TUTORIAL

ATMOSPHERICAL PARAMETERS
OPTICAL TURBULENCE AND ASTROCLIMATIC PARAMETERS
 POTENTIAL TEMPERATURE
 MIXING RATIO
 PRECIPITABLE WATER VAPOR
 RELATIVE HUMIDITY
 WIND DIRECTION
 C_{N}^{2}: CONSTANT OF THE STRUCTURE FUNCTION OF THE REFRACTIVE INDEX
 FRIED'S PARAMETER
 SEEING (ε)
 ISOPLANATIC ANGLE (θ_{0})
 WAVEFRONT COHERENCE TIME (τ_{0})
The potential temperature θ of an air parcel is defined as the temperature which the parcel of air would have if it were expanded or compressed adiabatically from its existing pressure and temperature to a standard pressure p_{0} (generally taken as 1000 mb).
$\theta = T\left( \frac{P_{0}}{P} \right)^{\frac{R}{cp}}$

(1)

The mixing ratio of the water vapor MR is:
$MR = \frac{m_{wp}}{m_{d}}.$

(2)

The PWV (mm) is the total atmospheric water vapor contained in a vertical column of unit crosssectional area extending between any two specified levels. It is commonly expressed in terms of the height to which that water substance would stand if completely condensed and collected in a vessel of the same unit cross section.
The total precipitable water is that contained in a column of unit cross section extending all of the way from the earth's surface to the top of the atmosphere.
$PWV = \frac{10^3}{\rho g} \int_{p_{sup}}^{p_{inf}} MR dp .$

(3)

The relative humidity (RH) with respect to water is the ratio (expressed in percentage) of the actual mixing ratio MR to the saturation mixing ratio MR_{s} with respect to water at the same temperature and pressure:
$RH = 100* \frac{MR}{MR_s},$

(4)

$MRs = \frac{m_{vs}}{m_d}.$

(5)

The convention used is 0°: wind blowing from the North; 90°: wind blowing from the East.
The optical turbulence is measured by the C_{N}^{2}, the constant of the structure function of the refractive index:
$D_n(d)=C_N^2 d^{2/3}\;\;\;\;\;\;\;\; where\; l_0 \leq d \leq L_0,$

(6)

The astroclimatic parameters are useful to describe specific characteristics of the optical turbulence and are measured in the visible band (λ = 0.5 microns).
The algorithms used in the numerical code for the ALTA Center project are those reported in Masciadri et al., 1999 [1].
The Fried’s parameter r_{0} is the typical size (diameter of a circular area for example) in which the perturbed wavefront is coherent i.e. the rms wavefront aberration due to the passage of a wavefront through the atmosphere is equal to 1 radian. It corresponds to the size of the equivalent telescope for an observation outside the earth’s atmosphere. The analytical expression of r_{0} normalized with respect to the zenith is:
$r_0=\left[0.423\left(\frac{2\pi}{\lambda}\right)^2\int_0^\infty C_N^2(h) dh\right]^{3/5}.$

(7)

The seeing (ε) is defined as the width at the half height of a star image (called Point Spread Function – PSF) at the focus of a ‘large’ telescope (D > r_{0}). The analytical expression of ε normalized with respect to the zenith is:
$\epsilon=0.98\frac{\lambda}{r_0},$

(8)

The isoplanatic angle θ_{0}^{(I)} is defined as the maximum angular separation of two stellar objects producing at the telescope entrance pupil a maximum rms between the two paths of 1 rad^{2} i.e. a surface inside which we have a coherent wavefront with an accuracy of 1 rad^{2}. The analytical expression of θ_{0} normalized with respect to the zenith is:
$\theta_{AO}=0.31\frac{r_0}{h_{AO}}.$

(9)

$h_{AO}=\left[ \frac{\int_0^\infty h^{5/3}C_N^2(h)dh}{\int_0^\infty C_N^2(h)dh} \right]^{3/5},$

(10)

$\theta_{AO}=0.057\cdot\lambda^{6/5}\left( \int_0^\infty h^{5/3}C_N^2(h)dh \right)^{3/5}.$

(11)

The coherence wavefront time τ_{0}^{(II)} is:
$\tau_{AO}=0.31\frac{r_0}{v_{AO}},$

(12)

$v_{AO}=\left[ \frac{\int_0^\infty \mathbf{V}(h)^{5/3}C_N^2(h)dh}{\int_0^\infty C_N^2(h)dh} \right],$

(13)

$\tau_{AO}=0.057\cdot\lambda^{6/5}\left[ \int_0^\infty \mathbf{V}(h)^{5/3} C_N^2(h)dh \right]^{3/5}.$

(14)

[1]  Masciadri, E., Vernin, J., Bougeault, P., 1999, A&ASS, 137, 185.
[2]  Roddier, F., Gilli, J. M., Vernin, J., 1982a, J. Optics (Paris), 13, 63.
[3]  Roddier, F. Gilli, J. M., Lund, G., 1982b, J. Optics (Paris), 13, 263.
(I) Two slightly different definition of θ_{0} called θ_{0,AO} and θ_{0,I} can be found in the literature [2]. The first one is applied to the Adaptive Optics and the constant factor is 0.31, the second is applied to the Interferometry and the constant factor is 0.36. We use θ_{0}=θ_{0,AO} in the web site of ALTA.
(II) Two slightly different definition of τ_{0} called τ_{0,AO} and τ_{0,I} can be found in the literature [3]. The first one is applied to the Adaptive Optics, in this case the constant factor is 0.31, the second is applied to the Interferometry, in this case the constant factor is 0.36. We use τ_{0}=τ_{0,AO} in the web site of ALTA.