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Consider a comet which passes through its aphelion at a
Chapter 8, Problem 8.20(choose chapter or problem)
Consider a comet which passes through its aphelion at a distance \(r_{\max }\) from the sun. Imagine that, keeping \(r_{\max }\) fixed, we somehow make the angular momentum \(\ell\) smaller and smaller, though not actually zero; that is, we let \(\ell \rightarrow 0\). Use equations (8.48) and (8.50) to show that in this limit the eccentricity \(\epsilon\) of the elliptical orbit approaches 1 and that the distance of closest approach \(r_{\min }\), approaches zero. Describe the orbit with \(r_{\max }\) fixed but \(\ell\) very small. What is the semimajor axis \(a\)?
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QUESTION:
Consider a comet which passes through its aphelion at a distance \(r_{\max }\) from the sun. Imagine that, keeping \(r_{\max }\) fixed, we somehow make the angular momentum \(\ell\) smaller and smaller, though not actually zero; that is, we let \(\ell \rightarrow 0\). Use equations (8.48) and (8.50) to show that in this limit the eccentricity \(\epsilon\) of the elliptical orbit approaches 1 and that the distance of closest approach \(r_{\min }\), approaches zero. Describe the orbit with \(r_{\max }\) fixed but \(\ell\) very small. What is the semimajor axis \(a\)?
ANSWER:Step 1 of 3
From Eq. (8.48), \(c=\ell^{2} / \gamma \mu\).
Therefore, as \(\ell \rightarrow 0\), the length \(c \rightarrow 0\).
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