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 stars using a 5-meter diameter telescope, we can use the concept of angular resolution.
The formula for calculating the angular resolution (θ) is given by θ = 1.22 * (λ / D), where λ is the wavelength of light being observed and D is the diameter of the telescope.
In this case, the wavelength of visible light (y) is given as 500 nm (or 500 * 10^-9 meters), and the diameter of the telescope (D) is 5 meters.
Plugging the values into the formula, we get:
θ = 1.22 * (500 * 10^-9 meters / 5 meters)
θ = 1.22 * 100 * 10^-9 radians
Simplifying, we find:
θ = 1.22 * 10^-7 radians
Now, to calculate the minimum separation distance (s), we can use the equation s = r * θ, where r is the distance to the binary star system.
In this case, r is given as 1,000 light years, which is equivalent to 10^3 parsecs or 10^3 * 3.09 * 10^16 meters.
Plugging in the values, we get:
s = 10^3 * 3.09 * 10^16 meters * 1.22 * 10^-7 radians
Simplifying and performing the multiplication, we find:
s ≈ 3.76 * 10^10 meters
Therefore, the minimum separation distance, perpendicular to the line of sight, for which the two stars can be resolved is approximately 37.6 billion meters.