Celestial MechanicsCelestial Mechanics

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

WonWon

Advisor: Dr. Steve SuraceAssistant: Jessica Kiscadden

http://www.akhtarnama.com/CCD.htm

What is Celestial Mechanics?What is Celestial Mechanics?► Calculating motion of Calculating motion of

heavenly bodies as seen heavenly bodies as seen from Earth. from Earth.

► 6 Main Parts6 Main Parts Geometry of an EllipseGeometry of an Ellipse Deriving Kepler’s LawsDeriving Kepler’s Laws Elliptical MotionElliptical Motion Spherical TrigonometrySpherical Trigonometry The Celestial Sphere The Celestial Sphere SundialSundial

Elliptical GeometryElliptical Geometry

► Planetary orbits are elliptical Planetary orbits are elliptical

► Cartesian form of ellipse Cartesian form of ellipse 2 2

2 2

( ) ( )1

x h y k

a b

2 2

2 2

( cos ) ( sin )1

r c r

a b

Sun

r

planet

r = (rcos θ, rsin θ)

►Shifting left Shifting left cc units and converting to polar form gives units and converting to polar form gives

Elliptical GeometryElliptical Geometry

► Solving for Solving for rr yields yields

2(1 )

1 cos

a er

e

Kepler’s Laws of Planetary Kepler’s Laws of Planetary MotionMotion

►1. Planetary orbits 1. Planetary orbits are elliptical with are elliptical with Sun at one focusSun at one focus

►2. Planets sweep 2. Planets sweep equal areas in equal areas in equal timesequal times

►3. 3. T T 22//a a 33 = = kk“Kepler’s got nothing on me.”

Kepler’s First LawKepler’s First Law

► Starting with Newton’s laws and Starting with Newton’s laws and gravitational force equationgravitational force equation

► Doing lots of math:Doing lots of math:

► Yields the equation of an ellipseYields the equation of an ellipse

= mF a 2

-GMm=

rGF

1 )

1

2a( - er =

+ecosθ

2 2 22

2 2[( ) cos sin ] - 2 sin cos

d x dθ d θ dr dθ d r= -r θ+ θ θ+ θ

dt dt dt dt dt dt2 2 2

22 2

[( ) sin cos ] + 2 cos sind y dθ d θ dr dθ d r

= -r θ - θ θ+ θdt dt dt dt dt dt

2 22

2cos [( ) cos sin ] - 2 sin cos

2

-GM dθ d θ dr dθ d r= -r θ+ θ θ+ θ

r dt dt dt dt dt

2 22

2sin [( ) sin cos ] + 2 cos sin

2

-GM dθ d θ dr dθ d r= -r θ - θ θ+ θ

r dt dt dt dt dt

2

20 2

d dr drdt dt dt

22

2( )

2

-GM dθ d r= -r +

r dt dt

2 constantdθ

h= r =dt

2 2

2 3 2

-GM -h d r= +

r r dt2 2

2 22 2

[ ( )]d r d d du dθ d u

= -h = -h udt dt d dθ dt dθ

2

2 2

-GM d u= u+

h dθ

2

2

cos sin

hGMr =

h1+ (A θ+B θ)

GM

2Ahe=

GM

1 cos

e

ar =+e θ

min max 21 cos 1- cos

e e

a ar r a+e θ e θ

1 2e

= a( - e )A

Kepler’s Second LawKepler’s Second Law

► Equal areas in equal times

► Area in polar coordinates

1

2

β 2

αA= r dθ

►Differentiating both sides yieldsDifferentiating both sides yields2

2

rdA d

dA dA dθ h= =

dt dθ dt 2

►Expanding with chain rule and Expanding with chain rule and substitutingsubstituting

► T T 22/a /a 33 = k = k► From constant of Kepler’s Second LawFrom constant of Kepler’s Second Law

► Substituting and simplifying yieldsSubstituting and simplifying yields

2 2

T

0

h hTA= dt = = πab

2 2

3

4=

a

T π

GM

Kepler’s Laws and Elliptical Kepler’s Laws and Elliptical GeometryGeometry

► Easier to work with circumscribed circle

► Use trigonometry

► or

coscos

1 cos

eE

e

1tan( ) tan( )

2 1 2

E e

e

Finding OrbitFinding Orbit

► Define M = E – e sin E► Differentiating and substituting

2dM

dt T

2 tM

T

►Solving differential equation with E =0 at t =0,

Spherical TrigonometrySpherical Trigonometry

► Studies triangles Studies triangles formed from three formed from three arcs on a spherearcs on a sphere

► Arcs of spherical Arcs of spherical triangles lie on triangles lie on great circles of great circles of spheresphere

Points A, B, & C connect to form spherical triangle ABC

Spherical TrigonometrySpherical Trigonometry

Given information Given information from spherefrom sphere

Derive Spherical Law of Cosines

Derive Spherical Law of Sines

Law of CosinesLaw of Cosines

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

2 2 2' ( tan ) ( tan ) 2( tan )( tan )cosa b a b

c r r r r Cr r r r

2 2 2 2 2 2 2 2 2 2 2 2 2' ( tan ) ( tan ) 2(cos ) ( tan ) ( tan )a b c a b

c r r r r r r r rr r r r r

Spherical Law of CosinesSpherical Law of Cosines

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

cos cos cos sin sin cosc a b a b

Cr r r r r

Spherical Law of SinesSpherical Law of Sines► Manipulated Spherical Law of Cosines intoManipulated Spherical Law of Cosines into

2 2 22

2 2 2 2

sin sin sin 2cos cos cos 2sin

sin sin sin sin

a b c a b cC r r r r r rc a b cr r r r

sin sin sin

sin sin sin

A B Ca b cr r r

►Equation is symmetric function, yielding Spherical Law of Sines.

ApplyingApplying

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

Where is the Sun?Where is the Sun?

► Next goal: Find equations for the Next goal: Find equations for the coordinates of Sun for any given daycoordinates of Sun for any given day

► DefinitionsDefinitions Right Ascension (Right Ascension (αα) = longitude) = longitude

► Measured in h, min, secMeasured in h, min, sec

Declination (Declination (δδ ) = latitude) = latitude► Measured in degreesMeasured in degrees

Where is the Sun?Where is the Sun?

►Using Spherical Law of Sines for Using Spherical Law of Sines for

this triangle, derived formula this triangle, derived formula calculating declination calculating declination of Sunof Sun sin sin δδ = (sin = (sin λλ )(sin )(sin ε ε )) On August 3, 2006On August 3, 2006

►λλ = 2.3026 = 2.3026►δδ = 17° 15’ 25’’ = 17° 15’ 25’’

Where is the Sun?Where is the Sun?

►Using Spherical Law of Cosines to find Using Spherical Law of Cosines to find formula for right ascension and its formula for right ascension and its value for Sunvalue for Sun

August 3, 2006August 3, 2006►λλ = 2.3026 = 2.3026►αα = 8h 57min 37s = 8h 57min 37s

cos - sin sin90 coscos =

cos cos

Predicting Sunrise and Predicting Sunrise and SunsetSunset

►HH = Sun’s path on certain date = Sun’s path on certain date On equator at vernal equinoxOn equator at vernal equinox

►Key realizationsKey realizations Angle Angle HH Draw the zenithDraw the zenith

Predicting Sunrise and Predicting Sunrise and SunsetSunset

► Find angle Find angle HH using Spherical Law of using Spherical Law of Cosines Cosines HH = 106.09 = 106.09° = 7 hours 4 minutes° = 7 hours 4 minutes

► Noon now: 1:00 PM (daylight savings)Noon now: 1:00 PM (daylight savings)

► Aug. 3, 2006Aug. 3, 2006 Sunrise - 5:56 AMSunrise - 5:56 AM Sunset - 8:04 PMSunset - 8:04 PM

By Golly Moses!That’s

Amazing!

Constructing a SundialConstructing a Sundial

Constructing a SundialConstructing 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 SundialConstructing a Sundial

►Coordinates in Rotated Axes

Stick (0, -Lcosφ, Lsinφ)

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

Constructing a SundialConstructing 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

tan tan sinx

y

Sundial ConstructedSundial Constructed

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

Time θ9:00 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°

Sundial Pictures!Sundial Pictures!

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