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

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

• Planetary orbits are elliptical

• Cartesian form of ellipse

Elliptical Geometry

• Solving for r yields

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 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

• Easier to work with circumscribed circle

• Use trigonometry

• or

Finding Orbit

• Define M = E – e sin E

• Differentiating and substituting

Spherical Trigonometry

• Studies triangles formed from three arcs on a sphere

• Arcs of spherical triangles lie on great circles of sphere

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!