orbit is equal to the length of the semi‐major axis. The eccentricity is calculated by dividing the distance between the two foci by the length of the major axis.

(Peter Bond)

In reality, most of the planets follow orbits that are only slightly elliptical. Their eccentricities are so small that they look circular at first glance. Pluto and Mercury are the main exceptions, with eccentricities exceeding 0.2.

Another key characteristic of an ellipse is its maximum width, known as the major axis. Half of the major axis is termed the semi‐major axis. The average distance of a planet from the Sun as it goes around its elliptical orbit is equal to the length of the semi‐major axis.

After intensive work on the implications of his discovery, Kepler eventually formulated his Three Laws of Planetary Motion.

 Kepler's First Law: The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. (Generally, there is nothing at the other focus.)

 Kepler's Second Law: The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. In order to do so, a planet must move faster along its orbit near the Sun and more slowly when it is far away. A planet's point of nearest approach to the Sun is termed perihelion; the furthest point from the Sun on its orbit is termed aphelion. Hence, a planet moves fastest when it is near perihelion and slowest when it is near aphelion.

 Kepler's Third Law: The square of a planet's sidereal (orbital) period is proportional to the cube of its mean distance (semi‐major axis) from the Sun. This means that the period, or length of time a planet takes to complete one orbit around the Sun, increases rapidly with its distance from the Sun. Thus, Mercury, the innermost planet, takes only 88 days to orbit the Sun, whereas remote Pluto takes 248 years to do the same.

Figure 1.14 Kepler's 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 Kepler's second law, the line joining a planet to the Sun sweeps out equal areas in equal times. In this diagram, the three shaded sectors, A₁, A₂, and A₃, all have equal areas. A planet takes as long to travel from A to B as from C to D and E to F, because it moves most rapidly when it is nearest the Sun (at perihelion) and slowest when it is farthest from the Sun (at aphelion).

(Peter Bond)

Figure 1.15 A graph showing the orbital periods of the planets plotted against their semi‐major axes, using a logarithmic scale. The straight line that connects the planets has a slope of 3/2, verifying Kepler's third law which states that the squares of the orbital periods increase with the cubes of the planetary distances. This law applies to any bodies in elliptical orbits, including Jupiter's four largest satellites (inset).

(Kenneth R. Lang, The Cambridge Guide to the Solar System)

This law can be used to make some useful, but fairly simple, calculations. For example, if the period is measured in Earth years and the distance is measured in astronomical units (AU), the law may be written in the simple form: P(years)² = R(AU)³.

This equation may also be written as: P(years) = R(AU)^3/2. Thus, if we know that Pluto's average distance from the Sun (semi‐major axis) is 39.44 AU, we can calculate that its orbital period P = (39.44)^3/2 = 247.69 years. Similarly, if we know that Mars takes 1.88 Earth years to orbit the Sun, we can calculate that its semi‐major axis R = (1.88)^2/3 = 1.52 AU.

Figure 1.16 The early stages of star and planet formation. (a) A Hubble Space Telescope view of five young stars in the Orion Nebula. Four are surrounded by gas and dust trapped in orbit as the stars formed. These are possibly protoplanetary disks, or “proplyds,” that might eventually produce planets. The bright proplyds are closest to the hottest stars of the parent star cluster, while the object farthest from the hottest stars appears dark. (C. R. O'Dell/Rice University; NASA) (b) This HST image shows Herbig‐Haro 30, a young star surrounded by a thin, dark disk. The disk extends 64 billion km, dividing the nebula in two. The central star is hidden from direct view, but its light reflects off the upper and lower surfaces of the disk to produce the pair of reddish nebulas. Gaseous jets (green) remove material from above and below the disk and transfer angular momentum outwards. (Chris Burrows/STScI, the WFPC2 Science Team and NASA) (c) A computer simulation showing how a protoplanetary disk surrounding a young star begins to fragment and form gas giant planets with stable orbits.

(Mayer, Quinn, Wadsley, Stadel, 2002, Science)

Observations of young star systems show that the gas disks that form planets usually have lifetimes of only 1 to 10 million years, which means the gas giant planets probably formed within this time frame. In contrast, Earth probably took at least 30 million years to form, and may have taken as long as 100 million years.

It is worth noting here that computer simulations of the early Solar System show that even the slightest differences in initial conditions can produce different planetary systems. Depending on exactly where each embryo started out, the orbital positions of new planets vary randomly from simulation to simulation. The total number of planets – and hence, their final masses – may also vary greatly. It seems that planet formation is a very chaotic process as evidenced by exoplanet systems which bear little resemblance to our Solar System (see Chapter 14).