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Topics in Mathematics: Scaling the Cosmos

Topics in Mathematics: Scaling the Cosmos. Mr. K NASA/GRC/LTP. Edited by Ruth Petersen. 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.

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Topics in Mathematics: Scaling the Cosmos

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  1. Topics in Mathematics: • Scaling the Cosmos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mr. K NASA/GRC/LTP . . . . . . . . . . . . . Edited by Ruth Petersen . . . . .

  2. 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?

  3. 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 1.4X106 km. If I were to use the Sun’s diameter as a UNIT OF MEASURE, what should I do?

  4. 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?

  5. 1. Our Cosmic Back Yard

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

  7. 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

  8. Most people’s models look something like this: Is yours similar? What made you choose the scale you used?

  9. 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. 30 Seconds Now: Redo your drawings based on this information.

  10. Your drawing should look more like this: How large would an Apollo spacecraft be on this scale? Think about Apollo 13!

  11. Does this result surprise you? Why or why not?

  12. 2. Somewhat Further Out

  13. . . . . . . Next, let’s think about the solar system! . . . . . . . . . . . . . . . . . . . . . . .

  14. 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

  15. Question: • How many km are in a L.M.? • How many km are in a L.H.?

  16. 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

  17. 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

  18. 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!

  19. 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.

  20. Our model becomes: 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

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

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

  23. Here’s what I got: 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

  24. 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!

  25. 3. Into the Infinite!

  26. . . . . . . . . . . . . . . So, what about the rest of the cosmos? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  27. 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 Messier’s Catalog of Celestial Objects

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

  29. Should we use the same scale as before? Why, or why not? 1.5 Minutes How far would Proxima be on this scale?

  30. 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 . . .

  31. 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 than1/6th the [actual] distance from Earth to Mars at opposition

  32. 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”

  33. 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!)

  34. 4. And Finally, Back to the Beginning

  35. 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.

  36. The further the galaxy, theredderits 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.

  37. 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

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

  39. 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  1,500,000W\words = 10,000 years of cosmic history. Here goes . . .

  40. 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!

  41. 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!!!

  42. 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?

  43. 5. Wrap-up

  44. 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.

  45. 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.

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

  47. For those interested in talking more, contact me at: joseph.c.kolecki@grc.nasa.gov

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