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The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. A planet's distance from the sun is 741,000,000 km at perihelion and 817,000,000 km at aphelion. Find an equation for the planet's orbit. (Place the origin at the center of the orbit with the sun on the x-axis. Use the following as necessary: x, y.)

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frika

Consider the attached ellipse. Let the sun be at the right focus. Then perihelion is at right vertex on the x-axis and aphelion is at the left vertex on the x-axis.

The distances:

  • from perihelion to the sun in terms of ellipse is a-c;
  • from aphelion to the sun in terms of ellipse is a+c.

Then

[tex]\left\{\begin{array}{l}a-c=741,000,000\\a+c=817,000,000\end{array}\right.[/tex]

Add these two equations:

[tex]2a=1,558,000,000 \\ \\a=779,000,000[/tex]

and subtract first equation from the second:

[tex]2c=76,000,000 \\ \\c=38,000,000.[/tex]

Note that [tex]b=\sqrt{a^2-c^2},[/tex] thus

[tex]b=\sqrt{779,000,000^2-38,000,000^2}=\sqrt{605,397,000,000,000,000}.[/tex]

The equation for the planet's orbit is

[tex]\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\Rightarrow \dfrac{x^2}{606,841,000,000,000,000}+\dfrac{y^2}{605,397,000,000,000,000}=1.[/tex]

ngarcia6054

Answer:

The shape of planetary orbits follows from the observed fact that the force of gravity between two objects depends on the square of the distance between them. ... Ellipses are closed so the planets we see in elliptical orbits stick around.

Step-by-step explanation:

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