Light Reflect Theory  

R = I_r / I_i = (c_1 - c_2) / (c_1 + c_2) * (1 - A) * f(λ) 

Lets use this modified equation to calculate the reflectance of a mirror used in a reflecting telescope to observe the Moon.

Here's an example:

Suppose we have a reflecting telescope with a mirror that has a diameter of 20 cm and a surface roughness that results in an absorptance of 5%. The telescope is pointed at the Moon, which has an average distance from Earth of about 384,400 km. We want to calculate the reflectance of the mirror at a specific wavelength of light, say 550 nm (green light).

First, we can calculate the speed of light in vacuum using the formula:

c = λ * f

where λ is the wavelength of light and f is its frequency. For green light with a wavelength of 550 nm, we have:

c = 550 nm * 5.45 × 10^14 Hz = 2.997 × 10^8 m/s

Next, we can calculate the speed of light in air (c_1) and in the mirror material (c_2). For simplicity, let's assume that the mirror is made of aluminum, which has a refractive index of about 1.3 for green light. Then we have:

c_1 = c = 2.997 × 10^8 m/s

c_2 = c / n = 2.997 × 10^8 m/s / 1.3 = 2.306 × 10^8 m/s

where n is the refractive index of the mirror material.

Next, we can calculate the fraction of incident light that is reflected by the mirror using the modified equation:

R = I_r / I_i = (c_1 - c_2) / (c_1 + c_2) * (1 - A) * f(λ)

Let's assume that the function f(λ) is a constant for green light, with a value of 0.85. Then we have:

R = (c_1 - c_2) / (c_1 + c_2) * (1 - A) * f(λ)

= (2.997 × 10^8 m/s - 2.306 × 10^8 m/s) / (2.997 × 10^8 m/s + 2.306 × 10^8 m/s) * (1 - 0.05) * 0.85

= 0.69

This means that about 69% of the incident green light from the Moon is reflected by the mirror, and the remaining 31% is either absorbed or scattered. Note that this is just an example, and the actual reflectance and other factors may vary depending on the specific telescope and mirror used, as well as the properties of the Moon's surface and the wavelength of the light observed.