Chapter 28, Problem 23E

Calculate the velocity, the distance, and the look-back time of the most distant galaxies in Figure 28.21 using the Hubble constant given in this text and the redshift given in the diagram. Remember the Doppler formula for velocity $\left(v=c\times \frac{\Delta \lambda }{\lambda }\right)$ and the Hubble law ( $v=H\times d$ , where d is the distance to a galaxy). For these low velocities, you can neglect relativistic effects.

Figure 28.21 Sloan Digital Sky Survey Map of the Large-Scale Structure of the Universe. This image shows slices from the SDSS map. The point at the center corresponds to the Milky Way and might say “You Are Here!” Points on the map moving outward from the center are farther away. The distance to the galaxies is indicated by their redshifts (following Hubble’s law), shown on the horizontal line going right from the center. The redshift $\text{z}=\Delta \lambda /\lambda$ , where Δ λ is the difference between the observed wavelength and the wavelength λ emitted by a nonmoving source in the laboratory. Hour angle on the sky is shown around the circumference of the circular graph. The colors of the galaxies indicate the ages of their stars, with the redder color showing galaxies that are made of older stars. The outer circle is at a distance of two billion light-years from us. Note that red (older stars) galaxies are more strongly clustered than blue galaxies (young stars). The unmapped areas are where our view of the universe is obstructed by dust in our own Galaxy. (credit: modification of work by M. Blanton and the Sloan Digital Sky Survey)