Atmospheric refraction

The direction of light as it passes through the atmosphere is also changed because of refraction since the index of refraction changes through the atmosphere.

Let $z$ be the true zenith distance, $\alpha$ be the observed zenith distance, $\alpha_n$ be the observed zenith distance at layer n in the atmosphere, $\mu$ be the index of refraction at the surface, and $\mu_n$ be the index of refraction at layer n. At the top of the atmosphere:

$${\sin z\over \sin \alpha_N} = {\mu_N \over 1}$$

At each infinitessimal layer:

$${\sin \alpha_n \over \sin \alpha_{n-1}} = {\mu_{n-1}\over \mu_n}$$

as so on for each layer down to the lowest layer:

$${\sin \alpha_1 \over \sin \alpha} = {\mu\over \mu_1}$$

Multiply these to get:

$$\sin z = \mu \sin \alpha$$

from which we can see that the refraction depends only the index of refraction near the earth's surface.

We define astronomical refraction, $r$, to be:

$$\sin (\alpha + r) = \mu \sin \alpha$$

so $r$ gives the error in the measured zenith distance.

In cases where $r$ is small (pretty much always):

$$\sin \alpha + r\cos\alpha = \mu \sin\alpha$$

$$r = (\mu - 1) \tan \alpha$$

$$\equiv R \tan \alpha$$

where $R$ is the ``constant of refraction''.

A typical value of the index of refraction is $\mu \sim 1.00029$, which gives R = 60 arcsec (red light).

The direction of refraction is that a star apparently moves towards the zenith. Consequently in most cases, star moves in both RA and DEC:

$$r_\alpha = r \sin q$$

$$r_\delta = r \cos q$$

where $q$ is the parallactic angle, the angle between N and the zenith:

$$\sin q = \cos \phi {\sin h\over \sin \alpha}$$

Note that the expression for $r$ is only accurate for small zenith distances ($z<45$). At larger $z$, can't use plane parallel approximation. Observers have empirically found:

$$r = A \tan \alpha + B \tan^3 \alpha$$

$$A = (\mu-1) + B$$

$$B \sim -0.07''$$

but these vary with time, so for precise measurements, you'd have to determine A and B on your specific night of observations.

Of course, the index of refraction varies with wavelength, so consequently does the astronomical refraction r;

$\lambda l$	R
3000	63.4
4000	61.4
5000	60.6
6000	60.2
7000	59.9
10000	59.6
40000	59.3

This gives rise to the phenomenon of atmospheric dispersion, or differential refraction. Because of the variation of index of refraction with wavelenth, every object actually appears as a little spectrum with the blue end towards the zenith. The spread in object position is proportional to $\tan\alpha$.

Note the importance of this effect for spectroscopy, and the consequent importance of the relation between a slit orientation and the parallactic angle.