A quasar emits familiar hydrogen lines whose wavelengths are 8.5% longer than what we measure in the laboratory.
(a) Using the Doppler formula for light, estimate the speed of this quasar.
(b) What result would you obtain if you used the “classical” Doppler shift discussed in Chapter 16?

Verified step by step guidance

1. Step 1: Understand the problem. The quasar's hydrogen lines are redshifted, meaning their wavelengths are longer than those measured in the laboratory. This redshift is due to the Doppler effect, which occurs when an object emitting light moves relative to the observer. We need to calculate the speed of the quasar using both the relativistic Doppler formula and the classical Doppler formula.
2. Step 2: Write down the relativistic Doppler formula for light. The formula is: $\frac{\sqrt{1+\frac{v}{c}}}{\sqrt{1-\frac{v}{c}}}=\frac{\lambda }{\lambda }$, where $\lambda$ is the observed wavelength, $\lambda \text{'}$ is the laboratory wavelength, $v$ is the speed of the quasar, and $c$ is the speed of light.
3. Step 3: Substitute the given information into the relativistic formula. The observed wavelength is 8.5% longer than the laboratory wavelength, so $\lambda =\lambda \text{'}\times 1.085$. Rearrange the formula to solve for $v$.
4. Step 4: Write down the classical Doppler formula for light. The formula is: $\frac{\mathrm{\Delta \lambda }}{\lambda }=\frac{v}{c}$, where $\mathrm{\Delta \lambda }$ is the change in wavelength ($\lambda -\lambda \text{'}$). Substitute the given information into this formula to solve for $v$.
5. Step 5: Compare the results from the relativistic and classical formulas. The relativistic formula accounts for effects at high speeds close to the speed of light, while the classical formula assumes lower speeds. Discuss why the relativistic formula is more accurate for astronomical objects like quasars.

Key Concepts

Doppler Effect

The Doppler Effect refers to the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. In the context of light, when a source moves away from an observer, the observed wavelength increases (redshift), while it decreases (blueshift) when the source approaches. This effect is crucial for understanding how the motion of astronomical objects, like quasars, affects the light we receive from them.

Redshift

Redshift is the phenomenon where light from an object is shifted to longer wavelengths due to the object's motion away from the observer. It is commonly used in astronomy to determine the speed and distance of celestial objects. The amount of redshift can be quantified and is directly related to the velocity of the object through the Doppler formula, making it essential for estimating the speed of quasars.

Classical vs. Relativistic Doppler Shift

The classical Doppler shift formula applies to speeds much less than the speed of light and assumes a simple relationship between the source and observer's velocities. In contrast, the relativistic Doppler shift accounts for the effects of special relativity, which become significant at high velocities, such as those of quasars. Using the correct formula is vital for accurate calculations of speed and wavelength shifts in high-velocity astrophysical contexts.