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Chapter 11: Problem 66

(a) What is the approximate velocity relative to us of a galaxy near the edge of the known universe, some \(10 \mathrm{Gly}\) away? (b) What fraction of the speed of light is this? Note that we have observed galaxies moving away from us at greater than \(0.9 c\).

Short Answer

Expert verified
(a) The approximate velocity of the galaxy near the edge of the known universe is \(v = (70 \frac{\mathrm{km/s}}{\mathrm{Mpc}}) \times (10 \times 0.306601\, \mathrm{Mpc})\). (b) The fraction of the galaxy's velocity relative to the speed of light is \(f = \frac{v}{c}\).

Step by step solution

01

Calculate the distance in Megaparsecs (Mpc)

To use Hubble's law, we first need to convert the distance from Gigalight-years (Gly) to Megaparsecs (Mpc). Using the conversion factors: 1 Gly = 1 billion light-years and 1 light-year = 0.306601 Mpc, we have: \(D_{Mpc} = 10 \mathrm{Gly} \times \frac{1\, \mathrm{billion\, light\text{-}years}}{1\, \mathrm{Gly}} \times \frac{0.306601\, \mathrm{Mpc}}{1\, \mathrm{light\text{-}year}}\)
02

Calculate the velocity of the galaxy using Hubble's Law

Hubble's law states that the velocity of a galaxy (v) is directly proportional to its distance (D) with Hubble's constant (H) as the proportionality factor. The approximate value of Hubble's constant is 70 km/s/Mpc. Thus, we can write: \(v = H \times D_{Mpc}\)
03

Compute the velocity in km/s for the given distance of 10 Gly

To compute the velocity of the galaxy, substitute the value of H and D into the equation: \(v = (70 \frac{\mathrm{km/s}}{\mathrm{Mpc}}) \times (10 \times 0.306601\, \mathrm{Mpc})\)
04

Calculate the fraction of the velocity relative to the speed of light

To find the fraction of the galaxy's velocity relative to the speed of light (c), divide the galaxy's computed velocity (v) by the speed of light (c ≈ 300,000 km/s): \(f = \frac{v}{c}\)
05

Summary of the two parts of the exercise

(a) Using Hubble's Law and the given distance, we have calculated the approximate velocity of the galaxy near the edge of the known universe. (b) We found the fraction of the galaxy's velocity relative to the speed of light to compare it with the observed velocity of galaxies moving away from us at greater than 0.9c.

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