A galaxy's hydrogen- \(\beta\) spectral line, normally at \(486.1 \mathrm{nm},\) appears at \(495.4 \mathrm{nm}\). (a) Use the Doppler shift of Chapter 14 to find the galaxy's recession speed, and (b) infer the distance to the galaxy. Is it appropriate to use Chapter 14 's nonrelativistic Doppler formulas in this case?

Edwin Hubble's groundbreaking discovery in the early 20th century changed our understanding of the universe. He found that galaxies are moving away from us, and the farther away they are, the faster they seem to be receding. This relationship is encapsulated in Hubble's Law, a cornerstone of cosmological theory.

Formally, Hubble's Law can be expressed as the equation: \[ v = H_0 \times d \] where \(v\) is the recession speed of the galaxy, \(H_0\) (Hubble's constant) is the proportionality constant, and \(d\) is the distance of the galaxy from the observer. His observations imply an expanding universe, with Hubble's constant being a measure of the rate of this expansion.

To apply Hubble's Law practically, as seen in the solved exercise, astronomers observe the shift in the spectral lines of distant galaxies. By measuring this shift, they can determine the recession speed of the galaxies, which when plied into Hubble's Law, provides an estimate of the distance to the galaxies.

Note that the value of Hubble's constant has been refined with new observations and techniques over time, and its precise determination is still a subject of ongoing research.

Imagine you're hearing an ambulance siren coming towards you and then moving away; the pitch of the siren seems to change. This change in frequency is similar to the redshift seen in the light from galaxies, which is used to measure their recession speed. As galaxies move away from us due to the expansion of the universe, the light they emit is stretched into longer wavelengths, shifting towards the red end of the spectrum.

To calculate the recession speed (\(v\)), we use the change in the observed wavelength (\(\Delta \lambda\)) from the known original wavelength (\(\lambda\)), applying the formula: \[ v = \frac{\Delta \lambda}{\lambda} \times c \] where \(c\) is the speed of light. This method assumes the galaxy's movement is mostly along the line of sight from the observer.

In the original exercise, the concept of recession speed is pivotal as it forms the basis for estimating the distance to the galaxy using Hubble's Law. The accuracy of this method, however, depends on the redshift being relatively small such that relativistic effects can be ignored.

The Doppler effect is a phenomenon observed when there is a relative motion between a source of waves (like sound or light) and an observer. In simplified, non-relativistic scenarios, where the speeds involved are much less than the speed of light, the Doppler effect for light can be applied to find the shift in wavelength or frequency using the formula given in the exercise.

The key is to ascertain whether the speeds are non-relativistic. This is crucial as relativistic speeds (speeds approaching the speed of light) require Einstein's theory of relativity for accurate results. Compared to the speed of light, if the galaxy's recession speed is very small (non-relativistic), then the simple Doppler shift equation used in the exercise is indeed appropriate.

To illustrate using our original exercise, a spectral line shift from \(486.1 nm\) to \(495.4 nm\) suggests a non-relativistic speed because the observed change is quite small relative to the speed of light. Thus, the non-relativistic Doppler effect formula provides a reasonably accurate measurement of the galaxy's recession speed, which can then feed into Hubble's Law for distance estimation.