The orbit of comet Hale-Bopp is an ellipse as shown below. The units are astronomical units (abbreviated AU). 1 AU ≈ 92,960,000 mi.
(a) Find an equation in standard form of the orbit of the comet. (Enter your answer in terms of AU.)
(b) The distance from the center, C, of the orbit to the sun is approximately 182.085 AU. Find the aphelion (the point at which the comet is farthest from the sun) in miles. Round to the nearest million miles.
mi
(c) Find the perihelion (the point at which the comet is closest to the sun) in miles. Round to the nearest hundred-thousand miles.
mi
To find the equation of the orbit in standard form, we need to identify the center of the ellipse, its major and minor axes, and the orientation of the axes.
(a) Equation of the Orbit:
Since the ellipse is centered at C, the equation of the orbit will have the form:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
where (h, k) represents the center of the ellipse, and a and b represent the lengths of the semi-major and semi-minor axes respectively.
From the diagram, we can see that the center of the ellipse is at the origin (0,0).
The distance from the center, C, of the orbit to the sun is given as 182.085 AU. This distance represents half of the major axis, so the length of the semi-major axis is 2 * 182.085 AU.
Therefore, a = 2 * 182.085 AU.
The minor axis of the ellipse represents the distance from the center to a point on the orbit perpendicular to the major axis. From the diagram, we can see that the length of the minor axis is 2 AU.
Therefore, b = 2 AU.
The equation of the orbit in standard form is:
x^2 / (182.085 AU)^2 + y^2 / (2 AU)^2 = 1
Simplifying the equation gives:
x^2 / 33150.926225 AU^2 + y^2 / 4 AU^2 = 1
Therefore, the equation of the orbit in standard form is:
x^2 / 33150.926225 + y^2 / 4 = 1
(b) Aphelion:
The aphelion is the point at which the comet is farthest from the sun. It occurs at the end of the major axis of the ellipse.
To find the aphelion in miles, we need to convert the AU distance to miles. 1 AU is approximately 92,960,000 miles.
The length of the semi-major axis is 2 * 182.085 AU, which is 2 * 182.085 * 92,960,000 miles.
Calculating this value gives:
2 * 182.085 * 92,960,000 = 33,802,880,000 miles.
Therefore, the aphelion is approximately 33,802,880,000 miles.
(c) Perihelion:
The perihelion is the point at which the comet is closest to the sun. It occurs at the opposite end of the major axis from the aphelion.
As we calculated in part (b), the length of the semi-major axis is 33,802,880,000 miles.
The perihelion occurs at the distance of one semi-major axis length from the center, C. So, the perihelion distance is 33,802,880,000 - (2 * 33,802,880,000) = -33,802,880,000 miles.
Therefore, the perihelion is approximately -33,802,880,000 miles.