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INTRODUCTION

Introduction

In our daily lives we use relationships between angular size of a known object and its distance all the time without really being aware of it. For example an average sized car looks larger when it is closer to our eyes. If the car is half as big as its normal size, it must therefore be twice as far away. This means that it subtends an angle half as large as another car which is in close proximity. The eye can measure only the angular separations or angular sizes of objects; it takes the mind to convert to real physical measurements. In science we want to make this process more systematic and quantitative, and use a simple and practical technique for measuring angles to measuring distances of various objects. In astronomy the distances are huge compared to the distances we are used to. Astronomers need to use different tools to find the distances of far away objects. This leads to what we call the cosmic distance ladder which we will explore in detail in the various lab exercises in this course.We will start with exploring the lower rungs of the distance ladder.

A journey through the universe – So how large is the universe?

In the previous lab course we built a scale model of the solar system. Because we spend most of our time as humans interacting with the world on a scale that is human-sized, things we can touch and see and investigate with our senses, it is easy to miss that we are part of a vast universe with objects both unimaginably large and incredibly small compared to the world of our senses. This short introductory exercise is designed to help you shrink down domains of the astronomical world to a size that will give you a sense of relative proportions and distances. We will explore the realms of earth and moon, sun, solar system, and galaxies by shrinking them in turn down to two inch circles.

In order to shrink each of the realms in a way that is consistent we need to use ratios. Ratios are ways of comparing two elements, in this case usually size of an object and distances between objects.

PRE-LAB ACTIVITY

Pre-Lab Activity

Before starting the lab exercise go to the following URL

http://heasarc.gsfc.nasa.gov/docs/cosmic/

and read about astronomical distances, what they mean with regard to the distances we are used to in our daily lives and how astronomers go about to measure them. At the bottom of the page you will find the entry point to your journey through the cosmic distance ladder. It is labeled “Begin”. After clicking on “Begin” you will see the first window labeled :”Our Earth”. You can choose to read about the earth by clicking on the appropriate link at the bottom or you can click on “Zoom out” and move on to the next object on the distance ladder. The reading also explains the following terms used in the lab handout: Parallax, luminosity, absolute magnitude and apparent magnitude – but you need to find them in the information provided by this NASA website. Remember science is about inquiry and your journey of discovery begins with this first exercise.

The web address below will lead you to a flash scale of the universe. This flash scale shows you how the sizes of the objects we know relate to each other. By moving the slider you can finally determine that we are not the center of the universe.

http://htwins.net/scale/

Distances and Models
Your Name___________________________
Objectives of this lab exercise:
After completing this exercise you should • Understand measuring tools
• Learn how astronomers are able to determine distances to celestial objects
• Understand the vast dimensions of the universe and how to scale those dimensions to build
a model
Part I – Exploring everyday measuring tools [6 points]
Science is all about measuring. One of the most important parts of doing astronomy is measuring
distances, angles, and luminosity of celestial objects. In Astronomy the distances are huge and
unimaginable. You might ask yourself the question “How can you gather information about an
object that is so far away that you cannot reach it? For example, how do we know if a star is far
away, but very bright, or close to Earth and relatively dim? Before we start with exploring
astronomical distances you will investigate how you measure things in everyday life, and what tools
you can use.
1. List at least 10 tools that you use for measuring things in your everyday life.
2. If you were going to measure the area of your room what kind of measuring tools might you
use?
Part II – Measuring distant objects [24 points]
Parallax as explained in the pre-lab activity, is an interesting way of measuring the distance of an
object by how much it appears to move when viewed against a much more distant background from
one location, then another (the distance between the locations is called the baseline) .
3. Print a paper meter stick and tape it to your arm so that your chin can rest on the “0” mark.
With your other hand hold a pencil or pen at the 50 cm mark. Then shut first one eye, then the
other and observe how the pencil moves against objects in the background. Describe what
happens.
4. Move the pencil closer, to the 25 cm mark. What happens to the motion of the pencil now
when you first close one eye and then the other? Can you quantify the difference? Is the
motion apparently twice as much as it was before? Five times as much? Write your answers
below.
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5. Now move the pencil to about the 100 cm end of the paper meter stick or as far as it can get
from your eyes. Again quantify the motion compared to when the pencil was at the 50 cm
position.
6. The pencil at 50 cm is shown below. Explain how the drawing would change for the 100 cm
and 25 cm situations. In particular describe how the parallax angle, the angle between the
two sight lines, changes as the distance of the pencil from the eyes changes. Does the angle
double when the distance doubles? To perform this task use the insert shapes feature in your
word program and draw the appropriate lines in the figure below. Use different colors for the
two cases.
Pencil 50 cm from eyes.
Distant
object
or wall
eyes
7. Parallax can be accurately measured if you have the right tools. What kinds of instruments
would improve your experiment?
8. Parallax is the first step on the ladder to measure distance to the stars. What kind of
baseline do you think astronomers use to measure distance to the nearby stars? List some
possibilities below.
Part III – Stellar magnitudes [10 points]
The apparent brightness of stars is rated on a scale based on that developed by Hipparchus about
150 years B.C.E. That system classified stars from 1 to 6. Stars classified 1 were the brightest in the
sky, 6 were near the limit of what can be observed with the naked eye. The modern scale includes
both much dimmer objects and brighter objects with negative magnitudes. The Sun’s apparent
magnitude is approximately -26.7.
Remember that apparent magnitude is just what we perceive. If your friend shines a flashlight at
you from across the room it will appear brighter than if you see the same flashlight across the
length of a football field. Astronomers use the letter “m” for apparent magnitude and the value of
m in this flashlight example would change depending on where your friend is in relation to you.
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Astronomers define another quantity as well, the absolute magnitude of a star (M) which is what
its magnitude would be if it were at a distance from us of 10 parsecs. The absolute magnitude
can be used to determine how much energy the star radiates – how luminous it is – and is an
important measure of what the star is like. For stars with known distance from parallax
measurements astronomers can figure out both apparent and absolute magnitude. There are other
methods that you will learn later in lab for determining absolute magnitude of some stars. If both
apparent and absolute magnitude are known, it is possible to calculate distances. The
relationship that connects distance (in parsecs, you can read about this measurement in your
astronomy text, but it is about 3.26 light years) and apparent and absolute magnitude is:
D = 10(m-M+5)/5
Use this relationship in the exercises below.
9. The star α-Orionis (Betelgeuse) in the image below has an apparent magnitude of m = 0.45
and an absolute magnitude of M = –5.14.
Find the distance D to Betelgeuse. Betelgeuse is the red star at the left shoulder of Orion (seen
from Earth) and is a red supergiant. When viewed with the naked eye, it has a clear orangered hue:
10. α-Cygni (Deneb) is the upper left star in the Summer Triangle (see photo below) and the main
star in the Swan. Its apparent magnitude is 1.25 and the distance to Deneb is 993 parsec.
Calculate the absolute magnitude for Deneb. What does this tell you about the nature of
Deneb?
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Part IV – How big is the universe?
Realm of the Earth and Moon [30 points]
We will begin by shrinking the Earth to a two – inch diameter circle, then scaling other objects close
to earth to see how they compare.
Earth has a diameter of approximately 12756 km. We will use the metric system measurements for
all astronomical objects today. Since one inch is 2.54 cm, it would be more consistent with the
metric system to think of this a 5 cm universe. However, since the United States still doesn’t use the
metric standard, most of us are used to thinking in terms of inches and feet and have a better idea
of what an inch is (about the length of your thumb from top joint to tip) than what a cm is.
Now do the math. In this realm 2 inches = 12756 km and the Moon is 3475 km in diameter, so how
many inches in diameter would it be in our two inch Earth/Moon system?
𝑺𝒊𝒛𝒆 𝒐𝒇 𝒆𝒂𝒓𝒕𝒉 𝒊𝒏 𝟐 − 𝒊𝒏𝒄𝒉 𝒎𝒐𝒅𝒆𝒍
𝒔𝒊𝒛𝒆 𝒐𝒇 𝑴𝒐𝒐𝒏 𝒊𝒏 𝟐 − 𝒊𝒏𝒄𝒉 𝒎𝒐𝒅𝒆𝒍
=
𝑺𝒊𝒛𝒆 𝒐𝒇 𝒆𝒂𝒓𝒕𝒉 𝒊𝒏 𝒌𝒊𝒍𝒐𝒎𝒆𝒕𝒆𝒓𝒔
𝑺𝒊𝒛𝒆 𝒐𝒇 𝑴𝒐𝒐𝒏 𝒊𝒏 𝒌𝒊𝒍𝒐𝒎𝒆𝒕𝒆𝒓𝒔
or, putting in the numbers
so 𝒙 = 𝟐 (
𝟑𝟒𝟕𝟓
𝟏𝟐𝟕𝟓𝟔
𝟐 𝒊𝒏
𝟏𝟐𝟕𝟓𝟔 𝒌𝒎
=
𝒙
𝟑𝟒𝟕𝟓 𝒌𝒎
) inches (the km units cancel).
In other words the moon’s diameter is about .54 inches, or about ¼ the diameter of the
Earth.
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Do the same process to calculate the distance the Moon is from Earth. In this case x is the distance
of Moon from Earth in the model, but you will now need to use the fact that the Moon is about
384403 km away from Earth on average when measuring from the centers of Earth and Moon.
Now we have:
𝟐 𝒊𝒏𝒄𝒉𝒆𝒔
𝒙
=
𝟏𝟐𝟕𝟓𝟔 𝒌𝒎
𝟑𝟖𝟒𝟒𝟎𝟑 𝒌𝒎
Doing the math we get x = 2 (384403/12756) or 60 inches. Since 1 foot = 12 inches that is about
5 feet away.
Other sizes and distances
In this model, where the Earth is 2 inches across, the Sun is about 18 feet in diameter (imagine a
round, glowing mini-van!) and is 1800 feet, or about 6 football fields away from Earth
Realm of the Sun
The next stage of this exercise is to shrink the Sun down to a 2 inch diameter. The Sun is
approximate 1400000 km in diameter. Write this number in scientific notation.
11. Sun’s diameter in scientific notation:
12. Estimate of Earth’s size and distance from the Sun in this model
13. Earth’s estimated diameter:
14. Earth’s estimated distance from the Sun:
We use the same routine to calculate Earth’s size in this model:
𝟐 𝒊𝒏
𝒙
=
𝟏𝟒𝟎𝟎𝟎𝟎𝟎 𝒌𝒎
𝟏𝟐𝟕𝟓𝟔 𝒌𝒎
Finish finding Earth’s diameter and then distance in this model:
15. Calculated size of Earth in model:
16. Calculated distance of Earth from Sun in this model:
17. Comparison of your estimate and results in this model:
Other objects and distances shrunk to this scale:
•
•
•
•
Earth Size – A grain of salt, with a dust-speck Moon 1/2 inch away from it
Sun-Earth Distance – 20 feet (5.5 m) away
Pluto’s Orbit – 2.5 soccer fields away from the 2-inch Sun
Nearest Star to Sun – 900 miles away (1500 km)
Realm of the Solar System
Although Pluto is no longer considered a planet, we can still use the average distance of Pluto from
the Sun as the boundary for our Solar System. In reality the solar system – the objects that are
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gravitationally controlled by the Sun – extends much farther, but let’s put the boundary for our
purposes today at Pluto.
Now shrink the solar system diameter to the size of a 2 inch circle so that Pluto would orbit around
the edge of the 2 inch circle. Do the appropriate calculations to find the distances and sizes of the
objects listed below.
18. Estimate of Earth’s distance from the Sun:
19. Show steps in calculation of distance of Earth from Sun in the 2 inch solar system model:
20. Comparison of estimate and calculated result:
Other sizes and distances:
•
•
Nearby Star Discovered to Have Orbiting Planets – 5 soccer fields away. Two planets have been
discovered around the star Epsilon Eridani, which is visible from the southern hemisphere.
Our Milky Way Galaxy – Size of North America. At this scale, our 2-inch Solar System is part of a
continent-sized system of 200 billion shining speck stars. These stars, spread 30 miles (50 km)
high, are generally separated from each other by more than 2 soccer fields.
Realm of the Galaxy
Finally take a look at the galaxy realm. Remember that in the 2 in solar system model the galaxy
was about the size of North America (roughly speaking of course!). But when it is re-scaled to the
size of a 2 inch circle we get a better perspective on the size of the universe it, and we, are
embedded in.
•
•
•
•
•
•
Size of Sun and Stars – Individual stars are invisible, smaller than atoms, at this 2-inch scale. The
bright specks in this galaxy image come from the added light of thousands of stars.
Location of Sun – 1/2 inch (about 1 cm) from edge of 2-inch galaxy image
Distance to Andromeda Galaxy, the Nearest Spiral – 5 feet (1.5 m) at this scale – hold the two
galaxy images apart with your arms spread wide.
Distance to Farthest Galaxies Observed by Hubble Telescope – 4 miles (6.5 km). In the Hubble
image of the “Ultra Deep Field” almost all the fuzzy spots of light are distant galaxies. Because
light takes time to travel through space, we see the farthest of these not as they are now, but as
they were 12 billion years ago.
Size of the Whole Universe? – No one knows…it could be infinite.
Light Travel Time – It would take 100,000 years for a beam of light to cross our galaxy and 2.5
million years for light to travel from the Andromeda Galaxy to us.
Conclusion questions [Grade points: 15]
1. The absolute magnitude, M, is defined as the apparent magnitude a star would have if it were
placed 10 parsecs from the Sun. But wouldn’t it be more correct to measure this distance from
the Earth? Does it make a difference whether we measure this distance from the Sun or from
the Earth?
2. Explain how this exercise has impacted your thinking about the size and importance of Earth
in the universe.
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3. Given what you have learned in the 2-inch universe exercise, what do you think the likelihood
is of interstellar travel by humans in spaceships? Explain your reasoning.
Submit this report as an attachment in the assignment drop box on Blackboard.
Before submitting your report, make sure that it is correctly formatted. Do not leave too
much space between answers and make it look professional and neat. Up to 15 additional
points will be given for a correctly formatted report.
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Approximate diameters and distances measured
Object
Distance relative to Sun
in units of 106 km
Sun
Earth
Mercury
Venus
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
1400000
12756
4880
12102
6794
142984
120536
51118
49528
2320
149.6
57.9
108.2
227.9
778.3
1427
2870
4497
5906
Distance of galactic center
relative to Solar System in
light years
Milky Way
center
Moon
ISS
Hubble
telescope
Diameter in km
Diameter of Milky Way Galaxy in light
years
26000
Distance from Earth in km
384000
400
570
100000
Diameter
3475 km
108 m
13.2 m
Additional Information
1 km = 1000 m
1 light-year
= 9460730472580800 metres (exactly)
≈ 9.461 petametres
≈ 5.878625 trillion miles
≈ 63241.077 astronomical units
≈ 0.306601 parsecs

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