
Index
9a. Earth orbits Sun? 9c. Copernicus to Galileo 10. Kepler's Laws Kepler's Laws (For teachers) 10a. Scale of Solar Sys. 11. Graphs & Ellipses 11a. Ellipses and First Law 12. Second Law 12a. More on 2nd Law 12b. Orbital Motion 12c. Venus transit (1) 12d. Venus transit (2) 12e. Venus transit (3) Newtonian Mechanics 13. Free Fall 
As already noted, other ways exist for labeling points in the plane. For instance, a point P may be labeled by its distance r from a central point O ("origin") and the angle f (or Greek φ) which the line OP makes with some standard direction. Such "polar coordinates" (drawing on the left, below) are the ones best suited for describing planetary motion. The Ellipse in Polar CoordinatesAgain, if all the values of (r,f) of a curve are related by some equation which can be symbolically written r = r(f) then the function r(f) is said to be the equation of the line, in polar coordinates. The simplest function is a constant number a, giving the line r = a The value of r equals a for any value of f. That gives a circle around the origin, its radius equal to a, shown in the drawing on the right above. The EllipseConsider next the curve whose equation is r = a(1– e^{2})/(1+ e cos f) where the eccentricity e is a number between 0 and 1. If e = 0, this is clearly the circle encountered earlier. 
How about other values? The function cos f represents a wavelike behavior (picture below), and as f goes through a full circle, it goes down, from +1 to 0, then –1, then up again to 0 and +1. The denominator also rises and falls as a wave, and it is smallest when cos f = –1 . Here is the table of the main values (360 is in parentheses, because it represents the same direction as 0 degrees): As already noted, other ways exist for labeling points in the plane. For instance, a point P may be labeled by its distance r from a central point O ("origin") and the angle f (or Greek φ) which the line OP makes with some standard direction. Such "polar coordinates" (drawing on the left, below) are the ones best suited for describing planetary motion. The Ellipse in Polar CoordinatesAgain, if all the values of (r,f) of a curve are related by some equation which can be symbolically written r = r(f) then the function r(f) is said to be the equation of the line, in polar coordinates. The simplest function is a constant number a, giving the line r = a The value of r equals a for any value of f. That gives a circle around the origin, its radius equal to a, shown in the drawing on the right above. The EllipseConsider next the curve whose equation is r = a(1– e^{2})/(1+ e cos f) where the eccentricity e is a number between 0 and 1. If e = 0, this is clearly the circle encountered earlier. How about other values? The function cos f represents a wavelike behavior (picture below), and as f goes through a full circle, it goes down, from +1 to 0, then –1, then up again to 0 and +1. The denominator also rises and falls as a wave, and it is smallest when cos f = –1 . Here is the table of the main values (360 is in parentheses, because it represents the same direction as 0 degrees):

f degrees  0  90  180  270  (360) 
cos f  1  0  –1  0  1 
1 + e cos f  1 + e  1  1 – e  1  1 + e 
As long as e is less than 1, the denominator is always positive. It is never zero, so that for any f one can name, one can always find a suitable r. In other words, the curve goes completely around the origin, it is closed. The expression (1 – e^{2}) can be factoredthat is, written as two expressions multiplied by each other ("the product of two expressions"). As discussed in the section on algebraic identities 1 – e^{2} = (1 – e)(1 + e) At some of the points on the above table, either (1 – e) or (1 + e) cancels the denominator, giving: 
f degrees  0  90  180  270  (360) 
r  a(1 – e)  a(1 – e^{2})  a(1 + e)  a(1 – e^{2})  a(1 – e) 
The distance of the line from the origin thus fluctuates between a(1 – e) and a(1 + e), and the result is a flattened circle or ellipse; the point O (the origin) is its focus. All planetary orbits resemble ellipses, each with its own value of e or eccentricity: the smaller e is, the closer the shape to a circle. The Earth's orbit is very close to a circle, with e = 0.0068, and other major planets (except for Pluto) have comparable eccentricities: if you saw a scale drawing of that orbit on a sheet of paper, your eye would not be able to tell it apart from a circle. The orbit of Comet Halley, on the other hand, has e quite close to 1. 
 As mentioned in the preceding section a second focus O' can be drawn symmetrical to O, and the ellipse can be defined (its original definition, in fact) as the collection of points for which the sum R_{1}+R_{2} of their distances from O and O' is always the same
The longest dimension of the ellipse, its width AB along the line connecting the two foci, is its "major axis." Suppose (R_{1},R_{2}) are the distances of A from the foci O and O'. Then R_{1} = OA =a(1–e) is the smallest distance of the ellipse from O, R_{2} = O'A = OB (by symmetry) is the largest and therefore equals a(1 + e). But, OA + OB = AB, hence 

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Author and Curator: Dr. David P. Stern
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Last updated: 12 December 2004