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The Eccentricity Story
Introduction
The concept of eccentricity, like the general equation Ax 2 ๎ Bxy ๎ Cy 2 ๎ Dx ๎ Ey ๎ F = 0 is a unifying concept for the conic sections: circle, ellipse, parabola, and hyperbola. One of the greatest uses of eccentricity is in astronomy. As we shall see, the circle has an eccentricity of zero while the eccentricity of an ellipse varies from 0 to 1, not inclusively. As the eccentricity of an ellipse approaches 1, it becomes more highly elongated. Planetary orbits all have eccentricities between 0 and 1. The eccentricity of the earth is approximately 0.0166666, indicating a very nearly circular orbit. At the other extreme is the eccentricity of the orbit of comet Kouhoutek which is 0.99993. This means it's orbit is an exceedingly elongated ellipse which requires the comet 200,000 years to complete! The figure above shows the relative scale of Kouhoutek's orbit, the ellipse, with the size of the solar system, the region inside the circle.
What is Eccentricity?
Definition: Let a fixed point F , called the focus, and a fixed line, L , called the directrix be given. Let S be the set of points such that given any point P in set S , the ratio formed by the distance from P to F divided by the distance from P to L is constant. The value of this ratio is called the eccentricity of set S. The diagram below illustrates this definition. Y D P Set S PF PD = e = eccentricity of set S F X directrix focus
Y
D (-q,y) Set S P (x, y) F (q, 0) x = -q X We will now apply the definition to the situation shown in the diagram above in which for convenience we have placed the focus at F (q, 0), and the directrix, x = -q, on opposite sides of the origin. Let point P be placed at (x, y).
Applying the Definition
According to the definition of the eccentricity, e , of the set of points P we require that PF PD = e. Using the distance formula we proceed to apply the definition to obtain an equation for set S. PF = e ๎ PD ๎ ๎๎^ x^ โ^ q ๎ 2 ๎ y 2 = e (^) ๎๎ x ๎ q ๎ 2 ๎ ๎ y โ y ๎ 2 ๎ x โ q ๎ 2 ๎ y 2 = e 2 ๎ x ๎ q ๎ 2 After squaring out and rearranging terms we get y 2 ๎ ๎ 1 โ e 2 ๎ x 2 โ 2 q ๎ 1 ๎ e 2 ๎ x = q 2 ๎ e 2 โ 1 ๎ (1)
e = 0
If we let e = 0, equation (1) becomes y 2 ๎ x 2 โ 2 qx ๎ q 2 = 0 y 2 ๎ ๎ x โ q ๎ 2 = 0 The graph of this equation is the single point (q, 0).
1 > e > 0
Now suppose e = 1. Substituting 1 for e in equation (1) results in
๎ x โ q ๎ 1 ๎ e 2 ๎ 1 โ e (^2) ๎ 2 ๎ 2 qe 1 โ e (^2) ๎
2 ๎^
y 2 ๎ 4 q 2 e 2 1 โ e (^2) ๎ = 1 (2) Compare this with ๎ x โ h ๎ 2 a
2 ๎^
๎ y โ 0 ๎ 2 b 2 =^1.^ Clearly^ a^ =^ 2 qe 1 โ e 2 ,^ and b 2 = 4 q 2 e 2 1 โ e 2 makes sense only if (1 โ^ e (^2) ) > 0 as b^2 is never negative. Therefore we require that 1 > e^2 which means that 0 < e < 1. Therefore equation (2) describes an ellipse whose eccentricity lies between 0 and 1.
e > 1
Let's take equation (2) and sneak a minus sign between the two terms on the left side, legally, of course with one other alteration done to the first term. ๎ x โ q ๎ 1 ๎ e 2 ๎ 1 โ e (^2) ๎ 2 ๎ 2 qe e 2 โ 1 ๎
2 โ^
y 2 ๎ 4 q 2 e 2 e 2 โ 1 ๎^ = 1 (3) Compare (3) with the standard form ๎ x โ h ๎ 2 a
2 โ^
๎ y โ 0 ๎ 2 b 2 =^1.^ We have a = 2 qe e 2 โ 1 , (^) and b^2 = ๎ 4 q 2 e 2 e 2 โ 1 ๎^ with both a being positive and b^2 being positive if (^) e^2 โ 1 ๎
0. In this case e^2 > 0 or e > 1 causes (3) to be a hyperbola.
Summary
We summarize the results in the table. Conic Section Eccentricity Ellipse 0 < e < 1 Parabola 1 = e Hyperbola 1 < e
What about the Circle?
We discuss the circle eccentricity by considering the ellipse. Look again at equation (2) which we repeat below in slightly altered form. ๎ x โ q ๎ 1 ๎ e 2 ๎ 1 โ e (^2) ๎ 2 ๎ 2 qe 1 โ e (^2) ๎
2 ๎^
y 2 ๎ 2 q e ๎^1 โ^ e (^2) ๎ 2 =^1 We have a^ =^ 2 qe 1 โ e 2 and^ b^ =^ 2 qe ๎^1 โ^ e 2.^ The graph is shown below. Y ๎ q ๎ 1 ๎ e 2 ๎ 1 โ e 2 ,^0 ๎ b a X If we let e approach 0, the ratio of b to a approaches 1. To see why, consider the following. b a
๎ 2 qe ๎^1 โ^ e (^2) ๎ ๎ 2 qe 1 โ e (^2) ๎
1 โ e 2 ๎^1 โ^ e 2 =^ ๎^1 โ^ e 2 As e ๎ 0 , b a ๎ 1 so that the size of b approaches the size of a, becoming equal to the radius of a circle. Thus the circle has eccentricity zero. The image on the next page shows seven graphs on the same coordinate system sharing the same focus and directrix. Eccentricities range from 0.04, 0.1, 0.4, 0.7, 1, 1.4, and 12. The second branch of the hyperbolas with eccentricities 1.4 and 12 are not shown.
Assorted Images of Interest
The eccentricity of the earth's orbit actually changes over time as shown in the graph below. Note the rhythmic pattern. Such changes, although small, do affect world temperatures. Haley's Comet is the most famous comet. Its orbit, shown relative to some of the planets is shown in the next image. All planetary orbits are conic sections with the sun at one focus. If a body comes close to the sun but travels at the critical speed, below which it would be captured, its path is a parabola. If moving a little faster, its path is one branch of a hyperbola. The last image
(yes, this has to end) shows these scenarios. New comets are carefully observed to determine if they have ever made a previous pass around the sun. Obviously if the path is a portion of a parabola or hyperbola, the comet is confirmed as being truly new.
