A certain telescope has an objective mirror with an aperture...

A certain telescope has an objective mirror with an aperture diameter of 200 $\mathrm{mm}$ and a focal length of 2000 $\mathrm{mm}$ . It captures the image of a nebula on photographic film at its prime focus with an exposure time of 1.50 $\mathrm{min}$ . To produce the same light energy per unit area on the film, what is the required exposure time to photograph the same nebula with a smaller telescope that has an objective with a diameter of 60.0 $\mathrm{mm}$ and a focal length of 900 $\mathrm{mm}$ ?

Key Concepts

Exposure Time Scaling

Exposure time scaling involves adjusting the duration of the exposure to ensure that the same light energy per unit area is delivered to the film or detector. Since the intensity on the focal plane depends on the square of the inverse of the focal ratio, differences in focal ratio between telescopes require the exposure time to be scaled appropriately. In practical terms, if one telescope has a higher f-number, it delivers less light per unit area, and a longer exposure time is necessary to achieve an equivalent exposure.

Image Surface Brightness

For extended objects, such as nebulae, the brightness recorded on the imaging medium is determined by the surface brightness of the image rather than the total amount of light gathered. This brightness is governed by the telescope’s focal ratio, making it a key concept in comparing different telescopes. The same object will appear with different brightness in the focal plane if observed with telescopes having different f-numbers, even if one gathers more total light.

Focal Ratio

The focal ratio of a telescope is defined as the ratio of its focal length to its aperture diameter. This parameter is crucial because it determines how concentrated the light is when forming an image. A telescope with a lower focal ratio (faster system) produces a brighter image for extended objects, enabling shorter exposure times, whereas a higher focal ratio (slower system) spreads the light more thinly, requiring longer exposures.

Please give Ace some feedback