B Astronomers and Units

The necessity to unify disparate measures. During Antiquity and the Middle Age, measures of weight, length and duration were varying from one place to another. The first government in History to try and homogenize measures was under king Henri III, in England. The 1297 version of the Magna Carta (originally signed in 1215), expressed the will to define standards for measuring weights and distances. “Article 25. One measure of Wine shall be through our Realm, and one measure of Ale, and one measure of Corn, that is to say, the Quarter of London; and one breadth of dyed Cloth, Russets, and Haberjects, that is to say, two Yards within the lists. And it shall be of Weights as it is of Measures.” The rest of the world kept using different yards and pounds, for several centuries.

Measuring the Earth. In France, the Académie royale des sciences was funded in 1666 by Jean-Baptiste COLBERT, under king Louis XIV, influenced by his secretary for sciences and arts, Charles PERRAULT (Débarbat & Quinn, 2019). In 1667, the royal observatory was created and astronomers were tasked with providing more accurate maps of the realm. Clergyman Jean PICARD performed a series of measures by triangulation, with a unique measuring board, and estimated the size of the Earth (Mesure de la Terre, 1671). During the XVIII

^{t h}

century, several scientific expeditions in Latin America or the North Pole (by La Condamine, Maupertuis, et al.) refined the measurement of the size of the Earth and confirmed its flattening around the poles.

The introduction of the metric system. It is only during the French revolution (1789–1799) that the metric system was introduced. It was consistent with its time. It was aimed at erasing differences in an abstract way, with new standards independent of the old human references (foot, inch, etc.). One of its important features was that it was a decimal system, simplifying calculations. It can be traced back to the months before the revolution, in 1789. The cahiers de doléances (register of grievances) expressed the wish to have unified measures throughout the realm. In 1790, Charles-Maurice DE TALLEYRAND-PÉRIGORD, a bishop elected at the recent national assembly, submitted a memoir to adopt a new system of weights and measures, contributed by Marie-Jean-Antoine CRITAT DE CONDORCET and Joseph-Jérôme LEFRANÇOIS DE LALANDE (Débarbat & Quinn, 2019). In 1792, Jean-Baptiste DELAMBRE and Pierre-François MÉCHAIN were charged with measuring the length of the meridian between Dunkerque and Barcelona (Alder, 2015). The definition of the meter was then

1 ∕ 10 000 000

of the distance between the North Pole and the equator. The kilogram was defined as the mass of one cubic decimeter of water.

Difficulty of adoption. The metric system was not adopted right away, even in France. It was mocked by Napoléon, although Laplace promoted the advantage of its decimal system to him. For a while, we kept “mesure usuelles”, which were a standardization of imperial units. In Germany, around 1830, Carl-Friedrich GAUSS formalized the metric unit system in physics, and proposed to add the seconds to meters and kilograms, leading to the CGS system (Centimetre-Gram-Second). During the 1851 World’s fair in London, France promoted the metric system to the world. It led to the “Treaty of the Metre”, signed in Paris in 1875, by seventeen countries adopting the metric system. Great Britain, the Netherlands and Portugal were opposed. England, in particular, felt that adopting the French system would be a political defeat. A diplomatic solution was proposed in adopting the Greenwich meridian, during the 1883 Geodetic Congress: “The Conference hopes that, if the whole world is agreed upon the unification of longitudes and hours in accepting the Greenwich meridian as the point of departure, Great Britain will find in this fact an additional motive to take on her side new steps in favour of the unification of weights and measures, by joining the Metrical Convention of May 20, 1875” (The Geodetic Congress, 1883). The Greenwich meridian was finally adopted during the 1884 International Meridian Conference, but England did not adopt the metric system…

The international system. Since 1960, the International System of units (SI) is the Meter-Kilogram-Second-Ampere system (MKSA or MKS). Astronomers are one of the last communities to use CGS units (and the Gaussian system for electrodynamics; Table B.4). The continued use of a mix between imperial units and the CGS system is counterproductive. The most dramatic example is the crash of the 1999 Mars Climate Observer probe, because of a conversion mistake between imperial and metric units (Popular Mechanics, 2017). Table B.1 gives the correspondence between the MKS and CGS systems, and Tables B.2 – B.3 list the fundamental constants in both systems. Table B.4 compares the MKSA and Gaussian systems for electrodynamics.


Quantity	International units (MKS)		Astronomer’s units (CGS)

GENERAL

Length	1	m (meter)	$1 0^{2}$	cm (centimeter)

Force	1	N (Newton)	$1 0^{5}$	dyn (dynes)

Energy	1	J (joule)	$1 0^{7}$	erg

Power	1	W (watt)	$1 0^{7}$	erg/s

Flux (1 Jansky)	$1 0^{- 26}$	W/m $^{2}$ /Hz	$1 0^{- 23}$	erg/s/cm $^{2}$ /Hz

ELECTROMAGNETISM

Charge	1	C (Coulomb)	$2.997 924 58 \times 1 0^{9}$	esu (electrostatic unit)

Current	1	A (Ampere)	$2.997 924 58 \times 1 0^{9}$	esu/s

Electric potential	1	V (volt)	$1 ∕ 299.792 458$	statV (statvolt)

Electric field	1	V/m	$1 ∕ 299 79.245 8$	statV/cm

Magnetic field	1	T (Tesla)	$1 0^{4}$	G (Gauss)

Magnetic flux	1	Wb (Weber)	$1 0^{8}$	G.cm $^{2}$

Auxiliary field H	1	A/m	$4 π \times 1 0^{- 3}$	Oe (Oersted)

ANGULAR DISTANCE

1 arcsec	$4.848 \times 1 0^{- 6}$	rad

1 arcsec at 1 Mpc	4.848	pc

Table B.1: Unit conversion.


Quantity	International units (MKS)		Astronomer’s units (CGS)

UNIVERSAL

Light speed, $c$	$2.999 792 458 \times 1 0^{8}$	m/s	$2.999 792 458 \times 1 0^{10}$	cm/s

Newton constant, $G$	$6.674 28 \times 1 0^{- 11}$	m $^{3}$ /kg/s $^{2}$	$6.674 28 \times 1 0^{- 8}$	cm $^{3}$ /g/s $^{2}$

Planck constant, $h$	$6.626 068 96 \times 1 0^{- 34}$	J.s	$6.626 068 96 \times 1 0^{- 27}$	erg.s
$ℏ \equiv h ∕ 2 π$	$1.054 571 628 \times 1 0^{- 34}$	J.s	$1.054 571 628 \times 1 0^{- 27}$	erg.s

Magnetic constant,
$μ_{0} \equiv 4 π \times 1 0^{- 7}$	$1.256 637 061 4 \times 1 0^{- 6}$	N/A $^{2}$

Electric constant,
$𝜖_{0} \equiv 1 ∕ μ_{0} c^{2}$	$8.854 187 817 \times 1 0^{- 12}$	F/m

Elementary charge, $e$	$1.602 176 487 \times 1 0^{- 19}$	C	$4.806 529 5 \times 1 0^{- 10}$	esu

ATOMIC

Electron mass, $m_{e}$	$9.109 382 15 \times 1 0^{- 31}$	kg	$9.109 382 15 \times 1 0^{- 28}$	g
$m_{e} c^{2}$	$8.187 104 38 \times 1 0^{- 14}$	J	$8.187 104 38 \times 1 0^{- 7}$	erg
$m_{e} c^{2} ∕ e$	$0.510 998 910$	MeV

Proton mass, $m_{p}$	$1.672 621 637 \times 1 0^{- 27}$	kg	$1.672 621 637 \times 1 0^{- 24}$	g
$m_{p} c^{2}$	$1.503 277 359 \times 1 0^{- 10}$	J	$1.503 277 359 \times 1 0^{- 3}$	erg
$m_{p} c^{2} ∕ e$	$0.938 272 013$	GeV

Rydberg,
$R_{\infty} \equiv α^{2} m_{e} c ∕ 2 h$	$10 973 731.568 527$	m $^{- 1}$	$109 737.315 685 27$	cm $^{- 1}$
$R_{\infty} c$	$3.289 841 960 361 \times 1 0^{15}$	Hz
$R_{\infty} h c$	$2.179 871 97 \times 1 0^{- 18}$	J	$2.179 871 97 \times 1 0^{- 11}$	erg
$R_{\infty} h c ∕ e$	$13.605 691 93$	eV

MACROSCOPIC

Boltzmann constant, $k$	$1.380 650 4 \times 1 0^{- 23}$	J/K	$1.380 650 4 \times 1 0^{- 16}$	erg/K
$k ∕ e$	$8.617 343 \times 1 0^{- 5}$	eV/K
$k ∕ e h$	$69.503 56 \times 1 0^{10}$	Hz/K

Atomic mass unit,
$m_{u} \equiv m (12 C) ∕ 12$	$1.660 538 782 \times 1 0^{- 27}$	kg	$1.660 538 782 \times 1 0^{- 24}$	g

Avogadro number, $N_{A}$	$6.022 141 79 \times 1 0^{23}$	mol $^{- 1}$

Molar gas constant,
$R \equiv k N_{A}$	$8.314 472$	J/mol/K	$8.314 472 \times 1 0^{7}$	erg/mol/K

Table B.2: Fundamental constants.


Quantity	International units (MKS)		Astronomer’s units (CGS)

GENERAL

Astronomical unit, $a . u . \equiv ⟨ ⊙ - \oplus ⟩$	$1.495 979 \times 1 0^{11}$	m	$1.495 979 \times 1 0^{13}$	cm

Parsec, $p c \equiv 1 a . u . ∕ 1^{″}$	$3.085 678 \times 1 0^{16}$	m	$3.085 678 \times 1 0^{18}$	cm

SOLAR SYSTEM

Solar radius, $R_{⊙}$	$6.959 9 \times 1 0^{8}$	m	$6.959 9 \times 1 0^{10}$	cm

Solar mass, $M_{⊙}$	$1.988 9 \times 1 0^{30}$	kg	$1.988 9 \times 1 0^{33}$	g

Solar luminosity, $L_{⊙}$	$3.846 \times 1 0^{26}$	W	$3.846 \times 1 0^{33}$	erg/s

Earth radius, $R_{\oplus}$	$6.378 140 \times 1 0^{6}$	m	$6.378 140 \times 1 0^{8}$	cm

Earth mass, $M_{\oplus}$	$5.974 \times 1 0^{24}$	kg	$5.974 \times 1 0^{27}$	g

GALAXY

Solar velocity around G.C., $Θ_{\circ}$	$220$	km/s

Distance sun-G.C., $R_{\circ}$	$8.0$	kpc

Local disk density, $ρ_{disk}$	$3 - 12 \times 1 0^{- 21}$	kg/m $^{3}$	$3 - 12 \times 1 0^{- 24}$	g/cm $^{3}$
$n_{disk}$	$1 - 5 \times 1 0^{6}$	m $^{- 3}$	$1 - 5$	cm $^{- 3}$

Local halo density, $ρ_{halo}$	$2 - 13 \times 1 0^{- 22}$	kg/m $^{3}$	$2 - 13 \times 1 0^{- 25}$	g/cm $^{3}$
$n_{halo}$	$10 - 60 \times 1 0^{4}$	m $^{- 3}$	$0.1 - 0.6$	cm $^{- 3}$

COSMOLOGY

Hubble expansion rate, $H_{0}$	$71$	km/s/Mpc

Critical density, $ρ_{c} \equiv 3 H_{0}^{2} ∕ 8 π G$	$1.399 062 \times 1 0^{11}$	$M_{⊙}$ /Mpc $^{3}$	$9.472 \times 1 0^{- 30}$	g/cm $^{3}$

Pressureless matter density,
$Ω_{M} \equiv ρ_{M} ∕ ρ_{c}$	$0.15 ≲ Ω_{M} ≲ 0.45$

Baryon density, $Ω_{B} \equiv ρ_{B} ∕ ρ_{c}$	$0.019 ≲ Ω_{B} ≲ 0.046$

Cosmological constant,
$Ω_{Λ} \equiv Λ_{c}^{2} ∕ 3 H_{0}^{2}$	$0.6 ≲ Ω_{Λ} ≲ 0.8$

Table B.3: Astronomical constants.


Quantity	Rationalized MKSA	Gaussian units


Lorentz Force	$\vec{F} = q (\vec{E} + \vec{v} \land \vec{B})$	$\vec{F} = q (\vec{E} + \frac{\vec{v}}{c} \land \vec{B})$
	$\frac{d \vec{F}}{d V} = ρ \vec{E} + \vec{j} \land \vec{B}$	$\frac{d \vec{F}}{d V} = ρ \vec{E} + \frac{\vec{j}}{c} \land \vec{B}$



Dielectric Constant & Permeability	$𝜖_{0} = \frac{1 0^{7}}{4 π c^{2}} μ_{0} = 4 π 1 0^{- 7}$	$𝜖_{0} = 1 μ_{0} = 1$



Displacement & Magnetic Field	$\vec{D} = 𝜖 \vec{E} + \vec{P}$	$\vec{D} = 𝜖 \vec{E} + 4 π \vec{P}$
	$\vec{H} = \frac{\vec{B}}{μ} - \vec{M}$	$\vec{H} = \frac{\vec{B}}{μ} - 4 π \vec{M}$



Maxwell Equations	$\vec{\nabla} . \vec{D} = ρ$	$\vec{\nabla} . \vec{D} = 4 π ρ$
	$\vec{\nabla} \land \vec{H} = \vec{j} + \frac{\partial \vec{D}}{\partial t}$	$\vec{\nabla} \land \vec{H} = \frac{4 π}{c} \vec{j} + \frac{1}{c} \frac{\partial \vec{D}}{\partial t}$
	$\vec{\nabla} \land \vec{E} + \frac{\partial \vec{B}}{\partial t} = \vec{0}$	$\vec{\nabla} \land \vec{E} + \frac{1}{c} \frac{\partial \vec{B}}{\partial t} = \vec{0}$
	$\vec{\nabla} . \vec{B} = 0$	$\vec{\nabla} . \vec{B} = 0$



Poynting Vector	$\vec{P} = \vec{E} \land \vec{H}$	$\vec{P} = \frac{c}{4 π} \vec{E} \land \vec{H}$



Electromagnetic Power	$P = \iint_{S} \vec{P} . \vec{d S}$	$P = \iint_{S} \vec{P} . \vec{d S}$



Energy Density	$U = \frac{1}{2} (𝜖 {\vec{E}}^{2} + \frac{{\vec{B}}^{2}}{μ})$	$U = \frac{1}{8 π} (𝜖 {\vec{E}}^{2} + \frac{{\vec{B}}^{2}}{μ})$

Table B.4: Classical electrodynamics.

B.2 Working with Units


	Central wavelength, $λ_{0}$	Zero-point flux, F $_{ν, 0}$

U	0.36 $μ m$	1884 Jy

B	0.44 $μ m$	4646 Jy

V	0.55 $μ m$	3953 Jy

R	0.66 $μ m$	2875 Jy

I	0.80 $μ m$	2241 Jy

J	1.25 $μ m$	1602 Jy

H	1.60 $μ m$	1010 Jy

K	2.18 $μ m$	630 Jy

L	3.45 $μ m$	278 Jy

M	4.75 $μ m$	153 Jy

N	10.6 $μ m$	36.3 Jy

Table B.5: The magnitude system.

Appendix B
Astronomers and Units

B.1 Brief History of Unit Systems

B.2 Working with Units

Appendix BAstronomers and Units

B.1 Brief History of Unit Systems

B.2 Working with Units

Appendix B
Astronomers and Units