# CONSTANTS IN ASTRONOMY

We define in this chapter the useful constants for the understanding of the motion and distance measurements in the solar system. However, we give the definition of the most fundamental units and constants of astronomy in general.

## The systems of units

Before defining constants, it is necessary to remember that for any measurement, you must first define a system of units. We present below the International System of Units (SI). Units are distinguished into 3 classes : basic units, derived units and supplementary units.
Remember the basic units of the international system (SI):

Quantity Unit Symbol
Length meter m
Mass kilogramme kg
Time second s
Intensity of
electric current
ampère A
Temperature
(thermodynamic)
kelvin K
Quantity of material mole mol
Intensity of light candela cd

The definition of these units has changed over years, each definition being compatible with the previous but allowing a more accurate realization. The latest definitions (the date is given in parentheses after each definition) published by the Bureau International des Poids et Mesures (BIPM) are as follows :

• unit of length : the meter is the length of path of the light during a duration of 1/299 792 458 second in vacuum (1983)

• unit of mass : the kilogramme is the unit of mass equal to the mass of the international prototype of the kilogramme made of platinum irridium kept at the Bureau International (1889, 1901)

• unit of time : the second is the duration of 9 192 631 770 cycles of the radiation corresponding to the transition between two hyperfine levels of the fundamental state of the atome of Cesium 133 (1967)

• unit of electric current : the ampère is the intensity of a constant current, maintained inside two conductors parallel, straight, infinitely long, of negligible circular section and placed at a distance of one meter from each other in space, which will produce a force between these conductors equal to 2 × 10 -7 newton per meter (1946, 1948)

• hydrodynamic temperature unit : kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water (1967)

• unit of quantity of material : the mole is the quantity of material of a system containing as elementary entities as there are atoms in 0.012 kilogram of carbon 12 not linked, at rest and in their fundamental state(1971)

• unit of intensity of light : the candela is the intensity of light, in a given direction, of a source emitting a monochromatic radiation of frequency 540 x 1012 hertz and the energetic intensity of which in that direction is 1/683 watt per steradian (1979).

The derived units are built by combination of basic units from algebraic relationships linking the corresponding quantities.

The supplementary units are the units of plane angle and solid angle :

• the radian is the plane angle determined by two radii which, on the circumference of a circle, cut an arc equal in length to the radius;

• the steradian is the solid angle which, having its summit at the center of a sphere, described at the surface of the sphere an area equal to that of a square with sides being the radius of the sphere

.

## The system of astronomical units

The basic units are :

• unit of length (prior to 2012): astronomical unit (au or ua). This is the semi-major axis of an orbit around the Sun that describe a planet of negligible mass, undisturbed, with a mean motion equal to k radians per day, k being the constant of Gauss, and the time and mass units being as follows. 1 ua = 1,495 978 7061 x 1011 meters (1992).

• unit of length (after 2012): astronomical unit (au or ua). 1 ua = 149 597 870 700 meters (2012).

• unit of time: the day, equal to 86400 seconds of the International System (SI).

• unit of mass: the mass of the Sun 1.9889 x 1030 kg (1992).

The auxiliary units are provided through their correspondence with the basic units in the tables below.

Units of time :

second day Julian year
1 second 1
1 day 86 400 1
1 Julian year 31 557 600 365.25 1
1 Julian century 3 155 760 000 36 525 100

Note on the units of time :
- the day is no more based upon the rotation of the Earth which is not uniform;
- the year and the century are Julian and not Gregorian.

Units of length :
in meters
(m)
in astronomical units
(ua)
in light years
(al)
in parsec
(pc)
1 meter (1976)
1 meter (1992)

1

1 au (1976)
1 au (2012)
149 597 870 000
149 597 870 700

1

1 ly (1976)
1 ly (1992)
9.460 730 472 5808 x 1015
9.460 730 472 5808 x 1015
63 241.077 38
63 241.077 10

1
0.306 601 395 22
0.306 601 393 87
1 pc (1976)
1 pc (1992)
3.085 677 5671 x 1016
3.085 677 5807 x 1016
206 264.806 248
206 264.806 248
3.261 563 7619
3.261 563 7763

1

the speed of light is 299 792 458 m/s

Note on the units of length :
- the light year is the distance traveled by the light into a Julian year in a vacuum space-time;
- the parsec is the distance of a star, the annual parallax of which being equal to one second of a degree, i.e. the distance from which an astronomical unit is seen under an angle of one second of a degree.

## The system of astronomical constants

We have the following classification for these constantes:

• the constants of definition the values of which are arbitrary. The system of constants depends on them;

• the primary constants the values of which are determined through observation;

• the derived constants coming from the previous constants through simple algebraic relationships.

The values of the main constants are given in the table below:

1976 (UAI) 1992 (IERS)
Constantes of definition
k, constant of Gauss (radian/day) 0.017 202 098 95 0.017 202 098 95
c, speed of light (meters/second)  - 299 792 458
Primary constants
c, speed of light (m s-1) 299 792 458  -
time of light for 1 au (seconds) 499.004 782 499.004 783 53
equatorial radius of the Earth (m) 6 378 140 6 378 136.3
GM, constante géocentrique de la gravitation (m3 s-2) 3,986 005 x 1014 3,986 004 418 x 1014
G, constant of the gravitation (m3 kg-1 s-2) 6.672 x 10-11 6.672 59 x 10-11
obliquity of the ecliptic for J2000 23° 26' 21".448 23° 26' 21".4119
Derived constants
astronomical unit (in meters) 149 597 870 000 149 597 870 610
mass of the Sun (in kg) 1.9891 x 1030  1.9889 x 1030

Notes :
- the constant of Gauss corresponds to an angular velocity of 0.985 607 668 601 425 degree/day;
- the speed of light which was a primary constant in 1976 is now a constant of definition since 1992;
- there are many other derived constants, such as the ratio of the mass of the Sun to the masses of the planets;
- some constants are, in fact, depending on time. In that case, the chosen date, named "origin of time" or "standard epoch" is J2000, i.e. January 1, 2000 at 12h of the used time scale.

Credit : Bureau des longitudes/IMCCE