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.
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 |
An ordinary ruler is a good example of a linear scale:
A logarithmic scale is labeled a bit differently:
Let's extend the logarithmic scale one space to the right as shown. What is the value of X?
Now, let's extend the logarithmic scale one space to the left as shown. What is the value of X?
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!
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.
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:
10^{6} = 1,000,000 | 10^{-6} = 1 ÷ 1,000,000 = 0.000,001 |
The general rule is that 10^{n}, where n is a positive whole number, is the product of n factors of 10:
10^{n} | = | 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 ÷ 10^{n} |
A FILM DEALING WITH
THE RELATIVE SIZE OF THINGS
IN THE UNIVERSE
AND THE EFFECT
OF ADDING ANOTHER ZERO
An on-line version of the movie by Charles & Ray Eames.
An interactive web page illustrating the same idea as the Powers of Ten movie.
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.
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.
We now know enough to chart the sizes of things in the Powers of Ten movie, using a logarithmic scale with a factor of 10^{5} 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 10^{21} times smaller than galaxies, and 10^{9} times smaller than stars, but 10^{10} times bigger than atoms.
Using the chart above, estimate how many times bigger galaxies are than stars.
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.
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.
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 Three hundred something million years Ice ages come one hundred fifty million years apart A venerable desert woodrat nest of twigs and shreds |
A spoken language works Hot summers every eight or ten, & a song might last four minutes, a breath is a breath. |
We use scientific notation to help with the arithmetic of large and small numbers.
1,230,000 = 1.23 × 10^{6} | 0.00000123 = 1.23 × 10^{-6} |
The same number can take different forms:
1,230,000 = 1.23 × 10^{6} = 12.3 × 10^{5} = 0.123 × 10^{7} |
The form 1.23 × 10^{6} is usually preferred, because the constant in front (1.23) is between 1 and 10.
To multiply, you add the exponents:
(1.2 × 10^{6}) × (2 × 10^{5}) = (1.2 × 2) × 10^{(6+5)} = 2.4 × 10^{11} |
To divide, you subtract the exponents:
(4.2 × 10^{12}) ÷ (2 × 10^{8}) = (4.2 ÷ 2) × 10^{(12-8)} = 2.1 × 10^{4} |
To add or subtract numbers in scientific notation, you have to make the exponents the same first:
(1.2 × 10^{6}) + (2 × 10^{5}) = (1.2 × 10^{6}) + (0.2 × 10^{6}) = 1.4 × 10^{6} |
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.
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:
(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'.
(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.
(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.
(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.
(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:
6 ft = 6 × (1 ft) = 6 × (0.3045 m) = (6 × 0.3045) m = 1.84 m.
165 lb = 165 × (1 lb) = 165 × (0.454 kg) = (165 × 0.454) kg = 75 kg.
43 yr = 43 × (1 yr) = 43 × (3.15 × 10^{7} sec) = (43 × 3.15 × 10^{7}) sec = 1.35 × 10^{9} sec.
A brief primer on scientific notation, and an on-line test you can take to check your understanding.
A brief primer on units of measurement, how to use them in calculations, and how to convert between them.
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 |