What is the size of the smallest craater on the moon that you can see with a telescope with a resolution of 2 arc seconds at a distance of 384,400 km?away
First, convert 2 arc seconds to radians.
2 arc sec*(1 degree/3600 arc sec)/57.3 degree/rad = 9.7*10^-6 rad
Finally, multiply that by the distance to the moon.
9.7*10^-6*384,400 = 3.7 km
is the smallest resolvable object size.
Two arc seconds is typical for a good earth-based telescope with about a 10 cm aperture under good seeing conditions.
To determine the size of the smallest crater visible with a telescope, we need to consider the resolution of the telescope and the distance to the moon.
The formula to calculate the angular resolution of a telescope is given by:
θ = 1.22 * (λ / D),
where θ is the angular resolution, λ is the wavelength of light, and D is the diameter of the telescope.
Assuming the wavelength of light is 550 nm (green light) and the diameter of the telescope is irrelevant, as we are given the resolution directly, we can rearrange the formula to solve for the diameter of the smallest crater:
D = 1.22 * (λ / θ).
Given that the resolution of the telescope is 2 arc seconds, which is approximately 0.00056 degrees or 0.0000098 radians, and the distance to the moon is 384,400 km, we can proceed with the calculation.
D = 1.22 * ((550 nm) / (0.0000098 rad)).
Converting the wavelength to meters:
D = 1.22 * ((550 × 10^(-9) m) / (0.0000098 rad)).
Calculating:
D ≈ 67 meters.
Therefore, with a telescope with a resolution of 2 arc seconds at a distance of 384,400 km, you can see craters on the moon that have a size of approximately 67 meters or larger.
To determine the size of the smallest crater on the moon that can be seen with a telescope, we need to consider the resolving power of the telescope and the distance to the moon.
The resolving power of a telescope can be calculated using the formula:
θ = 1.22 * (λ/D)
Where:
θ = angular resolution in radians
λ = wavelength of light being observed
D = diameter of the telescope's objective lens or primary mirror
In this case, we need to convert the resolving power from radians to arc seconds. There are approximately 206,265 arc seconds in a radian, so:
θ (arc seconds) = θ (radians) * 206,265
Given that the telescope has a resolution of 2 arc seconds, we can rearrange the formula to solve for D:
D = 1.22 * (λ/θ)
Now let's plug in the values:
θ = 2 arc seconds
λ = assumed visible light wavelength of 550 nm (nanometers)
Converting the wavelength of light from nanometers to meters:
λ = 550 * 10^(-9) m
Now, we can calculate the diameter of the telescope:
D = 1.22 * (550 * 10^(-9) m) / (2 arc seconds * (1 radian / 206,265 arc seconds))
Calculating this yields the approximate diameter of the telescope:
D ≈ 368,741 meters
So, with a telescope diameter of 368,741 meters and a resolution of 2 arc seconds, the smallest crater on the moon that can be seen is about that size.