---
title: "A satellite of mass \\(m\\) orbits a star of mass \\(M\\) (\\(M \\gg m\\)) in an ellipse with a fixed semi-major axis \\(a\\). The orbital eccentricity \\(e\\) can be varied over the interval \\(0 \\le e < 1\\), and the satellite's angular momentum about the star is given by \\(L = m\\sqrt{GMa(1-e^2)}\\). Which of the following statements correctly describes the orbital angular momentum \\(L\\), the radial turning points (\\(r_{\\text{min}}\\) and \\(r_{\\text{max}}\\)), and the orbital period \\(T\\) in the limit as \\(e \\to 1^-\\)?"
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url: "https://nerd-notes.com/ubq/124309/"
date_modified: "2026-09-28T14:04:43+00:00"
---

# A satellite of mass \(m\) orbits a star of mass \(M\) (\(M \gg m\)) in an ellipse with a fixed semi-major axis \(a\). The orbital eccentricity \(e\) can be varied over the interval \(0 \le e < 1\), and the satellite's angular momentum about the star is given by \(L = m\sqrt{GMa(1-e^2)}\). Which of the following statements correctly describes the orbital angular momentum \(L\), the radial turning points (\(r_{\text{min}}\) and \(r_{\text{max}}\)), and the orbital period \(T\) in the limit as \(e \to 1^-\)?

A satellite of mass \(m\) orbits a star of mass \(M\) (\(M \gg m\)) in an ellipse with a fixed semi-major axis \(a\). The orbital eccentricity \(e\) can be varied over the interval \(0 \le e < 1\), and the satellite's angular momentum about the star is given by \(L = m\sqrt{GMa(1-e^2)}\). Which of the following statements correctly describes the orbital angular momentum \(L\), the radial turning points (\(r_{\text{min}}\) and \(r_{\text{max}}\)), and the orbital period \(T\) in the limit as \(e \to 1^-\)?

![A horizontal coordinate axis with a filled black circle labeled M located to the right of the center. Two confocal ellipses are drawn centered along the horizontal axis, both sharing the exact same major axis length labeled 2a between their leftmost and rightmost endpoints. The wider ellipse is drawn with a thin dashed curve, and the inner, highly flattened ellipse is drawn with a thicker solid curve. Both ellipses share a focus at M. A horizontal double-headed dimension arrow extends above the shapes across the full width of the ellipses labeled 2a. Below the horizontal axis, a horizontal dimension arrow extends from M to the rightmost vertex labeled r_{\text{min}}, and another horizontal dimension arrow extends from M to the leftmost vertex labeled r_{\text{max}}. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604283-XIsTRl.jpg)

- **A.** \(L \to 0\), with turning points \(r_{\text{min}} \to 0\) and \(r_{\text{max}} \to 2a\), while the orbital period remains constant at \(T = 2\pi\sqrt{\dfrac{a^3}{GM}}\).
- **B.** \(L \to 0\), with turning points \(r_{\text{min}} \to 0\) and \(r_{\text{max}} \to \infty\), while the orbital period diverges such that \(T \to \infty\).
- **C.** \(L \to m\sqrt{GMa}\), with turning points \(r_{\text{min}} \to a\) and \(r_{\text{max}} \to a\), while the orbital period remains constant at \(T = 2\pi\sqrt{\dfrac{a^3}{GM}}\).
- **D.** \(L \to 0\), with turning points \(r_{\text{min}} \to 0\) and \(r_{\text{max}} \to 2a\), while the orbital period diverges such that \(T \to \infty\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124309/*