Earth tugs the Sun around as it orbits, but it has a much smaller effect (only \(0.09 \mathrm{m} / \mathrm{s}\) ) than that of any known extrasolar planet. How large a shift in wavelength does this effect cause in the Sun's spectrum at \(500 \mathrm{nm} ?\)

The Doppler effect is a great way to understand how wavelengths change when an object moves. Think about a police siren. As it speeds towards you, the sound pitch gets higher; as it moves away, the pitch lowers. This principle applies to light, too! When a source of light, like a star, moves towards or away from us, its light wavelength shifts.
In our exercise, we want to know how the Earth's pull on the Sun affects the Sun's light spectrum.
Using the Doppler effect formula: \[\frac{\Delta \lambda}{\lambda_{0}} = \frac{v}{c}\], we can find this shift.
Here, \( \Delta \lambda \) is the change in wavelength, \( \lambda_0 \) is the original wavelength (500 nm), \( v \) is the relative velocity (0.09 m/s), and \( c \) is the speed of light. By solving this formula, we determine how much the wavelength changes due to Earth's motion.

Relative velocity is crucial in understanding the Doppler effect. It measures how fast one object moves in relation to another. In astronomy, this concept helps determine how stars and other cosmic objects move.
Let's dive into the scenario from the exercise. The Earth's orbit around the Sun slightly tugs the Sun, causing a velocity of 0.09 m/s. Even though this speed is tiny, it can still influence the light we receive from the Sun.
Knowing the relative velocity, you can use it in the Doppler effect formula to calculate the wavelength shift. The smaller the velocity, the smaller the impact on the wavelength. In our case, because the relative velocity is 0.09 m/s, the Sun’s light spectrum alteration remains minuscule.

The speed of light, denoted as \( c \), is one of the fundamental constants in physics. It travels at \( 3 \times 10^{8} \) meters per second in a vacuum (which is extraordinarily fast!).
This constant plays a key role in many formulas, including the one for the Doppler effect. When calculating the shift in wavelength due to relative motion, the speed of light helps determine the magnitude of this shift.
In our exercise, plugging the speed of light into the Doppler effect formula: \[ \Delta\lambda = \frac{v}{c} \times \lambda_{0}\], allows us to calculate how much the wavelength of light changes as a result of the relative velocity between the Earth and the Sun.
Understanding the speed of light's role in these calculations reveals just how dynamic and interconnected astronomical observations can be.