It is desirable to be able to measure the radial velocity of stars (using the Doppler effect) to an accuracy of \(1 \mathrm{~km} / \mathrm{s}\) or better. One complication is that radial velocities refer to the motion of the star relative to the Sun, while the observations are made using a telescope on the Earth. Is it important to take into account the motion of the Earth around the Sun? Is it important to take into account the Earth's rotational motion? To answer this question, you will have to calculate the Earth's orbital speed and the speed of a point on the Earth's equator (the part of the Earth's surface that moves at the greatest speed because of the planet's rotation). If one or both of these effects are of importance, how do you suppose astronomers compensate for them?

The Doppler effect is a phenomenon used to measure the velocity of an object relative to an observer by observing the change in frequency or wavelength of a wave emitted by the object. For instance, when a star moves towards Earth, the wavelengths of the light it emits compress, making them bluer (blue shift); conversely, when a star moves away, wavelengths stretch, making them redder (red shift). This effect is a cornerstone of astronomical measurements, particularly when attempting to gauge the radial velocity of a star with respect to the Sun.

To ascertain the radial velocity of a star with precision, adjustments for the Doppler effect are essential. Astronomers usually account for the relative motion of Earth when they observe stars because this motion affects the frequency of the incoming light. This interplay between observation and motion is delicate; even the Earth's own speed can influence the observed data significantly, which is why understanding both Earth's orbital and equatorial speeds is paramount.

Understanding Earth's orbital speed is critical when measuring the radial velocity of stars to an accuracy of 1 km/s or better. Earth orbits the Sun at an average speed calculated by the formula
\( v = \frac{2\pi r}{T} \), where \( r \) is the radius of Earth's orbit and \( T \) is the time taken to complete one orbit. Given that Earth orbits at approximately 30 km/s, it's clear that this speed must be taken into account when measuring the radial velocity of stars.

This orbital motion induces a Doppler shift in the observed spectra of stars, and to obtain an accurate measurement, it's crucial to adjust for Earth's motion around the Sun. Astronomical observations are generally corrected for this orbital speed to ensure that the radial velocities reflect the stars' movements relative to the Sun and not contaminated by Earth's orbital dynamics.

Similarly, Earth's equatorial speed is a vital component to consider for high-precision astronomical measurements. The formula \( v = \frac{2\pi R}{t} \) is used to measure this speed, with \( R \) representing Earth's radius and \( t \) the time of one rotation— one day. Earth's rotation at the equator moves the surface at about 0.5 km/s. Despite being half the resolution of the target accuracy for radial velocity measurements, it's still significant enough to influence observations.

Astronomers need to take into account this rotational speed particularly when making observations from telescopes situated on Earth's surface. Adjustments can be made to account for the observer's motion due to the planet's rotation. For highly precise measurements required in modern astronomy, even the smallest speed can result in a noticeable Doppler shift. Thus, compensation for the equatorial speed is as important as compensating for the orbital speed.