Using Interference to Study Stars

Two telescopes, placed at a distance $D$ from one another, receive light of wavelength $\lambda$ from a distant star. The star is at an angle $\theta$ above horizon. Light from both telescopes is then transmitted to the sensor S which is located exactly halfway between them (see Fig. 6).

Figure: Stellar Interference.

(a)

What is the difference in the distance traveled by the light received by telescope A and telescope B?

(b)

What is the intensity of light measured by the sensor? Express your answer in terms of the angle $\theta$, distance $D$, wavelength $\lambda$, and the intensity $I_0$ that would be measured if only one of the telescopes was used.

(c)

What are the values of $\cos\theta$ for which the intensity is at a maximum? At a minimum?

(d)

If the star is moving, $\theta$ will change with time. By how much does $\cos\theta$ have to change to cause a large variation in the light intensity (e.g., from a maximum to the neighboring minimum)?

(e)

In a typical situation, $D\gg\lambda$ (typical numbers would be $D\approx 1$ m, $\lambda\approx 10^{-6}$ m). Explain how the setup described in this problem can be used to detect tiny shifts in star positions, unobservable by naked eye.