Problem 41 - Astronomy: At Play in the Cosmos

Question

The wavelength in the laboratory for a particular spectral line is \(580 \mathrm{nm} .\) It is observed in the spectrum of a galaxy at \(660 \mathrm{nm}\) What is the recession velocity of the galaxy, in kilometers per second?

Step-by-Step Solution

Answer

The recession velocity of the galaxy is approximately 41370 km/s.

Step 1: Understand the Doppler Effect

To find the recession velocity of the galaxy using the change in wavelength, we need to apply the principles of the Doppler Effect, which relates the observed change in wavelength to the velocity of the source relative to the observer.

Step 2: Set Up the Formula for Redshift

The formula for redshift (z) is given by \( z = \frac{\lambda_{observed} - \lambda_{rest}}{\lambda_{rest}} \), where \( \lambda_{observed} \) is the wavelength observed, and \( \lambda_{rest} \) is the rest wavelength. For this problem, \( \lambda_{observed} = 660 \) nm and \( \lambda_{rest} = 580 \) nm.

Step 3: Calculate the Redshift (z)

Plug the values into the formula: \( z = \frac{660 \text{ nm} - 580 \text{ nm}}{580 \text{ nm}} = \frac{80 \text{ nm}}{580 \text{ nm}} \approx 0.1379 \).

Step 4: Relate Redshift to Velocity

For relatively small velocities (non-relativistic), the redshift is approximately equal to the recessional velocity divided by the speed of light \( (c) \). This gives the formula \( z \approx \frac{v}{c} \).

Step 5: Solve for Velocity

Using \( z = 0.1379 \), and \( c \approx 3 \times 10^5 \) km/s, solve for \( v \): \[ v = z \cdot c \approx 0.1379 \times 3 \times 10^5 \text{ km/s} \approx 41370 \text{ km/s}. \]

Key Concepts

Redshift Calculation

Redshift is a key concept in astronomy and cosmology, illustrating how light stretches as objects move away from us. This phenomenon occurs due to the Doppler Effect, the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the waves.
In the context of galaxies, redshift is calculated using the formula:

• \[ z = \frac{\lambda_{observed} - \lambda_{rest}}{\lambda_{rest}} \]

Here, \( \lambda_{observed} \) is the wavelength of the spectral line observed from the galaxy, and \( \lambda_{rest} \) is the wavelength of the same line, measured in the laboratory when the source is at rest relative to us. In our example, when a galaxy emits light observed at 660 nm instead of its usual 580 nm, the redshift becomes visible. This increase in wavelength indicates that the galaxy is moving away from us.
For our specific problem, plugging the values into the redshift formula results in a redshift of approximately 0.1379. This dimensionless number effectively measures how much the universe has expanded since the light left the galaxy.

Galaxy Recession Velocity

When discussing galaxies, we often want to measure how fast they are moving away from us—this is their recession velocity. Due to the universe's expansion, almost all distant galaxies exhibit redshift, meaning they are retreating.
For small velocities, redshift and velocity are directly proportional, expressed by the formula:

• \[ z \approx \frac{v}{c} \]

Here, \( z \) is the redshift, \( v \) is the recessional velocity, and \( c \) is the speed of light, approximately \( 3 \times 10^5 \) km/s. This simplification is valid for non-relativistic speeds where velocities are much less than the speed of light.
Given a redshift of 0.1379, we calculate the galaxy's recession velocity as:

• \[ v = z \cdot c \approx 0.1379 \times 3 \times 10^5 \text{ km/s} \approx 41370 \text{ km/s} \]

Understanding these values helps astronomers gauge not only the rate at which galaxies are receding but also infer details about the universe's expansion.

Spectral Line Wavelength

Spectral lines serve as crucial tools in studying galaxies, providing insights into a galaxy’s motion and properties. Each element emits or absorbs light at specific wavelengths, which appear as lines in the spectrum.
When we observe spectral lines from distant galaxies, the wavelength can be shifted compared to when the source is measured at rest. This is relevant due to the effect of a galaxy’s velocity on the light it emits. Such shifts are either towards the red end of the spectrum when moving away (redshift) or towards the blue when approaching (blueshift).
In the exercise given, the spectral line that normally measures 580 nm lengthened to 660 nm, indicating that the galaxy is receding from us.

• This wavelength shift is measurable and provides indirect evidence of the universe's vast expansion.
• By examining spectral lines, astronomers can determine velocities, composition, and distances of galaxies, significantly contributing to our understanding of cosmic evolution.

Through continuous observation of such shifts, we gain vital clues to the very nature and scope of our universe.