Modeling the Motion of Orbiting Bodies Taran Lynn May 15, 2015 - - PowerPoint PPT Presentation

Modeling the Motion of Orbiting Bodies

Taran Lynn May 15, 2015

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Objective

The objective of this project is to model the position of an orbiting object as a function of time. This is achieved by using Newton’s law of universal gravitation and his third law of motion to find a second order ODE that models the relationship between position and time. I will also use Kepler’s laws to find initial conditions for the Newtonian model.

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Newtonian Model

S P # « r Fg d2 # « r dt2 d # « r dt

Figure: Orbiting Body Diagram

Force on Orbited Object Fg = Gmsmp |# « r |3 # « r Force on Orbiting Object mp d2 # « r dt2 Newton’s Third Law mp d2 # « r dt2 = −Fg mp d2 # « r dt2 = −Gmsmp |# « r |3 # « r d2 # « r dt2 = −Gms |# « r |3 # « r

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Multiple First Order ODEs

We can split the second order vector ODE ¨ # « r = −Gms |# « r |3 # « r into four first order scalar ODE’s. ˙ x = vx (1) ˙ vx = − Gmsx (x2 + y2)3/2 (2) ˙ y = vy (3) ˙ vy = − Gmsy (x2 + y2)3/2 (4)

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Example: Halleys Comet I

1 The mass of the sun is 1.99 × 1030 kg. 2 The comet’s distance from the sun at its perihelion is 8.78 × 1010 m. 3 Its speed at the perihelion is 5.46 × 104 m/s.

• 6×1012
• 5×1012
• 4×1012
• 3×1012
• 2×1012
• 1×1012

x (m )

• 5×1011

5×1011 y (m ) Sun

Figure: Halley’s Comet’s Path

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Example: Halleys Comet II

2 ×109 4 ×109 6 ×109 8 ×109 1 ×1010t (s)

• 6 ×1012
• 5 ×1012
• 4 ×1012
• 3 ×1012
• 2 ×1012
• 1 ×1012

1 ×1012 Comet Position

x (m ) y (m )

Figure: Halley’s Comet’s x and y Positions as a Function of Time

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Example: Halleys Comet III

2 ×109 4 ×109 6 ×109 8 ×109 1 ×1010t (s)

• 5000

5000 Comet Velocity

vx (m /s) vy (m /s)

Figure: Halley’s Comet’s x and y Velocities as a Function of Time

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Example: Halleys Comet IV

2×109 4×109 6×109 8×109 1×1010t (s) 5000 10000 15000 v (m /s)

Figure: Halley’s Comet’s Speed

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Initial Conditions

# « r (0) can be calculated using parallax. The mass of the orbited object can be found by sending a satellite into circular orbit around the orbiting body and using the equation M = 4π2r3 GT 2 . (5)

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Initial Conditions: Velocity

The velocity of the orbiting object can be found using the vis-viva equation v2 = GM 2 r − 1 a

• .

(6) S r P v

Figure: Path of an Orbiting Body

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Bibliography I

Eugene Butikov. Relative motion of orbiting bodies. research. St. Petersburg State University, St. Petersburg, Russia, n.d. Emily Davis. “Deriving Kepler’s Laws of Planetary Motion”. Presentation Slide. Kyriacos Papadatos. “The Equations of Planetary Motion and Their Solution”. In: The General Science Journal (n.d.). Office of Public Information. Comet Halley Summary. N.A.S.A. url: http://er.jsc.nasa.gov/seh/halley.html. Reymond Serway and John Jewett Jr. Physics for Scientists and Engineers with Modern Physics. Ninth. Brooks/Cole, 2014. David Surowski. Kepler’s Laws of Planetary Motion and Newton’s Law of Universal Gravitation. Retrieved from Surowski’s website.

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Bibliography II

• Wikipedia. Wikipedia, The Free Encyclopedia. 2015. url:

http://en.wikipedia.org/.

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