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lecture-04

# lecture-04 - These lecture notes were prepared for Rutgers...

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These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics by Prof. Chuck Keeton, and modified by Profs. Saurabh Jha and Eric Gawiser. All rights reserved. c 2011 Lecture 4: Deriving Kepler’s Laws This discussion is drawn from Chapters 1–2 of Carroll & Ostlie. I. Mathematical Prelude: Cartesian and Polar Coordinates In Cartesian coordinates the position vector (measured from the origin of the coordinate system) of an object in two dimensions is: ~ r = x ˆ x + y ˆ y This is sometimes written as ~ r = r x ˆ x + r y ˆ y or ~ r = x ˆ i + y ˆ j or ~ r = x ˆ e x + y ˆ e y , but we will try to stick with the notation above. ˆ x and ˆ y are unit vectors , i.e. | ˆ x | = | ˆ y | = 1, and they always point in the direction of the x- and y-axis, respectively, as shown in Figure ?? . r ! y x ! r ^ ! ^ ! x ^ y ^ ! ! 2 r ! 2 x ^ y ^ r ^ ! ^ Figure 1: A moving object with positions ~ r and ~ r 2 can be described by projection onto unit vectors in Cartesian (ˆ x and ˆ y ) and polar (ˆ r and ˆ θ ) coordinates. If our object moves in two dimensions as a function of time, then we can describe its motion ~ r ( t ) = x ( t x + y ( t y . In general I won’t explicitly write out the fact that ~ r , x , and y are functions of time. 1

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We can derive the velocity of the object: ~v d~ r dt = dx dt ˆ x + dy dt ˆ y Sometimes this is written ~v = ˙ ~ r = ˙ x ˆ x + ˙ y ˆ y , but again, we’ll try to stick with the notation above. We can also write the acceleration vector: ~a d~v dt = d 2 x dt 2 ˆ x + d 2 y dt 2 ˆ y The forms for the velocity and acceleration vectors in Cartesian coordinates are simple because the unit vectors ˆ x and ˆ y are fixed; they don’t change with time, and so ( d ˆ x/dt ) = ( d ˆ y/dt ) = 0. We can also write the position of the object in polar coordinates , in which we state the object’s radius r and the angle θ , taken by convention to be the angle between the x-axis and ~ r . Then we have x = r cos θ y = r sin θ r = p x 2 + y 2 tan θ = y x Similarly, we can define unit vectors in polar coordinates . The easier one to picture is ˆ r , which has unit length ( | ˆ r |
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