2. Astronomical Scales & Numbers

Reading: Chapter 2 + Appendix A

Review: Chapter 1 - particularly Section 1.4

To explore the universe we need to be familiar with very large numbers - so, we start with a reminder of how the arithmetic of large numbers works. We can then scale down the vast distances of the universe to something we can handle.

Scientific Notation:

It is very inconvenient to always have to write out the age of the universe as 15,000,000,000 years or the distance to the Sun as 149,600,000,000 meters. To save effort, powers-of-ten notation is used.

10⁰ = 1                      10^-1 = 1/10 = 0.1
10¹ = 10                    10^-2 = 1/100 = 0.01
10² = 100                   10^-3 = 1/1000 = 0.001
10³ = 1000                 10^-4 = 1/10000 = 0.0001
10⁴ = 10,000 .
10⁵ = 100,000 .
10⁶ = 1,000,000 .
. .
. .

Let's define two terms for a number in scientific notation. If we write a number as

1.23 x 10⁴

then the number 1.23 is called the mantissa and 4 is the exponent. In scientific notation, we always write the mantissa so it is between 1 and 10.

For example, 10 = 10¹. The exponent tells you how many times to multiply by 10. As another example, 10^-2 = 1/100. In this case the exponent is negative, so it tells you how many times to divide by 10. The only trick is to remember that 10⁰= 1.

Using powers-of-ten notation, the age of the universe is 1.5 x 10¹⁰ years and the distance to the Sun is 1.496 x 10¹¹ meters.

Oftentimes we will use English words for convenient powers of ten:

a factor of ten = 10¹ = 10	one tenth = 10^-1
a factor of one hundred = 10²	one hundredth = 10^-2
. . . of one thousand = 10³	one thousandth = 10^-3
. . . of one million = 10⁶	one millionth = 10^-6
. . . of one billion = 10⁹	one billionth = 10^-9

Converting to Scientific Notation:

Place the decimal point after the first non-zero digit, and count the number of places the decimal point has moved. If the decimal place has moved to the left then multiply by a positive power of 10; to the right will result in a negative power of 10.

Example: To write 3040 in scientific notation we must move the decimal point 3 places to the left, so it becomes 3.04 x 10³.

Example: To write 0.00012 in scientific notation we must move the decimal point 4 places to the right: 1.2 x 10^-4.

Addition and Subtraction with Scientific Notation:

When adding or subtracting numbers in scientific notation, their powers of 10 must be equal. If the powers are not equal, then you must first convert the numbers so that they all have the same power of 10.

Example:
(6.7 x 10⁹) + (4.2 x 10⁹) = (6.7 + 4.2) x 10⁹ = 10.9 x 10⁹ = 1.09 x 10¹⁰. (Note that the last step is necessary in order to put the answer in scientific notation.)

Example:
(4 x 10⁸) - (3 x 10⁶) = (4 x 10⁸) - (0.03 x 10⁸) = (4 - 0.03) x 10⁸ = 3.97 x 10⁸.

Now that you have seen these examples it is your turn (see Appendix A of the textbook for explanation of units ):

(1) Convert the cost of Hubble Space Telescope ($2,200,000,000) to scientific notation. _____________________________

(2) Write the mass of the sun (1.989 x 10³⁰ kg) out in `long hand'. __________________________

(3) How many dollars is a Megabuck? ____________________________

(4) (5.6 x 10⁶) - (2.3 x 10⁵) = ______________________________________________________

(5) (3.6 x 10²⁷) + (8.5 x 10²⁸) = ____________________________________________________

Multiplication and Division with Scientific Notation:

It is very easy to multiply or divide just by rearranging so that the powers of 10 are multiplied together.

Example:
(6 x 10²) x (4 x 10^-5) = (6 x 4) x (10²x 10^-5) = 24 x 10^2-5 = 24 x 10^-3 = 2.4 x 10^-2.

Example:
(9 x 10⁸) ÷ (3 x 10⁶) =

9 x 10⁸

3 x 10⁶

= (9/3) x (10⁸/10⁶) = 3 x 10^8-6 = 3 x 10².

(6) (5 x 10⁶) x (2 x 10⁵) = _________________________________________________________

(7) (3 x 10²⁷) ÷ (1.5 x 10⁶) = ______________________________________________________

Approximation with Scientific Notation

Because working with powers of 10 is so simple, use of scientific notation makes it easy to estimate approximate answers in your head. This is especially important when using a calculator since, by estimating answers in your head, it allows you to check whether your answers are reasonable. To make approximations, simply round the numbers in scientific notation to the nearest integer, then do the operations in your head.

Example:
Estimate 5,795 x 326. In scientific notation the problem becomes (5.795 x 10³) x (3.26 x 10²). Rounding each to the nearest integer makes the approximation (6 x 10³) x (3 x 10²), which is 18 x 10⁵, or 1.8 x 10⁶. (The exact answer is 1.88917 x 10⁶.)

Example:
Estimate (5 x 10¹⁵) + (2.1 x 10⁹). Rounding to the nearest integer this becomes (5 x 10¹⁵) + (2 x 10⁹). We see immediately that the second number is nearly 10¹⁵/10⁹, or one million, times smaller than the first. Thus, it can be ignored in the addition problem and our approximate answer is 5 x 10¹⁵. (The exact answer is 5.0000021 x 10¹⁵).

Significant Digits

Significant digits are important in science when you are making calculations. The reason is that measurements in science are not infinitely precise. There is a limit to how accurate our measurements can be and this limit is given by the number of significant digits. For example, let's say you measure a book with a ruler:

If your ruler has inches as the smallest division of length, then it is ridiculous to say the length of the book is 8.3856228311 inches in length. You cannot measure more accurately than one inch. At best, guessing at how much more than 8 inches the book is, you can say the book is 8.4 inches long. In this case, you cannot provide more than two significant digits to your measurement. If you had a ruler that was accurate up to .01 inch, and you were able to measure the book to be 8.39 inches in length, then you would have four significant digits in your measurement. The number of significant digits that you will work with will depend not only on how accurately you can measure, but the actual size of the measurement. If you were to use your ruler accurate to .01 inch to measure a button that was only 0.24 inches across, then you are back to two significant digits! Here are the rules for telling how many significant digits are in a number:

The leftmost digit which is not a zero is the most significant digit.
If the number does not have a decimal point, the rightmost digit which is not a zero is the least significant digit.
If the number does have a decimal point, the rightmost significant digit is the least significant digit, even if it's a zero.
Every digit between the least and most significant digits should be counted as a a significant digit.

For example, according to these rules, all of these numbers have three significant digits:

782
782,000,000
0.0782
7.82 x 10⁶
7.82
1.00

For numbers ending in zero, and using Rules #3 and #4 above, we can see that:

120 has two significant digits.
120.0 has four significant digits.
1.000000 has seven significant digits.
100000 has one significant digit.
1.20 x 10² is the way you write 120 if you want to make it clear that there are 3, not 2 significant digits in the number.

How many significant figures should one retain in the final answer to a problem?

The number of significant figures in the result should be equal to the number of significant digits in the least precise number (that is, the number with the fewest significant digits given in the problem).

4+0.00001=4  but 
4.00000+0.00001=4.00001

4.4 x 2.2 = 9.68 which we round 
up to 9.7 since there are only 2 sig. figs. in the 
input.

2,345,678 x 2.0 = 4,691,356 = 4.7 
x 10⁶

(8) Estimate the circumference of the Earth in kilometers. The circumference of a circle is 2 p R; the radius R of the Earth is 6374 km, and p is about 3.14. (a) Estimate in your head - to 1 significant figure : _____________________ (b) calculate with a calculator - keeping only an appropriate number of significant figures: _____________________

Astronomical Scales

To get a better sense of very big (or very small) things we often scale them or down - making them smaller (or bigger) so that we can view them better. For example, a globe is the Earth scaled down to human dimensions. A map of Rocky Mountain Park lets you put the Park in your pocket. Scaling in the other direction, photographic microscope takes pictures of tiny bugs or cells or such things - "blowing up" an image to make a tiny thing visible to human eyes. In astronomy we are generally trying to shrink the vastness of the universe down to human scales. We make maps of huge regions of space by scaling down the large distances of the "real size" universe by a scale factor.

(9) Look around your room or office and see if you can find "image" of something that has been scaled down to fit onto your wall or onto a book or mug. What is the scale factor by which the image has been shrunk from "lifesize"? A factor of 2? 10? 100? A million?

To help put the vast universe onto a scale that we can visualize and comprehend as physical entities (rather than as huge numbers) we scale astronomical objects to something meaningful. For the solar system we use the Earth-Sun distance. For more distant objects we use the speed of light. Here is a sketch of the Astronomical Scales of the solar system and of the Galaxy.

First, the solar system scale factor - the Astronomical Unit:

Distance of Earth to Sun
= 93 million miles
= 150 million kilometers
= 150 x 10⁶ km
= 1.50 x 10⁸ km

1 kilometer = 1000 meters

= 1.5 x 10⁸ x 10³ meters
= 1.5 x 10¹¹meters
= 1 Astronomical Unit or AU for short.

We shall be using the AU a lot.

Now let's think about the speed of light:

Speed of Light = 300,000,000 meters / second
= 3 x 10⁸ m/s

1 kilometer = 1000 meter

= 3 x 10⁸ km/s
1000
= 3 x 10⁵ km/s
= 3 x 10⁵ x 60 x 60 km/hour
= 1 x 10⁹ km/hour = 1 billion km/hour
(roughly half a billion miles-per-hour)

Next, we need to convert SPEED to DISTANCE:

You know that

SPEED = DISTANCE meters = m / s
TIME seconds

So, multiplying both sides by TIME we can get

DISTANCE = SPEED x TIME
(meters / sec) x (sec) = meters

Distance travelled by light in 1 year
= (speed of light) x (seconds in a year)
= (3 x 10⁸ m / s) x (60 x 60 x 24 x 365 s)
= 3 x 10⁸x 3 x 10⁷ meters
= 9 x 10¹⁵ meters
= 1 Light-year

Remember, a Light-Year (LY) is a unit of DISTANCE (NOT of time!!)

(10) The nearest star Alpha-Centauri is 4.3 Light-Years away. How many kilometers is this?

(11) If we look up at Alpha-Centauri tonight, how long ago did the light that is reaching our eyes today leave Alpha-Centauri?

Finally, we can take DISTANCE and SPEED to get TIME:

TIME = DISTANCE meters         = seconds
           SPEED meters/second
Time for Light to travel between Sun and Earth
= Distance between Sun and Earth
                Speed of Light
=     1.5 x 10¹¹m
        3 x 10⁸ m / s
= 15 x 10¹⁰x 10^-8 seconds
    3
= 5 x 10² seconds = 500 seconds
= 8 minutes

This means we can use a distance scale for the solar system of

1 Astronomical Unit = 8 Light-minutes

(12) As we shall be discussing later, radio signals travel at the same speed as light. The Galileo spacecraft sends data back to the Earth via radio signals. When Jupiter and Earth are closest to each other in their orbits around the Sun they are 4.2 AU apart. How long does a radio signal take to travel from the Galileo spacecraft to Earth?

Model answers to the comprehension questions.