The atmosphere of earth extends upto height H and its refractive index varies with depth y from the top as `mu = 1 + (y)/(H)` . Calculate the apparent thickness of the atmosphere as seen by an observer in space.

Answer

To solve the problem of finding the apparent thickness of the Earth's atmosphere as seen by an observer in space, we can follow these steps:

### Step 1: Understand the Refractive Index Variation
The refractive index \( \mu \) of the atmosphere varies with depth \( y \) from the top as:
\[ \mu = 1 + \frac{y}{H} \]
where \( H \) is the total height of the atmosphere.

### Step 2: Set Up the Relationship for Apparent Thickness
The apparent thickness \( h' \) of the atmosphere can be expressed in terms of the actual thickness \( dy \) and the refractive index \( \mu \):
\[ dh' = \frac{dy}{\mu} \]
This means that the infinitesimal change in apparent thickness is the actual thickness divided by the refractive index.

### Step 3: Substitute the Expression for \( \mu \)
Substituting the expression for \( \mu \) into the equation gives:
\[ dh' = \frac{dy}{1 + \frac{y}{H}} \]

### Step 4: Simplify the Expression
We can rewrite the equation as:
\[ dh' = \frac{dy}{\frac{H + y}{H}} = \frac{H \, dy}{H + y} \]

### Step 5: Integrate to Find Total Apparent Thickness
To find the total apparent thickness \( h' \), we need to integrate \( dh' \) from \( y = 0 \) to \( y = H \):
\[ h' = \int_0^H \frac{H \, dy}{H + y} \]

### Step 6: Perform the Integration
The integral can be computed as follows:
\[ h' = H \int_0^H \frac{dy}{H + y} \]
This integral can be solved using the natural logarithm:
\[ \int \frac{dy}{H + y} = \ln(H + y) \]
Thus, evaluating the limits:
\[ h' = H \left[ \ln(H + y) \right]_0^H = H \left[ \ln(2H) - \ln(H) \right] = H \ln(2) \]

### Final Result
The apparent thickness of the atmosphere as seen by an observer in space is:
\[ h' = H \ln(2) \]

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