(11a)  Ellipses and Kepler's First Law

   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 φ (or Greek f, also written 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 Coordinates

   Again, if all the values of (r, φ) of a curve are related by some equation which can be symbolically written

r = r(φ)

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 φ. That gives a circle around the origin, its radius equal to a, shown in the drawing on the right above.

The Ellipse

Consider next the curve whose equation is

r   =   a(1– e²) / (1+ e cos φ)

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 φ represents a wave-like behavior (picture below), and as φ 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 φ = –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 φ (Greek f) 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 Coordinates

   Again, if all the values of (r, φ) of a curve are related by some equation which can be symbolically written

r = r(φ)

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 φ. That gives a circle around the origin, its radius equal to a, shown in the drawing on the right above.

The Ellipse

Consider next the curve whose equation is

r = a(1– e²)/(1+ e cos φ)

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 φ represents a wave-like behavior (picture below), and as φ goes through a full circle, it goes down, from +1 to 0, continues to –1, then up again to 0 and +1. The denominator also rises and falls as a wave, and it is smallest when cos φ = –1 . Here is the table of the main values (360 is in parentheses, because it represents the same direction as 0 degrees):

   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 – e²) can be factored--that is, written as two expressions multiplied by each other ("the product of two expressions"). As discussed in the section on algebraic identities

1 – e² =  (1 – e)(1 + e)

At some of the points on the above table, either (1 – e) or (1 + e) cancels the denominator, giving:

   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 = 0.0168, 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.