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Chapter 7: Problem 39

The best current technology can measure radial velocities of about \(0.3 \mathrm{m} / \mathrm{s}\). Suppose you are observing a spectral line with a wavelength of 575 nanometers (nm). How large a shift in wavelength would a radial velocity of \(0.3 \mathrm{m} / \mathrm{s}\) produce?

Short Answer

Expert verified
\( \triangle \text{λ} = 5.75 \times 10^{-10} \text{ meters} \)

Step by step solution

01

- Understand the Doppler Shift Formula

The Doppler shift formula relates the change in wavelength (\triangle \text{λ}) to the original wavelength (λ) and the radial velocity (v). The formula is given by: \[ \frac{\triangle \text{λ}}{\text{λ}} = \frac{\text{v}}{\text{c}} \]where c is the speed of light, approximately \(3 \times 10^8 \) meters per second.
02

- Plug In Known Values

We know the original wavelength λ is 575 nm or 575 × 10^{-9} meters, the radial velocity v is 0.3 meters per second, and the speed of light c is \(3 \times 10^8\) meters per second. Plug these values into the Doppler shift formula: \[ \frac{\triangle \text{λ}}{575 \times 10^{-9}} = \frac{0.3}{3 \times 10^8} \]
03

- Solve for \triangle \text{λ}

Cross-multiply to solve for \triangle \text{λ}:\[ \triangle \text{λ} = \frac{0.3}{3 \times 10^8} \times 575 \times 10^{-9} \]Simplify the expression to find \[ \triangle \text{λ} = \frac{0.3 \times 575 \times 10^{-9}}{3 \times 10^8} \] \[ \triangle \text{λ} = 5.75 \times 10^{-10} \text{ meters} \]

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Key Concepts

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

Radial Velocity
Radial velocity refers to the speed at which an object moves towards or away from an observer. When observing celestial bodies, radial velocity helps us understand if these bodies are moving closer or drifting farther away. This concept is crucial in astronomy and becomes measurable thanks to the Doppler effect. If a star moves towards Earth, its spectral lines shift to shorter wavelengths (blue shift). If it moves away, they shift to longer wavelengths (red shift). Radial velocity is key in studying stellar dynamics, exoplanet detection, and the expansion of the universe.
Wavelength Shift
When an object emitting light moves relative to an observer, the observed wavelength of the light changes, creating a wavelength shift. This is a direct consequence of radial velocity. For example, in our exercise, we see how a tiny radial velocity (0.3 m/s) can cause a small wavelength shift. The shift is calculated using the Doppler shift formula: \( \frac{\triangle \text{\text{λ}}}{\text{\text{λ}}} = \frac{\text{v}}{\text{c}} \). Understanding wavelength shifts allows astronomers to determine the motion of distant stars and galaxies. Such shifts are typically very small but observable with precision instruments.
Speed of Light
The speed of light is a fundamental constant in physics, approximately \(3 \times 10^8 \) meters per second. It signifies the maximum speed at which information or matter can travel through the universe. In the Doppler shift formula, the speed of light (c) is crucial for relating the observed wavelength shift to the radial velocity. Because light travels so fast, even minor radial velocities in astronomical terms can be detected when measured against the speed of light. This constant helps us calculate the wavelength shift for moving celestial bodies.
Spectral Lines
Spectral lines are specific wavelengths of light emitted or absorbed by elements and compounds. These lines serve as a fingerprint for identifying the chemical composition and physical conditions of astronomical objects. When a star or galaxy moves, its spectral lines shift in wavelength due to the Doppler effect. Observing these shifts lets scientists measure the radial velocity of the source. The analysis of spectral lines is fundamental in fields like astrophysics, allowing for detailed study of stars, planets, and other celestial phenomena.
Doppler Effect
The Doppler effect is the change in frequency or wavelength of a wave concerning an observer moving relative to the wave source. This effect is responsible for the frequency shifts we see in light and sound waves. In astronomy, the Doppler effect explains why light from moving stars or galaxies appears shifted. For objects moving towards us, the light shifts to shorter wavelengths (blue shift), and for objects moving away, it shifts to longer wavelengths (red shift). The formula \( \frac{\triangle \text{λ}}{\text{λ}} = \frac{\text{v}}{\text{c}} \) quantifies this shift, enabling astronomers to measure celestial velocities.

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Most popular questions from this chapter

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