Zhao Chen, Jamie Dougherty, Charlene Grahn, Meghan Kane, Richard Li, David Pan, Matthew Salesi, Katelyn Seither, Akash Shah, Sanjeev Tewani, Robert Won


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Celestial Mechanics

  • Zhao Chen, Jamie Dougherty, Charlene Grahn, Meghan Kane, Richard Li, David Pan, Matthew Salesi, Katelyn Seither, Akash Shah, Sanjeev Tewani, Robert Won




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


Elliptical Geometry



Elliptical Geometry

  • Solving for r yields



Kepler’s Laws of Planetary Motion



Kepler’s First Law

  • Starting with Newton’s laws and gravitational force equation

  • Doing lots of math:

  • Yields the equation of an ellipse



Kepler’s Second Law

  • Equal areas in equal times

  • Area in polar coordinates



Differentiating both sides yields

  • Differentiating both sides yields



T 2/a 3 = k

  • T 2/a 3 = k

  • From constant of Kepler’s Second Law

  • Substituting and simplifying yields



Kepler’s Laws and Elliptical Geometry



Finding Orbit

  • Define M = E – e sin E

  • Differentiating and substituting



Spherical Trigonometry



Spherical Trigonometry



Law of Cosines

  • Solve for side c’ in triangles A’OB and A’B’C



Spherical Law of Cosines

  • c’ equations equated and simplified to obtain Spherical Law of Cosines



Spherical Law of Sines

  • Manipulated Spherical Law of Cosines into



Applying



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


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’’


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


Predicting Sunrise and Sunset

  • H = Sun’s path on certain date

    • On equator at vernal equinox
  • Key realizations

    • Angle H
    • Draw the zenith


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


Constructing a Sundial



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



Constructing a Sundial

  • Coordinates in Rotated Axes

  • Stick (0, -Lcosφ, Lsinφ)

  • Sun (-rsin15°, rcos15°sinφ, rcos15°cosφ)



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



Sundial Constructed

  • Finally, by plugging in different values for η, we arrive at the following chart.



Sundial Pictures!






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