Scientific Notation & Units

Introduction

Adapted from http://casswww.ucsd.edu/physics/ph7/Units.htmli (H.E. Smith)

Since Astronomy is one of the oldest sciences, it has developed many traditions over the years. While an astronomer would not hesitate for an instant to apply cutting-edge technology at the telescope or computer, the language and units of measurement in the field change much more slowly. So, as you set out to investigate the Universe with us, let us introduce you to a few words, units and short-hand that you'll encounter here.

Scientific Notation

As we have seen in class, the important numbers in Astronomy span almost 40 orders of magnitude in size. Consider the mass of the Sun:

MSun = 1,989,000,000,000,000,000,000,000,000,000,000 grams

It's cumbersome, to say the least, having to write out all of those zeros. Even kilograms (eliminate 3 zeros) or metric tons (eliminate 6 zeros) don't help much. Furthermore, we really don't know the Sun's mass beyond the accuracy of the fourth digit. All those zeros are just place-keepers, carrying no useful information. For this reason, scientists use a short-hand called Scientific Notation to express very large or very small numbers.
In scientific notation the Sun's mass becomes:
MSun = 1.989 x 10 33 gm.

The number above the ten, called the power of ten or exponent, stands for the number of decimal places. If it is positive, as in the mass of the Sun, the decimal places are in front ofthe decimal point. So, 1033 means "move the decimal point 33 places to the right and fill the empty places with zeros" (or, more mathematically, multiply by ten 33 times).

For very small numbers, such as the mass of the proton,

Mp+ = 0.000000000000000000000001673 grams

we use negative powers of 10. The mass of the proton becomes
Mp+ = 1.673 x 10 -24 grams

For negative exponents, the powers of 10 are after the decimal point; 10-24 means "move the decimal point 24 places to the left and fill in with zeros" (or divide by ten 24 times).

In both of the examples above, the coefficient (the part before the times sign), contains 4 digits. This means that there are 4 significant figures in the number. If, for example, we knew the Sun's mass to 6 significant figures, we would say that its mass is 1.9890033 grams.

There are several good web pages about Scientific Notation. If you would like to read a bit more, try out the University of Maryland's Astronomy Programs site, with a Scientific Notation Exercise and an Astronomical Distance Calculator.

Units

Adapted from http://casswww.ucsd.edu/physics/ph7/Units.htmli (H.E. Smith)

A centimeter is pretty small - not a very practical unit for the enormous distances in the Universe. If Astronomer A had to use centimeters to tell Astronomer B the distance to the Sun, it would look like this:

14,959,850,000,000 cm

There are three special units of distance used by astronomers. These are the astronomical unit (AU), the light-year and the parsec. The astronomical unit is the average distance of the Earth from the Sun shown above.

1 AU = 1.5 x 1013 cm = 150 million km = 93 million miles = 8.3 "light-minutes"

A light-year (ly) sounds like a measure of time, but it is a length - the distance light travels in one year.(We can use a light-year as a unit of measure because ALL light travels at the same speed; it is a fundamental constant of the Universe. More about this later...) So, in one year, light travels:

The name parsec comes from the technique of measuring distance called parallax, and will be introduced later.


Arithmetic Using Scientific Notation

From http://www.astro.lsa.umich.edu/Course/Bernstein102/Scinot/scinot.html

4.3 x 106 x 2 x 102 = 8.6 x 108

4.3 x 106 x 2 x 10-2 = 8.6 x 104

4.2 x 106 2 x 102 = 2.1 x 104

4.2 x 106 2 x 10-2 = 2.1 x 108

42 x 106 = 4.2 x 107

4200 x 106 = 4.2 x 109

42 x 10-6 = 4.2 x 10-5

0.42 x 106 = 4.2 x 105

0.000043 x 106 = 4.3 x 101

0.42 x 10-6 = 4.2 x 10-7

You should always adjust the decimal place in the coefficient so that the coefficient is always greater than one but less than ten. Mathematically it doesn't make any difference, but that is the standard practice, and it does make a number easier to read.

4.2 x 106 + 6.4 x 105 = 4.2 x 106 + 0.64 x 106 = 4.84 x 106

4.2 x 10-6 + 6.4 x 10-5 = 0.42 x 10-5 + 6.4 x 10-5 = 6.82 x 10-5

9.2 x 1011 + 9.4 x 1010 = 9.2 x 1011 + 0.94 x 1011 = 10.14 x 1011 = 1.014 x 1012

4.2 x 106 - 6.4 x 105 = 4.2 x 106 - 0.64 x 106 = 3.56 x 106

4.2 x 10-6 - 6.4 x 10-5 = 0.42 x 10-5 - 6.4 x 10-5 = -6.38 x 10-5

1.2 x 1011 - 9.4 x 1010 = 1.2 x 1011 + 0.94 x 1011 = 0.26 x 1011 = 2.6 x 1010