As you learned in the Spectroscopy lab, elements can emit light at specific wavelengths. Astronomers observe light from distance galaxies and identify these wavelengths. However, these specific wavelengths are shifted to longer wavelengths.

Astronomers observe a galaxy emitting light at
𝜆
= 9546 Å. From lab experiments on Earth, astronomers determine the expected, or rest-frame, wavelength to be
𝜆
𝑜
= 6364 Å.

Use the following equation to calculate the redshift for this galaxy.

𝑧
=
𝜆
−
𝜆
𝑜
𝜆
𝑜

Question

As you learned in the Spectroscopy lab, elements can emit light at specific wavelengths. Astronomers observe light from distance galaxies and identify these wavelengths. However, these specific wavelengths are shifted to longer wavelengths.

Astronomers observe a galaxy emitting light at 
𝜆
 =  9546 Å. From lab experiments on Earth, astronomers determine the expected, or rest-frame, wavelength to be 
𝜆
𝑜
= 6364 Å.

Use the following equation to calculate the redshift for this galaxy.

𝑧
=
𝜆
−
𝜆
𝑜
𝜆
𝑜

BlackTom AI · Accepted Answer

Redshift is defined by z = (λ - λ0) / λ0, which measures how much an observed wavelength λ has been stretched relative to its rest-frame wavelength λ0.

Relevant concepts:
- Rest-frame wavelength λ0 is the wavelength measured in the laboratory (here 6364 Å).
- Observed wavelength λ is what we measure from the distant galaxy (here 9546 Å).
- The redshift z is a dimensionless ratio; positive z means wavelengths are stretched to longer (redder) values.
- For moderate z, the observed-to-rest ratio equals 1 + z.

Step-by-step calculation:
1. Write the formula: z = (λ - λ0) / λ0.
2. Substitute the given values: λ = 9546 Å, λ0 = 6364 Å.
3. Compute the difference: λ - λ0 = 9546 Å - 6364 Å = 3182 Å.
4. Divide by λ0: z = 3182 Å / 6364 Å.
5. Cancel the units (Å/Å): z = 3182 / 6364 = 0.5.

Intermediate arithmetic: 6364 × 0.5 = 3182, so the division yields exactly 0.5. Also note that λ / λ0 = 9546 / 6364 = 1.5, which equals 1 + z; thus z = 1.5 - 1 = 0.5, confirming the same result.

Why this answer is correct:
The calculation follows the exact definition of redshift and uses precise substitution and arithmetic. The observed wavelength is 50% larger than the rest wavelength, so the redshift is 0.5.

Other options:
- No alternative choices were provided in this problem; any answer other than 0.5 would contradict the arithmetic shown above.

Conclusion:
The computed redshift is z = 0.5, which matches the provided correct answer.

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