Problem 31 - Physics

Question

A distant galaxy emits light that has a wavelength of \(434.1 \mathrm{nm}\). On earth, the wavelength of this light is measured to be \(438.6 \mathrm{nm}\). (a) Decide whether this galaxy is approaching or receding from the earth. Give your reasoning. (b) Find the speed of the galaxy relative to the earth.

Step-by-Step Solution

Answer

The galaxy is receding, and its speed is approximately \(3.12 \times 10^6 \text{ m/s}\).

Step 1: Understand the concept of redshift and blueshift

When a galaxy is moving away from the Earth, the light emitted from it experiences a phenomenon known as the redshift, where the wavelength becomes longer, moving towards the red end of the spectrum. Conversely, if the galaxy is approaching the Earth, the light experiences a blueshift, where the wavelength becomes shorter, moving towards the blue end.

Step 2: Compare the wavelengths to determine motion

The original wavelength of the light from the galaxy is given as \(434.1 \text{ nm}\), and it is measured on Earth as \(438.6 \text{ nm}\). Since the observed wavelength on Earth is longer than the emitted wavelength, the light is redshifted. Therefore, according to Step 1, the galaxy is receding from the Earth.

Step 3: Use the redshift formula to calculate speed

The formula for redshift \(z\) is \(z = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}}\). Substitute the given values: \(z = \frac{438.6 \text{ nm} - 434.1 \text{ nm}}{434.1 \text{ nm}} = \frac{4.5 \text{ nm}}{434.1 \text{ nm}} \approx 0.0104\).

Step 4: Relate redshift to speed using the Doppler effect

For speeds much less than the speed of light, the redshift \(z\) is approximately equal to the ratio of the galaxy's velocity \(v\) to the speed of light \(c\): \(z \approx \frac{v}{c}\). Therefore, \(v \approx z \cdot c\). Using \(c = 3 \times 10^8 \text{ m/s}\), we have \(v \approx 0.0104 \times 3 \times 10^8 \approx 3.12 \times 10^6 \text{ m/s}\).

Key Concepts

Redshift and Blueshift

In astronomy, understanding the Redshift and Blueshift is crucial for determining the motion of celestial objects relative to Earth. When a source of light, like a galaxy, is moving away from an observer, the light stretches out, leading to redshift. This means the wavelengths appear longer, shifting towards the red part of the spectrum.

On the other hand, when a galaxy is approaching, its light waves compress, creating a blueshift. Here, the wavelengths become shorter, moving towards the blue end of the spectrum.

• If observed wavelength > emitted wavelength: **Redshift** (Galaxy is moving away)
• If observed wavelength < emitted wavelength: **Blueshift** (Galaxy is approaching)

In the provided exercise, the wavelength of light from a distant galaxy was longer when measured on Earth than when emitted, indicating a redshift. This tells us that the galaxy is moving away from Earth.

Galaxy Motion

Galaxy motion refers to the way galaxies travel through space. Their movement can be determined by analyzing the redshift or blueshift of light they emit. Observed spectral lines shifting indicate a galaxy's velocity in relation to Earth.

In the exercise, since the galaxy's light is redshifted, we conclude that it is receding from us. The speed of this motion is calculated using redshift formulas, which connect the change in wavelength to velocity. By substituting wavelength values into the redshift formula, we calculated the galaxy's speed.

• Redshift indicates moving away
• Blueshift indicates approaching
• Formula: \( z = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}} \)

By understanding galaxy motion, scientists can map the expansion of the universe and study its dynamics.

Wavelength and Light

The concept of wavelength and light is essential in studying astronomical phenomena. Wavelength defines the distance between consecutive peaks of a wave. In the context of light, different wavelengths are perceived as different colors.

Light emitted by galaxies can change in wavelength as a result of motion, thanks to the Doppler Effect. When light waves are stretched (redshift), the object is moving away. Conversely, when they are compressed (blueshift), the object is moving closer.

In examining the exercise, the initial wavelength of light from a galaxy was noted as \(434.1 \text{ nm}\), and upon reaching Earth, became \(438.6 \text{ nm}\). This increase indicates a redshift. The change in wavelength provides critical information about the galaxy's velocity and direction.

• Greater wavelength = redshift
• Shorter wavelength = blueshift
• Wavelength changes help determine celestial motion

Understanding wavelength and its implications allows astronomers to infer much about the position and motion of the cosmos.