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From atmospheric to astronomical coefficients

This web page handles the conversion from atmospheric absorption/scattering coefficient to astronomical extinction coefficients. The astronomical extinction coefficients are closely related to the absorption/scattering coefficients. The main difference is that the atmospheric absorption/scattering coefficients ß_x [(km)^-1] are related to a path within the atmospheric (say between two earth bound objects), while the astronomical extinction coefficients k_x [-] are related to an object outside the atmospheric (like celestial objects). An accuracy of 10-20% of these coefficients looks to be oke.

Light intensity ratio in an astronomical sense I_obs/I_na is (Schaefer [2000]):

I_obs/I_na = 10^-0.4*ky*X

(1)

k_y = astronomical extinction coefficient [-]
X = airmass [-]
y = W(ater), a(erosol), R(ayleigh)

The airmass represents the ratio of the integrated mass of air encountered in a given direction referred to the zenithal integrated mass. And in the zenith direction X =1.

In a similar way in atmospheric optics we are defining the intensity ratio I_obs/I_na with (pers. comm. Aubé [2007]):

I_obs/I_na = e^-ßy^*Hy*X

(2)

ß_y = ground level atmospheric scattering/absorption coefficient [(km)^-1]
H_y = scale height [km]

If we combine the above two formula (1) and (2) and using X=1 (in zenith direction) we get:

k_y= -ln (e^-ß^y*Hy) /0.4/ln(10)
k_y= -ß_y*H_y/(0.4*ln(10))

k_y= 1.09*ß_y*H_y[-]

(3)

In the web site the following scale heights are used:
H_W = 3 [km] (Ricchiazzi [1997])
H_a = 3.75 [km] (Su [2003])
H_R = 8.52 [km] (Su [2003])

Standard Visibility Range

The Standard Visibility Range (V_r) (using epsilon=0.02) can be used to determine the total atmospheric absorption/scattering coefficient (pers. comm. Myers [2007]). The function is:

ß_Vr(550) = 3.912/V_r

(4)

(or ß_MOR(vis)~-ln(0.05)/MOR with MOR (Meteorological Optical Range, epsilon 0.05), and thus MOR~V_r/1.3).

The V_r related astronomical absorption/scattering coefficient is also (Ozone atmospheric absorption coeffcient is not important for this earth bound measured V_r):

ß_Vr= ß_W + ß_a + ß_R	(5)
ß_a= ß_Vr - ß_W - ß_R	(6)

Because ß_W and ß_R (calculated from k_W and k_R; Schaefer [2000]) are quite deterministic, one can determine from ß_Vr (calculated from the Visibility Range (V_r)) the ß_aand thusk_a.

A check: the k_Rschaefer is around 0.1066 [-] (Schaefer [2000]), and with ß_R = 0.0116 [(km)^-1] and H_R = 8.52 [km] (Su [2003]) one can calculate (using 3):
k_R= 1.09*ß_R*H_R = 1.09 * 0.0116 * 8.52 = 0.1077 [-]
So they are quite comparable: k_Rschaefer/k_R = 1.01! Around 1% of an error.

Visibility and atmospheric phenomena

Form Shepard, F.D., Reduced visibility due to fog on the highway, Synthesis of Highway Practice 228, 1996. Chapter Two, Table 1 "International Classification of Visibility", page 4; the following Description and Distance range are taken (the 'Astr. Ext. coeff range' is calculated with above formula):

Description	Distance range	Astr. Ext. coeff. (k_tot) range
Dense fog	<40 m
Thick fog	40 - 200 m
Fog	200 m - 1 km
Mist	1 - 2 km	16 - 8
Haze/Poor visibility	2 - 4 km	8 - 4
Moderate visibility	4 - 10 km	4 - 1.7
Good visibility	10 - 40 km	1.7 - 0.48
Excellent visibility	40 - 100 km	0.48 - 0.25
Exceptional visibility^*	100 - 350 km	0.25 - 0.13

* Exceptional visibility not mentioned in Shepard, the boundary of 350 km is very close to theoretical Rayleight boundary (400 km).

Acknowledgments

I would like to thank the following people for their help and constructive feedback: Martin Aubé, Jan Hollan, Daryl Myers, John Ogren, Keith Pickering, Bradley Schaefer and all other unmentioned people. Any remaining errors in methodology or results are my responsibility of course!!! If you want to provide constructive feedback, let me know.

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Last major content related changes: May 17, 2007