Problem 45 - Astronomy: At Play in the Cosmos

Question

An astronomer calculated that the recession velocity of a galaxy was \(65,000 \mathrm{km} / \mathrm{s},\) by noting the redshift of a spectral line whose rest wavelength is \(420 \mathrm{nm}\). At what wavelength, in nanometers, did he observe the spectral line? (See Going Further \(16.1 .\) )

Step-by-Step Solution

Answer

The spectral line was observed at approximately 511 nm.

Step 1: Understand Redshift and Recession Velocity

The redshift phenomenon occurs when the light from a celestial object is shifted towards the red end of the spectrum, indicating the object is moving away from us. The recession velocity is the speed at which the galaxy is moving away, calculated here as 65,000 km/s.

Step 2: Formula for Observed Wavelength

The observed wavelength can be calculated using the formula \( \lambda_{o} = \lambda_{r} (1 + z) \), where \( \lambda_{o} \) is the observed wavelength, \( \lambda_{r} \) is the rest wavelength, and \( z \) is the redshift, which is defined by \( z = \frac{v}{c} \) where \( v \) is the recession velocity and \( c \) is the speed of light (approximately 300,000 km/s).

Step 3: Calculate the Redshift (z)

Compute the redshift using the formula \( z = \frac{v}{c} = \frac{65,000}{300,000} = 0.2167 \).

Step 4: Calculate the Observed Wavelength

Substitute \( \lambda_{r} = 420 \mathrm{nm} \) and \( z = 0.2167 \) into the equation \( \lambda_{o} = \lambda_{r} (1 + z) \). This gives: \( \lambda_{o} = 420 (1 + 0.2167) = 420 \times 1.2167 = 511.014 \mathrm{nm} \).

Step 5: Conclusion

The observed wavelength of the spectral line is approximately 511 nm when rounded to the nearest nanometer.

Key Concepts

Recession Velocity

Recession velocity is a crucial concept in astronomy, especially when studying the motion of galaxies. It refers to the speed at which a galaxy moves away from the Earth, signifying the ever-expanding universe. When galaxies recede, they cause the light they emit to stretch towards longer wavelengths, a phenomenon known as redshift.

• Recession velocity indicates how fast a galaxy is moving away from us.
• The recession of galaxies is a key piece of evidence for the expanding universe theory.
• In this exercise, the given recession velocity is 65,000 km/s, showcasing a considerable speed away from us.

Understanding this concept allows astronomers to explore how galaxies are distributed across the universe and gather insights about the universe's expansion and history.

Observed Wavelength

The observed wavelength is the wavelength of light as measured after it has been affected by the motion of its source. When galaxies move away from us, their light stretches, and the observed wavelength becomes longer than its original, or rest, wavelength. This phenomenon is crucial in understanding the redshift effect. Here, the observed wavelength can be calculated using the formula:\[ \lambda_{o} = \lambda_{r} (1 + z) \]

• \( \lambda_{o} \) is the observed wavelength.
• \( \lambda_{r} \) is the rest wavelength.
• \( z \) is the redshift (a measure of how much the wavelength has increased).

The measured change in wavelength gives astronomers vital information about the motion and distance of galaxies.

Spectral Line Rest Wavelength

The spectral line rest wavelength is the wavelength of light emitted by an element when it is not influenced by any external forces or motions. This concept serves as a reference point to understand shifts in wavelength caused by the motion of celestial bodies.

• The rest wavelength is the baseline used to determine how much the wavelength changes during redshift.
• In our exercise, it's given as 420 nm, typical of the waves emitted by certain elements found in stars and galaxies.
• By comparing the observed wavelength with the rest wavelength, scientists calculate the redshift and ultimately, the galaxy's recession velocity.

Such comparisons allow us to study the universe's mechanics beyond our immediate observations.

Speed of Light

The speed of light is a fundamental constant, denoted by \( c \), and is critical in equations involving redshift and recession velocity. It is approximately 300,000 km/s in a vacuum, making it the fastest speed at which information can travel. In redshift calculations, the speed of light is needed to determine the redshift (\( z \)) using the formula:\[ z = \frac{v}{c} \]

• \( v \) is the recession velocity.
• \( c \) is the constant speed of light.

This constant enables astronomers to gauge how fast galaxies are moving in relation to the speed at which light travels, offering insights into the vast expanses of the cosmos. Understanding how the speed of light impacts our observations is a foundational element of astrophysics and cosmology.