The orbit of Halley's comet, last seen in 1986 and due to...

The orbit of Halley's comet, last seen in 1986 and due to return in $2062,$ is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is 36.18 AU. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles. $]$ Find a polar equation for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?

The orbit of Halley's comet, last seen in 1986 and due to
return in $2062,$ is an ellipse with eccentricity 0.97 and one
focus at the sun. The length of its major axis is 36.18 AU.
[An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles. $]$ Find a polar
equation for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?

Key Concepts

Aphelion Distance

The aphelion is the point on an orbital path where the object is farthest from the central body. In an elliptical orbit, the aphelion distance can be calculated using the formula a(1 + e), where a is the semi-major axis and e is the eccentricity. This concept is important when determining the maximum distance in an orbital path.

Polar Equation of a Conic Section

The polar equation for an ellipse with one focus at the origin is given by r = (a(1-e^2))/(1 + e cos ?), where a is the semi-major axis and e is the eccentricity. This formulation expresses the distance from the focus to any point on the ellipse as a function of the angle, making it a useful tool for describing orbital paths.

Ellipse

An ellipse is a conic section defined as the collection of points for which the sum of the distances from two fixed points (called foci) is constant. In orbital mechanics, many planetary and comet orbits are elliptical, with one of the foci occupied by the central body (such as the sun).

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being circular. It is a dimensionless number between 0 and 1 for ellipses, where 0 represents a perfect circle and values close to 1 indicate a highly elongated shape.

Semi-major Axis

The semi-major axis is half the length of the longest diameter of an ellipse. It establishes the scale of the ellipse and is fundamental in calculations related to orbital mechanics, such as determining the orbital period and the distances at different points along the orbit.

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