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a5ea549
constants: Add planetary and lunar masses M = GM/G
Wiljea Jun 1, 2026
abdda9d
constants: Add rotation angular velocity wrot = 2*pi/Prot
Wiljea Jun 1, 2026
b266018
constants: Add WGS-84/GPS Earth reference values and calendar years
Wiljea Jun 1, 2026
aa3d5ce
constants: Fix wrot radian units and WGS-84 eccentricity names
Wiljea Jun 1, 2026
3bbd48d
constants: fix rounding of computed astronomy constants (wrot, f-eart…
Wiljea Jun 3, 2026
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237 changes: 237 additions & 0 deletions doc/calc-help/constants.md
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Expand Up @@ -994,6 +994,14 @@ Mercury gravitational parameter

It is measured by radio tracking of spacecraft (Mariner 10, MESSENGER). [Reference 4](#reference-4)

### M☿ constant

Mercury mass

Mercury's mass, derived from the gravitational parameter `GM☿` and the measured
gravitational constant `G` as `GM☿/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Reference 4](#reference-4) [Reference 5](#reference-5)

### Req☿ constant

Mercury equatorial radius
Expand Down Expand Up @@ -1044,6 +1052,13 @@ Mercury's sidereal rotation period is a measured quantity. It is the true time
it takes to spin 360° on its axis. Mercury is in a 3:2 spin-orbit resonance with
the Sun, rotating three times for every two orbits. [Reference 22](#reference-22)

### ωrot☿ constant

Mercury rotation angular velocity

Mercury's sidereal rotation angular velocity, computed as `2π/Prot☿` from the
sidereal rotation period. The unit is radians per second (`r/s`). [Reference 22](#reference-22)

### ϵ☿ constant

Mercury axial tilt
Expand Down Expand Up @@ -1110,6 +1125,14 @@ Venus gravitational parameter
Venus's gravitational parameter, measured by radio tracking of spacecraft
(Magellan, Venus Express). [Reference 4](#reference-4)

### M♀ constant

Venus mass

Venus's mass, derived from the gravitational parameter `GM♀` and the measured
gravitational constant `G` as `GM♀/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Reference 4](#reference-4) [Reference 5](#reference-5)

### Req♀ constant

Venus equatorial radius
Expand Down Expand Up @@ -1161,6 +1184,15 @@ Measured. Venus's sidereal rotation period is the true time it takes to spin 360
on its axis. Venus rotates retrograde (opposite to its orbital motion), with a
rotation period longer than its orbital period. [Reference 22](#reference-22)

### ωrot♀ constant

Venus rotation angular velocity

Venus's sidereal rotation angular velocity, computed as `2π/Prot♀` from the
sidereal rotation period. The value is negative because Venus rotates retrograde
(opposite to its orbital motion); the uncertainty uses the positive magnitude.
The unit is radians per second (`r/s`). [Reference 22](#reference-22)

### ϵ♀ constant

Venus axial tilt
Expand Down Expand Up @@ -1223,6 +1255,14 @@ Earth gravitational parameter
Exact nominal value (IAU 2015). Earth's gravitational parameter. An exact nominal
value defined by the IAU (2015). [Particle Data Group 2023](#particle-data-group-2023)

### M♁ constant

Earth mass

Earth's mass, derived from the gravitational parameter `GM♁` and the measured
gravitational constant `G` as `GM♁/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Particle Data Group 2023](#particle-data-group-2023) [Reference 5](#reference-5)

### Req♁ constant

Earth equatorial radius
Expand Down Expand Up @@ -1274,6 +1314,13 @@ Earth sidereal rotation period
Measured. Earth's sidereal rotation period (one sidereal day), the time for one
rotation relative to the fixed stars. [Reference 24](#reference-24)

### ωrot♁ constant

Earth rotation angular velocity

Earth's sidereal rotation angular velocity, computed as `2π/Prot♁` from the
sidereal rotation period. The unit is radians per second (`r/s`). [Reference 24](#reference-24)

### ϵ♁ constant

Earth axial tilt
Expand Down Expand Up @@ -1333,6 +1380,74 @@ year). Value in JDN. It is the most recent point in its orbit when it was
closest to the Sun. [Materials 20](#materials-20)


### a♁GPS constant

Earth equatorial radius (WGS-84)

The semi-major axis of the WGS-84 reference ellipsoid, a defining constant of the
World Geodetic System 1984 used by GPS. Exact by definition. [WGS-84](#wgs-84)

### f♁GPS constant

Earth flattening (WGS-84)

The flattening of the WGS-84 reference ellipsoid, `1/298.257223563`, a defining
constant. Exact by definition. [WGS-84](#wgs-84)

### ω♁GPS constant

Earth nominal mean angular velocity (WGS-84)

The nominal mean angular velocity of the Earth in the WGS-84 system. Exact by
definition, and distinct from the sidereal value `ωrot♁` derived from `Prot♁`. [WGS-84](#wgs-84)

### GM♁GPS constant

Earth gravitational parameter (WGS-84)

The geocentric gravitational constant of the WGS-84 system, including the mass of
the Earth's atmosphere. Exact by definition, and distinct from the IAU nominal
`GM♁`. [WGS-84](#wgs-84)

### e12♁GPS constant

Earth first eccentricity squared (WGS-84)

The square of the first eccentricity of the WGS-84 reference ellipsoid, computed
from the flattening as `2·f♁GPS−f♁GPS²`. This is an ellipsoid (shape) eccentricity,
not the orbital eccentricity `e♁`. [WGS-84](#wgs-84)

### e22♁GPS constant

Earth second eccentricity squared (WGS-84)

The square of the second eccentricity of the WGS-84 reference ellipsoid, computed
from the first as `e12♁GPS/(1−e12♁GPS)`. An ellipsoid eccentricity, distinct from
the orbital eccentricity `e♁`. [WGS-84](#wgs-84)

### Ytrop♁ constant

Tropical year

The tropical (seasonal) year, equinox to equinox, about 365.24219 days at J2000.
It is distinct from `Porb♁` (the anomalistic year, perihelion to perihelion, used
by `T₀♁`) and from the calendar years `YJul♁` and `YGreg♁`. [Reference 24](#reference-24)

### YJul♁ constant

Julian year

The Julian year, exactly 365.25 days. It is the year used to define the
light-year. [Reference 24](#reference-24)

### YGreg♁ constant

Gregorian mean year

The mean year of the Gregorian calendar, exactly 365.2425 days
(365 + 1/4 − 1/100 + 1/400). [Reference 24](#reference-24)


## Moon constants

### GM☽ constant
Expand All @@ -1342,6 +1457,14 @@ Moon gravitational parameter
Measured. Moon's gravitational parameter, measured by lunar laser ranging
and spacecraft radio tracking. [Reference 4](#reference-4)

### M☽ constant

Moon mass

The Moon's mass, derived from the gravitational parameter `GM☽` and the measured
gravitational constant `G` as `GM☽/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Reference 4](#reference-4) [Reference 5](#reference-5)

### Req☽ constant

Moon equatorial radius
Expand Down Expand Up @@ -1393,6 +1516,14 @@ Measured. Moon's sidereal rotation period is the true time it takes to spin
360° on its axis. The Moon is tidally locked to Earth, so its rotation period
equals its orbital period. [Reference 22](#reference-22)

### ωrot☽ constant

Moon rotation angular velocity

The Moon's sidereal rotation angular velocity, computed as `2π/Prot☽` from the
sidereal rotation period. Because the Moon is tidally locked, this equals its
orbital mean motion. The unit is radians per second (`r/s`). [Reference 22](#reference-22)

### ϵ☽ constant

Moon axial tilt
Expand Down Expand Up @@ -1458,6 +1589,14 @@ Mars gravitational parameter
Measured. Mars system gravitational parameter, including the contribution of
its moons Phobos and Deimos, measured by spacecraft radio tracking. [Reference 4](#reference-4)

### M♂ constant

Mars mass

Mars's mass, derived from the gravitational parameter `GM♂` and the measured
gravitational constant `G` as `GM♂/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Reference 4](#reference-4) [Reference 5](#reference-5)

### Req♂ constant

Mars equatorial radius
Expand Down Expand Up @@ -1507,6 +1646,13 @@ Measured. Mars's sidereal rotation period is the true time it takes to spin
360° on its axis. A Martian day (sol) is very similar in length to an Earth day.
[Reference 22](#reference-22)

### ωrot♂ constant

Mars rotation angular velocity

Mars's sidereal rotation angular velocity, computed as `2π/Prot♂` from the
sidereal rotation period. The unit is radians per second (`r/s`). [Reference 22](#reference-22)

### ϵ♂ constant

Mars axial tilt
Expand Down Expand Up @@ -1570,6 +1716,14 @@ Jupiter gravitational parameter
Exact nominal value (IAU 2015). Jupiter system gravitational parameter.
An exact nominal value defined by the IAU (2015). [Particle Data Group 2023](#particle-data-group-2023)

### M♃ constant

Jupiter mass

Jupiter's mass, derived from the gravitational parameter `GM♃` and the measured
gravitational constant `G` as `GM♃/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Particle Data Group 2023](#particle-data-group-2023) [Reference 5](#reference-5)

### Req♃ constant

Jupiter equatorial radius
Expand Down Expand Up @@ -1623,6 +1777,13 @@ Measured. Jupiter's sidereal rotation period is the true time it takes to
spin 360° on its axis (System III, based on radio emissions from its
magnetosphere). [Reference 22](#reference-22)

### ωrot♃ constant

Jupiter rotation angular velocity

Jupiter's sidereal rotation angular velocity, computed as `2π/Prot♃` from the
sidereal rotation period (System III). The unit is radians per second (`r/s`). [Reference 22](#reference-22)

### ϵ♃ constant

Jupiter axial tilt
Expand Down Expand Up @@ -1688,6 +1849,14 @@ Saturn gravitational parameter
Measured. Saturn system gravitational parameter, measured by radio
tracking of the Cassini spacecraft. [Reference 4](#reference-4)

### M♄ constant

Saturn mass

Saturn's mass, derived from the gravitational parameter `GM♄` and the measured
gravitational constant `G` as `GM♄/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Reference 4](#reference-4) [Reference 5](#reference-5)

### Req♄ constant

Saturn equatorial radius
Expand Down Expand Up @@ -1740,6 +1909,13 @@ Measured. Saturn's sidereal rotation period is the true time it takes
to spin 360° on its axis (System III, based on Cassini radio
measurements). [Reference 22](#reference-22)

### ωrot♄ constant

Saturn rotation angular velocity

Saturn's sidereal rotation angular velocity, computed as `2π/Prot♄` from the
sidereal rotation period (System III). The unit is radians per second (`r/s`). [Reference 22](#reference-22)

### ϵ♄ constant

Saturn axial tilt
Expand Down Expand Up @@ -1805,6 +1981,14 @@ Uranus gravitational parameter
Measured. Uranus system gravitational parameter, measured by Voyager
2 radio tracking. [Reference 4](#reference-4)

### M⛢ constant

Uranus mass

Uranus's mass, derived from the gravitational parameter `GM⛢` and the measured
gravitational constant `G` as `GM⛢/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Reference 4](#reference-4) [Reference 5](#reference-5)

### Req⛢ constant

Uranus equatorial radius
Expand Down Expand Up @@ -1856,6 +2040,15 @@ Measured. Uranus's sidereal rotation period is the true time it takes
to spin 360° on its axis. Uranus rotates retrograde relative to its
orbital motion. [Reference 22](#reference-22)

### ωrot⛢ constant

Uranus rotation angular velocity

Uranus's sidereal rotation angular velocity, computed as `2π/Prot⛢` from the
sidereal rotation period. The value is negative because Uranus rotates retrograde
relative to its orbital motion; the uncertainty uses the positive magnitude. The
unit is radians per second (`r/s`). [Reference 22](#reference-22)

### ϵ⛢ constant

Uranus axial tilt
Expand Down Expand Up @@ -1923,6 +2116,14 @@ Neptune gravitational parameter
Measured. Neptune system gravitational parameter, measured by Voyager
2 radio tracking and Hubble Space Telescope astrometry of Triton. [Reference 4](#reference-4)

### M♆ constant

Neptune mass

Neptune's mass, derived from the gravitational parameter `GM♆` and the measured
gravitational constant `G` as `GM♆/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Reference 4](#reference-4) [Reference 5](#reference-5)

### Req♆ constant

Neptune equatorial radius
Expand Down Expand Up @@ -1974,6 +2175,13 @@ Measured. Neptune's sidereal rotation period is the true time it takes
to spin 360° on its axis (System III, from Voyager 2 radio measurements).
[Reference 22](#reference-22)

### ωrot♆ constant

Neptune rotation angular velocity

Neptune's sidereal rotation angular velocity, computed as `2π/Prot♆` from the
sidereal rotation period (System III). The unit is radians per second (`r/s`). [Reference 22](#reference-22)

### ϵ♆ constant

Neptune axial tilt
Expand Down Expand Up @@ -2040,6 +2248,14 @@ Pluto gravitational parameter
Measured. Pluto system gravitational parameter, measured by New Horizons
radio tracking. [Reference 4](#reference-4)

### M♇ constant

Pluto mass

Pluto's mass, derived from the gravitational parameter `GM♇` and the measured
gravitational constant `G` as `GM♇/G`. The relative uncertainty is carried as
`ⓇG` (the dominant term), so the mass self-corrects whenever `G` is updated. [Reference 4](#reference-4) [Reference 5](#reference-5)

### Req♇ constant

Pluto equatorial radius
Expand Down Expand Up @@ -2089,6 +2305,15 @@ Measured. Pluto's sidereal rotation period is the true time it takes to
spin 360° on its axis. Pluto rotates retrograde and is tidally locked to
its moon Charon. [Reference 26](#reference-26)

### ωrot♇ constant

Pluto rotation angular velocity

Pluto's sidereal rotation angular velocity, computed as `2π/Prot♇` from the
sidereal rotation period. The value is negative because Pluto rotates retrograde;
the uncertainty uses the positive magnitude. The unit is radians per second
(`r/s`). [Reference 26](#reference-26)

### ϵ♇ constant

Pluto axial tilt
Expand Down Expand Up @@ -2220,6 +2445,14 @@ by tracking surface features using Doppler techniques. It is the true
time it takes to spin 360° on its axis. [Reference 20](#reference-20) [Reference 21](#reference-21)


### ωrot☉ constant

Solar rotation angular velocity

The Sun's sidereal rotation angular velocity at the equator (System I), computed
as `2π/Prot☉` from the sidereal rotation period. The unit is radians per second
(`r/s`). [Reference 20](#reference-20) [Reference 21](#reference-21)

## Cosmology constants

### Λ constant
Expand Down Expand Up @@ -3319,6 +3552,10 @@ Particle Data Group (2024). "Review of Particle Physics — Astrophysical Consta

Particle Data Group 2023 Prša, A., et al. (2016). "Nominal values for selected solar and planetary quantities: IAU 2015 Resolution B3". The Astronomical Journal, 152(2), 41. arXiv:1605.09788 — DOI: 10.3847/0004-6256/152/2/41

### WGS-84

National Geospatial-Intelligence Agency (2014). Department of Defense World Geodetic System 1984: Its Definition and Relationships with Local Geodetic Systems, NGA.STND.0036_1.0.0_WGS84, 3rd ed. (Defining parameters of the WGS-84 reference ellipsoid used by GPS.)

### Reference 4

Park, R.S., et al. (2021). "The JPL Planetary and Lunar Ephemerides DE440 and DE441". The Astronomical Journal, 161(3), 105. DOI: 10.3847/1538-3881/abd414
Expand Down
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