Astronomical Constants

v.20220121


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]):


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
[7a],[7b]
rotation angle of the Earth on J2000.0 UT1
θ0 = 0.779 057 273 2640 rev.
[8b],[9]
change in rotation angle of the Earth
dθ/dUT1 = 1.002 737 811 911 354 48 rev./UT1-day
[8b],[9]

2.1 Natural measurable constants

Newtonion constant of gravitation
G = 6.674 28 (±67) ×10–11 m3/kg/s2
[10a]
newer value from CODATA 2018:
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
equatorial radius of the Earth
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:
ω = 7.292 115×10–5 rad/s
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
[11a],[12a]
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"
[11c],[12b]
polar flattening factor of the Earth (zero-frequency tide model)
f = 0.003 352 8197 = 1/298.256 42
[16],[17a]

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])

aequatorial radius
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
polar radius
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)
2.661 699 473 2866×10–6 rad/s
[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
[34a],[36c]
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°
[11d],[24a],[25a]
mean inclination of aequator on orbital plane
I' = 6°41'16" = 6,6878°?
[?]
rotation speed
13.176 358 15 °/d
[26b]
Physical:
mean radius
RM = 1 737.4   km
[26d]
mean radius in Watt's profiles
1 738.065 km
[11e]
nominal radius in Earth radii (for eclipse calculations)
k = 0.272 5076 ae
[23]
 
0.272 5076 × 6 378.140 = 1 738.092 km
[23],[11b]
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

aequatorial radius
RS = 696 000 km
[26c]
nominal radius
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
[29],[5b]
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"
[37],[38]
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]
cosmological
H0 = 67.4 (±0.5) km/s/Mpc
[47b]
radius of the observable universe (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])

References

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  2. IAU Transactions: https://www.iau.org/publications/iau/transactions_a/
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  4. IAU Current Best Estimates:
    1. Luzum B. e.a. (2011): The IAU 2009 system of astronomical constants: the report of the IAU working group on numerical standards for Fundamental Astronomy. Celest.Mech.Dyn.Astron. 110(4), pp.239..304; DOI: 10.1007/s10569-011-9352-4;
    2. [http://maia.usno.navy.mil/NSFA/NSFA_cbe.html] defunct
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    1. IAU Inter-Division A-G Working Group on Nominal Units for Stellar and Planetary Astronomy: IAU 2015 Resolution B2 on Recommended Zero Points for the Absolute and Apparent Bolometric Magnitude Scales.
      http://arxiv.org/abs/1510.06262
    2. IAU Inter-Division A-G Working Group on Nominal Units for Stellar and Planetary Astronomy: IAU 2015 Resolution B3 on Recommended Nominal Conversion Constants for Selected Solar and Planetary Properties.
      http://arxiv.org/abs/1510.07674
  6. 17e Conférence Générale des Poids et Mesures 1983, Resolution 1.
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    1. Mohr P.J., Taylor B.N., Newell D.B. (2008): CODATA Recommended Values of the Fundamental Physical Constants: 2006 . Rev.Mod.Phys. 80, pp.633..730; DOI: 10.1103/RevModPhys.80.633;
      http://arxiv.org/pdf/0801.0028v1.pdf
    2. Mohr P.J., Taylor B.N., Newell D.B. (2012): CODATA Recommended Values of the Fundamental Physical Constants: 2010 . Rev.Mod.Phys. 84, pp.1527..1605 (2012); DOI: 10.1103/RevModPhys.84.1527;
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    2. ibidem p.58 constant 4
    3. ibidem p.59 constant 14
    4. ibidem p.60
    5. ibidem p.66
  12. Seidelmann, P.K (1977): Numerical values of the constants of the Joint Report of the Working Groups of IAU Commission 4 . Celest.Mech. 16, pp.165..177; DOI: 10.1007/BF01228598;
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    1. ibidem p.165
    2. ibidem p.168
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  14. 3e Conférence Générale des Poids et Mesures 1901, Resolution 70 . See: Bureau International des Poids et Mesures: The International System of Units (SI), 8th ed. 2006, p.143;
    http://www.bipm.org/en/CGPM/db/3/2/
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  16. Groten, E. (2004): Geodesist's Handbook 2004 .
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    1. ibidem p.18
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    http://www.aanda.org/articles/aa/pdf/2003/30/aa3487.pdf
    1. ibidem, derived from eq. B1,B2 on p.1149 .
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    1. ibidem p.581 eq.(39).
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  21. IAU (2006) XXVIth General Assembly. Resolution No. B1: Adoption of the P03 Precession Theory and Definition of the Ecliptic.
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  22. WGS-1984
    1. derived; also see p.3-23 eqs. 3-63 and 3-64, and p.3-47 table 3.8 in: Defense Mapping Agency: Technical Report 3850.2, 1st ed. (1984): DoD World Geodetic System 1984, Ch.3;
      http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350.2-a/Chapter%203.pdf
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    2. ibidem table 5.4 p.701
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    2. ibidem table 2
    3. ibidem table 4
    4. ibidem table 5
  27. Kuiper G.P., Middlehurst B.M. (eds.): The Solar System. Vol.III Planets & Satellites. Univ. Of Chicago Press, 1961.
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  29. Kopp K., Lean J.L. (2010): A new, lower value of total solar irradiance: Evidence and climate significance. Geophys.Res.Lett. 38, L01706; DOI: 10.1029/2010GL045777;
    http://onlinelibrary.wiley.com/doi/10.1029/2010GL045777/pdf
  30. [https://sites.google.com/site/mamajeksstarnotes/basic-astronomical-data-for-the-sun] inaccessible
  31. Chapront-Touzé M., Chapront J. (1983): The lunar ephemeris ELP 2000 . Astron.&Astrophys. 124, pp.50..62;
    http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1983A%26A...124...50C&link_type=ARTICLE
    1. ibidem, p.51 table 1 and p.54 table 9 and text
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    1. ibidem p.351 table 9
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    1. ibidem par. 3.3 p.8
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    1. ibidem, derived from table 4 on p.704
    2. ibidem, from the sidereal motion n from W1 in table 4 p.704 according to a3 = GME×(1+μ)/n2
    3. ibidem, constant value in table 9 p.351 of ref.[32] scaled with corrections from table 2 p.703 and table 4 p.704 from ref.[36] on parameters from table 1 p.51 and table 9 p.54 from ref.[31]
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    1. WMAP best estimate of the Hubble constant:
      http://map.gsfc.nasa.gov/universe/uni_expansion.html
    2. WMAP best estimate of the age of the universe:
      http://map.gsfc.nasa.gov/universe/uni_age.html
    3. WMAP background radiation temperature:
    4. WMAP average density of the universe:
      http://map.gsfc.nasa.gov/universe/uni_matter.html
  44. Planck (2013):
    1. Planck Collaboration (2013): Planck 2013 results. I. Overview of products and scientific results. Astron.&Astrophys. vol.?, pp.?..? ; DOI: 10.1051/0004-6361/201321529 ;
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    3. Also see:
      http://scienceblogs.com/startswithabang/2013/03/21/what-the-entire-universe-is-made-of-thanks-to-planck/
  45. Planck (2015):
    1. Planck Collaboration (2015): Planck 2015 results. I. Overview of products and scientific results. Astron.&Astrophys. vol.?, pp.?..? ; DOI: ? ;
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    2. Planck Collaboration (2015): Planck 2015 results. XIII. Cosmological parameters. Astron.&Astrophys. vol.?, pp.?..? ; DOI: ? ;
      http://arxiv.org/abs/1502.01589v2 ; p.31 Table 4 column TT+lowP+lensing
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    4. Gillessen, S. e.a. (2016): An Update on Monitoring Stellar Orbits in the Galactic Center. Astrophys.J. vol.?:?, pp.?..? ; DOI: 10.3847/1538-4357/aa5c41 ;
  47. Planck 2018:
    1. Planck Collaboration (2018): Planck 2018 results. I. Overview and the cosmological legacy of Planck. Astron.&Astrophys. vol.?, pp.?..? ; DOI: ? ;
      https://arxiv.org/abs/1807.06205
    2. Planck Collaboration (2018): Planck 2018 results. VI. Cosmological parameters. Astron.&Astrophys. vol.?, pp.?..? ; DOI: ? ;
      https://arxiv.org/abs/1807.06209 ; p.15 Table 2 TT,TE,EE+lowE+lensing
    3. Also see:
      https://wiki.cosmos.esa.int/planck-legacy-archive/index.php/Cosmological_Parameters
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    2. Riess A.G. e.a. (2021): A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team. Astrophys.J. ? ;
      https://arxiv.org/abs/2112.04510
  49. Full Moon Cycle: http://arpeters.net/English_pages/Full_moon_cycle.htm
  50. Chaldean lunar cycle: See Claudius Ptolemaeus, Almagest IV 2 H272 . p.176 in: G.J. Toomer: "Ptolemy's Almagest", G. Duckworth & Co., London 1984 . ISBN 0-7156-1588-2