91Ó°ÊÓ

The period of a pulsating variable star may be estimated by considering the star to be executing radial longitudinal pulsations in the fundamental standing wave mode. That is, the star's radius varies periodically with time, with a displacement antinode at the star's surface. (a) Would you expect the center of the star to be a displacement node or antinode? (b) By analogy with a pipe with one open end, show that the period of pulsation \(T\) is given by $$T=\frac{4 R}{\langle v\rangle}$$ where \(R\) is the equilibrium radius of the star and \(\langle v\) ) is the average sound speed in the material of the star. (c) Typical white dwarf stars are composed of material with a bulk modulus of \(1.33 \times 10^{22} \mathrm{~Pa}\) and a density of \(10^{10} \mathrm{~kg} / \mathrm{m}^{3}\). They have radii equal to \(9.0 \times 10^{-3}\) solar radius. What is the approximate pulsation period of a white dwarf?

Step by step solution

01

Determining the Center of the Star

To determine whether the center of the star is a node or an antinode, consider the fundamental standing wave in a radial mode. In such modes, the star's surface, which is an antinode, suggests that the center must be a node, as this is the only wave pattern that fits the boundary conditions.

02

Understanding the Analogy with a Pipe

The star's fundamental mode of radial pulsation can be analogous to a pipe with one closed end (center of the star) and one open end (the surface). The closed end (center) is a displacement node, and the open end (surface) is an antinode.

03

Deriving the Period of Pulsation

Based on the properties of a pipe with one closed end, the fundamental frequency of oscillation is \( f = \frac{v}{4L} \) where \( v \) is the wave speed and \( L \) is the length of the pipe. For a star, \( L = R \) (radius of the star) and \( v \) is the average sound speed, \( \langle v \rangle \). The period \( T \) is the inverse of frequency, hence\[ T = \frac{4R}{\langle v \rangle} \]

04

Calculating the Average Speed of Sound

The average speed of sound \( \langle v \rangle \) in the material can be calculated using the formula \( \langle v \rangle = \sqrt{\frac{B}{\rho}} \), where \( B \) is the bulk modulus and \( \rho \) is the density. Substituting the given values for a white dwarf:\[ \langle v \rangle = \sqrt{\frac{1.33 \times 10^{22} \text{ Pa}}{10^{10} \text{ kg/m}^3}} = \sqrt{1.33 \times 10^{12} \text{ m}^2/\text{s}^2} \approx 1.15 \times 10^6 \text{ m/s} \]

05

Finding the Radius of the White Dwarf

The radius of a white dwarf star is given as \( R = 9.0 \times 10^{-3} \) solar radius. One solar radius is \( 6.957 \times 10^8 \text{ m} \), thus:\[ R = 9.0 \times 10^{-3} \times 6.957 \times 10^8 \text{ m} \approx 6.261 \times 10^6 \text{ m} \]

06

Computing the Pulsation Period

Using the derived formula \( T = \frac{4R}{\langle v \rangle} \), substitute \( R \) and \( \langle v \rangle \) calculated from previous steps:\[ T = \frac{4 \times 6.261 \times 10^6 \text{ m}}{1.15 \times 10^6 \text{ m/s}} \approx 21.7 \text{ s} \]

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Key Concepts

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

Pulsating Variable Star

Pulsating variable stars are types of stars that show variations in brightness over time due to periodic size changes. These stars expand and contract in a regular cycle, causing changes in their luminosity. Such pulsations can occur in different modes, but in many cases, radial pulsations dominate. This means that the entire star pulsates in and out in a symmetric manner.

Observing and studying these pulsations allow us to gain insights into the internal properties of stars, such as their radius and sound speed. By analyzing the period of pulsation, astronomers can infer important physical characteristics and processes occurring within these stars.

Radial Longitudinal Pulsations

Radial longitudinal pulsations refer to pulsations where the star's radius changes, causing the star to expand and contract uniformly. These pulsations must be understood in terms of waves that travel within the star. During these pulsations, the star oscillates between larger and smaller radii.

In these radial modes, standing waves form within the star. Standing waves are patterns of vibration that remain in a fixed position. Key points along these waves, known as nodes, experience no displacement, while antinodes, located at the star's surface, experience maximum displacement.

Fundamental Standing Wave Mode

The fundamental standing wave mode is the simplest form of standing wave that fits within the star's structure. In this mode, the surface of the star acts as an antinode (point of maximum displacement), while the center is a node (point of no displacement).

When considering the star analogous to a pipe with one closed end and one open end, the closed end (center of the star) must be a node, and the open end (surface of the star) must be an antinode. This analogy helps in understanding how the star vibrates and calculating the period of these oscillations.

Bulk Modulus

Bulk modulus is a measure of a material's resistance to uniform compression. It is a key property in determining how sound waves travel through a star. The bulk modulus, denoted as \( B \), relates to the star’s density (\( \rho \)) and the average speed of sound within the star’s material.

The formula connecting bulk modulus to sound speed is given by:
\[ \langle v \rangle = \sqrt{\frac{B}{\rho}} \]
This indicates that higher bulk modulus or lower density results in a faster sound speed. For typical white dwarf stars, the bulk modulus is quite high, contributing to a substantial sound speed across the star.

Average Sound Speed

The average sound speed (\( \langle v \rangle \)) in a star's material is crucial for determining the period of pulsation of a star. Sound waves travel through the star's matter, and their speed depends on the material's properties, like bulk modulus and density.

To find the average sound speed in a material, use the formula:
\[ \langle v \rangle = \sqrt{\frac{B}{\rho}} \]
Understanding the average sound speed allows astronomers to calculate the star's pulsation period. For instance, substituting typical values for a white dwarf's bulk modulus and density provides the necessary sound speed to use in further calculations of pulsation period. By grasping the concept of sound speed, we can better comprehend the dynamic behaviors of pulsating stars.