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Celestial Mechanics: Exploring the Motion of Heavenly Bodies

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Delve into celestial mechanics through calculating planetary orbits and elucidating Kepler’s laws. Uncover the significance of elliptical geometry, spherical trigonometry, and the celestial sphere in understanding planetary motion.

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Celestial Mechanics: Exploring the Motion of Heavenly Bodies

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  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

  2. Let's Do Some Math!!!!

  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 planet r r = (rcos θ, rsin θ) Sun • 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 • Equal areas in equal times • Area in polar coordinates

  9. Kepler's Second Law • Differentiating both sides yields • Expanding with chain rule and substituting

  10. Kepler's Third Law • 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 Spherical Trigonometry • 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. By Golly Moses! That’s Amazing! 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

  24. Constructing a Sundial

  25. Constructing a Sundial • The coordinates are: Stick: (0, 0, L) Sun: (-Rsin15°, Rcos15°, 0) • A 15o 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 PM 0° 2:00 PM 9.97° 3:00 PM 20.75° 4:00 PM 33.27° 5:00 PM 48.65°

  29. Sundial Pictures!

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

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