/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 64 The comet Hale-Bopp has an ellip... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 64

The comet Hale-Bopp has an elliptical orbit with an eccentricity of \(e \approx 0.995 .\) The length of the major axis of the orbit is approximately 500 astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun?

Short Answer

Expert verified
The comet Hale-Bopp has its orbit described by the polar equation \(r = \frac{2.49375}{1 - 0.995 \cos\(\theta\)}\). The closest it comes to the sun is approximately 498.75 astronomical units.

Step by step solution

01

Calculate the semi-major axis length

The length of the major axis of the comet Hale-Bopp's orbit is 500 astronomical units. This means that the semi-major axis length is half of this length, so \(a = 500 / 2 = 250\) astronomical units.
02

Substitute the values

Plugging the values into the equation yields \(r = \frac{250(1 - 0.995^2)}{1 - 0.995 \cos\(\theta\)}\). Simplifying this yields \(r = \frac{250(1 - 0.990025)}{1 - 0.995 \cos\(\theta\)}\). This can be further simplified to \(r = \frac{250(0.009975)}{1 - 0.995 \cos\(\theta\)}\), yielding the polar equation of \(r = \frac{2.49375}{1 - 0.995 \cos\(\theta\)}\). This is the polar equation that describes the orbit of the comet Hale-Bopp.
03

Calculate the closest distance to the sun

Recall that the polar coordinates (r, 胃) of any point are related to the Cartesian coordinates (x, y) by the equations x = r cos 胃 and y = r sin 胃. Therefore, the distance between the comet and the sun, at any point on the orbit, is given by \(r = x / \cos\(\theta\)\). At its closest approach to the sun, the comet is at the perihelion of its orbit, where \(\theta = 0\). Thus, \(r = 2.49375 / (1 - 0.995 \cdot \cos\(0\))\). Computing this value results in \(r = 2.49375 / (1 - 0.995) = 2.49375 / 0.005 = 498.75\) astronomical units, which represents the closest distance of the comet Hale-Bopp to the sun.

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