# Presentation: 313535

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***Topics in Mathematics:***
***Scaling the Cosmos***

***Mr. K***
***NASA/GRC/LTP***

***Edited by Ruth Petersen***

**Notes:**
Scaling the Cosmos
We often read that the distances between objects in space are vast.
How vast?
Beyond imagination?
Or can we build models to help us visualize the distances?

This presentation demonstrates how units of measure may be selected to make scale models and explores:
 The questions shown above,
What these models teach us,
How models can be used, and 
Models’ limitations.

## 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?***

**Notes:**
Getting Started
First, though, please work through these preliminary exercises. They will provide a nice base of ideas with which we can work...

## If I divide 36 feet by 3 feet, what answer do I get? “12 feet,” “12 yards,” or just plain “12”?

***If I divide 36 feet by 3 feet/yard, what is the correct answer? Why?***
***Write the number 1,234,567,890 in scientific notation.***
***Round the answer of Question 9 to three figures.***
***The Sun has a diameter of 1.4X10******6****** km. If I were to use the Sun’s diameter as a UNIT OF MEASURE, what should I do?***

## HINTS:

*** ***
***What must I do to the number 1.4X10******6****** 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?***

## 1. Our Cosmic Back Yard

**Notes:**
Our Cosmic Back Yard
OK!!! Now that you have completed the preliminary exercises, you have some of the essential tools to proceed. Let’s begin small and work our way outward!

## Have you every looked at the moon and wondered, “How far away is it really?”

***Have you every looked at the moon and wondered, *****“*****How far away is it really?”***

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**Notes:**
The moon is practically in our cosmic back yard. We see it often throughout the course of every month. But have you ever asked, “ How far away is it really ? ”

## 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***

**Notes:**
Again, before you go on, try your intuition and complete the exercise given above. You may find intuition an accurate guide. Or, you might be very surprised...

## Most people’s models look something like this:

***Is yours similar? What made you choose the scale you used?***

**Notes:**
Well, how did you do?

## Now, let’s calculate. From a table, I find: 

***Moon’s diameter: 3,500 km***
***Earth’s 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.***

***Now: ***_***Redo***_*** your drawings based on this information.***

***30 Seconds***

**Notes:**
Now we will roll up our sleeves, take out our calculators, and put on our thinking caps! Please select a unit of measure that you feel is convenient for making a scale drawing, then do the necessary calculations.

## Your drawing should look more like this:

***How large would an Apollo spacecraft be on this scale? Think about Apollo 13!***

**Notes:**
To many people, the above drawing comes as a shock. After all, the moon often appears to be just over the horizon, warm and close, an object of romantic paintings and poetry. There is the chill of loneliness inherent in the drawing above. And we might already be asking ourselves: “ If this is true of the earth and its moon, what about the rest of the solar system ? ”

## Does this result surprise you?

***Does this result surprise you?***
***Why or why not?***

**Notes:**
Were you surprised? Please take a few moments to think about and/or discuss your reactions...

## 2. Somewhat Further Out

**Notes:**
OK... so what about the solar system?

## .

***Next, let’s think about the solar system!***

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**Notes:**
We know that our Sun wields its gravitational influence over a family of nine planets, numerous moons and ring systems, a swarm of asteroids, a hoard of comets, and uncountable specks of cosmic dust, ice, and other debris. The objects of interest in this exercise are the Sun and the nine planets.

## Here is some data to work with:

***Sun’s diameter: ******1,400,000 km***
***Earth’s 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***

**Notes:**
Here here is some data on the solar system. Please notice that different units have been used to denote different distances. One important lesson in scientific calculations is: Always use a consistent set of units. Therefore, let’s do some conversions before we go any further...

## Question:

***How many km are in a L.M.?***
***How many km are in a L.H.?***

**Notes:**
If you already know the answer, you may skip ahead to slide number 18.

## 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***

**Notes:**
The light-distances are acquired from the speed of light, a universal constant in modern physics. A light second is simply the distance that light travels in a second. A light minute is the distance light travels in a minute. And so on.
You should also understand that distance is speed multiplied by time.
With these ideas in hand, please take a few minutes to set up and solve the problems presented above.

## 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***

**Notes:**
Here are the numbers required to fill in the blanks, and the resulting distances in km.

## 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!***

**Notes:**
If the earth’s diameter makes a useful unit of measure for modeling the earth-moon system, what might you use as a unit for modeling the solar system? Why?

## I chose the Sun’s 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.***

**Notes:**
The Sun is the largest object in the solar system, and its diameter provides a useful choice. If we divide all of the planetary distances (in km) by the Sun’s diameter (in km), we will obtain a set of dimensionless ratios representing distances of each of the planets from the Sun in units of Sun-diameters. We may then (arbitrarily) assign units (such as cm or mm) to these numbers for the purposes of making scale drawings. Architects do much the same thing when they use an architect’s scale.
Please take a few minutes to do the calculations on your own.

## Sun’s diameter: 1 unit

***Earth’s 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***

***Our model becomes:***

**Notes:**
Your results should look like this.

## Now, let’s replace our “unit” with a metric. We might choose feet, inches, mm, whatever. What do you recommend?

**Notes:**
OK...Now we choose an arbitrary unit for making our drawing. What should it be???

## I chose mm. The model still becomes quite large!

**Notes:**
Since the numbers for the outer planets are quite large, and since we would like to contain the model in our room, we want a very small unit of measure for our drawing. The mm unit works nicely for this purpose.

## Sun’s diameter: 1 mm

***Earth’s 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***

***Here’s what I got:***

**Notes:**
Your table should now look like this. 
(Please notice that I used mm, cm, and m to keep my numbers from getting too large.)
Now you are ready to draw!

## But 4.3 m = 14 feet!

***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!***

**Notes:**
Even at this scale, you will need a VERY large sheet of paper. I suggest that a paper roll might work best...

## 3. Into the Infinite!

**Notes:**
Well, how alone do you think that we are now that you’ve finished Part 2? Do we dare go on?

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***So, what about the rest of the cosmos?***

**Notes:**
There is so much in the cosmos (everything, in fact!!!) that we must limit our attention to only a few selected objects. You may pick any that you wish...I have selected a few of my own to use in the slides that follow...

## 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***

***Again, here is some data to work with:***

***“******M” – From Messier’s Catalog of Celestial Objects***

**Notes:**
I chose a star, our galaxy, and our closest galactic neighbor.

## Why do you think that Charles Messier (1730 - 1817) cataloged celestial objects?

***Messier’s 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***

**Notes:**
(I also enjoy little historical footnotes. This one is for your general edification and may be skipped if not interesting...)

## Should we use the same scale as before?

***Why, or why not?***

***How far would Proxima be on this scale?***

***1.5 Minutes***

**Notes:**
Here we go again. Let’s repeat and apply what we have learned in Parts 1 and 2!

## 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***

***Using the same scale as before (solar diameter = 1 mm), our model becomes:***

***	******Which means . . .***

**Notes:**
Using the solar diameter yields some pretty big numbers! Let’s stick with them and interpret what we have learned.

## 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 earth’s circumference***
***M 31:****** ******More than****** ******1/6******th****** the [actual] distance from Earth to Mars at opposition****** ***

**Notes:**
We certainly are alone!!!

## And yet, these are only the closest objects! The universe is certainly a vast and empty place. With NASA’s 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”***

**Notes:**
Scaling can be fun. Here, I use numbers from basic physics to show that there are more molecules in a mole of water (18 grams of water) than there are stars estimable in the observable universe. Nothing profound, perhaps, but certainly fascinating!

## If we had used the solar system’s diameter as a unit of measure rather than the Sun’s diameter, what do you think your model would look like?

***Redo the model with the solar system’s diameter scaled to 1 mm. (This one’s for you to do on your own!)***

**Notes:**
Here’s one to try on your own!

## 4. And Finally, Back to the Beginning

**Notes:**
Well, we’ve learned how to scale space. Now, let’s take a look at time...

## 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.

**Notes:**
Here’s a little capsule summary of current scientific thinking regarding the age and the development of the universe. Please take a few moments to read it...

## 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******.***

## Universe began: 15 B years ago

***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***

***One last time, here is some data to work with:***

**Notes:**
OK. One more time, here is a data set to work with. Do you know what to do?

## Can you suggest a model to represent these ages of time?

## Let’s 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 . . .***

**Notes:**
This model is classic. You will find something similar in Carl Sagan’s book, Cosmos. 
This exercise shows how the model is derived.

## 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!***

**Notes:**
And...here are some results...

## 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.****** 20******th****** Century begins - Pg. 3000 -- 0.01 word from the end!!!***

## 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?

**Notes:**
If we are alone in the universe--separated from even our nearest neighbors by ENORMOUS distances--then we are also very much newcomers to this world! Think about this fact in light of historical developments in the last 150 years or so.

## 5. Wrap-up

## 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.******  ***

**Notes:**
I hope that you enjoyed this little numerical romp through the cosmos. Please keep on thinking, reading, and calculating. These are essential keys to understanding.
Ciao!!!

## 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.***

**Notes:**
Please try these exercises on your own. If you have questions, please feel free to contact me by e-mail!

## What else would YOU like to make scale models of?

***HAVE FUN!!!!!!!!!***

## For those interested in talking more, contact me at:

***joseph.c.kolecki@grc.nasa.gov ***