Mountaintop Observatory Site Lab: Airmass, Extinction, Fried Parameter and Seeing

Presets

Live readouts

Graphs

Cn2(h) above the site, log scale (dashed: inversion)

Seeing FWHM and 10 x extinction k against site altitude

Limiting magnitude against zenith angle (this site vs sea level)

Short exposure (speckles)

Long exposure (seeing disk)

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Numerical checks

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How to use

What to try: drag Site altitude from 100 m to 4 km and watch extinction, r0 and the speckle pattern change; raise the inversion above the site to see the turbulence layer pass below the telescope. Increase the jet stream to shrink r0 and tau0. Compare the short-exposure speckle picture with the long-exposure disk. Try 3.5 m aperture at one arcsecond seeing, or the I band against B. Switch to the full-moon preset to watch the limiting magnitude collapse.

The science

Why site matters. Starlight crosses the whole atmosphere. Two things happen on the way: the light is dimmed (extinction) and its wavefront is corrugated by turbulence (seeing).

Airmass. The path length through the air grows as the star gets lower. Plane-parallel: X = sec z. The page uses the Kasten-Young (1989) fit X(z) = 1 / [cos z + 0.50572 (96.07995 - z)^-1.6364] which stays finite at the horizon and is checked against a numerical column through a spherical exponential atmosphere and against Pickering (2002).

Extinction. m_obs = m_0 + k X, k = 1.0857 (tau_R + tau_A + tau_O3) Rayleigh scattering tau_R = 0.00857 lambda^-4 (...) exp(-h/8434 m), aerosols tau_A = tau_0 (lambda/0.55)^-1.3 integrated only over the air above the site with the aerosol load confined below the inversion, and a fixed ozone term. A site above the inversion sheds almost all aerosol: that is the sea-of-clouds effect.

Turbulence. The refractive-index structure constant follows the Hufnagel-Valley model with a ground layer and an inversion layer: Cn2(h) = 0.00594 (w/27)^2 (1e-5 h)^10 e^(-h/1000) + 2.7e-16 e^(-h/1500) + A e^(-(h-h0)/100) + C_inv e^(-((h-h_inv)/60)^2) with the wind speed w computed from a Bufton jet-stream profile. Only the turbulence above the telescope counts, so a high site removes the ground layer and the inversion.

Seeing quantities. r0 = [0.423 k^2 sec z * Int Cn2 dh]^(-3/5) FWHM = 0.98 lambda / r0 theta0 = [2.914 k^2 Int Cn2 h^(5/3) dh]^(-3/5) tau0 = 0.314 r0 / v The von Karman outer scale L0 reduces the seeing disk by sqrt(1 - 2.183 (r0/L0)^0.356). The delivered image width adds the diffraction core 1.03 lambda/D in quadrature. Scintillation uses a Young/Dravins-type empirical fit. r0 grows as lambda^(6/5), so infrared images are sharper.

The speckle picture. A Kolmogorov phase screen (FFT method with subharmonics) is drawn across a circular aperture of D/r0 pixels, translated by the wind (frozen flow), and Fourier transformed: PSF = |FFT(P(x) exp(i phi(x)))|^2. Short exposures show speckles, long exposures average to the seeing disk; the page measures the long-exposure FWHM of the simulation and compares it with 0.98 lambda/r0.

Limiting magnitude. The CCD equation for a 5-sigma detection inside an aperture of radius 0.67 FWHM: N* = [S^2 + sqrt(S^4 + 4 S^2 n_B)] / 2, n_B = sky photons + npix (RN^2 + dark) with the sky from a Bortle-class dark background, a van Rhijn airmass factor, the moon and twilight. Zero-point photon rates and detector numbers are typical, rounded values.

Honest simplifications. Plane-parallel turbulence scaling with sec z; no water-vapour band, no dome or mirror seeing, no adaptive optics; one frozen turbulent screen; the 3D mountain, sea and cloud layer are illustrative (an island volcano). The sky-glow, Bortle and moon numbers are illustrative fits, not measurements. Literature values (HV 5/7: r0 = 5 cm, theta0 = 7 microradians) are design values, not a particular site.

Live checks. HV 5/7 r0 and theta0 vs the published design numbers, the rms Bufton wind vs 21 m/s, Kasten-Young vs a numerical spherical-atmosphere column (60, 80 degrees) and vs Pickering, the FFT screen structure function (6.88 (r/r0)^(5/3)), and the simulated long-exposure FWHM.