Two stars are 3.80 1011 m apart and are equally distant from the earth. A telescope has an objective lens with a diameter of 1.19 m and just detects these stars as separate objects. Assume that light of wavelength 550 nm is being observed. Also assume that diffraction effects, rather than atmospheric turbulence, limit the resolving power of the telescope. Find the maximum distance that these stars could be from the earth.

Angular separation

= 1.22 (wavelength)/(aperture dia)
= 5.64*10^-7
= 3.8*10^11/X

X = distance from Earth, in meters
= 6.7*10^17 m

To find the maximum distance that these stars could be from the Earth, we need to consider the resolving power of the telescope and its ability to distinguish between two close objects. Resolving power is determined by the diameter of the telescope's objective lens and is given by the formula:

θ = 1.22 * (λ / D)

Where:
θ is the angular resolution (in radians)
λ is the wavelength of light being observed
D is the diameter of the objective lens

In this case, the angular resolution corresponds to the angle between the two stars that the telescope can distinguish.

Given:
λ = 550 nm = 550 * 10^-9 m
D = 1.19 m

Let's calculate the angular resolution:

θ = 1.22 * (550 * 10^-9 m / 1.19 m)
θ ≈ 5.63 * 10^-7 radians

Now, let's consider the distance and geometry of the problem. The angle formed by the two stars at the Earth is the same angle formed by the two stars at the telescope. This angle can be calculated using the formula:

α = d / r

Where:
α is the angle formed by the two stars (in radians)
d is the distance between the two stars
r is the distance from the telescope to the stars (which we want to find)

Given:
d = 3.80 * 10^11 m
α = 5.63 * 10^-7 radians

Rearranging the formula, we can solve for r:

r = d / α
r ≈ (3.80 * 10^11 m) / (5.63 * 10^-7 radians)
r ≈ 6.75 * 10^17 m

Therefore, the maximum distance that these stars could be from the Earth is approximately 6.75 * 10^17 meters.