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Chapter 10: Problem 7

Suppose a planetary nebula is a spherical shell whose thickness is \(10 \%\) of its radius. Compare the length of the longest line of sight with that through the center, and relate your answer to the appearance of the planetary nebula.

Short Answer

Expert verified
The longest line of sight is 10% longer than the line through the center, giving the nebula a ring-like appearance.

Step by step solution

01

- Define the Variables

Let the radius of the planetary nebula be denoted by \( R \). The thickness of the shell is \(0.1 R \). The length of the longest line of sight through the shell would correspond to the diameter of the spherical shell, which is \( 2R \).
02

- Calculate Length Through the Center

Since the thickness of the shell is \(0.1 R \), the inner radius of the shell can be calculated as \( R_{inner} = R - 0.1R = 0.9R \). The line of sight through the center spans the entire diameter of the shell: \( 2R \).
03

- Calculate the Maximum Length Through the Shell

The longest line of sight within the shell will be along the diameter of the inner surface, corresponding to \(2(0.9R) = 1.8R \).
04

- Compare the Lengths

Compare the longest line of sight \(2R \) with the line of sight through the center \(1.8R \). The ratio of these lengths is \( \frac{1.8R}{2R} = 0.9 \).
05

- Relate to Appearance

The appearance of the planetary nebula will be more extended along the plane of the sky compared to the line of sight through the center, making it appear brighter and larger along the shell’s edges, showing a ring-like structure.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

spherical shell geometry
To understand planetary nebulae, it's essential to grasp the concept of spherical shell geometry. Imagine a three-dimensional shape that resembles a hollow ball or shell. A typical characteristic of this geometry is having a thin outer layer compared to its overall radius.
For our nebula, its thickness is only 10% of its radius.
With radius represented by \(R\), the thickness of the shell is \(0.1R\).
This thickness lightly influences observations, such as how light travels through different parts of the shell. Geometry like this helps to model celestial objects more accurately.
astronomical observations
Astronomers study and gather information from planetary nebulae using various observations. These observations help us understand their structures, compositions, and behaviors. By observing the light passing through different parts of the nebulae, scientists can determine how dense or thick its parts are.
For example, light traveling through the center of a nebula journeys a distance of \(2R\) because it covers the whole diameter.
Such observations can tell us a lot about the nebula's physical properties and dimensions. In our calculation, we saw that other lines of sight are shorter, proving that light traveling along different paths offers varied information.
planetary nebula structure
Planetary nebulae are intriguing because of their unique structures. Often resembling spherical shells, these cosmic entities consist of ejected material from dying stars. For our problem, understanding the shell's thickness (10% of its radius) helps visualize how light travels through it.
By comparing the different paths light can take, we learn that the nebula appears more extensive along the outer edges than through the center.
This structural detail is key to forming that beautiful ring-like appearance we often associate with planetary nebulae.
optical astronomy
Optical astronomy plays a vital role in studying planetary nebulae. By analyzing the light (visible spectrum) emitted or absorbed by these objects, astronomers can evaluate their properties.
For example, instruments like telescopes capture the light to produce images and spectra, which are crucial for studying the nebula's chemical makeup and structure.
The visual appearance, like the bright ring-shaped edges we calculated, help confirm the nebula's spherical shell structure. Optical astronomy thus provides deep insights into these fascinating celestial phenomena.

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Most popular questions from this chapter

The momentum of a photon is \(E / c .\) (a) Calculate the momentum per second delivered to the outer layers of a \(10^{2} \mathrm{L}_{\odot}\) star if all the photons are absorbed in that layer. (b) How does the force on the layer compare with the gravitational force on the layer if the layer has a radius \(R=100 R_{\odot}\) and a mass \(M=0.1 \mathrm{M}_{\odot}\) and the rest of the star has a mass of \(1 \mathrm{M}_{\odot}\)

Use the fact that \(P \alpha \rho^{-1 / 2}, \rho \propto M / R^{3}\) and \(L \propto R^{2} T^{4}\) to derive a relationship between period and luminosity

(a) What is the kinetic energy in a \(0.1 \mathrm{M}_{\odot}\) planetary nebula, expanding at \(10^{3} \mathrm{km} / \mathrm{s} ?\) (b) Compare that with the gravitational potential energy of the shell when it was at \(R=100 R_{\odot},\) assuming that there is \(1 \mathrm{M}_{\odot}\) of material left behind.
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