Two stars are 2.4 × 1011 m apart and are equally distant from the earth. A telescope has an objective lens with a diameter of 1.72 m and just detects these stars as separate objects. Assume that light of wavelength 590 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.
To find the maximum distance that these stars could be from the Earth, we can use the formula for the resolving power of a telescope:
θ = 1.22 * (λ / D),
where:
- θ is the angular resolution,
- λ is the wavelength of light being observed, and
- D is the diameter of the objective lens of the telescope.
In this case, we're given the wavelength of light (590 nm) and the diameter of the objective lens (1.72 m). Since the telescope just detects the stars as separate objects, we can set the angular resolution equal to the angular separation between the stars:
θ = 2 * d / D,
where d is the linear separation between the stars.
Rearranging the equation, we can solve for d:
d = θ * D / 2.
Therefore, the linear separation between the stars is:
d = (1.22 * λ * D) / (2 * D).
Substituting the given values:
d = (1.22 * (590 nm) * (1.72 m)) / (2 * (1.72 m)).
Calculating this expression gives us the linear separation d.
Now, we know that the distance between the two stars is 2.4 × 10^11 m, and they are equally distant from the Earth. Therefore, we can express the maximum distance that these stars could be from the Earth as:
distance = (d / 2) / tan(θ),
where θ is the angular separation between the stars.
Substituting the known values:
distance = ((2.4 × 10^11 m) / 2) / tan(θ).
Now, we have all the information needed to calculate the maximum distance. Calculate the value of tan(θ) using the earlier equation. Calculate the value of d using the second formula. Finally, substitute those values into the distance formula to find the maximum distance.