Earth sciences

1 mm of rain on 1 m² is 1 litre. The runoff coefficient C is the share that can be collected: about 0.8 for a metal or tiled roof, 0.1–0.3 for grassland, nearly 1 for paving.

Use: A 50 m² roof in a climate with 600 mm a year yields about 24,000 litres: drinking and cooking water for a family of five at 13 litres a head a day, if stored through the dry months. Size gutters, tanks and drains from the heaviest rain, not the average.

First: Perrault measured in 1674 that rain over the Seine basin exceeds the river's flow; Halley showed evaporation from the sea is enough to supply the rain, 1687 (Philosophical Transactions).

Flow Q through ground of cross-section A is its hydraulic conductivity K times the slope of the water table, Δh/L (written i). The water itself moves at v, faster by the inverse of the porosity n. K is about 10⁻² m/s for gravel, 10⁻⁴ for clean sand, 10⁻⁶ for silt and below 10⁻⁹ for clay.

Use: With a slope of 1 in 100 and porosity 0.3, water crosses 30 m of clean sand in about 100 days but 30 m of gravel in about a day. The 30 m rule between a latrine and a well protects in sand and silt; in gravel, fissured limestone or cracked rock it does not (separate waste from water). Also sets the flow of a slow sand filter and a well's yield (wells).

First: Henry Darcy, Les fontaines publiques de la ville de Dijon, 1856 (Internet Archive).

Air pressure falls by a factor e every scale height H. Near sea level it drops about 1 hPa for every 8 m of climb; at 3,000 m it is about 70 % of sea level, so each breath holds 30 % less oxygen and water boils at about 90 °C.

Use: Altitude sickness is common above about 2,500 m: climb slowly, and descend at once for confusion or breathlessness at rest. A barometer measures height; a falling barometer at one place means a low is coming.

First: Pascal had a barometer carried up the Puy de Dôme, 1648 (MacTutor); Halley gave the law of pressure with height, 1686 (MacTutor).

Rising air expands and cools: dry air by 9.8 °C per km, saturated air by about half that, and the average atmosphere by 6.5 °C per km. The temperature minus the dew point, times about 125 m, gives the height where rising air starts to form cloud.

Use: A hill 1,000 m up averages 6.5 °C colder than the valley below it. On still clear nights the rule reverses near the ground: cold air drains downhill and pools, so frost strikes valley floors first and fruit trees belong on slopes (choose the ground).

First: William Thomson (Lord Kelvin), on the convective equilibrium of the atmosphere, 1862 (MacTutor).

How cold the wind makes bare skin feel, from the air temperature T in °C and the wind speed V in km/h at 10 m. At −10 °C a 30 km/h wind feels like −20 °C. At a wind chill of about −28 °C exposed skin can freeze within 30 minutes.

Use: Decides when to cover the face and hands and cut time outside; the same body loses heat as fast whether working, waiting or injured. Hypothermia is in the first-aid guide.

First: Adopted by the US and Canadian weather services, 2001 (US National Weather Service).

A mineral that held only the parent isotope P when it formed accumulates the daughter D at a fixed rate λ; the ratio D/P gives its age. Uranium–lead, potassium–argon and rubidium–strontium pairs date rocks from thousands to billions of years; carbon-14 dates once-living material up to about 50,000 years.

Use: Dates the rocks that hold ores, coal and oil, and so the order in which they formed. The Earth is 4.55 billion years old, from meteorites that formed with it.

First: Rutherford proposed it, 1905; Holmes made the first geological time scale with it, 1913; Patterson's age of the Earth, 1956 (Geochimica et Cosmochimica Acta).

The Earth's surface is a dozen or so rigid plates moving at 2–10 cm a year, about as fast as fingernails grow. New ocean floor forms at mid-ocean ridges and records the Earth's magnetic field as it cools, in stripes that reverse every few hundred thousand years; old floor sinks back at trenches.

Use: Plate edges are where earthquakes, volcanoes and many ore deposits are: copper and gold above sinking plates, for example. GPS stations now measure each plate's motion directly, to a millimetre a year.

First: Wegener's continental drift, 1912–1915; Vine and Matthews read the magnetic stripes as sea-floor spreading, 1963 (Nature).

An earthquake's size is its seismic moment M₀ in newton-metres: the rock's rigidity μ times the area A of fault that slipped times the slip D. Each step of one in magnitude is about 32 times the energy; two steps, 1,000 times.

Use: The magnitude 9.1 Sumatra earthquake of 2004 released about 1,000 times the energy of a magnitude 7.1. Building codes set the shaking to design for from the largest quake expected at a site; in quake zones, light timber or reinforced frames survive where unreinforced masonry falls (zoning).

First: Richter's magnitude scale, 1935 (Bulletin of the Seismological Society of America); the moment magnitude, Hanks and Kanamori, 1979 (Journal of Geophysical Research).

A wave much longer than the water is deep travels at the square root of g times the depth d. Over 4,000 m of ocean that is 200 m/s, 720 km/h; in 10 m of water near shore only 10 m/s, so the wave slows, shortens and rises as it arrives.

Use: A tsunami from a quake 100 km offshore arrives in minutes. Strong or long shaking near a coast, or the sea suddenly drawing back, means go to high ground or far inland at once, on foot, without waiting for a warning, and stay there for hours: the first wave is often not the largest.

First: Lagrange derived the speed of long waves in shallow water, 1781 (MacTutor).

With the Sun's 1,361 W/m² spread over the whole globe (the factor 4) and 30 % reflected (the albedo α), the Earth would radiate away what it absorbs at 255 K, −18 °C. The surface averages about 15 °C: the 33 °C difference is the natural greenhouse effect.

Use: Explains why the Earth is habitable, why snow and ice (which raise α) amplify cooling, and why adding infrared-absorbing gases must warm the surface.

First: Fourier saw that the atmosphere keeps the Earth warm, 1824 (MacTutor); Tyndall measured infrared absorption by CO₂ and water vapour, 1859; Arrhenius calculated the warming from CO₂, 1896 (Philosophical Magazine).

The extra heat trapped grows with the logarithm of the CO₂ concentration: each doubling adds about 3.7 W/m². The IPCC's best estimate of the long-term warming from a doubling is 3 °C, likely between 2.5 and 4 °C. CO₂ rose from about 280 ppm before industry to 427 ppm in 2025 (NOAA).

Use: Sets the scale of the change any long-term plan must allow for: sea level, rainfall, heat and the crops a place can grow (decide what not to rebuild, energy without the carbon bill).

First: Arrhenius, 1896; the logarithmic forcing, Myhre and colleagues, 1998 (Geophysical Research Letters); the sensitivity, IPCC Sixth Assessment, 2021 (IPCC).