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Chapter 11: Problem 67

(a) Calculate the approximate age of the universe from the average value of the Hubble constant, \(H_{0}=20 \mathrm{km} / \mathrm{s} \cdot \mathrm{Mly} .\) To do this, calculate the time it would take to travel \(0.307 \mathrm{Mpc}\) at a constant expansion rate of \(20 \mathrm{km} / \mathrm{s} .\) (b) If somehow acceleration occurs, would the actual age of the universe be greater or less than that found here? Explain.

Short Answer

Expert verified
The approximate age of the universe, calculated using the given Hubble constant of \(H_{0} = 20 \frac{km}{s \cdot Mly}\), is found to be \(Age = \frac{0.307 * 3.0857 * 10^{19} km}{20 \frac{km}{s} * 3.1536 × 10^7 s}\) years. If acceleration occurs, the actual age of the universe would be greater than the calculated age, as the expansion rate was slower in the past, making it take longer for the universe to reach its present size.

Step by step solution

01

Convert units from Mpc to km

To calculate the time it takes to travel a certain distance, we must first convert the 0.307 Mpc to kilometers. 1 Mpc = 3.0857 ×10^19 km So, 0.307 Mpc = 0.307 × 3.0857 × 10^19 km
02

Calculate the travel time

Now that we have the distance in km, we can calculate the time it would take to travel that distance at a rate of 20 km/s. Time = Distance / Speed Time = (0.307 * 3.0857 * 10^19 km) / (20 km/s)
03

Convert time to years

The time we obtained previously is in seconds. To have a better understanding of the age, let's convert it to years. 1 year = 3.1536 × 10^7 seconds Time (years) = Time (s) / 3.1536 × 10^7
04

Part (a) result - Approximate age of the universe

By calculating the previous steps, we have now found the approximate age of the universe, according to the provided Hubble constant, to be: Age = Time (years)
05

Part (b) - Acceleration and the actual age of the universe

If acceleration occurs, it means that the expansion rate of the universe was slower in the past than it is currently. As a result, it would have taken a longer time for the universe to reach its present size. Therefore, the actual age of the universe would be greater than the calculated age based on a constant expansion rate. The actual age would consider the changing expansion rate, which has been accelerating, making it longer than when assuming a constant rate over time.

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