Final answer:
A 0.8 m telescope would require approximately 253 minutes to make an image of the same quality as a 9 m telescope that takes an exposure of 2 minutes, due to the larger telescope's greater light collection capability.
Step-by-step explanation:
To determine how long you would need to observe the same galaxy with a 0.8 m telescope compared to a 9 m telescope to make an image of the same quality, we can use the square of the ratio of their diameters to find the difference in light collecting area. The amount of light collected by a telescope, and thus the exposure time needed for a comparable image quality, is proportional to the area of the aperture (which is πd²/4, with d being the diameter).
For the 9 m telescope, which has an area of π*(9²)/4, we can compare this to the 0.8 m telescope (π*(0.8²)/4), by calculating the ratio of their areas:
π*(9²)/4 divided by π*(0.8²)/4 equals (9/0.8)², which simplifies to (81/0.64), or approximately 126.56.
This means the 9 m telescope collects roughly 126.56 times more light than the 0.8 m telescope. Since the smaller telescope collects less light, it will require more exposure time to collect the same amount of light. Therefore, if the 9 m telescope required 2 minutes of exposure, the 0.8 m telescope would need:
2 minutes * 126.56 ≈ 253.12 minutes, or approximately 253 minutes for an equivalent exposure.