The Sizes of Things in Astronomy

One of the more difficult concepts for students of  Astronomy 110 to comprehend is the great range in sizes of the many different types of objects in our Universe. As we wind our way through the semester, we will talk about the smallest subatomic particles, and we will discuss the largest object--the entire Universe. Before we start on this path, however, it is important for you to become comfortable with the way we will express very large and very small numbers using something called "scientific notation".

To start this exercise, let's look at some familiar numbers: ten, one hundred, and one thousand.

The number ten is written as: 10
The number one hundred is written as: 100
The number one thousand is written as: 1,000

All three of these numbers are multiples of 10:

10 = 1 X 10
100 = 10 X 10
1,000 = 10 X 100 = 10 X 10 X 10

Back in high school you learned about the squares of numbers, for example 2 = 2 X 2 = 4. Here are some other examples:

3 = 3 X 3 = 9
4 = 4 X 4 = 16
5 = 5 X 5 = 25
10 = 10 X 10 = 100
etc.

The superscript two in these expressions is called the "exponent". The exponent can actually be any real number, but in this class we will be dealing with integers. Here are some other examples of integer exponents:

3 = 3 X 3  X 3 = 27
10 = 10 X 10 X 10 = 1,000
4 = 4 X 4 X 4 X 4 =  256
5 = 5 X 5 X 5 X 5 X 5 X 5 = 15,625

Now we see that we can express 1,000 as 103   and 100 as 102. How do we express 10? This way: 101.  Scientific notation breaks every number down to two components, one of which is a power of 10 (that is 10 with an exponent). For example, let's look at the number 200:

200 = 2 X 10 X 10 = 2 X 102

Here are some other numbers:

500 = 5 X 10 X 10 = 5 X 102
3,000 = 3 X 10 X 10 X 10 = 3 X 103
6,000,000 = 6 X 10 X 10 X 10 X 10 X 10 X 10 = 6 X 106  (six million)

Ok, we have been dealing with very easy numbers. What about a number like 3,100? Well, simply break this number down:

3,100 = 310 X 10 = 31 X 10 X 10 = 31 X 102      .... but note that there is still another power of 10 we could remove:

3,100 = 310 X 10 = 31 X 10 X 10 = 3.1 X 10 X 10 X 10 = 3.1 X 103    because 3.1 X 10 = 31! Here are some other examples:

250 = 2.5 X 102
4,300,000 = 4.3 X 106
5,123 = 5.123 X 103

Ok, now we can handle the big numbers we are going to find in Astronomy 110, how about very small numbers? This is a little more confusing. Let's start with some easy examples: 0.1, 0.01, and 0.001. That is 1/10, 1/100, and 1/1,000:

0.1 = 1/10
0.01 = 1/100
0.001 = 1/1,000

Just like earlier, we can break-up the denominator (the bottom number of a fraction) into powers of 10:

0.1 = 1/(10)
0.01 = 1/(10 X 10)
0.001 = 1/(10 X 10 X 10)

The scientific notation way of expressing these numbers is through the use of exponents that are negative numbers:

0.1 = 1/(10) = 10-1
0.01 = 1/(10 X 10) = 10-2
0.001 = 1/(10 X 10 X 10) = 10-3

Alright, how do we express small numbers that are not even powers of 10? Just like their big cousins:

0.2 = 2 X 1/(10) = 2 X 10-1
0.013 = 13 X 1/(10 X 10 X 10) = 1.3 X 10 X 1/(10 X 10 X 10) = 1.3 X 10-2
0.0001587 = 1.587 X 10-4

That is all there is to scientific notation. Now, why do we need it? Because the numbers in astronomy span such a large range that it gets difficult to express them. For example the Sun is located 93,000,000 miles from Earth, and this is actually a very small distance compared to most of the distances we will encounter in astronomy. For example, the nearest star is located about 12,000,000,000,000 miles away!  It is very cumbersome to write down all those zeroes, so we would use scientific notation to say that the Sun is located 9.3 X 107miles away from the Earth, while the nearest star is 1.2 X 1013 miles from Earth. Soon, we will find that we are too lazy to even write down these numbers, and will switch to measuring such distances using other "units" (such as "light years"). Speaking of "units", we next need to talk about the metric system.

Metric System

Scientists exclusively use the metric system. The reason for this is that it is easy! The main unit of distance in the metric system is the "meter". In the English system you are familiar with, there are a variety of distance units, including inches, feet, and miles. Remember that a mile has 5,280 feet, and one foot has 12 inches, so one mile has 63,360 inches. Very hard to convert from one unit to the other. Not so in the metric system, as the bigger and smaller units are based on powers of ten:

1 mm = 1 millimeter = 1/1000 of a meter = 10-3 meters (or 1 meter = 1,000 mm)
1 cm  = 1 centimeter = 1/100 of a meter  = 10-2 meters (or 1 meter = 100 cm)
1 km = 1 kilometer = 1,000 meters = 103  meters

So, how many centimeters are in a kilometer?: 100 cm/m X 1,000 = 105  cm = 1 km

While there are a number of other named length/distance units in the metric system, we will mostly be using the mm, cm, m, and km. It is relatively straight forward (using a calculator) to convert between the English system you commonly use and the metric system with these simple conversion factors:

1 inch = 25.4 mm
1 foot =  304.8 mm = 0.3048 m
1 mile = 1,609.34 m = 1.609 km

The other metric system unit of interest to us for astronomy is the metric unit for mass, the "kilogram", which is 1,000 grams. In the English system we commonly associate "pounds" with kilograms, but this is not really correct. The pound is a unit of force caused by the pull of the Earth's gravity. Thus, while we normally say one kilogram = 2.2 lbs, this is not true everywhere! For example, you may have heard of "weightless" astronauts in Earth orbit, or that an astronaut only weighs 34 lbs on the Moon. But the mass of an astronaut does not change no matter what planet or moon he is on, or even if drifting out in space. I know this is confusing, but it is an important concept. We will talk about the masses of objects in this class, and not their "weight". For example, the mass of the Sun is 2 X 1030  kg, but to measure the weight of the Sun in pounds we need to put the Sun on a bathroom scale that is located on the surface of the Earth!

A Trip through the Universe

We are about to head off on a voyage of discovery this semester that will take us very far away and, as we will soon find out, very far back in time. To use what we have learned today, let's hop into our car and head out into the Universe to see where we are going, and to appreciate the size of the Universe in which we live. Let's say there is no speed limit and we can drive at 125 mph. 125 mph is equal to 200 kph. New Mexico is about 450 km from top to bottom. At 200 kph, it would take 2.3 hours to drive from Raton to El Paso (if the road was straight, and had no traffic!). If we wanted to drive our car to Washington, D.C, which is located about 3,200 km (2,000 miles)  from Las Cruces, it would take 16 hours. If we wanted to drive our car all the way to Santiago, Chile (about 8,800 km from Las Cruces) it would take 44 hours--of course, the road to Chile is not straight, so it would probably take three times longer!

This is too slow, let's switch to a jet. A typical jet airliner travels at a speed of 1,000 kph. Thus it would only take 8.8 hrs to fly from El Paso to Santiago. Let's now fly our jet to the Moon (of course this is impossible with a normal jet--but pretend anyway). The Moon is 384,000 km from the Earth. It would take our jet 384 hours, or 16 days to fly to the Moon. Mars is one of the Earth's closest neighbors in the solar system, and can get as close as 35 million km (like summer, 2003). Let's hop in the interplanetary jet and take a trip to Mars. We better bring lots to eat, as this trip will take us 35,000 hrs, or 1,458 days = 4 yrs! Mars will have made two trips around the solar system in that time. Pluto is about 25 times further away, so it would take our jet 100 years to get to Pluto.

We must go faster. Ok, let's attach some big rocket engines to our jet. Using those on the space shuttle, we should be able to get to about 20,000 kph. Now a trip to Mars only takes 73 days, and Pluto is a mere 5 years away! Let's go to the nearest star. We already stated that it is 1.2 X 1013 miles from Earth, or 1.9 X 1013  km. With our new super-duper rocket jet traveling at 20,000 kph, it will take us 109,589 yrs to get to Alpha Centauri. The center of our Milky Way galaxy is 6,000 times further away than Alpha Centauri, it would take 657 million years (6.57 X 106 yr) to get to the center of the Milky Way in our rocket jet.

This is too slow! We need to go faster. Let's go at the fastest speed possible, the speed of light: 3.0 X 105 km/s (note that we will find out this semester that it is impossible to travel this fast, but it is ok for our fantasy trip). Now it would only take 4 yrs to travel to Alpha Centauri (we say that Alpha Centauri is 4 light-years away.... one of those new units we will encounter later this semester!). The center of our galaxy is 26,000 light years away, so it would take us 26,000 years to get there! The nearest big galaxy like the Milky Way (the "Andromeda Galaxy") is more than 2 million light years away. Thus, even traveling at the fastest speed possible (at least in our Universe), it would take two million years to get to the nearest galaxy like our own. Since it takes two million years for light to travel from Andromeda to here, we see the Andromeda galaxy in the distant past, when the ancestors of human beings were still roaming the plains of Africa. We now know of galaxies in our Universe that are thousands of times further away than the Andromeda galaxy. We are going on a long trip, both in distance and in time!