Tools of the Trade

Section 3 of 5

Unit Conversions

2.3 Tool 3: Unit Conversions

Why Astronomers Use CGS

Astronomy runs on the CGS system — centimeter, gram, second — not the SI system you met in intro physics.

Base Unit	CGS	SI
Length	cm	m
Mass	g	kg
Time	s	s

The derived units differ too: energy in erg (not joule), force in dyne (not newton), power in erg/s (not watt).

CGS

The centimeter–gram–second system of units, standard throughout astrophysics. Derived units include the erg (energy) and dyne (force).

DigressionWhy CGS stuck

CGS became the astronomical standard in the 19th century and never left. One practical payoff: stellar luminosities and masses come out with comparable exponents — the Sun’s luminosity is $3.8 \times 10^{33}$ erg/s, conveniently close to its mass $2 \times 10^{33}$ g.

We will work in CGS for the rest of the course.

Scientific Notation: The Language of Extremes

Astronomical numbers span more than 40 orders of magnitude, so we write them as $a \times 10^{n}$ : a coefficient $a$ between 1 and 10, times a power of ten. Positive $n$ means a big number; negative $n$ , a small one.

The exponent rules are the whole game:

Operation	Rule	Example
Multiplication	Add exponents	$10^3 \times 10^5 = 10^8$
Division	Subtract exponents	$10^{33}/10^{27} = 10^6$
Powers	Multiply exponents	$(10^3)^2 = 10^6$

Scientific notation

Expressing a number as $a \times 10^n$ with $1 \le a < 10$ . Arithmetic reduces to adding, subtracting, or multiplying the exponents.

SI Prefixes You’ll Use

Prefix	Symbol	Power	Astronomy Example
giga	G	$10^{9}$	GHz (radio frequencies)
mega	M	$10^{6}$	Mpc (galaxy distances)
kilo	k	$10^{3}$	km, kpc
centi	c	$10^{-2}$	cm (the CGS base)
milli	m	$10^{-3}$	mm
micro	μ	$10^{-6}$	μm (infrared)
nano	n	$10^{-9}$	nm (visible light)

The Conversion Method: Multiplying by 1

The trick is to write every conversion as a fraction equal to 1. Since $1\,\text{m} = 100\,\text{cm}$ ,

$\frac{100\,\text{cm}}{1\,\text{m}} = 1$

Multiplying by 1 never changes the physics — only how the number is written. Chain these unit-fractions so the units you don’t want cancel.

Worked Example 4Converting Speed: km/s to cm/s

Problem

Convert Earth’s orbital speed, 30 km/s, to cm/s.

StepSet up unit-fractions equal to 1

$\dfrac{10^3\,\text{m}}{1\,\text{km}} = 1$ and $\dfrac{10^2\,\text{cm}}{1\,\text{m}} = 1$ .

StepChain them so km and m cancel

$30\,\dfrac{\text{km}}{\text{s}} \times \dfrac{10^3\,\text{m}}{1\,\text{km}} \times \dfrac{10^2\,\text{cm}}{1\,\text{m}} = 30 \times 10^{5}\,\text{cm/s}$ .

Dimensional check

The km cancels against km and m against m, leaving only cm/s — a speed, as required ✓.

Result

$30 \times 10^5 = 3 \times 10^{6}\,\text{cm/s}$ .

Worked Example 5Converting Luminosity: watts to erg/s

Problem

The Sun’s luminosity is $L_\odot = 3.8 \times 10^{26}$ W. Express it in CGS (erg/s).

StepRecall the bridging conversions

$1\,\text{W} = 1\,\text{J/s}$ and $1\,\text{J} = 10^{7}\,\text{erg}$ .

StepSubstitute

$3.8 \times 10^{26}\,\text{J/s} \times 10^{7}\,\text{erg/J} = 3.8 \times 10^{33}\,\text{erg/s}$ .

Dimensional check

Joules cancel against the erg/J factor, leaving erg/s — a power ✓.

Result

$L_\odot = 3.8 \times 10^{33}\,\text{erg/s}$ .

Key CGS Values to Memorize

Quantity	CGS Value
1 parsec	$3.09 \times 10^{18}$ cm
1 AU	$1.50 \times 10^{13}$ cm
Solar mass $M_\odot$	$2.0 \times 10^{33}$ g
Solar radius $R_\odot$	$7.0 \times 10^{10}$ cm
Solar luminosity $L_\odot$	$3.8 \times 10^{33}$ erg/s
Speed of light $c$	$3.0 \times 10^{10}$ cm/s
Gravitational constant $G$	$6.67 \times 10^{-8}$ cm³ g⁻¹ s⁻²