## 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 ## Planetary orbits are elliptical
## Elliptical Geometry
## Kepler’s Laws of Planetary Motion ## 1. Planetary orbits are elliptical with Sun at one focus ## 3. *T 2*/*a 3* = *k*
## Kepler’s First Law ## Starting with Newton’s laws and gravitational force equation ## Doing lots of math: ## Yields the equation of an ellipse
## Equal areas in equal times ## Area in polar coordinates
## 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 ## Arcs of spherical triangles lie on great circles of sphere
## Spherical Trigonometry
## 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? ## Definitions - Right Ascension (
*α*) = longitude - Declination (
*δ *) = latitude
## Where is the Sun? ## 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
*H* = Sun’s path on certain date
- On equator at vernal equinox
## Key realizations
## Predicting Sunrise and Sunset *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 ## 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, -Lcos*φ*, Lsin*φ*) ## Sun (-rsin15°, rcos15°sin*φ*, rcos15°cos*φ*)
## 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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