Tuesday, November 13, 2012

Two Pegs and a String

Click here for source document. Under gravity's influence, an object moves in an orbit shaped like an ellipse.  To consider an ellipse in more detail; use two pegs and a string.
First, Stretch the String!
When stretched taunt (perhaps by two pegs), a length of string is a straight line which can be modeled by a simple equation.
Straight Line Equation:
y = mx + b
Line's slope, m, is the tangent of the angle that it makes with the x-axis. The intercept on the y-axis is b.
Three Lines and a Triangle
Three intersecting lines can create a triangle. This particular set of lines include a vertical line (x=0), a horizontal line (y=0) and the line from above (with slope, m; and x intercept, b).


These particular lines create a right triangle (one angle = 90°) as shown.  Let lengths a and b be arbitrary; then, compute triangle's base, c, with a Pythagorean relationship:
c = (a2 - b2)
Reverse line's slope (change m to "-m") to obtain another right triangle as shown.  These two right triangles are congruent (same sides and angles).

Combine these two right triangles to form an isosceles (two sides of equal length, a) triangle such that an equal half is on either side of the y-axis. Triangle's base would have a length of 2c.
Next, Loop the String!
Take a continuous loop of string and encompass two pegs.
Let the pegs be separated by distance, 2c.

Then, Stretch It Again!!!
To draw an isosceles triangle with a string and two pegs, move pencil directly above center of base until string is completly taut. The two nonbase sides could both have length, a, while the base remains 2c in length.
Move the pencil to the left to form a new triangle.  Keep string taut to restrict new triangle's shape. It can no longer be isosceles; instead, it will be scalene (three different sides). However, the sum of the two nonbase sides (Side1 + Side2) will still equal 2a.
Replace the pencil with an asteroid to continue making imaginary triangles such that two nonbase sides equal 2a. At the apex of these triangles, the asteroid draws an elliptical orbit around the Sun. Two pegs represent the two foci of the orbit.  Replace one peg with Sol while other focus retains an imaginary peg.  In this partially drawn orbit, note that radius, r, gradually grows to max distance, Q, orbit's aphelion, farthest distance from Sol.
This method of drawing an ellipse provides us with a formal definition, which we shall adopt in this chapter, of an ellipse, namely:
An ellipse is the locus of a point that moves such that the sum of its distances from two fixed points
called the foci is constant familiar equation to the ellipse  In the theory of planetary orbits, the Sun will be at one focus. Let us suppose it to be at F and both have defensible positions), and it is equal to Q = a(1+ e). 2.3.9
A line parallel to the minor axis and passing through a focus is called a
latus rectum (plural: latera recta). The length of a semi latus rectum is commonly denoted by l (sometimes by p). Its length is obtained by putting x = ae in the equation to the ellipse, and it will be readily found that l = a(1e2 ). 2.3.10
The length of the semi latus rectum is an important quantity in orbit theory.

It will be found, for example, that the energy of a planet is closely related to the semi major axis a of its orbit, while its angular momentum is closely related to the semi latus rectum.
The circle whose diameter is the major axis of the ellipse is ca


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