Final answer:
The Hubble constant corresponding to an approximate age of the universe of 10¹⁰ years, assuming a constant expansion rate, is approximately 30 km/s·Mly. For a universe approximately 2 x 10¹⁰ years old, the corresponding Hubble constant would be approximately 15 km/s·Mly.
Step-by-step explanation:
Calculating the Hubble Constant from Universe Age
To find the Hubble constant that corresponds to a universe age of 10¹⁰ years, we assume a constant expansion rate. We must determine the velocity at which two galaxies 1 million light-years apart move away from each other to achieve that separation over the said period. The Hubble constant is typically represented as the velocity per distance, with usual units of km/s per Mly (million light-years).
First, calculate the speed necessary for two galaxies to be separated by 1 Mly after 10¹⁰ years (10¹⁰ years = 10¹⁰ * 3.1536 * 10⁷ seconds = 3.1536 * 10¹⁷ seconds). Next, we divide the distance (1 Mly) by time (in seconds) to obtain the speed in light-years per second, and then convert it to km/s (1 light-year = 9.461 * 10¹ km).
Dividing 1 Mly by the total number of seconds gives a speed in Mly/s, which is then converted into km/s by considering how many kilometers are in a Mly:
Speed = 1 Mly / (3.1536 * 10¹⁷ s) = (9.461 * 10¹ km) / (3.1536 * 10¹⁷ s) ≈ 30 km/s
This speed of 30 km/s for every Mly is the Hubble constant for a universe aged 10¹⁰ years. Thus, the correct answer to the student's question is (c) 30km/s·Mly.
Calculating the Hubble Constant for a Different Universe Age
For a universe age of 2 x 10¹⁰ years, we conduct a similar calculation:
Speed = 1 Mly / (2 x 10¹⁰ years * 3.1536 * 10⁷ seconds/year) ≈ 15 km/s
The corresponding Hubble constant for a universe approximately 2 x 10¹⁰ years old would be 15 km/s·Mly, reflecting a slower expansion rate over a longer time span.