LAB #1
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111
Subject
Astronomy
Date
Dec 6, 2023
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AST 111: Lab #1
Introduction to Solar System Astronomy
Objectives
To become familiar with the proper usage of scientific notation and
common astronomical units
To use ratios and small objects to model the scale of the solar system
References
Conceptual Astrophysics (1
st
edition),
Sirola
Materials
Calculator
Tape measures (metric)
Softball
Marbles
Pinheads
Introduction
Astronomy, including solar system astronomy, is the science of the very
big (and perhaps surprisingly, sometimes the science of the very small). Because
of this, people often have trouble imagining the size and scale of astronomical
objects and systems.
Astronomers approach these difficulties in a variety of ways. We will show
how to express “extreme” numbers in more user
-friendly manners and to use
more appropriate units of measurement. It also helps many students to see just
how various large numbers relate to each other
–
as an example, we will create a
scale model of the solar system by using a softball to represent the Sun.
Activity #1: Scientific Notation
Very large or very small numbers may be represented in a compact form
known as
scientific notation.
Scientific notation uses our decimal (i.e. base 10)
numbering system to make large and/or small numbers more manageable.
Consider a few examples:
2
The mass of the planet Earth, in kilograms
1
, is
5,970,000,000,000,000,000,000,000.
If we had to write this number out every time we needed it, we would
quickly get wrist cramps! More importantly, it impedes rather than furthers
understanding
–
even professional astronomers struggle to make sense of
numbers this large. Let’s use scientific notation to
better interpret this number.
First, notice that we have written the decimal point at the end of the
number. This usually isn’t done, but here we want it to stand out.
Move the decimal point until it is behind the first non-zero digit of the
number. In this case, that number is 5. Next, count up the number of columns the
decimal point moved; here, the decimal point moved 24 places. Now we can
restate the number in scientific notation:
24
10
97
.
5
The “24” is called the
exponent.
If we write out 10
24
in long form, we would
see that it is equal to 1,000,000,000,000,000,000,000,000, or a 1 followed by 24
zeroes. Scientific notation has recast the number as
24
10
97
.
5
, or “five point nine
seven times ten to the twenty-
fourth power”. A
lso, as an important aside, note we
only kept the non-zero digits in front
–
5.97, not 5.970 or 5.9700 etc. The zeroes
in the long-form number are merely spacers; they represent columns of ten, not
actual numbers. In general, we need only keep the first three or four digits of a
number.
The above was an example of a very large number. What about a very
small number? For example, the mass of an electron, again in kilograms, is
0.000 000 000 000 000 000 000 000 000 000 911
which is a tiny number indeed! Le
t’s apply scientific notation to this
number.
Move the decimal point behind the first non-zero digit as we did earlier.
For this example, the decimal point moves 31 places and the first non-zero digit
is 9. Because the number is less than one
–
or, equivalently, the decimal point
moved to the right instead of the left
–
the exponent is negative. Our new
representation of the number is now
31
10
11
.
9
It is clear that we have gained a great deal in clarity and understanding by
rewriting these numbers in scientific notation.
1
A kilogram is equivalent to 2.2 pounds on the surface of the Earth. The comparison is complicated
because kilograms are units of mass (the amount of “stuff” an object has) and pounds are units of weight,
which depends on mass and gravity both.
3
Below in Table 1-1 are examples of numbers you might encounter in
astronomy. For a number given in long form, write its equivalent in scientific
notation. For a number given in scientific notation, write its equivalent in long
form.
Table 1-1. Numbers in Long Form and Scientific Notation.
Activity #2: Units of Measurement
Even when numbers can be expressed in the more compact form of
scientific notation, they still may not mean much to a reader. The distance from
the Earth to the Sun is a good example:
km
000
,
700
,
149
km
10
497
.
1
8
Rounding off, the Earth is about 150 million kilometers (or 93 million miles)
from the Sun. To put this in perspective, consider a car owner who puts 15,000
km (or 9300 miles) on the car’s odometer each year. How many years would it
take to drive 150 million kilometers? Write your answer below.
150 million km / (15,000 km per year) = __________ years
One way to deal with unwieldy numbers is to change how the numbers are
expressed
–
in other words, to change the units. The most important unit in solar
system astronomy is the
Astronomical Unit
, or the
AU
, defined as the average
distance of the Earth from the Sun:
km
10
497
.
1
AU
1
8
This allows astronomers to make better sense of the scale of the solar
system. Instead of Earth being about 150 million kilometers from the Sun, we can
say the Earth is 1 AU from the Sun. The planet Mars is 228 million kilometers
from the Sun
–
or 1.52 AU from the Sun… and so on.
1.097
X
10
7
Number in Long Form
Number in Scientific Notation
297,000
5.67
X
10
-8
0.000 000 000 066 7
2.898
X
10
7
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4
How did we determine that Mars is 1.52 AU from the Sun? Below we show
how quantities can be converted from one system of measurement (units) to
another:
AU
52
.
1
km
10
150
AU
1
km
10
228
Sun
the
from
Mars
of
Distance
6
6
For each of the planets, look up its average distance from the Sun in your
textbook in both millions (i.e. 10
6
) of kilometers and in Astronomical Units. Write
these numbers in Table 1-2 below.
Table 1-2. Distances of Planets from the Sun in Different Units.
(a) Which unit is more
appropriate
(for the sake of clarity), the kilometer or
the Astronomical Unit?
(b) Which unit
correctly
expresses the distances?
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
57.9
108.2
149.7
1 (exactly)
Planet
Distance from
Sun (10
6
km)
Distance from
Sun (AU)
228
1.52
778.4
1427
2871
4498
5906
5
Activity #3: The Scale of the Solar System
Although helpful, using scientific notation and appropriately scaled units is
not always enough. The inherent difficulties in relating large numbers to each
other are illustrated by the scale of the solar system itself.
True scale pictures of the solar system
–
where the planets and their
spacings are to the same scale
–
are almost never shown. This is even true of all
astronomy textbooks, including our own. We shall see why upon completing this
activity.
We will construct a true scale model of the solar system. Imagine the Sun
is the size of a softball
–
about 5 centimeters (2 inches) in radius. The actual
radius of the Sun is 696,000 km. We can set up a
scale
by taking the ratio of the
actual radius to the model radius:
cm
5
km
000
,
696
scale
or 1 cm (in the model) is equivalent to 139,200 km (in space). This is exactly the
same notion as (for instance) saying that 1 inch “equals” 100 miles on a map.
(a) Why did we specify that 1 cm applied
in the model
and that 139,200
km applied
in space
?
In order
to find the radius of a planet, or a planet’s average distance from
the Sun, we simply divide the actual value by the scale.
How far is the Earth from the Sun in our model? Recall the actual average
distance from the Earth to the Sun is 149,700,000 km; the model distance from
the Earth to the Sun is therefore
cm
1080
km/cm
200
,
139
km
000
,
700
,
149
We can further modify the answer by changing from centimeters to meters
(divide by 100) to get 10.8 meters.
Table 1-3 lists the major objects in the solar system (the planets, the Sun,
and the Moon). For each object, record its actual radius and actual distance from
the Sun (see your textbook). Then calculate its model radius and model distance
from the Sun. Indicate the units as well as the numerical answers in the table.
Extra notes: the actual radius of the Moon is 1738 km. Also, use the actual
distance from the Moon to the Earth in the table, which is 384,000 km.
6
Table 1-3. Scale Model of the Solar System.
(b)
Alpha Centauri A (“
Cen A” in Table 1
-3) is the largest star in the
nearest solar system to our own. We can model
Cen A with another
softball. How far away from the Sun softball is the
Cen A softball?
Find the model distance in terms of meters, kilometers, and miles.
Show your work below. Note there are 1.609 kilometers per mile.
Sun
Mercury
Venus
Earth
Moon
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Cen A
696,000 km
2440 km
0 km
57.9
x
10
6
km
108.2
x
10
6
km
5 cm
6052 km
6378 km
0 m
150 x 10
6
km
10.8 m
1738 km
3394 km
71,492 km
60,268 km
25,559 km
24,766 km
1137 km
696,000 km
5 cm
4.0 x 10
13
km
5906
x
10
6
km
4498
x
10
6
km
2871
x
10
6
km
1427
x
10
6
km
778.4
x
10
6
km
227.9
x
10
6
km
384,000 km
Object
Name
Radius of Object
- Actual
- Model
Distance from Sun
- Actual
- Model
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7
(c) Space out the three model objects. Is the Moon significantly closer to
(or further from) the Sun than the Earth?
Activity #4: Pacing out the Scale of the Solar System
The lab instructor will assign each lab team an object from Table 1-3.
Each team should check its answers for its model radius & distance with the
instructor. Then each team will retrieve a model object from the instructor to
represent its actual object.
The class will gather outside at the high end of the street near the building;
if weather is bad, the class will gather at the end of the hall next to the door to the
astronomy laboratory. A volunteer will hold the model Sun (a softball) for
everyone to see. Then the distance for each object, starting with Mercury, will be
paced out wi
th the metric tape measure. At each object’s position, its student
team will stop & hold up its object for the others to see. The class will pace out
objects until the end of the street is reached (any remaining student teams will
return to the position of the model Sun).
(a) Why
don’t
textbooks show pictures of the true scale of the solar system
(i.e. for both the sizes of the planets and their distances from the Sun)?
(b) President G. W. Bush once proposed to send a manned mission to
Mars. Detractors say the distances are too great to overcome. Given
what we have learned in this lab regarding the scale of the solar
system, should we go to Mars or not? Briefly defend your answer.
8
Activity #5: The scale of the Earth-Moon-Sun system
As we discussed earlier, it is very difficult to have both astronomical
objects and their distances from each other on the same scale - either the
objects become too small, or the objects are too far apart. Nevertheless, it will be
useful for us to briefly review the scale of the solar system, concentrating on the
relationships between the Earth, Moon, and Sun.
From our previous work, if the Sun is modeled with a softball, than the
Earth-Sun distance is 10.8 meters. Fill in the appropriate spaces in Table 1-4
below from Activity #3:
Table 1-4. Model of the Earth-Moon-Sun System (Overall View).
(a) The instructor has placed several objects at the front that could serve
as models for the Earth and Moon. Which objects would be the best
choices and why?
(b) Does the Moon orbit the Earth or does the Moon orbit the Sun? Briefly
defend
both
answers.
-
Sun
Earth
Moon
696,000 km
6378 km
-
150 x 10
6
km
(from Earth
to Sun)
5 cm
1738 km
10.8 m
384,000 km
(from Earth
to Moon)
Object
Name
Radius of Object
- Actual
- Model
Distance
- Actual
- Model
Softball
Model
Object
9
Now let’s zoom in on the Earth. This time, we will model the Earth with the
softball. To construct this new scale, note the actual radius of the Earth is 6378
km and again assume the softball’s radius is exactly 5 cm:
cm
5
km
6378
scale
__________km/cm
Use this scale to calculate new model radii and distances for the Earth,
Moon & Sun and write the results in Table 1-5:
Table 1-5. Model of the Earth-Moon-Sun System (Close-Up View).
(c) The instructor has placed several objects at the front that could serve
as a model Moon. Which object would be the best choice and why?
(d) Can the Sun now fit in the classroom? Why or why not? List all the
possible reasons for your answer.
-
Sun
Earth
Moon
696,000 km
6378 km
-
150 x 10
6
km
(from Earth
to Sun)
5 cm
1738 km
384,000 km
(from Earth
to Moon)
Object
Name
Radius of Object
- Actual
- Model
Distance
- Actual
- Model
Model
Object
Softball
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10
(e)
How many “Earths” away is the Moon from the Earth? In other words,
if you could place the Earth side by side many times, about how many
Earths would be needed to reach the Moon? Show your work below.
…
Earth
Moon
(f)
Why don’t textbooks show the sizes of the planets and their distances
from the Sun to the same scale?