Problem 78 - Fundamentals of Physics

Question

In the red shift of radiation from a distant galaxy, a certain radiation, known to have a wavelength of \(434 \mathrm{nm}\) when observed in the laboratory, has a wavelength of \(462 \mathrm{nm}\). (a) What is the radial speed of the galaxy relative to Earth? (b) Is the galaxy approaching or receding from Earth?

Step-by-Step Solution

Answer

(a) The radial speed is approximately 19,350,000 m/s. (b) The galaxy is receding from Earth.

Step 1: Understand the Doppler Effect for Light

When an astronomical object like a galaxy moves relative to an observer on Earth, the observed wavelength of light from the galaxy is shifted due to the Doppler effect. If the galaxy is moving away, the wavelength is shifted to the red end of the spectrum (redshift). Conversely, if it is moving closer, the shift is towards the blue end.

Step 2: Use the Redshift Formula

For light, the redshift (z) can be calculated using the formula: \[ z = \frac{\Delta \lambda}{\lambda_0} = \frac{\lambda - \lambda_0}{\lambda_0} \] where \( \lambda \) is the observed wavelength and \( \lambda_0 \) is the original wavelength observed in the laboratory.

Step 3: Calculate the Redshift (z)

Using the given wavelengths, \( \lambda = 462\, \text{nm} \) and \( \lambda_0 = 434\, \text{nm} \): \[ z = \frac{462 - 434}{434} = \frac{28}{434} \approx 0.0645 \]

Step 4: Determine the Radial Velocity (v)

For small redshifts, the radial velocity of the galaxy (v) can be related to the redshift by the formula: \[ v = zc \] where \( c \) is the speed of light, approximately \( 3 \times 10^8 \) m/s. Substitute \( z = 0.0645 \): \[ v = 0.0645 \times 3 \times 10^8 \approx 1.935 \times 10^7 \text{ m/s} \].

Step 5: Interpret the Sign of the Radial Velocity

A positive redshift value indicates that the galaxy is moving away from Earth, as the observed wavelength is longer than the rest wavelength.

Key Concepts

Doppler Effect

The Doppler effect is a phenomenon that occurs when a source of waves, such as sound or light, moves relative to an observer. In the context of astronomy, the Doppler effect helps us understand how the movement of galaxies affects the light we see from them. When a galaxy moves away from us, the light it emits shifts toward the red end of the spectrum, known as "redshift." This shift in wavelength affects how astronomers calculate the speed at which the galaxy is moving relative to Earth.
The principle behind this is similar to what we experience with sound waves. For instance, the pitch of a siren changes as an ambulance passes by. With light, red wavelengths are longer and indicate a galaxy is receding, while blue wavelengths, being shorter, would suggest it is approaching.

Wavelength

Wavelength is a measurement that describes the distance between consecutive peaks of a wave. In the case of light, different wavelengths correspond to different colors visible in the spectrum.

• Shorter wavelengths correspond to blue and violet light.
• Longer wavelengths are typically red and orange.

In the original exercise, wavelengths of light observed in a laboratory setting and from a distant galaxy help to determine if the galaxy is moving toward or away from us. The radiant emission known to have a wavelength of 434 nm in the laboratory shifts to 462 nm when observed in light from the galaxy. Such a change signifies a redshift, which is used to measure cosmic movements.

Radial Velocity

Radial velocity refers to the speed at which an object moves closer to or farther away from an observer. It's crucial in astronomy for understanding the motion of galaxies and other celestial bodies. Radial velocity can be determined by the amount of red or blue shift seen in the light spectrum. For small redshifts, the radial velocity can be calculated using a simple relation: \[ v = zc \] where \( z \) represents the redshift and \( c \) is the speed of light. If the redshift \( z \) is positive, it indicates that the object is moving away from the observer.

Speed of Light

The speed of light is a constant that plays a fundamental role in the physics of our universe. It is approximately \( 3 \times 10^8 \text{ m/s} \). This constant is used across various applications in physics, particularly in equations dealing with the behavior of light. In our exercise, the speed of light is used as part of the formula to calculate the radial velocity of a galaxy. When combined with the redshift \[ v = zc \] this helps astronomers determine how fast celestial objects are moving in space. Its significance is not just limited to calculations like radial velocity but also extends to understanding many universal phenomena.