Units and Dimensions are very important when we are working with numbers. We need to make sure that our numbers have the correct units and dimensions before we divide them or multiply them.
A number without units doesn't tell you very much. If you went shopping in Japan you may see the price tag on a shirt read 1500.00 and you might freak out. That's because the store did not put the units of Yen (¥) on the tag even though the cost of the shirt still had the dimensions of money. So, put units on your answers so that nobody freaks out! To help you understand the difference between the concepts of units and dimensions, think about the following.
A person usually eats three meals per day. Meals per day are the units on the number 3. The dimensions are food per unit time. We could also say a person eats 21 meals per week. Again we have the dimensions of food per unit time, but with different units. The dimensions that you will be become very familiar with during the semester are:

In this course, we will be mostly working with the metric system.

Check out how one can express
Darth Vader's height with different units:

 

 

 

Meters | Yards | Feet
CHOOSE A UNIT OF LENGTH
 

Distances and Angles

Look at the following animation

You can see that if we measure the distance from A to B along the straight line we would get a different measurement than if we measured from A to B along the perimeter of the circle. But notice that the angular measurement, C, is the same in both cases. When it comes to objects in the sky, the angle C is all we can measure directly.

 

 

 

 

Units for Angles

We have inherited through the Greeks a sexagesimal (60-based) system of measuring angles and time. We know that there are 60 minutes of time in 1 hour and 60 seconds of time in 1 minute.

Note the similarities in the angular system of measurement:

 1 degree = 1° = 1/360th of a circle = .017 radians

60 minutes of arc (arcminutes) = 60' = 1°

60 seconds of arc (arcseconds) = 60" = 1'

 Radians are simply another unit of angular measurement. There are 360 degrees in a circle. There are 6.28, or 2π, radians in a circle.

Now to get a better idea of what 1 second of arc looks like, imagine a tennis ball seen at a distance of about 8 miles. The tennis ball subtends 1 second of arc. The word "subtends" refers to the angle covered by an object.

Another useful approximate conversion for future labs is

1 radian = 206,000 arcseconds

A useful rule of thumb (literally!) is that your thumbnail, held at arm's length, subtends about 1 degree. The disks of the Sun and Moon each subtend ½ a degree.

Likewise, if something is moving across the sky, if we don't know how far away it is, all we can measure is its changing angle. For instance, you could measure how long it takes for the moon to move one moon diameter (or 0.5°) across the sky. Such "angular speeds" we might measure in degrees per hour, or arcsec per year, and astronomers frequently need to measure them. In the Parallax lab, we'll see how to relate the angle something subtends to its actual size (in km, say) and its distance from us.

Temperature


In the metric system, we measure temperature on the Celsius scale, which is defined such that the freezing point of water is 0°C and the boiling point is 100°C. The conversion from familiar Fahrenheit temperatures to Celsius is:

 T°C = 5 / 9 x (T°F - 32°F)

So that 68 ° F (room temperature) is 20 ° C. Even more fundamental is the Kelvin (K) temperature scale. The conversion from Celsius is easy:

 T (K) = T°C + 273°C

A temperature of 0 K, or -273 ° C, is called absolute zero, and is the temperature at which all motion comes to a stop. Although an ice cube in your freezer may not look like it is moving, the water molecules in it are still jiggling around because the temperature is above absolute zero.
 
Working with Numbers
Now that we know more about units, we need to know how to work with numbers that have units. Look at the following example:

Problem

    An astronomy tutor meets with students in 15 minute sessions. She is paid $10.00 per hour. If she teaches 25 sessions, how much money will she make?

Solution

    First, we need to figure out how many minutes 25 sessions is:

     25 sessions x 15 minutes / 1 session = 375 minutes
    Next, we need to determine how many hours 375 minutes is:

     375 minutes x 1 hour / 60 minutes = 6.25 hours
    Finally, we can determine how much money the tutor made:

     6.25 hours x $10.00 /1 hour = $62.50

    With each calculation above, we changed the units into something we would work with. We converted sessions into minutes. Then we converted minutes into hours. And finally hours into dollars by using her hourly pay rate.
 

Prefixes for Powers of Ten
You may have seen the Powers of Ten film in this class or your lecture class. Below is a table which summarizes the information in the video. On the left is the object, and to the right is the scale of the object. We are just looking at the order of magnitude of the object. For example, Darth Vader is on the order of 1 meter tall. Order of magnitude really means "the closest power of ten". Thus 2 meters is closer to 1 meter than it is to 10 meters (the latter is the second closest power of ten). Pay attention to how the scale changes between objects.
O B J E C T
S C A L E (Order of Magnitude)
Darth Vader 1e0 m or 1 m
The Earth 1e7 m
Jupiter 1e8 m
the Sun 1e9 m
Inner Solar System 1e11 m
Outer solar System 1e12 m
Star Cluster 1e16 m
Galaxy 1e20 m
Cluster of Galaxies
 1e24 m
You should become familiar with the different size scales in astronomy. The Powers of Ten film and table are good tools for learning the orders of magnitude of the different size scales. However, the orders of magnitude may also be expressed as a units prefix rather than as an exponent.

When different prefixes are put in front of a "root" unit, then the value of unit changes. The most familiar prefix is probably "kilo." Kilometer and kilogram are two common uses of this prefix. A kilometer is equal to 1000 meters, and a kilogram is equal to 1000 grams - so "kilo" means a thousand. The two root units are "meter" and "gram." The following is a table of many of the prefixes that we will be using during the semester:
P R E F I X
O R D E R   O F
M A G N I TU D E
E X A M P L E S
milli (m) 10 -3 or 0.001 milliliter or mL
centi (c) 10 -2 or 0.01 centimeter or cm
kilo (k) 10 3 or 1000 kilogram or kg
mega (M) 10 6 or 1,000,000 Megabytes or MB
giga (G) 10 9 or 1,000,000,000 GigaWatts or GW
Notice that capitalizing a prefix makes a big difference!  A lowercase "m" is the symbol for "milli" while an uppercase "M" is the symbol for "mega".

Problem

    How many lightyears are in 14 Megalightyears?

Solution

    There are fourteen million lightyears in 14 Megalightyears.

    This may seem obvious now but as a simple, common sense sanity check during the semester just think about the math before you do it. For instance, you know in advance that there should be more lightyears than Megalightyears in your answer because lightyears are smaller than Megalightyears - just like on the last page since feet are smaller than meters you had to stack more "feet" blocks than "meter" blocks to express Darth Vader's height. Just keep your head screwed on straight and you won't go circling an answer like

    14 Megalightyears = 0.000014 lightyears     WRONG!