Astronomy 161: An Introduction to Solar System Astronomy Prof. Richard Pogge, MTWThF 2:30

# Lecture 2: Astronomical Numbers

## Key Ideas:

Scientific Notation
• Express numbers using powers of 10
• Standard Prefixes (kilo-, mega-, etc.)
Metric System of Units
• Units of Length
• Units of Time
• Units of Mass
• Weight versus Mass

## Big Numbers

Astronomical Numbers are, well, Astronomical!

Examples:

Average distance of the Earth from the Sun: 149,597,900 kilometers

Mass of the Sun: 1,989,000,000,000,000,000,000,000,000,000 kilograms

Age of the Earth: 4,550,000,000 years (4.55 Billion years)
Other big numbers, while not "astronomical" for comparison:
Number of OREO cookies sold to date: 490,000,000,000 [source: Nabisco]

US National Debt: \$9,016,288,006,279.21 (as of 2007 Sept 16). [source: U.S. Treasury Department, Bureau of the Public Debt].
(Makes you think we should be talking about "economical" numbers instead of "astronomical"...)

Because the numbers we will encounter in this course range from the very larger to the very small, we need a way of dealing with such numbers sensibly so we don't go crazy counting zero's, risking factor of 10 or greater mistakes at every turn.

## Scientific Notation

Scientific Notation is a compact and convenient way of expressing very large and very small numbers using powers of 10. You've all probably encountered scientific notation before. I hope the examples below are reminders for those who haven't used it in a while. If you need a detailed review, please see Section 1-6 of Kaufmann & Freedman.

Examples of Scientific Notation:

The Mass of the Sun:
1,989,000,000,000,000,000,000,000,000,000 kilograms = 1.989x1030 kilograms

The Size of a Hydrogen Atom:
0.0000000000106 meters = 1.06x10-11 meters

In each case, use of scientific notation eliminates most of the zeros which are just place-holders for factors of ten, letting us concentrate on the significant figures. In a field such as astronomy that deals with scales ranging from subatomic particles to the entire universe, this notation is a great simplification!

## The Metric System

Astronomers use the Metric System exclusively. The basic units of the metric system are:
• Length in Meters
• Mass in Kilograms
• Time in Seconds
The metric system is also known as the International System of Units (or "SI" for "Systeme Internationale"). At present, only the United States, Liberia & Myanmar (aka Burma) have not adopted SI units as their primary system of units. For more information on the SI units, see the SI Units page at the US National Institute of Standards & Technology (NIST).

## Standard Prefixes

In everyday use, we often add a prefix to the base unit to indicate common powers of ten. A brief listing of some of the more common is given below:

 Factor Prefix Examples 103 kilo- kilogram, kilometer, kilobyte 106 mega- megawatt, megayear, megabyte, megaton 109 giga- gigayear, gigaton, gigabyte 1012 tera- terawatt, terabyte 10-2 centi- centimeter 10-3 milli- millimeter, millisecond, milliliter 10-6 micro- microsecond, micron 10-9 nano- nanosecond, nanometer

Common Examples

Length:
1 kilometer = 103 meters (1000 meters)
1 centimeter = 10-2 meters (1/100th of a meter)
1 millimeter = 10-3 meters (1/1000th of a meter)
1 micron = 10-6 meters (short for "micrometer")

Time:
1 nanosecond = 10-9 s (1 billionth of a second)
1 Gigayear = 109 years (1 Billion years)
1 Megayear = 106 years (1 Million years)

## Units of Length

The basic unit of length is the meter (m)
1 ten-millionth the distance from the North Pole to the Equator of the Earth.

Modern Definition:
The distance traveled by light in a vacuum in 1/299792458th of a second.
We will most commonly encounter meters and kilometers.

## Astronomical Units of Length

Meters and kilograms are fine for most terrestrial applications, but when we start talking about the enormous distances between the planets, or between stars and galaxies, we need to define special units to keep the numbers from getting too big. The most important of these for our purposes in this course are:

The Astronomical Unit (AU):

1 AU is the Mean Distance from the Earth to the Sun:

1 AU = 1.496x108 kilometers
The AU is used for expressing the distances between planets.

In round numbers, you can use "1 AU = 150 Million km" for the purposes of this class.

The Light Year (ly):

1 Light Year (ly) is the Distance Traveled by Light in 1 Year:

1 ly = 9.46x1012 kilometers
The light year is used for expressing the distances between stars.

## Space is BIG

For example, what is the distance between the Earth and:
• The Moon: 384,000 kilometers
• The Sun: 1 AU (149,600,000 km)
• Alpha Centauri (nearest star): 4.2 light years (266,000 AU)
• Center of the Milky Way Galaxy: 26,000 light years (1.65x109 AU)
As you can see, if you only use kilometers or meters, the numbers would get out of hand very fast.

## Units of Time

The basic unit of time is the second (s):
1/86400th of the mean solar day.

Modern Definition:
9,192,631,770 oscillations of a 133Cesium atomic clock.
We will usually measure time in units of seconds, minutes, hours, and years.

## Units of Mass

The basic unit of mass is the kilogram (kg):
1 kilogram is the mass of 1 liter of pure water.

Modern Definition:
1 kilogram = mass of the international prototype of the kilogram.
This is a piece of platinum-iridium alloy kept at the International Bureau of Weights & Measures in Sèvres, France.
We will be most often use masses in kilograms.

## Mass versus Weight

Mass and Weight are NOT the same!

Strictly speaking:

Mass is the amount of matter in an object.

Weight is the force of gravity on an object.

Of the two, Mass is the more fundamental quantity.
• Mass is the same everywhere regardless of the strength of the local gravitational field

• Weight depends on the strength of the local gravity field (i.e., it is different on the Earth and Moon for the same mass.)

## Mass and Weight in Everyday Units

Metric:
Mass in kilograms
Weight in Newtons

English Units:
Mass in slugs
Weight in pounds
In commercial and everyday usage, "weight" is synonymous with "mass", and the metric unit of the kilogram is used. In engineering and physics, however, we need to be careful to make the distinction, and so weight takes units of Netwons, and mass units of kilograms.

This double meaning often causes confusion: a subtlety not often appreciated is that the conversion between pounds (weight) and kilograms (mass) that you can find in tables of weights and measures is only strictly true at the surface of the earth (and then only for an assumed mean gravitational force at the sea-level)!

On the Moon or Mars, where the gravity is different at the surface, the conversion is also different!

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Updated: 2007 September 16