I was studying about mirage when i noticed that there was no (at least I could not find any) quantitative study of mirage. Most places (websites and textbooks) I looked gave only a qualitative overview of mirage (why it occurs, how it occurs, etc.). So I formulated a quantitative question on the topic.

Question: Imagine you are in a massive desert (assume it to be flat). There is a tower of height $H$. A light source at the top of the tower can emit light in all directions. You are a person whose eyes are at a height $h$ ($h<H$) above the ground. The refractive index of the air varies as $$\mu=kd$$ where $k$ is a positive constant and $d$ is the distance above the ground.

  1. What is the minimum distance at which you will still be able to see the mirage.
  2. What is the minimum possible distance between you and the mirage.
  3. If the ray starts off making an angle $\alpha$ below the horizontal what is the condition that a mirage is created(the ray does not strike the ground)

(The 3 questions are independent of each other)

My approach: I started off by assuming an element of height $dx$ at an elevation $x$ above the ground. Then I assumed that the ray of light is incident at an angle of $\theta$ and that the angle of refraction is $\theta+d\theta$. the refractive index changes from $\mu (x+dx) $ to $\mu (x) $. So,
$$\mu (x+dx) \cdot\sin(\theta) = \mu (x)\cdot\sin(\theta + d\theta)$$

Solving this gave me weird results. I think it was because I was unable to incorporate Total Internal Reflection into it.

If somebody could either point me in the right direction for solving the question or point out an error (if there is one) in the question then I would be grateful.