Suppose you observe two galaxies: one at a distance of \(10.7 \mathrm{Mpc}\) with a recessional velocity of \(580 \mathrm{km} / \mathrm{s}\), and another at a distance of 337 Mpc with a radial velocity of 25,400 \(\mathrm{km} / \mathrm{s}\). a. Calculate the Hubble constant ( \(H_{0}\) ) for each of these two observations. b. Which of the two calculations would you consider to be more trustworthy? Why? c. Estimate the peculiar velocity of the closer galaxy. d. If the more distant galaxy had this same peculiar velocity, how would your calculated value of the Hubble constant change?

Recessional velocity is the speed at which an astronomical object, such as a galaxy, is moving away from an observer. This concept is fundamental to Hubble's Law, where it is found that distant galaxies are generally moving away from us, and their recessional velocity increases with distance. This provides evidence of the expanding universe. Recessional velocity is measured by observing the redshift in the light emitted from the galaxy. Redshift occurs because the wavelengths of light stretch as the galaxy moves away, shifting toward the red end of the spectrum. By measuring this redshift, astronomers can determine the galaxy's recessional velocity.

In the exercise, two galaxies at different distances (10.7 Mpc and 337 Mpc) have recessional velocities of 580 km/s and 25,400 km/s, respectively. These values were used to calculate the Hubble constant.

Peculiar velocity refers to the motion of a galaxy relative to the overall expansion of the universe. Essentially, it's the velocity of an object that is not due to the expansion but rather to its local gravitational influences and interactions with nearby galaxies. This can cause a galaxy to move toward or away from us or other galaxies for reasons unrelated to the universe's expansion.

In the exercise, for the closer galaxy at 10.7 Mpc with a recessional velocity of 580 km/s, we calculated an expected recessional velocity using a more trustworthy Hubble constant (75.37 km/s/Mpc). This expected velocity was 806.46 km/s. The difference between the observed velocity and expected velocity gave us the peculiar velocity:

\[ v_{peculiar} = v_{observed} - v_{expected} \] \[ v_{peculiar} = 580 \text{ km/s} - 806.46 \text{ km/s} = -226.46 \text{ km/s} \]
Thus, the peculiar velocity is approximately -226.46 km/s.

The Hubble constant, denoted by \( H_0 \), is a crucial value in cosmology that describes the rate at which the universe is expanding. It relates the recessional velocity of galaxies to their distance from us. This relationship is expressed in Hubble's Law:

\[ v = H_0 \times d \]where:

• \( v \) is the recessional velocity (\text{km/s}),
\( d \) is the distance to the galaxy (\text{Mpc}),
• and \(H_0 \) is the Hubble constant.

In the exercise, we calculated the Hubble constant for two galaxies:

• For the first galaxy at 10.7 Mpc with a recessional velocity of 580 km/s, the Hubble constant was:
  \[ H_{0_1} = \frac{580 \text{ km/s}}{10.7 \text{ Mpc}} = 54.21 \text{ km/s/Mpc} \]
• For the second galaxy at 337 Mpc with a recessional velocity of 25,400 km/s, the Hubble constant was:
  \[ H_0_2 = \frac{25,400 \text{ km/s}}{337 \text{ Mpc}} = 75.37 \text{ km/s/Mpc} \]

Typically, the value of the Hubble constant obtained at larger distances is considered more accurate because the impact of peculiar velocities is less significant. Thus, in this exercise, the calculation for the second galaxy (\( H_0_2 = 75.37 \text{ km/s/Mpc} \) ) was determined to be more reliable.

If the second galaxy had a peculiar velocity similar to the first galaxy, it would affect the Hubble constant calculation. Recalculating with an adjusted recessional velocity (\( v_{adjusted} = 25,626.46 \text{ km/s} \)) gave a new Hubble constant:
\[ H_{0_{adjusted}} = \frac{25,626.46 \text{ km/s}}{337 \text{ Mpc}} = 76.04 \text{ km/s/Mpc} \] This slight difference illustrates how peculiar velocities can impact Hubble constant measurements.