Astronomy Principles and Practice Fourth Edition.pdf

where ZOL = r.
But ZOL = ζ .AlsoAB is parallel to OX, hence i = z and
Atmospheric refraction 115
sin z = n sin ζ. (10.3)
Let R, definedby
R = z − ζ (10.4)
be the angle of refraction, the correction that has to be applied to the apparent zenith distance ζ to
obtain the true zenith distance z.
Eliminating z from equations (10.4) and (10.3), we obtain, on expanding,
Now R is a small angle, so that we may write
sin R cos ζ + cos R sin ζ = n sin ζ. (10.5)
sin R = R cos R = 1. (10.6)
If R is expressed in seconds of arc, equation (10.5) becomes, using (10.6),
R cos ζ = 206 265(n − 1) sin ζ
or
R = 206 265(n − 1) tan ζ.
If we let k = 206 265(n − 1), it is found that k is 60·′′ 3, at the standard temperature and pressure,
0 ◦ C and 1000 mbar (760 mm Hg); the value of k for other pressures and temperatures can be found
from formulas or tables. Then z, the true zenith distance, is given by
z = ζ + R. (10.7)
The formula
R = k tan ζ (10.8)
is valid for zenith distances less than 45 ◦ and is a fairly good approximation up to 70 ◦ . Beyond that,
a more accurate formula taking into account the curvature of the Earth’s surface is required, while for
zenith distances near 90 ◦ special empirical tables are used.
A further problem related to accurate corrections to positional measurements is the fact that
the refractive power, (n − 1), of the atmospheric air exhibits dispersion, i.e. its value is wavelength
dependent. At wavelengths of 400, 500, 600 and 700 nm the corresponding values of k are
approximately 60·′′ 4, 57·′′ 8, 57·′′ 4 and 57·′′ 2. By substituting these values in equation (10.8) it is
immediately apparent that, at a zenith distance of 45 ◦ , a stellar image which should be essentially
point-like appears as a tiny spectrum with a length ∼3 arc sec, with the blue end towards the zenith
and the red end towards the horizon. Under good seeing conditions (see section 19.7.3), this effect is
clearly visible when telescopic stellar images are inspected by eye.
For radio measurements, refraction depends strongly upon the frequency employed. The lower
atmosphere produces refraction effects approximately twice as large as the optical effect, decreasing
rapidly with increasing angle of elevation. The ionosphere also refracts radio waves due to induced
motion of charged particles in amounts dependent on the air-density gradient. If N is the electron
density per cubic metre and ν is the frequency in cycles per second (Hz), then the local effective
dielectric constant n (which varies throughout the ionosphere) may be expressed by
n =
(
1 − 81N
ν 2 ) 1/2
.