Question

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately 0.9670.967. The length of the major axis of the orbit is approximately 35.8835.88 astronomical units. (An astronomical unit is about 9393 million miles.)

Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the xx-axis.

Solution

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Given:

e=Eccentricity=0.9672a=Length major axis=35.88 astronomical units\begin{aligned} e&=\text{Eccentricity}=0.967 \\ 2a&=\text{Length major axis}=35.88\text{ astronomical units} \end{aligned}

We need to determine an equation for the ellipse, assuming that the origin corresponds with the center and the major axis is horizontal.

'slader'

Standard form of the equation of an ellipse (0<b<a0<b<a):

(xh)2a2+(yk)2b2=1Horizontal major axis(xh)2b2+(yk)2a2=1Vertical major axis\begin{aligned} \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}&=1 &\textcolor{#4559AC}{\text{Horizontal major axis}} \\ \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}&=1 &\textcolor{#4559AC}{\text{Vertical major axis}} \end{aligned}

  • The length of the major axis is 2a2a.
  • Center is (h,k)(h,k)
  • Eccentricity: e=cae=\frac{c}{a} with c2=a2b2c^2=a^2-b^2

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