Problem 54 - University Physics with Modern Physics

Question

Extrasolar Planets. In the not-too-distant future, it should be possible to detect the presence of planets moving around other stars by measuring the Doppler shift in the infrared light they emit. If a planet is going around its star at 50.00 \(\mathrm{km} / \mathrm{s}\) while emitting infrared light of frequency \(3.330 \times 10^{14} \mathrm{Hz},\) what frequency light will be received from this planet when it is moving directly away from us? (Note: Infrared light is light having wavelengths longer than those of visible light.)

Step-by-Step Solution

Answer

The observed frequency is approximately \(3.329 \times 10^{14}\) Hz.

Step 1: Understand the Doppler Effect Formula

The Doppler Effect formula for frequency shift when an object moves away is given by: \( f' = \frac{f}{1 + \frac{v}{c}} \) where \( f' \) is the observed frequency, \( f \) is the emitted frequency, \( v \) is the speed of the object, and \( c \) is the speed of light (\( 3.00 \times 10^8 \) m/s).

Step 2: Convert Speed to Consistent Units

The speed of the planet is given as 50.00 km/s. We need to convert this to meters per second for consistency with the speed of light: \( 50.00 \times 10^3 \) m/s.

Step 3: Plug Values into the Doppler Effect Formula

Substitute \( f = 3.330 \times 10^{14} \) Hz, \( v = 50.00 \times 10^3 \) m/s, and \( c = 3.00 \times 10^8 \) m/s into the Doppler Effect formula: \[ f' = \frac{3.330 \times 10^{14}}{1 + \frac{50.00 \times 10^3}{3.00 \times 10^8}} \]

Step 4: Calculate the Denominator Part

Calculate the fraction \( \frac{v}{c} = \frac{50.00 \times 10^3}{3.00 \times 10^8} \), which equals approximately \( 1.667 \times 10^{-4} \).

Step 5: Calculate the Observed Frequency

Now calculate \( f' = \frac{3.330 \times 10^{14}}{1 + 1.667 \times 10^{-4}} \). This results in an observed frequency \( f' \approx 3.329 \times 10^{14} \) Hz.

Key Concepts

Extrasolar Planets

Extrasolar planets, often referred to as exoplanets, are planets that exist beyond our solar system. They orbit stars other than our own sun. Detecting these distant worlds involves sophisticated techniques, as we can't simply "see" them with traditional telescopes due to their vast distance and the brightness of their host stars.
One of the most exciting methods for discovering extrasolar planets is through the Doppler Effect. By observing the tiny wobble in a star's light spectrum caused by gravitational pull from an orbiting planet, astronomers can infer the presence of these unseen worlds. This wobble causes a shift in the light's frequency—a lower frequency when the star moves away from us, and a higher frequency when it moves towards us.
Studying extrasolar planets allows scientists to better understand worlds beyond our own, potentially varying in size, composition, and atmosphere. Some may even reside in the "habitable zone," where conditions are right for liquid water and life as we know it.

Infrared Light

Infrared light is a type of electromagnetic radiation with wavelengths longer than visible light but shorter than radio waves. It is invisible to the human eye, but we experience it as heat.

• Every object, including planets and stars, emits infrared light to some degree. This emission helps astronomers study celestial bodies that are not easily observed in visible light.
• Infrared light is particularly useful in studying extrasolar planets. Since many of these planets are located at great distances, their emitted infrared radiation can still reach our telescopes, allowing us to detect them indirectly.

Infrared observations can reveal information about a planet's temperature, atmosphere, and surface composition. They serve as a crucial tool in the broader pursuit of understanding the universe.

Frequency Shift

Frequency shift is a pivotal concept when studying the Doppler Effect, especially in astronomy. It describes the change in frequency (or wavelength) of a wave in relation to an observer who is moving relative to the wave source.

• When an object, like a planet, is moving away from an observer, the light it emits shifts to a lower frequency. This is known as a "redshift," and it's essential in determining the speed and motion of celestial objects.
• The frequency shift helps scientists measure the velocity of an object in space without needing physical contact. By analyzing the amount of shift in the frequency, astronomers can deduce the object's speed and direction of movement.

Mathematically, the frequency shift is calculated using the formula: \( f' = \frac{f}{1 + \frac{v}{c}} \), where \( f \) is the original emitted frequency, \( v \) is the object's velocity, and \( c \) is the speed of light.
This formula is vital in the work of astronomers, allowing precise measurements that drive understanding of cosmic phenomena.