Chapter 1.4, Problem 28E

Hubble's Constant Astronomers believe that the universe is expanding and that stellar objects are moving away from us at a radial velocity $V$ proportional to the distance $D$ from Earth to the object.

a. Write $V$ as a function of $D$ using $H$ as the constant of proportionality.

b. The equation in part a was first discovered by Edwin Hubble in 1929 and is known as Hubble's law. The constant of proportionality $H$ is known as Hubble's constant. The currently accepted value of Hubble's constant is $70$ kilometers per second per megaparsec. (One megaparsec is about $3.086\times {10}^{19}$ kilometers.) With these units for $H$, the distance $D$ is measured in megaparsecs, and the velocity $V$ is measured in kilometers per second. The galaxy $\text{G}2237+305$ is about $122.7$ megaparsecs from Earth. How fast is $\text{G}2237+305$ receding from Earth?

c. One important feature of Hubble's constant is that scientists use it to estimate the age of the universe. The approximate relation is

$y=\frac{{10}^{12}}{H}$

Where $y$ is time in years. Hubble's constant is extremely difficult to measure, and Edwin Hubble's best estimate in 1929 was about $530$ kilometers per second per megaparsec. What is the approximate age of the universe when this value of $H$ is used?

d. The calculation in part c would give scientists some concern, since Earth is thought to be about $4.6$ billion years old. What estimate of the age of the universe does the more modern value of $70$ kilometers per second per megaparsec give?