Astronomical units and constants 2.7.2



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2.7.2 Astronomical units and constants

The IAU and IERS systems of astronomical units and constants

The current IAU system of values of the principal constants of the Solar System was adopted by the International Astronomical Union in 1976, and was introduced in the principal national and international almanacs from 1984 onwards. The system includes adopted relationships between SI units and the astronomical units of length, mass and time that are used in almost all theories of the motions of the members of the Solar System. The 1976 system differs in one fundamental respect from the previous 1964 system in that the astronomical unit of time is defined as one day (D) of 86 400 SI seconds, rather than as an ephemeris day. The definitions of the other astronomical units were not changed: the astronomical unit of mass is the mass of the Sun (S), while the astronomical unit of length is that length (A) for which the Gaussian gravitational constant (k) takes the value 0.017 202 098 095 in these units. The dimensions of k² are those of the constant of gravitation, G. The length A is approximately the mean distance of the Earth from the Sun.

The high precision that has been achieved in recent years in the monitoring of the rotation of the Earth has led the International Earth Rotation Service to introduce a comprehensive set of ‘IERS Standards’ that specify the numerical models to be used in the reduction of the observations. The values of some of the constants in the 1992 version of the IERS standards differ from those in the current IAU system and are given in the third column of the table.

Constant	IAU (1976) system	IERS (1992) standards (in same units)

Defining constants
Gaussian gravitational constant, k	0.017 202 098 95	same
Speed of light, c	299.792 458 Mm s⁻¹	same
Primary constants
Light-time for 1 au, τ_A	499.004 782 s	499.004 783 53
Equatorial radius of Earth, a_e	6.378 140 Mm	6.378 136 3
Dynamical form factor for
Earth, J₂	0.001 082 63	0.001 082 636 2
Geocentric gravitational
constant, GE	3.986 005 × 10¹⁴m³s⁻²	3.986 004 415
Constant of gravitation, G	6.672 × 10⁻¹¹m³kg⁻¹s⁻²	6.672 59
Ratio of mass of Moon to mass
of Earth, μ	0.012 300 02	0.012 300 034
General precession in longitude,
per Julian century at 2000.0	5029".0966	same
Obliquity of ecliptic at 2000.0	23° 26′ 21".448	23° 26′ 21".4119
Mean angular velocity of Earth, ω	—	0.000 072 921 15 rad s⁻¹
Derived constants
Length of 1 au, cτ_A= A	149.597 870 Gm	149.597 870 61
Solar parallax, arcsin (a_e/A) = π₀	8".794 148	8".794 142
Constant of aberration (for 2000.0),	20".495 52	—
Flattening factor Earth, f	0.003 352 81 = 1/298.257	same
Heliocentric gravitational
constant, A³k²/D²= GS	1.327 124 38 ×10²⁰m³s^{− 2}	1.327 124 40
Ratio of mass of Sun to mass of
Earth, S/E	332 946.0	332 946.045
Mass of Sun, S	1.989 1 ×10³⁰kg	1.988 9

Notes: (1) For the definitions of J₂ and f see section 2.7.4.
(2) A Julian century contains 36 525 days.
(3) The epoch 2000.0 is 2000 January 1.5.

Other units and the standard epoch

The unit of time in the fundamental formulae for precession (and in similar expressions) is the Julian century of 36 525 days (the tropical century is not to be used). The new standard epoch is designated J2000.0 and is the calendar date 2000 January l^d.5, which is the Julian date JD 245 1545.0.

An alternative unit for use in Newtonian dynamics of the solar system and binary stars is the Gaussian year, which is the sidereal period of a particle of negligible mass moving around the Sun in an orbit with a mean distance of 1 au; it is equal to 2π/k ( = 365.256 898) days. Kepler’s law for the relative motion of two isolated particles of mass M and m is then, simply,

a³ = P²(M + m)

where a is the semi-major axis of the orbit in au, P is the period in Gaussian years, and the unit of mass is the mass of the Sun.

A convenient unit for the measurement of the distances of nearby stars is the parsec (pc); this is the distance at which 1 au subtends an angle of 1 second of arc. Hence

1 parsec = l/(sin 1″) au = 2.063 × 10⁵ au = 3.086 × 10¹⁶ m

The multiple units kiloparsec (kpc) and megaparsec (Mpc) are more appropriate for most galactic and extragalactic objects respectively. The ‘light-year’ is normally only used in popular astronomical texts:

1 light-year = 0.307 pc = 6.32 × 10⁴ au = 9.46 × 10¹⁵ m

Constants relating to time

The following relationships hold in all time systems:

      1 day = 24 hours = 1440 minutes = 86 400 seconds
      1 Julian year = 365.25 days = 8766 hours = 525 960 minutes = 31 557 600 seconds
      1 century = 100 years

   The following values of the lengths of the year, month and day are expressed in units of 1 day of 86 400 SI seconds; T is measured in Julian centuries from 2000.0.

	1 tropical year (equinox to equinox)	= 365^d.242 193 − 0^d.000 0061 T
		= 365^d05^h48^m45^s.5 −0^s.53 T
	1 sidereal year (fixed star to fixed star)	= 365^d.256 360 + 0^d.000 0001 T
		= 365^d 06^h 09^m 09^s.5 + 0^s.01 T
	1 mean synodic month (new moon)	= 29^d.530 859 = 29^d 12^h 44^m 02^s.9
	1 mean tropical month (equinox)	= 27^d.321 582 = 27^d 07^h 43^m 04^s.7
	1 mean sidereal month (fixed star)	= 27^d.321 661 = 27^d 07^h43^m 11^s.5
	1 mean solar day (current)	= l^d.000 000 04 = 86 400^s.003
	1 mean sidereal day (current)	= 0^s.997 269 60 = 86 164^s.094
	1 mean period of rotation of Earth (current)	= 0^d.997 269 70 = 86 164^s. 102

The rate of rotation of the Earth is

72 921 151.467 − 0.844 ΔD picoradians/second,

where ΔD is the difference measured in milliseconds between the duration of the mean solar day and 86 400 SI seconds. The current value of ΔD is about 3.
Although the Earth's period of rotation is variable when measured in SI units the following relationships are almost independent of the variability.

	1 mean solar day
		= interval between successive transits of the fictitious mean sun
		= l^d.002 737 909 = 24^h 03^m56^s.555 3 = 86 636^s.555 3 of mean sidereal time
	1 mean sidereal day
		= interval between successive transits of the mean equinox
		= 0^d.997 269 566 = 23^h 56^m 04^s.090 5 = 86 164^s.090 5 of mean solar time
	1 period of rotation of the Earth with respect to an inertial reference frame
		= I^d.000 000 097 = 24^h 00^m 00^s.008 4 = 86 400^s.008 4 of mean sidereal time
		= 0^d.997269663 = 23^h 56^m 04^s.098 9 = 86 164^s.098 9 of mean solar time

Universal time is related to Greenwich mean sidereal time through the expression:

GMST at 0^h UT = 6^h 41^m 50^s.548 41 + 8640 184^s.812 866T_u + 0^s.093 104T_u² − 6.2 × 10⁻⁶T_u³

where T_u is measured in centuries of 36 525 mean solar days from 2000 January 1 at 12^h UT (JD 245 1545.0 UT).

Constants for precession and nutation

The precessional motion of the celestial equator (plane normal to the axis of rotation of the Earth) around the ecliptic (mean plane of the Earth's orbit around the Sun) has a period of about 25 725 years, and gives rise to the luni-solar precession in celestial longitude of 5037 per year. There is also a slow precessional motion of the ecliptic due to planetary perturbations; this gives both a change in the obliquity (inclination of the ecliptic to the equator) and a motion of the equinox (direction of the line of intersection of equator and ecliptic) along the equator of 0''.12 per year. The combined effect is known as the general precession.

Obliquity of ecliptic = 23° 26′ 21″.45 - 46″.81 T = 23°.439 291 − 0°.013 00 T
Annual general precession in longitude = 50″.2910 + 0".0222 T

Where T is measured in Julian centuries from 2000.0. For a star with right ascension α and declination δ the annual precessions in right ascension and declination are m + n sin α tan δ and n cos α, respectively, where

m = 3^s.074 96 + 0^s.001 86 T, n = 1^s.336 21 − 0^s.000 57 T= 20″.0431 − 0″.0085 T

The nutation of the axis of rotation of the Earth is normally specified by its effects in celestial longitude and obliquity. The principal terms in the trigonometric series for the nutation are:

In longitude

In obliquity

Period (days)


− 17''.200 − 0''.017 T	+ 9''.202 + 0''.001 T	6798
+ 0.206	− 0.090	3399
− 1.319	+ 0.574	183
− 0.227	+ 0.098	13.7

References

The Explanatory Supplement to the Astronomical Almanac (ed. P. K. Seidelmann, 1992, University Science Books, Mill Valley, California, ISBN 0-935702-68-7) contains detailed accounts of astronomical time and coordinate systems and of the IAU system of astronomical units and constants.
The IERS Standards (1992) (ed. D. D. McCarthy) have been published as IERS Technical Note 13 by the Central Bureau of the International Earth Rotation Service (Observatoire de Paris, 61 ave de l’Observatoire, F-5014 Paris, France).

G.A.Wilkins

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