A 5 meter diameter telescope is used to observe binary star systems. What is the minimum separation distance (perpendicular to the line of sight) for which the two stars can be resolved if they are at a distance of 1,000 light years = 10 m and they are observed in visible light (y=500 nm)?

To determine the minimum separation distance for resolving the binary star system using a 5 meter diameter telescope, we can make use of a concept called angular resolution.

Angular resolution is the smallest angle at which two objects can be distinguished from one another. It is related to the wavelength of light being used and the diameter of the telescope.

The formula to calculate the angular resolution (θ) is given by:

θ = 1.22 * (λ / D)

Where:
- θ is the angular resolution in radians
- λ is the wavelength of light
- D is the diameter of the telescope

In this case, we are observing the binary star system in visible light, which has a wavelength (λ) of 500 nm (or 500 * 10^-9 meters). The diameter of the telescope (D) is 5 meters.

Plugging these values into the formula, we get:

θ = 1.22 * (500 * 10^-9) / 5

Simplifying, we have:

θ = 1.22 * 10^-7 radians

Now, the minimum separation distance can be calculated using trigonometry. The distance (d) is related to the angular resolution (θ) and the distance to the binary star system (r) by the following formula:

d = r * θ

In this case, the distance to the binary star system is 1,000 light years = 10 meters. Plugging in the values, we have:

d = 10 * 1.22 * 10^-7

Simplifying, we get:

d = 1.22 * 10^-6 meters

Therefore, the minimum separation distance for which the two stars can be resolved is approximately 1.22 * 10^-6 meters.