Lab 1: Scale of the Universe

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This lab is meant to give us a basic understanding of the scale of the Universe and the objects within it.

Part I: Introduction

Section I: Scale Factor On Earth we understand scales when we think about objects that are big and small.

A basketball is bigger than a ping ball. How much smaller is a ping pong ball than a basketball? If we increase the size of the ping pong ball to the size of the basketball and increased the size of the basketball by the same amount how big would it be and what could we use to represent it? Let’s use this as an example.

Diameter of basketball ' 24 cm Diameter of ping pong ball ' 4 cm

To figure out how much smaller the ping pong ball is than a basketball, we need to take a ratio of the sizes of the two objects. The resultant number is called the scale factor.

diameter of bigger object

diameter of smaller object = scale factor (1)

Using this equation for our basketball and ping pong ball we get something that looks like this:

diameter of basketball

diameter of ping pong ball =

24 ��cm

4 ��cm = 6 (2)

Pay attention to the units here! When you have the the same unit on the top and bottom of the ratio they cancel each other out giving us a number with no unit. What does this mean?

diameter of bigger object = 6× diameter of small object

The scale factor “6” times the diameter of the ping pong ball is equal to the size of the basketball! We can get this by multiplying the scale factor by the size of the smaller object. (This works in the opposite direction as well, if we want to shrink the basketball to the size of the ping pong ball we divide by the scale factor.)

If we were to increase the size of the basketball by that same amount how big would it be? Multiply the scale factor by the diameter of the basketball.

scale factor× diameter of basketball = 6× 24cm = 144cm

How big is 144 centimeters? That can be hard to visualize. The next section will go over how to convert to a different unit that will hopefully help in understanding

Section II: Scientific Notation and Unit Conversions

In astronomy we deal with extremely large numbers. For instance, the mass of the Earth can be written as 6 million million million million kilograms (kg). More awkwardly, we could write this number as 6,000,000,000,000,000,000,000,000 kg (6 followed by 24 zeroes). Scientific notation saves us from this mess by succinctly expressing the same number as 6× 1024 kg.

Conversely, you can write a decimal as 0.0000000000000006 kg (with 15 zeroes after the decimal) and 6 × 10−16 kg in scientific notation (notice the negative sign in the exponent).

*Count how many places you move the decimal so that it is placed after the first number. If you move the decimal to the left you will get a positive exponent and if you move the decimal to the right, you’ll get a negative exponent. *

A few more examples:

4, 200, 000, 000km = 4.2× 109 57, 400, 000, 000, 000m = 5.74× 1013m

0.000000042km = 4.2× 10−8 0.00000000000574m = 5.74× 10−12m

Let’s practice writing the following numbers in scientific notation.

300 = 0.3 =

14,000 = 0.00034 =

Metric units work well with scientific calculations because each unit can be expressed as another unit multiplied by a power of ten. For example, to express kilometers (km) in units of meters (m), you simply multiply by 1000, or 103 since there are 1000 meters in a kilometer. Below are some unit relations:

• 1000 m = 1 km

• 100 cm = 1 m

• 10 mm = 1 cm

We now can use the above relations to convert our new, larger basketball of 144 cm from one unit to another. Say we want to convert from centimeters (cm) to units of meters (m):

144 cm = 144 ��cm

1 × 1 m

100 ��cm = 1.44 m

A meter is a similar distance to a yard, so when we multiplied our scale factor times the diameter of the basketball we get something that’s close to 4 feet! This could be something like a bean bag chair!

Notice that we wanted to use the conversion factor with cm in the denominator in order to cancel with the cm we started with (144 cm).

To write 144 cm in terms of km, we just continue the chain started above:

1.44 m = 1.44 ��m

1 × 1 km

1000 ��m = 0.00144 km = 1.44× 10−3 km

A Note on using Scientific Notation: The previous example divides three numbers: 144, 1000, and 100. Instead of multiplying this out the long way, rewrite 100 cm as 102

cm and 1000 as 103 cm (just count the zeroes to get the exponent).

The rules of exponents state when multiplying two numbers with exponents that share the same base (the base is 10 in this case), their exponents add.

102 × 103 = 102+3 = 105

Likewise, if you’re dividing two numbers with exponents that share the same base, their exponents subtract, bottom from top.

103

102 = 103−2 = 101 = 10

Practice: Now try for yourself and write 2 km in units of mm without using a calculator. Don’t forget units and show all of your work.

Part II: Sizes of objects and the scale of the Universe

Section I: Objects in the Universe

Before going into the scales of different objects in the universe, let’s talk a little more about what some of these objects are. One object that we see often is the Moon. The Earth has one Moon while some of the larger planets in our solar system have more moons. We can define a moon as an object that orbits (goes around) a planet and they are generally smaller than their host planet. Check out the image below to see the true size and scale of the Earth and Moon.

Figure 1: Image Source: http://photojournal.jpl.nasa.gov/cgi-bin/PIAGenCatalogPage- .pl?PIA00559, Image credit: NASA/Jet Propulsion Laboratory/Arizona State University

A planet is another object in space. The Earth is a rocky planet and is fairly small compared to the gas giant planets farther out in our solar system. In order to be a planet, an object needs to fulfill certain requirements. One major requirement is that a planet needs to be small enough to not burn its material like a star. Jupiter is the largest planet in our solar system and we can call it a planet because it is not creating energy like the way our Sun is.

The Sun is the largest object in our solar system and produces a lot of energy through fusion. The Sun is a star and a star can be a wide range of sizes. Almost every bright point you see in the night sky is a star and is producing energy like our Sun. The image below shows the sizes of the planets in our Solar System to scale. The distances are NOT to scale.

Figure 2: Image Source: NASA/Lunar and Planetary Institute

Our star is one of millions of stars within the Milky Way Galaxy. A galaxy is a collection of stars (stars that may have planets orbiting them), gas and dust that orbit around a central point, held together by gravity. Our Milky Way galaxy is one of billions of galaxies in the Universe. Figure 3 below is a picture of our closest spiral galaxy neighbor, M31, also known as The Andromeda Galaxy.

Figure 3: Image Source: Wikipedia

The image below was taken by the Hubble Space Telescope. It focused and stared at one tiny part of the sky and saw thousands of these galaxies.

Figure 4: Image Source: https://spacetelescope.org/images/heic0406a/

Section II: Scaling the Universe

Now that we have all of that in our tool box, we can more easily understand the scale of our universe. Let’s start with scaling down the Sun to the size of a small watermelon. You should not need a calculator!

Diameter of Sun ' 1.4× 109 m Diameter of Watermelon ' 14 cm

Question 1: Find the scale factor! How much smaller is the watermelon to the real Sun?

First we need to notice the units are not the same. We need to convert one to match the unit of the other. Let’s convert 14 cm to m.

diameter of watermelon = 14cm = 14 ��cm

1 × 1 m

100 ��cm =

Now we can plug the diameter of the Sun and diameter of the watermelon into our ratio below (hint: it will help to put your diameter of the watermelon in scientific notation):

Scale Factor = diameter of bigger object

diameter of smaller object =

1.4× 109 m =

We now have a scale factor! This means we can now shrink any astronomical object to scale! Make sure your answer matches with the scale factor calculated by your instructor: Scale Factor = 1× 1010 or just 1010.

Question 2: Shrink the Earth by the same scale as if the Sun were the size of a small watermelon (hint: we are shrinking something, so we need to divide by our scale factor). Show your work!

Diameter of Earth ' 13, 000 km

If your answer is in kilometers (km) convert to millimeters (mm). What object around your house could represent the Earth in this model?

Question 3: How far away should your model of the Earth be from the watermelon model of the Sun? Convert your answer to meters (hint: you can again use our scale factor to shrink the real distance from the Earth to the Sun).

Earth-Sun distance ' 150 million km = 1.5× 108 km

Can you fit our scale model inside your house? Explain.

Question 4: Now shrink the Solar System to fit our scale model.

Diameter of our Solar System ' 300 billion km = 3× 1011 km

Use the internet to convert your answer to miles. Is this bigger or smaller than the city of San Francisco? Explain your answer.

Question 5: Finally, shrink the Milky Way Galaxy down to fit our scale model and again use the internet to convert your answer from km to miles.

Diameter of the Milky Way Galaxy ' 1× 1018 km

How big is this? Is it bigger or smaller than the Earth? Explain your answer.