The second problem with the geocentric model was that planets became dimmer and brighter depending on their orbital position with respect to the sun. We now know that it was because they were closer or further from Earth; however, this was difficult to explain if you thought that they were orbiting in perfect circles around Earth.

To account for retrograde motion and the brightness and dimming of planets, Ptolemy proposed that Mars and other planets were on a second sphere or rim (epicycle) attached to a primary rim, which caused it to move backward periodically (Figure 1‑6). He also proposed that Earth was orbiting around an imaginary point.

During the Renaissance, Nicolaus Copernicus (1473-1543) spent three decades plotting the positions of the planets against the background of the fixed stars. He spent night after night measuring their angular positions with a triquetrum (Figure 1‑7). He then used mathematics to calculate the paths of the planets over time. Copernicus realized that the geocentric model was incorrect and that phenomena such as the backward movement of Mars made sense if Earth and Mars orbit the Sun.

Copernicus thought that the planets orbited the sun on celestial spheres (perfect circles). Because the planets orbit the sun in imperfect circles or ellipses, his mathematical calculations of planetary positions on perfect circles did not match the observed paths of the planets. Another weakness with Copernicus' work was that he used instruments that did not have enough precision to enable him to precisely estimate the positions of the planets. The slight error in his theory and lack of precision in his instruments justifiably led to uncertainty. Copernicus wrote a small handwritten pamphlet on the heliocentric theory for a few colleagues in 1514. He finally published his book: De Revolutionibus Orbium Coelestium (On the Revolutions of the Celestial Spheres) in 1543, the year of his death, in which he described his heliocentric model (Figure 1-8).

Tycho Brahe (1546-1601) was a Danish astrologer and astronomer. His major contribution to the study of the solar system was his precise plotting of the positions of the planets over a period of 40 years. The Danish king built an estate and observatory for Brahe (Figure 1‑9) on an island in 1576. Annual funding for the observatory amounted to 1% of the Danish national budget. The king even allowed Brahe to tax the citizens of the island on which the observatory was located. His instruments were enormous in comparison to those of Copernicus, and he was able to generate much more accurate tables of planetary positions; however, Brahe could not accept the heliocentric model, so he concluded that the sun orbits Earth, and Venus and Mercury orbit the sun.

Scientists used the principle of parallax to determine distances to planets and other objects. Using the principle of parallax, Tycho looked at a supernova explosion from different locations on Earth and observed that its position did not change (no parallax), which proved that it was beyond the moon. Previously, people had thought that all the celestial spheres beyond the moon were unchanging. Brahe also noted that a comet appeared to move across the supposed celestial spheres, which showed that there were no impermeable spheres in the sky.

Brahe fell out of favor with the new king of Denmark in 1597. He immigrated to Prague, where he became the imperial astronomer for the Austro-Hungarian Holy Roman Emperor King Rudolph II. Brahe died shortly after that in 1601, but not before the great mathematician Johannes Kepler (Figure 1‑10) joined him in 1600.

Much to the chagrin of Brahe’s family, Kepler (1571-1630) took possession of Brahe’s data on planetary positions when Brahe died. Kepler then analyzed the data for 20 years. Kepler discovered that planets do not orbit the sun in perfect circles, that planets speed up when they are closer to the sun, and that planets closer to the sun have faster orbits in general. With the realization that the motion of the planets was elliptical, the equations that described the motions of the planets worked perfectly. Unlike Brahe, Kepler thought that Copernicus was correct and that all of the planets orbited the sun. In addition to his mathematics, Kepler is often credited for beginning the scientific revolution. He realized that the patterns of movement of the planets indicated that a physical force moved them through the sky rather than angels. Why would angels push at a faster rate when planets were closer to the sun? This led to the search for natural causes in physics and other sciences and initiated the scientific revolution.

Galileo improved the design of the spyglass and developed an 8X magnification astronomical telescope by which he observed moons orbiting Jupiter and the phases of Venus as it orbited the sun. The fact that Venus had phases and was larger during the "new" phase was strong proof of the heliocentric model. The fact that the moons of Jupiter orbited Jupiter disproved the geocentric model because it showed that everything did not revolve around Earth. After Galileo published his drawings of the phases of Venus and Jupiter’s moons in 1610 in The Starry Messenger, he became a celebrity, and everyone wanted to look through his telescope (Figure 1‑11). However, the fun was short-lived. Galileo encountered resistance from those who believed in geocentrism. He was put on trial by the church and ordered not to promote the heliocentric model. This was only the beginning of his problems.

Although Brahe’s measurements and Kepler’s equations were precise enough to provide mathematical proof of the orbit of the planets around the sun, there still was not a physical explanation for the orbit of the planets. Isaac Newton considered Kepler’s laws of planetary motion and an apple that fell from a tree and conceived of the law of gravity, providing a physical explanation for planetary motion. He realized that gravitational force would decrease with distance and increase with mass. This would cause the planets to move in elliptical orbits and move faster when they were closer to the sun. Newton determined that the principle works on the earth and anywhere in space. The force of gravity between any two objects is proportional to the mass of the two objects and the distance between them squared.