Mathematics and astronomy

Summary: Mathematics is used in astronomy to measure and model celestial bodies.

Astronomy is the science that deals with celestial objects. It is divided into two disciplines: positional astronomy (or “astrometry”), which deals with the positions and movements of celestial objects; and astrophysics, which deals with their chemical and physical properties.

Positional astronomy began as a practical science. The first astronomers, before the invention of writing, dealt with such questions as the proper time of the year to plant crops and the proper dates for religious festivals. As their understanding improved over the ages, astronomers tackled other practical problems such as how to predict eclipses, how to tell time within a day, and how to navigate at sea.

Ancient people could take simple observations of the sun and moon and observe the patterns they made. From there it was a short leap to predicting future patterns. They would first record (or, before writing, memorize) the observations, and then perform a mathematical analysis—even if the analysis were nothing more than counting (for example, discovering there were about 365 days between winter solstices).

A much more sophisticated accomplishment was working out the complicated cycles on which lunar and solar eclipses occurred. An eclipse can be terrifying for a people who are not expecting it. If astronomers (many of whom doubled as priests) could predict eclipses, they could warn people in advance and reduce the collective fear.

A number of ancient peoples, including Mayans, Chinese, and Babylonians, developed elaborate calendar systems and tracked the movements of the planets. The Chinese constructed star charts, kept records starting possibly as early as 4000 b.c.e., and developed astronomical instruments. The Babylonians mapped constellations and introduced 60-minute hours and 60-second minutes. Both the Chinese and the Babylonians were able to predict eclipses. By 2500 b.c.e, Egyptians had measured star positions well enough to orient the pyramids to face celestial north. Polynesians traveled throughout the Pacific Ocean using stars as navigational aids.

The Greeks

The ancient Greeks effectively applied mathematics to astronomy. Eratosthenes (c. 200 b.c.e.) used geometry to calculate the size of Earth. Hipparchus (c. 161–126 b.c.e.) discovered the precession of the equinoxes and created the most accurate Greek tables of lunar motion. Like some other Greek astronomers, he held that Earth revolved around the sun. Claudius Ptolemy (c. 120–150 c.e.) combined observations from Hipparchus and others with his own observations to propose a model of how the solar system worked—assuming Earth was at the center. By using epicycles (circles revolving on circles), he produced what was by far the best model of the heavens until Nicolaus Copernicus.

The Greeks did not only conduct astronomical calculations by hand but used a computer as well. Though not much is currently known about it, a mechanical analog computer was built somewhere in the Greek world about 100 b.c.e., called the “Antikythera mechanism” after the place it was found. This remarkably sophisticated computer was able to show both solar and lunar calendars, track the complicated path of the moon using Hipparchus’s results, and predict eclipses for years into the future.

The Renaissance

During the Middle Ages, Arabs, Persians, and Jews, as well as European Christians (after c. 1000 c.e.), continued the work of the Greeks, including making new tables of planetary positions to update Ptolemy’s, and keeping track of the precession of the equinoxes. In 1543, Copernicus’s book on the solar system was published. Through a mathematical analysis of Ptolemy’s work and later observations, Copernicus showed that a system in which the sun was the center of the solar system led to simpler and more accurate analysis than Ptolemy’s.

Johannes Kepler used Tycho Brahe’s careful naked eye observations of the planets to show that Mars went around the sun in an ellipse, not a circle as the Greeks had assumed. Kepler stated his three laws, which relate the speed of a planet to the shape of its orbit, but he could not explain why these laws worked. Isaac Newton was the first to explain Kepler’s laws. He was able to show that any object affected by gravity would move in one of the conic sections: Kepler’s ellipse, a line, a circle, a parabola, or a hyperbola. The one exception was the planet Uranus, which did not follow its Newtonian orbit.

It was not until the 1800s that Urbain Leverrier, in France, and John Couch Adams, in England, (unknown to each other) made the assumption that the discrepancies were because of the gravitational pull of an unknown planet. The planet Neptune was discovered in 1846 using Leverrier’s prediction. Neptune was found by the consideration of the three components, P_x, P_y, and P_z, of Neptune’s position and the three components, V_x, V_y, and V_z, of Neptune’s velocity.

Until 1821, Uranus was moving faster in its orbit than expected—more than 4 planetary diameters ahead of its predicted position. After 1821, Uranus moved slower than expected. Obviously, Uranus moved past Neptune around 1821. If one adjusted the coordinate system so that P_x = 0 was Uranus’s position in its orbit in 1821 and examined how far Uranus was pulled above or below its expected orbit, then one can tell whether Neptune was above or below Uranus in 1821, which gives P_y, and also whether Neptune was moving up or down, which gives V_y. If we have V_z, which represents Neptune’s distance from the sun in 1821, then Kepler’s laws can be used to find the two remaining parameters: V_x and V_z. Leverrier and Adams used a shortcut to find P_z. Both used Titius-Bode’s law, an empirical formula, to predict the next planet beyond Uranus to be 38.8 times Earth’s distance from the sun. These predictions were good enough to find Neptune, although Neptune is only 30.1 times Earth’s distance from the sun.

Leverrier later examined the orbit of Mercury and found a discrepancy of 43 seconds of arc (which sounds small but is twice the discrepancies used to find Neptune). He computed the orbit of a hypothetical planet, called “Vulcan,” which would explain this 43-second variation. Vulcan has never been found, and Einstein’s general theory of relativity also explains this discrepancy.

Parallax

The ancient Greeks made attempts using parallax (the difference in the angle to a distant body measured from two different locations, also called triangulation) to find the size of the solar system. Being restricted to naked-eye observations, their results were inaccurate. Using telescopes, a much more accurate measurement was made in 1761 in which observers scattered across Earth found the parallax of Venus when it passed in front of the sun. The observations gave a value of 95.25 million miles from Earth to the sun (the modern estimate is just under 93 million miles). A much more difficult problem was to find the distances of stars by their parallax when viewed from opposite sides of Earth’s orbit, first accomplished by Friedrich Bessel in the 1830s. Is space Euclidean or non-Euclidean? If measurably non-Euclidean, this would show up in stellar parallax measurements. No such effect has yet been observed, so one can say—except for relativistic considerations—that space is Euclidean for hundreds of light-years from Earth.

Astrophysics

Astrophysical questions date to the ancient Chinese, who discovered sunspots, and Hipparchus (c. 190–120 b.c.e), who worked on the magnitude (or brightness) of stars. His magnitudes, much refined, are still in use today. However, astrophysics as a discipline can be said to have started with Joseph von Fraunhofer, who in 1815 devised a spectroscope and catalogued the various lines (known as the Fraunhofer lines) that can be seen in the solar spectrum. In the 1850s, Gustav Kirchhoff and Robert Bunsen determined that these lines belonged to different chemical elements. Thus, by examining the spectrum of a star, its chemical composition can be determined. In addition, it was discovered that magnetic fields caused broadening and splitting of Fraunhofer lines, allowing the magnetic fields of stars to also be investigated.

Over the course of the twentieth century, astrophysicists went from studying the spectrum of visible light to studying every frequency of electromagnetic waves—from gamma rays to radio waves. There is now no known radiation from a star that is not being used to help find answers to the questions of what stars are, and how they operate.

Bibliography

Freeth, Tony. “Decoding an Ancient Computer.” Scientific American 301, no. 6 (December 2009).

Gould Jr., Benjamin Althrop. Report to the Smithsonian Institution on the History of the Discovery of Neptune. Washington DC: Smithsonian, 1850.

Hester, Jeff, Bradford Smith, George Blumenthal, and Laura Kay. 21st Century Astronomy. New York: W. W. Norton, 2010.