Problem 100 - Fundamentals of Physics

Question

Some of the familiar hydrogen lines appear in the spectrum of quasar \(3 \mathrm{C} 9,\) but they are shifted so far toward the red that their wavelengths are observed to be 3.0 times as long as those observed for hydrogen atoms at rest in the laboratory. (a) Show that the classical Doppler cquation gives a relative velocity of recession greater than \(c\) for this situation. (b) Assuming that the relative motion of \(3 \mathrm{C} 9\) and Earth is due entirely to the cosmological expansion of the universe, find the recession speed that is predicted by the relativistic Doppler equation.

Step-by-Step Solution

Answer

(a) Classical Doppler gives \( v = 2c \), which exceeds light speed. (b) Relativistic Doppler gives \( v = 0.8c \).

Step 1: Understanding the Doppler Shift

The classical Doppler effect relates the observed frequency (or wavelength) change to the relative velocity of the source. The classical formula for the Doppler shift of wavelengths is given by:\[\frac{\Delta \lambda}{\lambda} = \frac{v}{c},\] where \( \Delta \lambda = \lambda_{observed} - \lambda_{emitted} \), \( v \) is the velocity of the source relative to the observer, and \( c \) is the speed of light.

Step 2: Applying the Classical Doppler Equation

Given that the observed wavelength, \( \lambda_{observed} \), is 3 times the rest wavelength \( \lambda_{emitted} \), we can write:\[\frac{\Delta \lambda}{\lambda} = \frac{(3\lambda_{emitted} - \lambda_{emitted})}{\lambda_{emitted}} = \frac{2\lambda_{emitted}}{\lambda_{emitted}} = 2.\] Now substitute this into the Doppler equation:\[2 = \frac{v}{c} \implies v = 2c.\] This implies a recession velocity that is twice the speed of light according to the classical Doppler equation.

Step 3: Understanding the Limitation

Since the derived speed, using the classical equation, is greater than \( c \), it suggests a limitation in using classical mechanics for relativistic speeds, which occurs when velocities approach the speed of light. Therefore, the classical equation is not suitable for this scenario.

Step 4: Using the Relativistic Doppler Equation

The relativistic Doppler shift for a receding source is given by:\[\frac{\lambda_{observed}}{\lambda_{emitted}} = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}.\]Given \( \frac{\lambda_{observed}}{\lambda_{emitted}} = 3 \), substitute into the equation:\[3 = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}.\]

Step 5: Solving the Relativistic Equation

Square both sides to eliminate the square root:\[9 = \frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}.\] Cross multiply to solve for \( v \):\[9(1 - \frac{v}{c}) = 1 + \frac{v}{c}.\]Expand and simplify:\[9 - 9\frac{v}{c} = 1 + \frac{v}{c}.\]Combine like terms:\[8 = 10\frac{v}{c} \implies \frac{v}{c} = \frac{8}{10} = 0.8.\]Therefore, the recession speed \( v = 0.8c \).

Key Concepts

redshift

Redshift occurs when light or other electromagnetic radiation from an object undergoes an increase in wavelength, or shift toward the red end of the spectrum. This typically happens due to the object moving away from the observer.
Redshift is a key concept in astrophysics as it helps measure the speed and distance of galaxies moving away from us due to the expanding universe.

• The amount of redshift can tell us how fast the object is moving away.
• Measured by the change in wavelength, compared to the light emitted by the object in a stationary state.

The formula for redshift (\(z\)) is given by:\[z = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}}\]where \(\lambda_{observed}\) is the observed wavelength and \(\lambda_{emitted}\) is the emitted wavelength in a non-moving frame. This ratio helps astrophysicists to determine how the universe is expanding by observing distant quasars and galaxies.

recession velocity

Recession velocity describes the speed at which an astronomical object is moving away from the observer. It is a key measure in cosmology for understanding the expansion rate of the universe.
The recession velocity leads to the observed redshift due to the Doppler Effect.

• In classical physics, recession velocity can be calculated using the simple Doppler shift formula.
• However, as seen in our exercise, it can sometimes suggest speeds faster than light, which is not physically possible.

For such high velocities, relativistic physics comes into play to provide an accurate description. Instead of being limited by the speed of light, recession velocity in an expanding universe is based on an understanding from Einstein's theory of relativity.

relativistic physics

Relativistic physics deals with conditions where speeds approach the speed of light, requiring adjustments to classical mechanics. In our context, this especially influences how we calculate recession velocity.
In relativistic physics, the Doppler effect is modified to accommodate the high speeds by using the relativistic Doppler equation: \[\frac{\lambda_{observed}}{\lambda_{emitted}} = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}\]where \(v\) is the velocity of the source, and \(c\) is the speed of light. This formula avoids implying any object can move faster than light.

• It combines time dilation with the classical Doppler effect.
• Ensures velocity calculations remain within the bounds set by relativity.

Thus, relativistic physics provides tools to continue analyzing objects in motion near or at the speed of light, offering a more accurate understanding of cosmic expansion and high-speed astronomical phenomena.