Topics in Mathematics:

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Topics in Mathematics Scaling the Cosmos
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Mr. K NASA/GRC/LTP
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Edited by Ruth Petersen
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• Getting Started
• Use a good dictionary and research what it means
  to make a scale drawing or model of something.
• Give an example of a scale model from everyday
  life.
• How does a model compare with the real thing?
• Why are scale models useful?
• How many yards are in 36 feet?
• How did you obtain your answer? Why?

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1. If I divide 36 feet by 3 feet, what answer do I
   get? 12 feet, 12 yards, or just plain 12?
2. If I divide 36 feet by 3 feet/yard, what is the
   correct answer? Why?
3. Write the number 1,234,567,890 in scientific
   notation.
4. Round the answer of Question 9 to three figures.
5. The Sun has a diameter of 1.4X106 km. If I were
   to use the Suns diameter as a UNIT OF MEASURE,
   what should I do?

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• HINTS
• What must I do to the number 1.4X106 km to
  produce the number 1?
• What must I do to all other dimensions I wish to
  scale to this new unit of measure?
• What IS a unit of measure?

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1. Our Cosmic Back Yard
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Have you every looked at the moon and wondered,
How far away is it really?
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Let me ask you that very question Just using
your intuition, draw your best guess at how the
Earth-Moon system would appear from space. Then,
hold up your drawings so that we may see them!
30 Seconds
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Most peoples models look something like this
Is yours similar? What made you choose the scale
you used?
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Now, lets calculate. From a table, I find
Moons diameter 3,500 km Earths diameter
12,800 km Earth-Moon separation 385,000 km
The moon is about 0.25 the size of the earth and
is about 30 earth-diameters away.
30 Seconds
Now Redo your drawings based on this information.
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Your drawing should look more like this
How large would an Apollo spacecraft be on this
scale? Think about Apollo 13!
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Does this result surprise you? Why or why not?
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2. Somewhat Further Out
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Next, lets think about the solar system!
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Here is some data to work with
Suns diameter 1,400,000 km Earths diameter
13,000 km Mercury 3.3 Light Minutes from the
Sun Venus 6 L.M. from the Sun Earth 8.3 L.M.
from the Sun Mars 12.7 L.M. from the
Sun Jupiter 43 L.M.from the Sun Saturn 1.3
Light Hours from the Sun Uranus 2.7 L.H. from
the Sun Neptune 4.2 L.H. from the Sun Pluto
5.5 L.H. from the Sun
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Question How many km are in a L.M.? How many km
are in a L.H.?
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c 300,000 km/sec. Thus 1 L.M. 300,000 km/sec
X ( ? ) km 1 L.H. 300,000 km/sec X ( ?
) km
15 Seconds
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1 L.M. 300,000 km/sec X 60 sec/min km 1.8 X
10 7 km 1 L.H. 300,000 km/sec X 3600 sec/hr km
1.1 X 10 9 km
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Previously, we expressed the Earth-Moon
dimensions in terms of earth-diameters. What
scale shall we choose for a solar system
model? 1 Minute - and explain your choice!
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I chose the Suns diameter as a unit of
length. If we divide all of the given distances
by 1,400,000 km, we will obtain them as
dimensionless values. The RATIOS of any pair of
values will remain unaltered.
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Our model becomes
Suns diameter 1 unit Earths diameter 0.01
units Mercury 42.4 units from the Sun Venus 77
units from the Sun Earth 107 units from the
Sun Mars 163 units from the Sun Jupiter 553
units from the Sun Saturn 1,021 units from the
Sun Uranus 2121 units from the Sun Neptune 3300
units from the Sun Pluto 4321 units from the Sun
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Now, lets replace our unit with a metric. We
might choose feet, inches, mm, whatever. What do
you recommend?
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I chose mm. The model still becomes quite large!
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Heres what I got
Suns diameter 1 mm Earths diameter 0.01
mm Mercury 4.2 cm from the Sun Venus 7.7 cm
from the Sun Earth 10.7 cm from the Sun Mars
16.3 cm from the Sun Jupiter 55.3 cm from the
Sun Saturn 1.02 m from the Sun Uranus 2.1 m
from the Sun Neptune 3.3 m from the Sun Pluto
4.3 m from the Sun
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But 4.3 m 14 feet! Just Think!!! With the Sun
no bigger in diameter than the thickness of a
DIME, the RADIUS of our model is still 14 feet!
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3. Into the Infinite!
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So, what about the rest of the cosmos?
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Again, here is some data to work with
Proxima Centauri 4.3 light years from the
Sun The Milky Way 10,000 light years diameter,
2,000 light years thick M-31 in Andromeda 2
million light years from the solar system
M From Messiers Catalog of Celestial Objects
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Messiers Catalog 103 nebulous celestial
objects 33 galaxies 55 star clusters 11 true
nebulae 1 double star 1 asterism 1 patch of the
Milky Way 1 duplicate observation
Why do you think that Charles Messier (1730 -
1817) cataloged celestial objects?
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Should we use the same scale as before? Why, or
why not?
1.5 Minutes
How far would Proxima be on this scale?
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Using the same scale as before (solar diameter
1 mm), our model becomes
Proxima Centauri 29 km (18 miles) from the
Sun The Milky Way 680,000 km (420,000 miles) in
diameter 136,000 km (84,000 miles) thick M31
14,000,000 km (8,700,000 miles) from the solar
system
Which means . . .
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• A 29 km radius sphere placed around the Sun
  planets has Proxima Centauri on its surface. All
  the rest of the universe is outside the sphere!
• The Milky Way Diameter is twice the actual
  distance from the earth to the moon thickness is
  greater than 3 times the earths circumference
• M 31 More than 1/6th the actual distance from
  Earth to Mars at opposition

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And yet, these are only the closest objects! The
universe is certainly a vast and empty place.
With NASAs Hubble Telescope, we can see out to
about 10 billion light years!! We now estimate
that there are about 5 sextillion stars in the
observable universe! But 5 sextillion 5 X 10
21. A mole of water contains 6 X 10 23
molecules--more than 100 times the number of
stars! Talk about A world in a grain of sand!
William Blake, Auguries of Innocence
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If we had used the solar systems diameter as a
unit of measure rather than the Suns diameter,
what do you think your model would look
like? Redo the model with the solar systems
diameter scaled to 1 mm. (This ones for you to
do on your own!)
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4. And Finally, Back to the Beginning
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According to modern astrophysics, the universe
began 15 billion years ago with an immense
explosion called the Big Bang, the echoes of
which can still be detected by radio telescopes
today, and even seen in the snow on the blank
screens of our televisions. It expanded from an
initial singularity, passed through several
developmental stages, including a rapid
inflation very early on, and is still
expanding. We know this from observations made of
distant galaxies whose light is reddened by the
expansion.
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The further the galaxy, the redder its light. By
carefully measuring this reddening, the rate of
expansion may be inferred. We also believe that
our solar system is about 4.6 billion years old
and that modern humanity arose about 10,000 years
ago.
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One last time, here is some data to work with
Universe began 15 B years ago Solar System
formed 4.6 B years ago Devonian Era began 300 M
years ago Dinosaurs became extinct 65 M years
ago Age of Mammals began 65 M years ago Great
Ice Age began 1 M years ago Great Ice Age ended
and Modern Humans appeared 10 K years ago First
Cities appeared 6 K years ago 20th Century
began 100 years ago
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Can you suggest a model to represent these ages
of time?
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Lets map all of cosmic history onto a 3,000 page
book with 500 words per page The entire book
contains 3,000 ? 500 1,500,000 (1.5 million)
words. Each word represents 15,000,000,000 years
? 1,500,000 W\words 10,000 years of cosmic
history.
Here goes . . .
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1. Universe began - Pg. 1. 2. Solar system is
formed - Pg. 2080 3. Devonian Era began - Pg.
2940 4. Dinosaurs became extinct - Pg. 2987 5.
Age of Mammals begins - Pg. 2987 6. Great Ice Age
began - Pg. 3000 Only 100 words from the end!
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7. Great Ice Age ended and modern humans appeared
- Pg. 3000 -- 1 word from the end 8. First cities
appear - Pg. 3000 -- 0.6 word from the end And
finally . . . 9. 20th Century begins - Pg. 3000
-- 0.01 word from the end!!!
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Well, THAT puts our entire 20th Century at the
end-of-the-last-dot-at-the-end-of-the-last-word-at
-the-end-of-the-last-page of the 3,000 page book.
Think about that! And think about the impact we
have had and are having on our planet!!! Do you
find this a little unsettling?
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5. Wrap-up
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We have scaled the cosmos, from our back yard,
outward into infinite space, and backward into
past geological time. In the end, we found an
unexpected vision of ourselves as a people,
having recently arrived on a cosmic speck
called Earth, and rapidly become a
globe-encircling planetary species. Was it worth
our effort? I think so-- provided that we can
acquire from the experience a greater measure of
self awareness and genuine compassion.
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• Moving Onward
• What have you learned about the size of our
  universe?
• Is time any less fascinating than space?
• Did you ever think about your world this way
  before?
• Now use what you have learned and make a scale
  model of a living cell, such as a skin cell.
• Do the same for a Hydrogen atom.

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• What else would YOU like to make scale models of?
• HAVE FUN!!!!!!!!!

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For those interested in talking more, contact me
at joseph.c.kolecki_at_grc.nasa.gov