﻿ Astronomical Constants

Astronomical Constants

Units

The units meter (m), second (s) and kilogram (kg) are the units of length, time and mass in the International System of Units (SI) (ref.[1]).

The astronomical units of length, time and mass are (ref.[11]):

• A = astronomical unit of length; approximately equal to the mean radius of the orbit of the Earth.
• D = astronomical unit of time; equal to one day of 86400 SI seconds.
• S = astronomical unit of mass; equal to the mass of the Sun.

IAU Standards

The International Astronomical Union (IAU) maintains a system of definitions and current best estimates of astronomical constants. See the Transactions (ref.[2]) and Resolutions (ref.[3]). Below is a selection of best values as known at the end of 2021 (ref.[4]). For astrophysical use fixed rounded nominal values have been established (ref.[5]).

1.1 Natural defining constant

speed of light
c = 299 792 458 m/s
[6]

1.2 Auxiliary defining constants

astronomical unit
au = A = 149 597 870 700 m
rotation angle of the Earth on J2000.0 UT1
θ0 = 0.779 057 273 2640 rev.
change in rotation angle of the Earth
dθ/dUT1 = 1.002 737 811 911 354 48 rev./UT1-day

2.1 Natural measurable constants

Newtonion constant of gravitation
G = 6.674 28 (±67) ×10–11 m3/kg/s2
[10a]
6,674 30 (±15) ×10–11 m3/kg/s2
[10]

2.3 Solar System Body constants

solar mass parameter (heliocentric gravitational constant) GMS
TDB compatible value:
= A3k2/D2 = 1.327 124 400 41×1020 m3/s2
ae = 6 378 136.6 m
dynamical form factor of the Earth (zero-frequency tide model)
J2 = 0.001 082 6359
secular change of J2 per century
dJ2 = −3.0×10–9/cy
geocentric gravitational constant GME
TT compatible value:
GME = 3,986 004 415×1014 m3/s2
TDB compatible value:
= 3,986 004 356×1014 m3/s2
potential of the geoid
W0 = 62 636 856.0 J/kg = m2/s2
nominal mean angular velocity of the Earth
TT compatible value:
mass ratio of the Moon to the Earth
μ = MM/ME = 0.012 300 0371
[7a]

Derived and older constants

Gaussian gravitation constant
k = 0,017 202 098 95
light time for 1 a.u.
τA = A/c = 499.004 783 84 s
ratio of mass of the Sun to the Earth
(GMS)/(GME) = S/E = 332 946.0487
ratio of the mass of the Earth to the Moon
ME/MM = 1/μ = 81.300 5678
ratio of the mass of the Sun to the combined mass of Earth and Moon
(S/E)/(1+μ) = 328 900.5596
Solar mass
(GMS)/G = S = 1.9884×1030 kg
Earth mass
(GME)/G = E = 5.9722×1024 kg
standard acceleration of gravity
gn = (GME)/(RE2)= 9.806 65 m/s2
[14]
solar parallax
arcsin(ae/A) = π = 8.794 143"
constant of aberration for epoch J2000.0
κ = 20.495 52"
polar flattening factor of the Earth (zero-frequency tide model)
f = 0.003 352 8197 = 1/298.256 42

Other Constants and Formulas

In the formulae below, T is the time passed since J2000.0 (= JD 2 451 545.0 TDB) measured in Julian centuries of 36 525 days.

Precession

Sources: [19a], [20], [21].

annual general precession
pA = 50.287 961 95" + 0.022 108 696" ×T
annual precession in right ascension
m = 3.074 773 605s + 0.001 855 4463s×T
annual precession in declination
n = 20.041 919 03" − 0.008 589 868" ×T
obliquity of the ecliptic
εA = 84 381.406" − 46.836 769"×T
[4]

Mean periods

Source: [18a]

mean solar day expressed in mean sidereal time:

1.002 737 909 344 99d + 59.0107d×10-12×T
= 24h03m56.555 367s + 0.000 0510s×T
mean sidereal day expressed in mean solar time:

0.997 269 566 334 86d − 58.6888d×10-12×T
= 23h56m04.090 531s − 0.000 0507s×T
sidereal rotation period of the Earth expressed in mean solar time:
dUT1/dθ =
0,997 269 663 237 157d
= 23h56m04.098 904s

Source: [36a]

sidereal month (from fixed star to same fixed star):

27.321 661 554d + 0.000 000 216d×T
= 27d07h43m11.558s + 0.019s×T
anomalistic month (from perigee to perigee):

27.554 549 886d − 0.000 001 007d×T
= 27d13h18m33.110s − 0.087s×T
tropical month (from aequinox to same aequinox):

27.321 582 252d + 0.000 000 182d×T
= 27d07h43m04.707s + 0.016s×T
draconitic month (from node to same node):

27.212 220 815d + 0.000 000 414d×T
= 27d05h05m35.878s + 0.036s×T
synodic month (from a phase to the same phase):

29.530 588 861d + 0.000 000 252d×T
= 29d12h44m02.878s + 0.022s×T
Julian year:

365.25d
= 365d06h00m00.000s
sidereal year (from fixed star to same fixed star):

365.256 362 95d + 0.000 000 11d×T
= 365d06h09m09.759s + 0.010s×T
anomalistic year (from apside to same apside):

365.259 635 77d + 0.000 003 12d×T
= 365d06h13m52.531s + 0.270s×T
tropical year (mean, from aequinox to same aequinox):

365.242 190 42d − 0.000 006 15d×T
= 365d05h48m45.252s − 0.531s×T
ecliptic year (from lunar node to same lunar node):

346.620 074 49d + 0.000 032 38d×T
= 346d14h52m54.436s + 2.798s×T
period of the lunar nodes:

6 793.476 501d + 0.012 400d×T
= 18.600 years
period of the lunar apsides:

3 233.605 425d + 0.016 894d×T
=  8.853 years
Full Moon Cycle:

beat period of anomalistic and synodic months
= 411.78443d
[49]

14 synodic = 15 anomalistic months
= 413.4d
saros cycle:

223 synodic = 242 draconitic = 239 anomalistic months = 16 Full Moon Cycles
= 6585.3d = 18y + 11d
cycle of Meton:

235 synodic months = 19 years
= 6939.7d
Chaldean lunar cycle:

251 synodic = 269 anomalistic months = 18 Full Moon Cycles
= 7412.2d = 20y + 107d
[50]

the Earth

(WGS-1984/EGM-1996 [16],[22b])

a = 6 378 137 m
nominal
RNeE = 6.3781×106 m
[5b]
aequatorial circumference
2πa = 40 075 017 m
flattening
f = 1/298.257 223 563
b = (1-f)a = 6 356 752.31 m
nominal
RNpE = 6.3568×106 m
[5b]
polar circumference
π×{3(a+b)−√[(3a+b)×(a+3b)]} = 40 007 862 m
[external site]
volume
π×4/3×(a2b) = 1.0832×1021 m3
[Wikipedia]
geocentric gravitational constant
GM = 3 986 004.418 (±8)×108 m3/s2
nominal
(GM)NE = 3 986 004               ×108 m3/s2
[5b]
original (for GPS):
3 986 005               ×108 m3/s2
(nominal) mass
(GM)NE/G = 5.9722×1024 kg
nominal mean angular velocity
ω = 7 292 115×10−11 rad/s
geopotential coefficient (C2,0)
derived:
−484.166 774 985×10–6
original (defining):
−484.166 85         ×10–6
dynamical form factor (J2) from GRS80
108 263×10–8
average density
5 513 kg/m3
gravitational acceleration (in mgal = 10-5 m/s2):

g(φ) = 978 032.677 14 + 5 185.960×sin2(φ) − 5.736×sin2(2φ) − 0.3086×h
[22a]
in which: φ = geodetic latitude on the WGS-84 ellipsoid; h = altitude above the ellipsoid in meters.
escape velocity
√(2GM/a) = 11.18 km/s

the Moon

Orbit (mean values at J2000.0):
mean aequatorial horizontal parallax
π = 3 422.608" = 0.950 7244°
[25b]
mean distance (for ae = 6 378.140 km [11b])
ae / sin(π) = 384 399.7 km
Keplerian orbital axis
aM = 384 747.964 km
[36b]
mean orbital angular velocity (derived)
[36a]
mean orbital velocity (derived)
1024 m/s
time averaged orbital parameters (from ELP: refs. [31],[32],[33],[34],[36]):
distance
rM = 385 500.560 km
excentricity (from constant E)
e = 0.054 9006
orbital inclination on ecliptic (from constant Γ)
i = 5.145 35°
mean osculating orbital parameters (ref.[35]):
axis
<a> = 383 397.7725 km
excentricity
<e> = 0.055 545 526
inclination on ecliptic
<i> = 5.156 689 83°
Rotation:
mean inclination of aequator on ecliptic
I = 1°32'32.7" = 1.542 24°
mean inclination of aequator on orbital plane
I' = 6°41'16" = 6,6878°?
[?]
rotation speed
13.176 358 15 °/d
[26b]
Physical:
RM = 1 737.4   km
[26d]
1 738.065 km
[11e]
k = 0.272 5076 ae
[23]

0.272 5076 × 6 378.140 = 1 738.092 km
apparent diameter at mean distance
2×arcsin(RM/rM) = 0.516 45° = 30'59.2"
selenocentric gravitational constant
GMM = μ×GME = 4.902 800 2×1012 m3/s2
mass
(GMM)/G = 73.458×1021 kg
average density
3 344 kg/m3
gravity at surface
(GMM)/(RM)2 = 1.624 m/s2 = 0.166×gn
escape velocity
√(2GMM/RM) = 2.38 km/s
magnitude of the Moon at mean distance
V0 = −12.74
[27a]
,,
= −12.72
[28]
magnitude at 1 AU at phase angle 0
V(1,0) = +0.21
[27a]
,,
= +0.23
[28]
colour index
(B−V) = +0.85
[28]
geometric albedo
11.5%
[27a]
,,
11.3%
[28]
Bond albedo
6.7%
[27a]
,,
6.9%
[28]

the Sun

RS = 696 000 km
[26c]
RN = 695.7×106 m
[5b]
apparent diameter at 1 AU
2×arcsin(RS/A) = 0.533 14° = 31'59.3 "
apparent diameter of photosphere at 1 AU
2×959.176" = 0.532 876° = 31'58.35"
[30]
nominal mass parameter of the Sun
(GM)N = 1.327 124 4×1020 m3/s2
[5b]
(nominal) mass
(GM)N/G = 1.9884×1030 kg
average density
1 408 kg/m3
gravity at surface
(GM)N/(RN)2 = 274.2 m/s2 = 27.96×gn
escape velocity
√(2GMN/RN) = 617.7 km/s
sidereal rotation period (convention according to Carrington, at B = ±26°)
25.38d
[26a]
synodic rotation period
= 1 / (1/25.38 − 1/365.256 36) = 27.2752 d
inclination aequator on ecliptic (derived)
7.252°
[26a]
longitude of ascending node aequator on ecliptic for aequinox and ecliptic of date (derived)
75.766° + 1.397°×T
nominal solar constant (average over 11y cycle)
SN = 1 361 W/m2
nominal luminosity
LN = 4πA2×SN = 3.828×1026 W
nominal effective surface temperature
from T4 = LN/σ×4π(RN)2 → TNeff☉ = 5 772 K
[5b]
apparent magnitude
V(1,0) = −26.71
[30]
absolute magnitude
MV =  +4.862
[30]
absolute bolometric magnitude
(defining) MBol☉ = +4.74
[5a]

(older) Mbol = +4.7554
[30]
apparent bolometric magnitude (at 1 AU)
mbol☉ = −26.832
[5a]
colour index
(B−V) = +0.653
[30]
spectral type
G2V
[30]
age of solar system
4 572 (±4)×106 years
[30]

Units of Length

light year
1 ly = y×D×c = 9.4607×1012 km = 63 241 AU = 0.306 60 pc
parsec
1 pc = A/tan(1") = 206 265 AU = 30.857×1012 km = 3.2616 ly

the Milky Way

pole of galactic plane (J2000.0)
α = 12h51m26.28s ; δ = +27°07'41.7"
direction of galactic longitude 0 (J2000.0, derived)
α = 17h45m37.20s ; δ = −28°56'10.2"
position of galactic center Sgr A* (epoch 2006, J2000.0)
α = 17h45m40.0360s ; δ = −29°00'28.170"
[46a]
in galactic coordinates (derived):
l = 359.9442°; b = −0.0462°
[46b]
distance of Sun to galactic center
8.32 ±0.14 kpc = 27.1×103 ly
[46d]
distance of Sun to galactic plane
8 pc = 26 ly
[?]
orbital velocity of the Sun
225 ±9 km/s
[39]
orbital period of the Sun (derived)
202 (±10)×106y
[39]
motion of the Sun w.r.t. the "Local Standard of Rest" (in the direction of the apex); source [39] :
U0 =
7.5 ±1.0 km/s
V0 =
13.5 ±0.3 km/s
W0 =
6.8 ±0.1 km/s
total:
16.9 ±1.0 km/s
apex Sun (derived)
l = 61° ; b = +24°

α = 18h05m ; δ = +35°

the Universe

Hubble constant from Cepheids
H0 = 73.04 (±1.04) km/s/Mpc
[48b]
Hubble constant from cosmology
H0 = 67.4 (±0.5) km/s/Mpc
[47b]
radius of the observable universe (cosmological Hubble length)
c/H0 = 4.45 Gpc = 14.5×109 ly

age
13.80 (±0.04)×109 y
[44a]
temperature
2.72548 (±0.00057) K
[40]
density
9.9×10-30 g/cm3 = 9.9×10-27 kg/m3
[43d]
baryonic mass fraction
Ωb = 4.95 (±0.03)%
[47b]
of which the original Helium fraction:
25.34 (±0.83)%
[41]
cold dark mass fraction
Ωc = 26.6 (±0.3)%
[47b]
dark energy fraction
ΩΛ = 68.5 (±0.5)%
[47b]
Also see the older results from WMAP(2012) (refs.[43a],[43b][43c],[43d]), Planck(2013) (refs.[44a],[44b],[44c]), Planck(2015) (refs.[45a],[45b])