1. Celestial Mechanics Zhao Chen, Jamie Dougherty, Charlene Grahn, Meghan Kane, Richard Li, David Pan, Matthew Salesi, Katelyn Seither, Akash Shah, Sanjeev Tewani, Robert Won Advisor: Dr. Steve Surace Assistant: Jessica Kiscadden http://www.akhtarnama.com/CCD.htm

3. What is Celestial Mechanics? ► Calculating motion of heavenly bodies as seen from Earth. ► 6 Main Parts  Geometry of an Ellipse  Deriving Kepler’s Laws  Elliptical Motion  Spherical Trigonometry  The Celestial Sphere  Sundial

4. Elliptical Geometry ► Planetary orbits are elliptical ► Cartesian form of ellipse Sun r planet r = (rcos θ, rsin θ) ► Shifting left c units and converting to polar form gives

5. Elliptical Geometry ► Solving for r yields

6. Kepler’s Laws of Planetary Motion ► 1. Planetary orbits are elliptical with Sun at one focus ► 2. Planets sweep equal areas in equal times ► 3. T 2 /a 3 = k “Kepler’s got nothing on me.”

7. Kepler’s First Law ► Starting with Newton’s laws and gravitational force equation ► Doing lots of math: ► Yields the equation of an ellipse

8. Kepler’s Second Law Kepler’s Second Law ► ► Equal areas in equal times ► ► Area in polar coordinates

9. ► Differentiating both sides yields ► Expanding with chain rule and substituting

10. ► T 2 /a 3 = k ► From constant of Kepler’s Second Law ► Substituting and simplifying yields

11. Kepler’s Laws and Elliptical Geometry ► ► Easier to work with circumscribed circle ► ► Use trigonometry ► ► or

12. Finding Orbit ► ► Define M = E – e sin E ► ► Differentiating and substituting ► Solving differential equation with E =0 at t =0,

13. Spherical Trigonometry ► Studies triangles formed from three arcs on a sphere ► Arcs of spherical triangles lie on great circles of sphere Points A, B, & C connect to form spherical triangle ABC

14. Spherical Trigonometry Given information from sphere  Derive Spherical Law of Cosines  Derive Spherical Law of Sines

15. Law of Cosines ► Solve for side c’ in triangles A’OB and A’B’C

16. Spherical Law of Cosines ► c’ equations equated and simplified to obtain Spherical Law of Cosines

17. Spherical Law of Sines ► Manipulated Spherical Law of Cosines into ► Equation is symmetric function, yielding Spherical Law of Sines.

18. Applying  Real world application-Calculating shortest distance between two cities  Given radius and circumference of Earth and latitude and longitude of NYC and London we found distance to be 5701.9 km

19. Where is the Sun? ► Next goal: Find equations for the coordinates of Sun for any given day ► Definitions  Right Ascension ( α ) = longitude ► Measured in h, min, sec  Declination (δ ) = latitude ► Measured in degrees

20. Where is the Sun? ► Using Spherical Law of Sines for this triangle, derived formula calculating declination of Sun  sin δ = (sin λ )(sin ε )  On August 3, 2006 ► λ = 2.3026 ► δ = 17° 15’ 25’’

21. Where is the Sun? ► Using Spherical Law of Cosines to find formula for right ascension and its value for Sun  August 3, 2006 ► λ = 2.3026 ► α = 8h 57min 37s

22. Predicting Sunrise and Sunset ► H = Sun’s path on certain date  On equator at vernal equinox ► Key realizations  Angle H  Draw the zenith

23. Predicting Sunrise and Sunset ► Find angle H using Spherical Law of Cosines  H = 106.09° = 7 hours 4 minutes ► Noon now: 1:00 PM (daylight savings) ► Aug. 3, 2006  Sunrise - 5:56 AM  Sunset - 8:04 PM By Golly Moses! That’s Amazing!

24. Constructing a Sundial

25. ► ► The coordinates are: Stick: (0, 0, L) Sun: (-Rsin15°, Rcos15°, 0) ► ► A 15 o change in the sun’s position implies a change in 1 hour

26. Constructing a Sundial ► ► Coordinates in Rotated Axes Stick (0, -Lcosφ, Lsinφ) Sun (-rsin15°, rcos15°sinφ, rcos15°cosφ)

27. Constructing a Sundial ► ► Solving for the equation of the line passing through the sun and the stick tip, we have ► ► Where η is the arc degree measure of the sun with respect to the tilted y axis

28. Sundial Constructed ► ► Finally, by plugging in different values for η, we arrive at the following chart. Time θ 9:00 AM-48.65° 10:00 AM-33.27° 11:00 AM-20.75° 12:00 PM-9.97° 1:00 PM0° 2:00 PM9.97° 3:00 PM20.75° 4:00 PM33.27° 5:00 PM48.65°

29. Sundial Pictures!

30. Once you’ve seen one equation, you’ve seen them all. - Dr. Miyamoto [Math] is real magic, not like that fork-bending stuff. - Dr. Surace