Positional Astronomy:
Refraction

The speed of light changes as it passes through a medium such as air.
We define the refractive index of any transparent medium as 1/v,
where v is the speed of light in that medium.

The speed of light in air depends on its temperature and its pressure,
so the refractive index of the air varies in different parts of the atmosphere.

Make a simple model of the atmosphere as n layers of uniform air above a flat Earth,
with a different velocity of light vi for each layer (i from 1 to n).
Apply Snell's Law of Refraction at each boundary.

At the first boundary, sin(i₁) / sin(r₁) = v₀ / v₁ .
At the next boundary, sin(i₂) / sin(r₂) = v₁ / v₂ , and so on.

But, by simple geometry, r₁ = i₂, r₂ = i₃ and so on.

So we have
sin(i₁) = (v₀ / v₁) sin(r₁)
          = (v₀ / v₁) sin(i₂)
          = (v₀ / v₁) (v₁ / v₂) sin(r₂)
          = (v₀ / v₂) sin(r₂)
          = ..........
          = (v₀ / v_n) sin(r_n)

In other words, the refractive indices of the intervening layers all cancel out.
The only thing that matters is the ratio between v₀
(which is c, the speed of light in vacuum)
and v_n (the speed in the air at ground level).

Now r_n is the apparent zenith distance of the star, z',
and i₁ is its true zenith distance, z.
So sin(z) = (v₀ / v_n) sin(z').

Refraction has no effect if a star is at the zenith (z=0).
But at any other position, the star is apparently raised; the effect is greatest at the horizon.

Define the angle of refraction R by R = z - z'.
Rearrange this as z = R + z'.
Then sin(z) = sin(R) cos(z') + cos(R) sin(z').

We assume R will be small, so, approximately,
sin(R) = R (in radians), and cos(R) = 0.

Thus, approximately,
     sin(z) = sin(z') + R cos(z').
Divide throughout by sin(z') to get
     sin(z)/sin(z') = 1 + R/tan(z')
which is to say,
     v₀/v_n = 1 + R/tan(z').
So we can write
     R = (v₀/v_n - 1) tan(z')
We write this as
     R = k tan(z')
where k = (v₀/v_n - 1)

Here v₀ is c, the velocity of light in a vacuum, which is constant.
But v_n depends on the temperature and pressure of the air at ground level.
At "standard" temperature (0°C = 273K) and pressure (1000 millibars),
     k = 59.6 arc-seconds.
The formula in the Astronomical Almanac is
     k = 16.27" P/(273+T)
where P is in millibars, and T is in °C.

Exercise:

A star is at Right Ascension 5h 0m and declination +26°20'.
The latitude φ is +56°20'.
Local Sidereal Time is 5h 0m.
Atmospheric pressure is 1050 millibars,
and the temperature is +5°C.

What is the star’s true altitude?

How much will the star’s image be shifted by atmospheric refraction,
and in which direction?

What will be the star’s Right Ascension and declination,
corrected for refraction?

Click here for the answer.

At large zenith distances, this model is inadequate.
The amount of refraction near the horizon is actually determined observationally.
At standard temperature and pressure,
refraction at the horizon (horizontal refraction) is found to be 34 arc-minutes.

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