Constants library — formula.expert

Astronomical Unit exact

\mathrm{au} = 1.495978707 \times 10^{11}\ \text{m}

mThe Sun–Earth yardstick, fixed by the IAU in 2012 as exactly 149 597 870 700 m and no longer tied to Earth's actual orbit.

Light-Year exact

\mathrm{ly} = 9.4607304725808 \times 10^{15}\ \text{m}

mDistance light travels in one Julian year of 365.25 days — exactly 9 460 730 472 580 800 m, since both c and the year are defined.

Parsec exact

\mathrm{pc} = 3.085677581491367 \times 10^{16}\ \text{m}

mDistance at which one au subtends one arcsecond — 648000/π au exactly, about 3.26 light-years, the working unit of stellar astronomy.

Solar Mass measured

M_\odot = 1.98841 \times 10^{30}\ \text{kg}

kgMass of the Sun, about 333 000 Earths and 99.86 per cent of all matter in the solar system — the yardstick for every stellar mass.

Nominal Solar Radius exact

\mathcal{R}^{\mathrm{N}}_\odot = 695,700,000\ \text{m}

mIAU nominal solar radius, exactly 6.957 × 10⁸ m — a fixed convention, since a gaseous Sun has no true surface to measure.

Nominal Solar Luminosity exact

\mathcal{L}^{\mathrm{N}}_\odot = 3.828 \times 10^{26}\ \text{W}

WIAU nominal solar luminosity, exactly 3.828 × 10²⁶ W — the conventional unit in which every other star's power output is quoted.

Solar Effective Temperature exact

\mathcal{T}^{\mathrm{N}}_{\mathrm{eff}\odot} = 5772\ \text{K}

KIAU nominal effective temperature of the Sun, 5772 K — the blackbody temperature that radiates the solar luminosity from the solar radius.

Solar Constant (Total Solar Irradiance) measured

S_0 = 1361\ \text{W/m}^{2}

W/m²Total solar irradiance above the atmosphere at one au, about 1361 W/m² — the input to every climate model and solar panel estimate.

Standard Gravitational Parameter of the Sun measured

GM_\odot = 1.32712440041 \times 10^{20}\ \text{m}^{3}\text{/s}^{2}

m³/s²The heliocentric gravitational constant GM⊙, known to ten digits from planetary radar — far better than the Sun's mass in kilograms.

Schwarzschild Radius of the Sun measured

r_{s,\odot} = 2953.25\ \text{m}

mRadius to which the Sun would have to be crushed to become a black hole, 2GM⊙/c² — a shade under three kilometres.

Solar Wind Speed (typical)

v_{\mathrm{sw}} = 400,000\ \text{m/s}

m/sTypical speed of the solar wind at Earth's orbit, about 400 km/s; the slow and fast streams range from roughly 300 to 800 km/s.

Mass of the Earth measured

M_\oplus = 5.9722 \times 10^{24}\ \text{kg}

kgMass of the Earth, 5.9722 × 10²⁴ kg — the unit in which rocky exoplanets are weighed, and limited in precision only by G.

Earth Equatorial Radius exact

a_\oplus = 6,378,137\ \text{m}

mSemi-major axis of the WGS 84 reference ellipsoid, exactly 6 378 137 m — the equatorial radius every GPS receiver is built around.

Earth Polar Radius exact

b_\oplus = 6356752.314245\ \text{m}

mSemi-minor axis of the WGS 84 ellipsoid, 6 356 752.3 m — 21.4 km shorter than the equatorial radius because the Earth is spinning.

Earth Mean Radius

R_\oplus = 6371008.8\ \text{m}

mMean radius (2a + b)/3 of the WGS 84 ellipsoid, about 6371 km — the single figure used when a spherical Earth is good enough.

Standard Gravitational Parameter of the Earth measured

GM_\oplus = 3.986004418 \times 10^{14}\ \text{m}^{3}\text{/s}^{2}

m³/s²The geocentric gravitational constant GM⊕, 3.986 004 418 × 10¹⁴ m³/s², known to nine digits and used by every GPS satellite.

Earth Mean Orbital Speed

v_\oplus = 29,780\ \text{m/s}

m/sMean speed of the Earth along its orbit, about 29.78 km/s — roughly 107 000 km/h, and it varies with distance from the Sun.

Earth Orbit Semi-Major Axis

a_{\oplus,\mathrm{orb}} = 1.495982612 \times 10^{11}\ \text{m}

mEarth's actual mean orbital distance, 1.000 002 61 au — close to the astronomical unit but a measured quantity, not the definition.

Earth Orbital Eccentricity

e_\oplus = 0.0167086\ \text{—}

—Eccentricity of Earth's orbit at J2000, 0.0167 — nearly circular, yet enough to vary sunlight at the top of the atmosphere by 6.8 per cent.

Earth Axial Tilt (Obliquity of the Ecliptic)

\varepsilon = 0.4090928\ \text{rad}

radObliquity of the ecliptic at J2000, 23.4393° or 0.409 rad — the tilt of Earth's spin axis that causes the seasons.

Escape Velocity of the Earth

v_{\mathrm{esc},\oplus} = 11,186\ \text{m/s}

m/sSpeed needed to break free of Earth's gravity from the surface, about 11.19 km/s, ignoring atmospheric drag and the planet's rotation.

Sidereal Day

T_{\mathrm{sid}} = 86164.0905\ \text{s}

sEarth's rotation period relative to the fixed stars, 23 h 56 min 4.09 s — about four minutes shorter than the solar day.

Mean Solar Day exact

d = 86,400\ \text{s}

sThe civil day of exactly 86 400 SI seconds — a defined unit that Earth's actual rotation now overruns by a millisecond or two.

Julian Year exact

a_{\mathrm{J}} = 31,557,600\ \text{s}

sExactly 365.25 days of 86 400 SI seconds — the conventional astronomical year that defines the light-year and the Julian century.

Sidereal Year

T_{\mathrm{sid}}^{\mathrm{yr}} = 31558149.8\ \text{s}

sOne orbit of the Sun relative to the fixed stars, 365.2564 days — about 20 minutes longer than the tropical year of the seasons.

Tropical Year

T_{\mathrm{trop}} = 31556925.2\ \text{s}

sEquinox to equinox, 365.2422 days — the year the seasons follow, and the quantity every calendar reform has tried to approximate.

Mass of the Moon measured

M_{\mathrm{Moon}} = 7.3459 \times 10^{22}\ \text{kg}

kgMass of the Moon, 7.346 × 10²² kg — 1.23 per cent of Earth's, the largest satellite-to-planet mass ratio in the solar system.

Mean Radius of the Moon measured

R_{\mathrm{Moon}} = 1,737,400\ \text{m}

mVolumetric mean radius of the Moon, 1737.4 km — just over a quarter of Earth's radius, and barely 0.3 per cent from a perfect sphere.

Mean Earth–Moon Distance

a_{\mathrm{Moon}} = 384,400,000\ \text{m}

mSemi-major axis of the lunar orbit, 384 400 km centre to centre; the actual distance ranges from 356 500 to 406 700 km.

Surface Gravity of the Moon measured

g_{\mathrm{Moon}} = 1.62\ \text{m/s}^{2}

m/s²Gravitational acceleration at the lunar surface, 1.62 m/s² — one sixth of Earth's, the value the Apollo crews had to learn to walk in.

Escape Velocity of the Moon

v_{\mathrm{esc},\mathrm{Moon}} = 2380\ \text{m/s}

m/sSpeed needed to leave the Moon's gravity from its surface, about 2.38 km/s — roughly a fifth of Earth's escape velocity.

Mass of Mars measured

M_{\mathrm{Mars}} = 6.4171 \times 10^{23}\ \text{kg}

kgMass of Mars, 6.417 × 10²³ kg — about 10.7 per cent of Earth's, small enough that the planet lost most of its atmosphere.

Mean Radius of Mars measured

R_{\mathrm{Mars}} = 3,389,500\ \text{m}

mVolumetric mean radius of Mars, 3389.5 km; the equatorial radius is 3396.2 km and the polar 3376.2 km, a 20 km flattening.

Surface Gravity of Mars measured

g_{\mathrm{Mars}} = 3.71\ \text{m/s}^{2}

m/s²Equatorial surface gravity on Mars, 3.71 m/s² — 38 per cent of Earth's, the figure every Mars lander design is built around.

Escape Velocity of Mars

v_{\mathrm{esc},\mathrm{Mars}} = 5030\ \text{m/s}

m/sSpeed needed to escape Mars from the surface, about 5.03 km/s — less than half Earth's, which is why a return mission is even thinkable.

Mass of Mercury measured

M_{\mathrm{Mercury}} = 3.3011 \times 10^{23}\ \text{kg}

kgMass of Mercury, 3.301 × 10²³ kg — the smallest planet, yet the second densest, with an iron core filling most of its volume.

Mass of Venus measured

M_{\mathrm{Venus}} = 4.8675 \times 10^{24}\ \text{kg}

kgMass of Venus, 4.8675 × 10²⁴ kg — 81.5 per cent of Earth's, making it our closest twin in bulk and nothing like it in climate.

Mass of Jupiter measured

M_{\mathrm{J}} = 1.8982 \times 10^{27}\ \text{kg}

kgMass of Jupiter, 1.898 × 10²⁷ kg — 318 Earths, and more than twice all the other planets combined; the unit for weighing exoplanets.

Equatorial Radius of Jupiter measured

R_{\mathrm{J}} = 71,492,000\ \text{m}

mJupiter's equatorial radius at the 1-bar level, 71 492 km; the polar radius is 66 854 km, a 6.5 per cent flattening from fast rotation.

Mass of Saturn measured

M_{\mathrm{Saturn}} = 5.6834 \times 10^{26}\ \text{kg}

kgMass of Saturn, 5.683 × 10²⁶ kg — 95 Earths spread so thinly that its mean density, 687 kg/m³, is less than that of water.

Mass of Uranus measured

M_{\mathrm{Uranus}} = 8.681 \times 10^{25}\ \text{kg}

kgMass of Uranus, 8.681 × 10²⁵ kg — 14.5 Earths of hydrogen, helium and icy volatiles, tipped on its side at 98 degrees.

Mass of Neptune measured

M_{\mathrm{Neptune}} = 1.02413 \times 10^{26}\ \text{kg}

kgMass of Neptune, 1.024 × 10²⁶ kg — 17.1 Earths, the densest of the giant planets and the one found with mathematics before a telescope.

Hubble Constant measured

H_0 = 67.4\ \text{km/(s}{\cdot}\text{Mpc)}

km/(s·Mpc)Present expansion rate of the universe, 67.4 km/(s·Mpc) from the cosmic microwave background — a galaxy 1 Mpc away recedes at 67 km/s.

Age of the Universe measured

t_0 = 4.354 \times 10^{17}\ \text{s}

sTime since the Big Bang, 13.797 billion years or 4.35 × 10¹⁷ seconds, from fitting the ΛCDM model to the microwave background.

Critical Density of the Universe measured

\rho_c = 8.53 \times 10^{-27}\ \text{kg/m}^{3}

kg/m³Density 3H₀²/8πG that makes the universe spatially flat, about 8.5 × 10⁻²⁷ kg/m³ — some five hydrogen atoms per cubic metre.

Cosmic Microwave Background Temperature measured

T_{\mathrm{CMB}} = 2.72548\ \text{K}

KTemperature of the relic radiation from the Big Bang, 2.725 48 K — the most perfect blackbody spectrum ever measured, anywhere.

Chandrasekhar Limit measured

M_{\mathrm{Ch}} = 2.86 \times 10^{30}\ \text{kg}

kgMaximum mass a white dwarf can support by electron degeneracy pressure, about 1.44 solar masses or 2.86 × 10³⁰ kg.