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ATMOSPHERICAL PARAMETERS

OPTICAL TURBULENCE AND ASTROCLIMATIC PARAMETERS

• POTENTIAL TEMPERATURE

• 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 p0 (generally taken as 1000 mb).  $\theta = T\left( \frac{P_{0}}{P} \right)^{\frac{R}{cp}}$ (1)
where T is the absolute temperature in Kelvin, R is the gas constant and cp is the specific heat capacity at constant pressure. Typically R/cp=0.286 for air.

• MIXING RATIO

• The mixing ratio of the water vapor MR is:  $MR = \frac{m_{wp}}{m_{d}}.$ (2)
where mwp is the mass of the water vapor and md is the corresponding mass of dry air. The mixing ratio is generally expressed in grams of water vapor per kilograms of air (gr/kg). This is the unit used in this website.

• PRECIPITABLE WATER VAPOR

• The PWV (mm) is the total atmospheric water vapor contained in a vertical column of unit cross-sectional 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)
where MR is the mixing ration of the water vapor in (kg/kg), p is the atmospheric pressure in (Pa), pinf is the pressure at 20m a.g.l., psup is the pressure at 20km a.g.l., ρ is the water density in (kg/m3), g is the standard gravity in (m/s2) and PWV is in (mm).

• RELATIVE HUMIDITY

• 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 MRs with respect to water at the same temperature and pressure:  $RH = 100* \frac{MR}{MR_s},$ (4)
where MRs with respect to water is defined as the ratio of the mass mvs of water vapor in a given volume of air saturated with respect to a plane surface of water to the mass md of dry air:  $MRs = \frac{m_{vs}}{m_d}.$ (5)
RH can equivalently be expressed as RH = 100*e/es, where e is the water vapor partial pressure and es is the saturation water vapor partial pressure.

• WIND DIRECTION

• The convention used is 0°: wind blowing from the North; 90°: wind blowing from the East.

• CN2: CONSTANT OF THE STRUCTURE FUNCTION OF THE REFRACTIVE INDEX

• The optical turbulence is measured by the CN2, 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)
where l0 is the inner scale (size of the smallest turbulent eddies) and L0 is the outer scale (size of the largest turbulent eddies) of the inertial range in which the Kolmogorov model is valid. The Kolmogorov model describes how the turbulence energy is injected in the inertial range, how it is transferred to eddies of different size and how it is dissipated. The turbulence is injected at the size of the largest eddies, the rate of turbulent energy for unit of mass and time is conserved and transferred to smaller and smaller eddies up to be dissipated at the size of the smallest eddies. The CN2 is a 3D function. All the astroclimatic parameters depend on the integral of the CN2 but weighted by different weighting functions that are therefore a 2D functions.
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 .

• FRIED'S PARAMETER

• The Fried’s parameter r0 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 r0 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)
where λ is the wavelength (λ = 0.5 microns), h the height above the ground.

• SEEING (ε)

• 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 > r0). The analytical expression of ε normalized with respect to the zenith is:  $\epsilon=0.98\frac{\lambda}{r_0},$ (8)
where λ is the wavelength (λ = 0.5 microns).

• ISOPLANATIC ANGLE (θ0)

• The isoplanatic angle θ0(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 rad2 i.e. a surface inside which we have a coherent wavefront with an accuracy of 1 rad2. The analytical expression of θ0 normalized with respect to the zenith is:  $\theta_{AO}=0.31\frac{r_0}{h_{AO}}.$ (9)
where h0 is:  $h_{AO}=\left[ \frac{\int_0^\infty h^{5/3}C_N^2(h)dh}{\int_0^\infty C_N^2(h)dh} \right]^{-3/5},$ (10)
where h is the height above the ground. Replacing Eq.7 in Eq.6 we obtain:  $\theta_{AO}=0.057\cdot\lambda^{6/5}\left( \int_0^\infty h^{5/3}C_N^2(h)dh \right)^{-3/5}.$ (11)
where λ is the wavelength (λ = 0.5 microns).

• WAVEFRONT COHERENCE TIME (τ0)

• The coherence wavefront time τ0(II) is:  $\tau_{AO}=0.31\frac{r_0}{v_{AO}},$ (12)
where  $v_{AO}=\left[ \frac{\int_0^\infty |\mathbf{V}(h)|^{5/3}C_N^2(h)dh}{\int_0^\infty C_N^2(h)dh} \right],$ (13)
where V(h) is the horizontal wind velocity vector. Replacing Eq. 10 in Eq. 9 we obtain:  $\tau_{AO}=0.057\cdot\lambda^{6/5}\left[ \int_0^\infty |\mathbf{V}(h)|^{5/3} C_N^2(h)dh \right]^{-3/5}.$ (14)
where λ is the wavelength (λ = 0.5 microns).

 - Masciadri, E., Vernin, J., Bougeault, P., 1999, A&ASS, 137, 185.
 - Roddier, F., Gilli, J. M., Vernin, J., 1982a, J. Optics (Paris), 13, 63.
 - Roddier, F. Gilli, J. M., Lund, G., 1982b, J. Optics (Paris), 13, 263.

(I) Two slightly different definition of θ0 called θ0,AO and θ0,I can be found in the literature . 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 θ00,AO in the web site of ALTA.
(II) Two slightly different definition of τ0 called τ0,AO and τ0,I can be found in the literature . 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 τ00,AO in the web site of ALTA.