Astronomical Constants

v.20240327

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]
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])

References

  1. BIPM: The International System of Units: https://www.bipm.org/en/measurement-units/
  2. IAU Transactions: https://www.iau.org/publications/iau/transactions_a/
  3. IAU Resolutions: http://www.iau.org/administration/resolutions/general_assemblies/
  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
  5. IAU Resolutions 2015: http://www.iau.org/static/resolutions/IAU2015_English.pdf
    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.
    Bureau International des Poids et Mesures: The International System of Units (SI), 8th ed. 2006, p.112
  7. Definition of the Astronomical Unit:
    1. Pitjeva E.V, Standish E.M (2009): Proposals for the masses of the of the three largest asteroids, the Moon-Earth mass ratio and the astronomical unit. Celest.Mech.Dyn.Astron. 103, pp.365..372; DOI: 10.1007/s10569-009-9203-8;
      http://adsabs.harvard.edu/abs/2011CeMDA.110..293L
    2. IAU (2012) 28th General Assembly, Resolution B2: http://www.iau.org/static/resolutions/IAU2012_English.pdf
  8. IAU (2000) XXIVth General Assembly. https://www.iau.org/static/resolutions/IAU2000_French.pdf
    1. ibidem, Resolution No. B1.6: IAU precession-nutation model 2000 .
    2. ibidem, Resolution No. B1.8: Definition and use of the celestial and terrestial ephemeris origins.
  9. p.403 eq.(22) in: Capitaine N., Guinot B., McCarthy D.D. (2000): Definition of the Celestial Ephemeris Origin and of UT in the International Celestial Reference Frame. Astron.&Astrophys. 355, pp.398..405 .
    http://adsabs.harvard.edu/abs/2000A%26A...355..398C
  10. CODATA: http://physics.nist.gov/cuu/Constants/index.html
    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;
      http://arxiv.org/pdf/1203.5425v1.pdf
    3. Mohr P.J., Newell D.B., Taylor B.N. (2015): CODATA Recommended Values of the Fundamental Physical Constants: 2014 .
      http://arxiv.org/pdf/1507.07956.pdf
  11. p.31, pp.52..67 in: IAU (1976): Proceedings of the 16th General Assembly, Transactions of the IAU XVI B, West R.M. (ed.), D.Reidel, Dordrecht 1977 .
    1. ibidem p.58 constant 1
    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;
    http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1977CeMec..16..165S&link_type=ARTICLE
    1. ibidem p.165
    2. ibidem p.168
  13. Lederle, T. (1980): The IAU (1976) System of Astronomical Constants. Mitt.Astron.Ges. 48, 59..65;
    http://articles.adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1980MitAG..48...59L&link_type=ARTICLE
  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/
  15. Groten, E. (2000): Parameters of Common Relevance of Astronomy, Geodesy, and Geodynamics. J.Geod. 74, pp.134..140 .
    Also see Geodesist's Handbook 2000, part 4: http://www.gfy.ku.dk/~iag/HB2000/part4/groten.htm
  16. Groten, E. (2004): Geodesist's Handbook 2004 .
    1. Groten, E. (2004): Fundamental parameters and current (2004) best estimates of the parameters of common relevance to astronomy, geodesy, and geodynamics. J.Geod. 77(10..11), pp.724..731; DOI: 10.1007/s00190-003-0373-y ;
      http://www.gfy.ku.dk/~iag/HB2004/part5/51-groten.pdf
    2. IAG: Geodesist's Handbook - 2004, part 5.1 . J.Geod. 78(9..12);
      http://www.iag-aig.org/attach/e354a3264d1e420ea0a9920fe762f2a0/51-groten.pdf
  17. Petit G., Luzum B. (2010): IERS Conventions 2010. IERS Technical Note No. 36;
    http://www.iers.org/TN36/
    1. ibidem p.18
  18. Capitaine N., Wallace P.T., McCarthy D.D. (2003): Expressions to implement the IAU definition of UT1 . Astron.&Astrophys. 406, pp.1135..1149; DOI: 10.1051/0004-6361:20030817;
    http://www.aanda.org/articles/aa/pdf/2003/30/aa3487.pdf
    1. ibidem, derived from eq. B1,B2 on p.1149 .
  19. Capitaine N., Wallace P.T., Chapront J. (2003): Expressions for IAU 2000 precession quantities. Astron.&Astrophys. 412, pp.567..586; DOI: 10.1051/0004-6361:20031539;
    http://www.aanda.org/articles/aa/pdf/2003/48/aa4068.pdf
    1. ibidem p.581 eq.(39).
  20. Hilton J.A. e.a. (2006): Report of the International Astronomical Union Division I Working Group on Precession and the Ecliptic. Celest.Mech.Dyn.Astron. 94, pp.351..367 . DOI 10.1007/s10569-006-0001-2;
    https://ui.adsabs.harvard.edu//#abs/2006CeMDA..94..351H/abstract
  21. IAU (2006) XXVIth General Assembly. Resolution No. B1: Adoption of the P03 Precession Theory and Definition of the Ecliptic.
    https://www.iau.org/static/resolutions/IAU2006_Resol1.pdf
  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
    2. National Imagery and Mapping Agency: Technical Report 3850.2, 3rd ed. (1997): DoD World Geodetic System 1984;
      http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf
  23. Resolution C 10, p.51 in: IAU (1982): Proceedings of the 18th General Assembly, Transactions of the IAU XVIII B, West R.M. (ed.), D.Reidel, Dordrecht 1983;
    http://www.iau.org/static/resolutions/IAU1982_French.pdf
  24. The Improved IAU System. Astronomical Almanac 1984, pp.S5..S39
    1. ibidem p.S8
  25. Explanatory Supplement to the Astronomical Almanac. Seidelmann P.K. (ed.), USNO, University Science Books 1992.
    1. ibidem p.400 and p.697
    2. ibidem table 5.4 p.701
  26. Archinal B.A. e.a. (2011): Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2009. Celest.Mech.Dyn.Astron. 109(2), pp.101..135; DOI: 10.1007/s10569-010-9320-4;
    1. ibidem table 1
    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.
    1. ibidem, Ch.8 p.289: Harris D.L.: Photometry and Colorimetry of Planets and Satellites.
  28. Lane A.P., Irvine W.M. (1973): Monochromatic phase curves and albedos for the lunar disk. Astron.J. 78(3), pp.267..277;
  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
  32. Chapront-Touzé M., Chapront J. (1988): ELP 2000-85 - A semi-analytical lunar ephemeris adequate for historical times. Astron.&Astrophys. 190(1-2), pp.342..352;
    http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1988A%26A...190..342C&link_type=ARTICLE
    1. ibidem p.351 table 9
  33. Chapront-Touzé, M; Chapront, J.: Lunar Tables and Programs from 4000 B.C. to A.D. 8000. Willmann-Bell, Richmond, VA, USA 1991 . ISBN 0943396336
    http://www.willbell.com/math/mc9.htm
  34. Chapront J., Chapront-Touzé M., Francou G. (Dec.1997): Nouvelles expressions des termes seculaires dans Lunar Tables and Programs from 4000 B.C. to A.D. 8000 . Notes Scientifiques et Technologiques du Bureau des Longitudes S055, ISSN 1243-4272, ISBN 2-910015-15-7
    1. ibidem par. 3.3 p.8
  35. Simon J.L. e.a. (1994): Numerical expressions for precession formulae and mean elements for the Moon and the planets. Astron.&Astrophys. 282(2), pp.663..683;
    http://articles.adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994A%26A...282..663S&link_type=ARTICLE
  36. Chapront J., Chapront-Touzé M., Francou F. (2002): A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR parameters. Astron.&Astrophys. 387(2), pp.700..709; DOI: 10.1051/0004-6361:20020420;
    http://www.aanda.org/articles/aa/pdf/2002/20/aa2201.pdf
    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]
  37. Blaauw A., Gum C.S., Pawsey J.L., Westerhout G. (1960): Report of the I.A.U. Sub-Commission 33b: The new I.A.U. system of galactic coordinates (1958 revision). Mon.Not.R.Astron.Soc. 121(2), pp.123..131;
    http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1960MNRAS.121..123B&data_type=PDF_HIGH&whole_paper=YES&filetype=.pdf
  38. p.329 in: Murray C.A. (1989): The transformation of coordinates between the systems of B1950.0 and J2000.0, and the principal galactic axes referred to J2000.0 . Astron.Astrophys. 218, pp.325..329;
    http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1989A%26A...218..325M&data_type=PDF_HIGH&whole_paper=YES&filetype=.pdf
  39. Francis C., Anderson E. (2009): Calculation of the Local Standard of Rest from 20 574 Local Stars in the New Hipparcos Reduction with Known Radial Velocities. New Astronomy 14(7), pp.615..629; DOI: 10.1016/j.newastr.2009.03.004;
  40. Fixen D.J. (2009): The temperature of the cosmic microwave background. Astrophys.J. vol.707(2), pp.916..920 ; DOI: 10.1088/0004-637X/707/2/916 ;
    http://iopscience.iop.org/0004-637X/707/2/916/pdf/apj_707_2_916.pdf
  41. Aver E., Olive K.A., Skillman E.D. (2012): An MCM determination of primordial helium abundance. J.CAP vol.1204, pp.004.. ;
    http://arxiv.org/abs/1112.3713
  42. Efstathiou, G. (2014): H0 revisited. MNRAS 440, 1138 ;
    http://arxiv.org/abs/1311.3461v2
  43. WMAP (2012)
    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 ;
      http://arxiv.org/abs/1303.5062v2
    2. Planck Collaboration (2013): Planck 2013 results. XVI. Cosmological parameters. Astron.&Astrophys. vol.?, pp.?..? ; DOI: 10.1051/0004-6361/201321591 ;
      http://arxiv.org/abs/1303.5076v3
    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: ? ;
      http://arxiv.org/abs/1502.01582v2 ; p.27 Table 9
    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
  46. Galactic Center:
    1. Petrov, L. e.a. (2011): The Very Long Baseline Array galactic plane survey - VGaPS. Astron.J. 142(2), pp.35..?
    2. SgrA* (SIMBAD DB): http://simbad.u-strasbg.fr/simbad/sim-id?Ident=Sgr+A%2A
    3. Boehle, A. e.a. (2016): An Improved Distance and Mass Estimate for Sgr A* from a Multistar Orbit Analysis. Astrophys.J. vol.830:17, pp.?..? ; DOI: 10.3847/0004-637X/830/1/17 ;
    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
  48. SH0ES team:
    1. Riess A.G. e.a. (2018): Milky Way Cepheid Standards for Measuring Cosmic Distances and Application to Gaia DR2: Implications for the Hubble Constant. Astrophys.J. ? ;
      https://arxiv.org/abs/1804.10655
    2. Riess A.G. e.a. (2022): 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.Lett. 934, 1;
      https://iopscience.iop.org/article/10.3847/2041-8213/ac5c5b
    3. Riess A.G. e.a. (2024): JWST Observations Reject Unrecognized Crowding of Cepheid Photometry as an Explanation for the Hubble Tension at 8σ Confidence. Astrophys.J.Lett. 962 L17 ;
      https://iopscience.iop.org/article/10.3847/2041-8213/ad1ddd
  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