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Computing differential refraction

Version: 1.0 2003-03-12 Ivo Saviane

The differential refraction $\Delta R$ between two different wavelengths $\lambda_{1}$ and $\lambda_{2}$ is computed (in arcseconds) as

$\Delta R=\frac{206264.80625}{10^{6}}\times[(n_{\lambda_{1}}-1)\times10^{6}-(n_{{\rm\lambda_{2}}}-1)\times10^{6}]\times\tan Z$

where $n_{\lambda}$ is the refraction index of air at wavelength $\lambda$ , and $Z$ is the zenith distance. Since $n_{\lambda}$ is always close to $1$ , $(n_{\lambda}-1)\times10^{6}$ is used instead, in order to make computations easier. The computation of the refraction index is streamlined by first computing the index for dry air in standard conditions, then correcting for a different pressure and temperature, and finally adding the contribution of humidity.

1.1 Standard conditions

Standard pressure, temperature and humidity conditions are defined as $P_{{\rm s}}=1013.25\times10^{2} {\rm Pa}$ , $T_{s}=288.15 {\rm K}$ (i.e. $15 {\rm C}$ ), and dry air. At high altitudes the pressure is smaller, and one can guess the value of pressure as $P=P_{{\rm s}}\times e^{-h/h_{0}}$ , where $h_{0}$ is the scale height of the atmosphere (typically $h_{0}=7 {\rm km}$ ). However, for La Silla we have actual measurements, just have a look at the ``meteomonitor''. Thus a typical value can be $P=775\times10^{2} {\rm Pa}$ .

1.2 Refraction index in standard conditions

We call $n_{{\rm s}}$ the refraction index of air, in standard conditions, and compute it as:

$(n_{{\rm s}}-1)\times10^{6}=64.328+\frac{29498.1\times10^{-6}}{146\times10^{-6}-(1/\lambda)^{2}}+\frac{255.4\times10^{-6}}{41\times10^{-6}-(1/\lambda)^{2}}$
where the wavelength is expressed in nm.

1.3 Refraction index in other conditions

Then for other temperatures and pressures, the refractive index in dry air is given by

$(n_{{\rm d}}-1)=(n_{{\rm s}}-1)\times\frac{P}{T}\times\frac{T_{{\rm s}}}{P_{{\rm s}}}$

1.4 Refraction index in presence of water vapor

If there is water vapor in the air, and its pressure is $P_{{\rm w}}$ , then the refractive index is smaller, and it is computed as

$(n-1)\times10^{6}=(n_{{\rm d}}-1)\times10^{6}-43.49\times(1-7.956\times10^{3}\times(1/\lambda)^{2})\times\frac{P_{{\rm w}}}{P_{{\rm s}}}$

Normally one has readings of relative humidity $H$ , not of water vapor pressure, so the latter is computed as

$P_{{\rm w}}=P_{{\rm w,sat}}\times\frac{H}{100}\times10^{2}$

where $10^{2}$ is used to convert from millibar to Pa, and $P_{{\rm w,sat}}$ is the saturation pressure of water vapor, which in turn is computed (in millibar) as

$P_{{\rm w,sat}}=6.11\times10^{(7.5\times t)/(t+237.7)}$

where $t$ is the tempeature in Celsius (so $t=T-273.15$ ). A typical vapor pressure is $550 {\rm Pa}$ .

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