Scales of Space and Time

Next: 2. Everyday Astronomy

One thing many people know about astronomy is that it deals with unimaginably large distances and absurdly long times. A simple trick allows astronomers to think about vast amounts of time and space without getting confused. We will learn this trick and use it to gain an overview of the size and age of things in the universe.

Topics

Linear & Logarithmic Scales
Powers of Ten
The Sizes of Things
Cosmic Time
Scientific Notation
Working With Units

Reading

	Figure It Out 1.2: Scientific Notation		p. 4
	Figure It Out 1.1: Keeping Track of Space and Time		p. 3
	A Closer Look 1.1: A Sense of Scale: Measuring Distances		p. 12
	1.1 Peering Through the Universe: A Time Machine		p. 2
	1.4 How do You Take a Tape Measure to the Stars		p. 11

Linear Scales

An ordinary ruler is a good example of a linear scale:

Marks are spaced equally along the scale
Numbers increase by a constant AMOUNT
(in this case by 1)

Logarithmic Scales

A logarithmic scale is labeled a bit differently:

Marks are still spaced equally along the scale
Numbers now increase by a constant FACTOR
(in this case a factor of 10)

Question 1.1

Let's extend the logarithmic scale one space to the right as shown. What is the value of X?

1,000,001
2,000,000
10,000,000
100,000,000
none of the above

Question 1.2

Now, let's extend the logarithmic scale one space to the left as shown. What is the value of X?

0
0.1
0.2
0.9
none of the above

What's Wrong With Linear Scales?

A linear scale is fine for comparing sizes of different kinds of fruit:

It's also fine for comparing diameters of different planets:

But it's useless if you want to compare fruit and planets on the same scale!

Why Do We Need Logarithmic Scales?

Using a logarithmic scale, we can easily plot fruit and planets together:

From this chart, we can see at a glance that a grape is smaller relative to other kinds of fruit than the Moon is relative to other planets.

Powers of Ten

Counting zeros when working with very large or very small numbers is tedious and leads to mistakes, so we will use powers-of-ten notation; for example:

10⁶ = 1,000,000

10^-6 = 1 ÷ 1,000,000 = 0.000,001

The general rule is that 10ⁿ, where n is a positive whole number, is the product of n factors of 10:

10ⁿ	=	10 × 10 × . . . × 10
		-- n copies of 10 --

while if n is negative then you divide 1 by the number you would get using the absolute value of n:

10^-n = 1 ÷ 10ⁿ

Powers of Ten: the Movie

A FILM DEALING WITH
THE RELATIVE SIZE OF THINGS
IN THE UNIVERSE

AND THE EFFECT
OF ADDING ANOTHER ZERO

Web Resources

Powers of Ten
An on-line version of the movie by Charles & Ray Eames.
Secret Worlds: The Universe Within
An interactive web page illustrating the same idea as the Powers of Ten movie.

Reading Between the Marks. I

On linear scale it's pretty clear what we mean when a value is plotted exactly between two numbered marks, as X is here:

Here X = 3.5; you can compute that by averaging the marked values:

X = (3 + 4) ÷ 2 = 7 ÷ 2 = 3.5

What about Y, which is exactly between 3 and X? All you need to do is average those values:

Y = (3 + 3.5) ÷ 2 = 6.5 ÷ 2 = 3.25

You can fill in the rest of the scale by using this rule over and over.

Reading Between the Marks. II

What about a logarithmic scale? What value does X have here?

Instead of adding, you multiply, and instead of dividing by 2, you take the square root:

X = √ (1,000 × 10,000) = √ 10,000,000 = 3,162.3

What about Y, which is exactly between 1000 and X? Apply the same rule again:

Y = √ (1,000 × 3,162.3) = √ 3,162,300 = 1,778.3

Again, you can fill the scale by using this rule over and over.

The Sizes of Things

We now know enough to chart the sizes of things in the Powers of Ten movie, using a logarithmic scale with a factor of 10⁵ between marks:

At a glance this shows us where the physical scale we're familiar with fits into the Universe as whole; we're about 10²¹ times smaller than galaxies, and 10⁹ times smaller than stars, but 10¹⁰ times bigger than atoms.

Question 1.3

Using the chart above, estimate how many times bigger galaxies are than stars.

10⁶
10⁹
10¹²
10¹⁵
10¹⁸

Cosmic Time. I

The book uses this chart to illustrate the history of the Universe.

This is a linear time scale, so it's hard to see where human beings fit in.

Cosmic Time. II

We can use a logarithmic scale to show how human time scales compare to cosmic time.

Actually, this chart, like the one before, is more about the history of our planet than the history of the universe as a whole. To really appreciate the history of the universe, we need to chart time since the Big Bang.

Excerpt from Old Woodrat's Stinky House

From Mountains and Rivers Without End, © 1996 by Gary Snyder.
My thanks to Gary for permission to quote this poem.

Us critters hanging out together
something like three billion years.

Three hundred something million years
the solar system swings around
with all the Milky Way -

Ice ages come one hundred fifty million years apart
last about ten million
then warmer days return -

A venerable desert woodrat nest of twigs and shreds
plastered down with ambered urine
a family house in use eight thousand years,
& four thousand years of using writing equals
the life of a bristlecone pine -

A spoken language works
for about five centuries,
lifespan of a douglas fir;
big floods, big fires, every couple hundred years,
a human life lasts eighty,
a generation twenty.

Hot summers every eight or ten,
four seasons every year
twenty-eight days for the moon
day/night the twenty-four hours

& a song might last four minutes,

a breath is a breath.

Scientific Notation

We use scientific notation to help with the arithmetic of large and small numbers.

1,230,000 = 1.23 × 10⁶

0.00000123 = 1.23 × 10^-6

The same number can take different forms:

1,230,000 = 1.23 × 10⁶ = 12.3 × 10⁵ = 0.123 × 10⁷

The form 1.23 × 10⁶ is usually preferred, because the constant in front (1.23) is between 1 and 10.

To multiply, you add the exponents:

(1.2 × 10⁶) × (2 × 10⁵) = (1.2 × 2) × 10⁽⁶⁺⁵⁾ = 2.4 × 10¹¹

To divide, you subtract the exponents:

(4.2 × 10¹²) ÷ (2 × 10⁸) = (4.2 ÷ 2) × 10^(12-8) = 2.1 × 10⁴

To add or subtract numbers in scientific notation, you have to make the exponents the same first:

(1.2 × 10⁶) + (2 × 10⁵) = (1.2 × 10⁶) + (0.2 × 10⁶) = 1.4 × 10⁶

Question 1.4

A neutron star contains as many atoms as an ordinary star, but each atom has been squashed by gravity to the size of its central nucleus. Using the chart above, estimate the diameter of a neutron star.

10³ meters
10⁴ meters
10⁵ meters
10⁶ meters
10⁷ meters

Working With Units

Units are a valuable tool; careful attention to the units at each step of a calculation can help you fix mistakes. The idea is to work with units as if they were symbols like those in algebra. For example:

Multiply units along with numbers:
(5 m) × (2 sec) = (5 × 2) × (m × sec) = 10 m sec.

The units in this example are meters times seconds, pronounced `meter seconds' and written `m sec'.
Divide units along with numbers:
(10 m) ÷ (5 sec) = (10 ÷ 5) × (m ÷ sec) = 2 m/sec.

The units in this example are meters divided by seconds, pronounced `meters per second' and written `m/sec'; these are units of speed.

Cancel when you have the same units on top and bottom:
(15 m) ÷ (5 m) = (15 ÷ 5) × (m ÷ m) = 3.

Here the result has no units of any kind! This is a `pure' number. It has the same value no matter what units were used for the measurements.
To add or subtract, convert both numbers to the same units first:
(5 m) + (2 cm) = (5 m) + (0.02 m) = (5 + 0.02) m = 5.02 m.

Recall that 1 cm = 0.01 m, so 2 cm = 0.02 m.
You can't add or subtract two numbers unless you can convert them both to the same units:
(5 m) + (2 sec) = ???

Meters and seconds are different kinds of quantities; one is length, and the other is time. We can't convert one to the other, so there is no way to add them.

Converting between different units is not hard if you remember to treat units like symbols; replace the original unit with its equivalent in the unit desired, and do the necessary arithmetic. For example:

Convert feet to meters using the equality 1 ft = 0.3045 m:
6 ft = 6 × (1 ft) = 6 × (0.3045 m) = (6 × 0.3045) m = 1.84 m.
Convert pounds to kilograms using the equality 1 lb = 0.454 kg:
165 lb = 165 × (1 lb) = 165 × (0.454 kg) = (165 × 0.454) kg = 75 kg.
Convert years to seconds using the equality 1 yr = 3.15 × 10⁷ sec:
43 yr = 43 × (1 yr) = 43 × (3.15 × 10⁷ sec) = (43 × 3.15 × 10⁷) sec = 1.35 × 10⁹ sec.

Web Resources

Scientific Notation
A brief primer on scientific notation, and an on-line test you can take to check your understanding.
Working With Units
A brief primer on units of measurement, how to use them in calculations, and how to convert between them.
Just What is a Logarithm, Anyway?
A simple explanation of linear and logarithmic scales as well as an introduction to logarithms. (Note: available as a pdf file only.)

Next: 2. Everyday Astronomy

Joshua E. Barnes (barnes@ifa.hawaii.edu)

Last modified: August 23, 2006
http://www.ifa.hawaii.edu/~barnes/ast110_06/sosat.html