Astronomy

The year of the seasons is 365.2422 days. A leap day every fourth year, dropped in century years unless they divide by 400, averages 365.2425: the calendar drifts one day in about 3,300 years. Twelve lunar months of 29.53 days fall 11 days short of the year.

Use: Keeps planting and festival dates fixed to the seasons (reading the year). A calendar kept by the Moon alone drifts through the seasons by 11 days a year and needs an extra month every two or three years.

First: Introduced by Pope Gregory XIII in 1582 on the calculations of Lilius and Christopher Clavius (MacTutor).

Measured against the stars the Earth turns once in 23 h 56 min 4 s; measured against the Sun, which moves a degree a day along its path, it takes 24 h. So each star rises about 4 minutes earlier every night, 2 hours earlier each month.

Use: Why the night sky changes with the seasons, and how a star's rising tells the time of year; a telescope's drive must turn once per sidereal day to follow a star.

First: Known to Babylonian astronomers; Hipparchus measured the year and found the slow drift of the equinoxes, about 130 BC (MacTutor).

With the noon Sun due south, latitude φ is 90° minus its altitude a plus its declination δ, how far north of the equator it is that day. δ swings between +23.44° in June and −23.44° in December; N is the day of the year from 1 January. The approximation is good to about a degree.

Use: Latitude anywhere, on any clear day, with a stick and a protractor; Polaris gives it directly at night in the north. The full method, including the Sun due north, is in navigation.

First: Eratosthenes measured the Earth's circumference from the noon Sun's angle at two cities, about 240 BC (MacTutor).

H₀ is the hour angle of sunrise and sunset; the Sun crosses 15° of hour angle an hour. At the equator every day is 12 hours. At 52° N midsummer gives about 16½ hours and midwinter about 7½; above 66.6° the Sun does not set in midsummer.

Use: When crops will flower (many respond to day length), how many hours of work light a season gives, and how much a solar panel can collect in winter at a given latitude (energy without the carbon bill).

First: Spherical astronomy from Ptolemy's Almagest, about AD 150 (MacTutor).

The Earth turns 15° an hour, so the difference between local noon and the time at a reference meridian, carried on a clock, gives the longitude difference: 4 minutes of time per degree.

Use: A clock 1 minute wrong puts a ship 15′ of longitude out, 28 km at the equator. That is why seagoing navigation waited for a chronometer good to seconds a day, and why GPS needs clocks good to billionths of a second (seagoing ships).

First: John Harrison's H4 sea watch, tested in 1761–1762 (Royal Museums Greenwich).

Planets move in ellipses with the Sun at one focus, sweep out equal areas in equal times (fastest when nearest), and the square of the period T grows with the cube of the orbit's average radius a. Round the Sun, T in years is √(a³) with a in astronomical units: Mars at 1.52 AU takes 1.88 years.

Use: Predicts where every planet and moon will be, and so eclipses and the seasons' exact lengths; the same law, with the Earth's mass, sizes every satellite orbit.

First: Johannes Kepler: the first two laws in Astronomia nova, 1609, the third in Harmonices mundi, 1619 (MacTutor); derived from gravitation by Newton, 1687.

A circular orbit at distance r from the centre needs the speed shown, and leaving for good √2 times as much. Near the Earth's surface that is 7.8 km/s to orbit and 11.2 km/s to escape. At 42,164 km from the centre (35,786 km up) an orbit takes one sidereal day, so the satellite hangs over one spot.

Use: Geostationary satellites for communications and weather, GPS satellites at 20,200 km in 12-hour orbits, and the energy cost of reaching orbit, which is why rockets are mostly fuel.

First: Newton's cannonball fired from a mountain, Principia, 1687 (Gutenberg).

The Sun's power spread over a sphere the size of the Earth's orbit gives 1,361 W on every square metre facing it, above the atmosphere. At the ground at noon on a clear day about 1,000 W/m² arrives; averaged over day, night and weather, much less.

Use: The budget for all solar energy: a 1 m² panel converting 20 % gives about 200 W in full sun. Panels tilted toward the equator at about the latitude angle catch the most over a year (energy without the carbon bill).

First: Measured from space, 1,360.8 ± 0.5 W/m² (Kopp and Lean, 2011); Pouillet made the first estimate, 1838.

As the Earth moves round the Sun, a near star shifts against far ones by its parallax angle p. A star with a parallax of one arcsecond is one parsec away. The nearest, Proxima Centauri, has 0.77″, so 1.30 pc, 4.2 light-years.

Use: The base of every distance in astronomy. The same triangle, scaled down, is how a surveyor fixes a distance from a measured baseline and two angles.

First: Friedrich Bessel measured the parallax of 61 Cygni, 1838 (MacTutor); the Gaia satellite has since measured it for about 1.5 billion stars (Gaia DR3).

Brightness is measured in magnitudes: five magnitudes is a factor of 100, and higher numbers are fainter. A star's apparent magnitude m minus its true magnitude M gives its distance d. Stars whose true brightness is known, such as Cepheid variables whose pulse period sets their brightness, are standard candles.

Use: Measures distances far beyond parallax, to other galaxies; also why a light twice as far away looks a quarter as bright.

First: Henrietta Swan Leavitt's period–luminosity law for Cepheids, 1912 (MacTutor); Pogson fixed the magnitude scale, 1856.

Light from a source moving away is stretched to longer wavelengths, toward the red; from one approaching, compressed toward the blue, by the fraction v/c. Sound does the same at the speed of sound.

Use: A star wobbling a few metres per second reveals a planet pulling on it: the first planet round a Sun-like star was found this way in 1995 (Mayor and Queloz). Radar speed guns and weather radar use the same shift.

First: Christian Doppler, 1842 (MacTutor).

Distant galaxies recede at speeds proportional to their distance: the universe is expanding. Run backwards, the expansion gives its age, 13.8 billion years from the cosmic microwave background (Planck 2018).

Use: Fixes the age and scale of the universe. The two best ways of measuring H₀ disagree, about 73 against 67, one of the largest open problems in physics (physics since 2000).

First: Georges Lemaître, 1927 (MacTutor); Edwin Hubble's observations, 1929 (PNAS).