Two stars are 4.40 1011 m apart and are equally distant from the earth. A telescope has an objective lens with a diameter of 1.03 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.

Question

Two stars are 4.40 1011 m apart and are equally distant from the earth. A telescope has an objective lens with a diameter of 1.03 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.

Answer 1

Using the Bessel equation estimate

theta= 1.22 wavlength/diameter

but for small angles, theta=separation/distance

so
distance=separation*diameter/ 1.22*wavelength

which is about= 4.4E11*1.03/1.22*550E-9
=about 4E18/6= .7E18m

check my math, you do it more accurately.

Answer 2

To find the maximum distance that these stars could be from the earth, we need to use the formula for resolving power.

Resolving power (R) is given by the equation:

R = 1.22 * (wavelength / D)

where R is the resolving power, wavelength is the wavelength of light being observed, and D is the diameter of the objective lens of the telescope.

In this case, the resolving power of the telescope is just enough to detect the stars as separate objects. This means that the resolving power should be equal to 1. Let's substitute the given values into the equation and solve for the maximum distance (d):

1 = 1.22 * (550 nm / D)

First, we need to convert the wavelength from nanometers to meters:

550 nm = 550 x 10^-9 m

Now, rearranging the equation to solve for D:

D = 1.22 * (550 x 10^-9 m)

D ≈ 6.71 x 10^-7 m

Now we have the diameter of the objective lens (D). To proceed, we will use the formula for angular diameter (θ) of an object:

θ = tan^−1 (d/D)

where θ is the angular diameter, d is the actual diameter of the object, and D is the distance to the object.

Let's solve for the angular diameter of the stars:

θ = tan^−1 (4.40 x 10^11 m / d)

Since the stars are equally distant from the earth, their angular diameter will be the same:

θ = θ = tan^−1 (4.40 x 10^11 m / d)

Since the resolving power is just enough to detect the stars as separate objects, the angular diameter should be equal to the resolving power, which is 1 radian.

1 = tan^−1 (4.40 x 10^11 m / d)

Now, solve for the maximum distance (d):

d = (4.40 x 10^11 m) / tan(1 rad)

Using a calculator, the maximum distance is approximately 7.79 x 10^11 meters.

Therefore, the maximum distance that these stars could be from the earth is approximately 7.79 x 10^11 meters.