Question

Electromagnetic radiation from a star is observed with an earth-based telescope. The star is moving away from the earth at a speed of 0.520c. If the radiation has a frequency of 8.64 \(\times\) 10\(^{14}\) Hz in the rest frame of the star, what is the frequency measured by an observer on earth?

Short answer

The frequency measured by an observer on Earth is approximately \(4.85 \times 10^{14}\text{ Hz}\).

Step-by-step solution

Identify the Doppler Effect Formula for Light

To find the frequency measured by an observer on Earth when a star is moving away, we can use the relativistic Doppler effect formula for electromagnetic waves: \[ f' = f \sqrt{\frac{1 - \beta}{1 + \beta}} \] where \( f' \) is the observed frequency, \( f \) is the emitted frequency, and \( \beta = \frac{v}{c} \) is the ratio of the star's velocity to the speed of light. Here, \( v = 0.520c \), so \( \beta = 0.520 \).

Insert Given Values into the Formula

We are given \( f = 8.64 \times 10^{14} \text{ Hz} \) and \( \beta = 0.520 \). Substitute these into the Doppler effect formula: \[ f' = 8.64 \times 10^{14} \text{ Hz} \times \sqrt{\frac{1 - 0.520}{1 + 0.520}} \].

Calculate the Frequency Ratio

Calculate the expression inside the square root: \[ \sqrt{\frac{1 - 0.520}{1 + 0.520}} = \sqrt{\frac{0.480}{1.520}} \].

Simplify the Expression

Further simplify the ratio: \( \frac{0.480}{1.520} = 0.3158 \). Then calculate the square root: \( \sqrt{0.3158} \approx 0.5618 \).

Calculate the Observed Frequency

Multiply the emitted frequency by the calculated ratio: \[ f' = 8.64 \times 10^{14} \text{ Hz} \times 0.5618 = 4.85 \times 10^{14} \text{ Hz} \]. This is the frequency of the radiation as observed on Earth.

Key concepts

Electromagnetic Radiation

Electromagnetic radiation is a form of energy that travels through space as waves. These waves include visible light, radio waves, X-rays, and more. They all move at the speed of light, which is about 299,792 kilometers per second.

Understanding electromagnetic radiation is key in astronomy since it allows us to observe distant objects in the universe. Different types of electromagnetic waves provide us with information about the composition, temperature, and motion of celestial objects.

When a star emits light, it does so in its rest frame, meaning the frequency of the light is specific to that star. However, due to the motion either towards or away from us (the observer), we detect this radiation differently. This is where the Doppler effect comes into play, altering the frequency we observe.

Relative Velocity

Relative velocity is the speed of an object as observed from another moving frame of reference. For instance, when a star moves away from Earth, it does so at a certain velocity. This relative motion affects how we perceive the light it emits.

In astrophysics, particularly with the Doppler effect, it's crucial to consider this velocity relative to the speed of light, especially when dealing with high speeds close to light's speed. This is critical because speeds approaching that of light require relativistic corrections to accurately describe their effects.

In the context of the exercise, the star's velocity relative to Earth is a fraction of the speed of light, specifically 0.520c. This means that the star is moving at 52% of the speed of light, leading to noticeable changes in the observed frequency of the electromagnetic radiation.

Observed Frequency

Observed frequency refers to the frequency of a wave as measured by an observer in a different frame of reference than the source of the wave. Due to the Doppler effect, the frequency of light from moving celestial bodies changes based on their velocity and direction relative to us.

If a star moves away from Earth, the light waves stretch out, a phenomenon known as redshifting. This decreases the frequency of the light, making it lower than what is emitted in the star's rest frame. Conversely, if the star were moving towards us, the light would blueshift, increasing its frequency.

In our example, the star's emitted light has a frequency of 8.64 × 10\(^{14}\) Hz. However, because the star is moving away, the frequency observed on Earth decreases to approximately 4.85 × 10\(^{14}\) Hz.

Relativistic Doppler Effect

The relativistic Doppler effect accounts for the changes in wavelength and frequency when light sources move at velocities close to the speed of light. The standard Doppler effect equations need modification at such high speeds to incorporate principles from Einstein's theory of relativity.

In essence, the relativistic Doppler effect considers the time dilation—that is, time appears to run slower for objects moving rapidly relative to an observer. This phenomenon results in significant changes in observed frequencies of light when compared to the classical approach.

Using the relativistic Doppler effect formula \[ f' = f \sqrt{\frac{1 - \beta}{1 + \beta}} \] where \( \beta = \frac{v}{c} \), we correct the classical interpretation to accommodate high-speed scenarios like the star in question moving at 0.520c away from Earth. This adjustment provides a more accurate observed frequency, demonstrating the importance of relativistic effects in astrophysical observations.