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In Vol. II, No. 2 of KRONOS, Rose and Vaughan (1) presented a rather detailed analysis of Babylonian observations of the "appearances" and "disappearances" of the planet Venus. The intent of their paper was to demonstrate that differences between the Babylonian measurements of the invisibility periods of Venus and comparable modern measurements might indicate that the orbits of Venus and Earth were perturbed in some way after the Babylonian measurements were made (or possibly during the period of the Babylonian observations). In their analysis they identified thirteen independent parameters having to do with orbital elements of Earth and Venus, temporal synchronization of calendars, etc. One of the parameters involved in the analysis is called the arcus visionis, which is the angle at which the Sun must be below the observer's horizon for the skyglow to be low enough to allow Venus to "shine through". This parameter alone of the thirteen is affected by conditions of the Earth's atmosphere and therefore this parameter by itself might indicate "upheavals of the Earth", if such upheavals are associated with large amounts of dust loading of the atmosphere, e.g. from volcanic eruptions, strong winds blowing over dry areas, from extraterrestrial sources (meteor dust, comet tail dust), etc.

The effect of atmospheric dust loading on the sighting of Venus (or any other heavenly body except the Sun) when the Sun is illuminating the upper atmosphere is easy to understand qualitatively. The basic idea is that the atmospheric dust (and air molecules themselves) scatters sun rays in all directions and creates a skyglow. The amount of skyglow at a given angle from the Sun increases with dust loading. When there is little dust in the atmosphere Venus may be visible on the horizon when the Sun is, say, five degrees below the horizon (arcus visionis = 5). However, with an increase in the dust load of the upper atmosphere it may be necessary for the Sun to be six or seven or more degrees below the horizon before Venus is visible. It is the intent of this paper to indicate qualitatively how upper atmospheric dust may increase the arcus visionis by using a simple model of dust loading and solar illumination of the atmosphere. The model is satisfactory for an observer looking in the direction of sunset or sunrise.


A direct calculation of the arcus visionis from atmospheric parameters using an assumed brightness ratio of the Sun and Venus in its various phases is possible in principle. However, in practice, the atmospheric parameters are not known with sufficient accuracy to warrant any more than a highly approximate calculation. Typical parameters are: the type(s) of atmospheric dust particles (here "dust" refers to any particles, liquid or solid, such as sulfuric acid droplets known to be concentrated at an altitude of about 18 km and meteoric dust particles which fall toward the Earth from altitudes up to 100 km see ref. (2); the scattering properties of the various types; and the numbers per unit volumes (densities) of the various types at various altitudes. These parameters vary widely from point to point over the Earth's surface and also they change considerably with time. Therefore I will present an analysis in which I basically ignore these parameters and use only assumed heights of dust layers to estimate the arcus visionis. I assume that if the layers of dust above a certain altitude are sufficiently dense, the dust will scatter enough sunlight to obscure Venus. With this assumption it is quite simple to calculate a value of the arcus visionis, ga using only an altitude, ho, above which are the layers of illuminated dust. The calculation is described further on in the paper. However, before presenting the relation between dust layer height and the arcus visionis, I will summarize the result of an approximate calculation of the value, within an order of magnitude, of the volume scattering which would be necessary for the dust to scatter enough light so that Venus wouldn't be visible. In order to do this calculation one needs first an estimate of how bright the sky would have to be relative to the brightness of Venus for Venus to be virtually invisible.

According to data in ref. (3), for twilight viewing (sky brightness about 0.1 cd/m2) a source of light that is so small that it can't be resolved by the naked eye (a "point" source) has an apparent visual size of about 2 minutes of arc (5.8 x 10-4 radians) and it must be about twice as bright as the surrounding sky for it to be detected with a 50% probability. Since the illumination on the Earth's surface caused by light from Venus is about 10 orders of magnitude (10-10, not ten "magnitudes") lower than that of the Sun when Venus is within 10 degrees of the Sun, Venus will be virtually undetectable against the sky if the radiation scattered into the eye of an observer by illuminated dust is about 4 times 10-10 times the solar radiation illuminating the dust cloud. Under these conditions the volume scattering coefficient, g, averaged over the atmospheric path that is illuminated by the Sun, is given by the approximate relation g > 5 x 10-10/A2 q2Z, where A is the average atmospheric transmission of light through the atmosphere, q is the apparent angular size of the source (about 5.8 x 10-4 radians), and Z is the length, measured along the sighting line to Venus, of the part of the atmosphere that is directly illuminated by the Sun. The value of A depends upon the scattering and absorption of light along a path tangential to the Earth's surface from outside the atmosphere to the surface of the Earth. A is estimated to be larger than 10-2. However, to avoid underestimating g, I use A = 10-2. Thus , g 15/Z with units of meters-l steradians-l when Z is in meters. The equation for Z is Z (2 hm Re)l/2 Re tan (ga/2), which is an approximation that is satisfactory for hm less than 100 km (see Fig. 1). In this equation Re = 6378 km and hm is the maximum altitude for which there may be appreciable dust. Assuming hm = 50 km, which is the normal maximum altitude for dust, and letting ga equal the "standard" value(1) of 5.75, Z is about 480 km. With Z = 480 km 5 x 105 m, the value of g averaged over the whole path is about 15/(5 x 105) = 3 x 10-5/m sterad., which is comparable with measured values of g for scattering angles less than 10 degrees at sea level in "clear" air.(3) Under unusual conditions of atmospheric dust loading g would increase (as might hm for a short time). Under such conditions Z could have a smaller value and still have the averaged g large enough to make Venus invisible. For example, if sufficient dust were added to the upper atmosphere to make ga increase to 10, and with hm = 50 km, Z would be about 240 km and the required average g would be about 6 x 10-5/m sterad., which is only twice as great as the value calculated before. It does not seem unreasonable that g could increase to even 10-4/m sterad. in the upper atmosphere if an extraterrestrial source of dust should exist (other than the normal meteoric source), or in the event of very strong volcanic eruptions. Thus it seems reasonable to estimate the arcus visionis using the simple formulation, based on altitudes of dust layers, that is presented in the next section and to thereby avoid the difficulties inherent in attempting a detailed calculation based on scattering properties of the atmosphere.




Figure 1 illustrates the geometry of sighting Venus through the Earth's atmosphere. The atmosphere above a minimum height, ho, is illuminated by direct rays of the Sun. The radiation illuminating the layers just above ho is reduced considerably in intensity below the level of radiation reaching the upper layers near the maximum height, hm because of attenuation by passage through the Earth's atmosphere (This attenuation of the radiation that reaches the lower layers is the source of the factor A in the previously used equation for g.). Ray bending due to atmospheric refraction is ignored in this calculation. The previously mentioned distance, Z, is noted on the figure along with ho, hm, and ga Under normal conditions Venus can be seen when ga is about 5.75, according to Rose and Vaughan.(1) (Note: Langdon, et al..(4) point out that Ptolemy assigned 5 to the arcus visionis for Venus. In a perfectly clear atmosphere the average g would be around 10-6 and the average A might be about 5 x 10-2 which would result in ga of about 3. If the Earth had no atmosphere the arcus visionis would be about 1/4, which is half the angular diameter of the Sun.) Using the trigonometric expression presented in Figure 2, I find that this corresponds to a minimum illuminated height of about ho = 8 km (and Z = 480 km with hm = 50 km). All components of the atmosphere above this height contribute to the scattered sunlight which "obscures" the light from Venus. As ga increases ho increases. When ga = 14.3, ho = 50 km. Since there is not expected to be any appreciable atmospheric dust above 50 km, it appears that 14 is an approximate upper bound for ga. An unusual extraterrestrial source of dust (comet tail?) might increase the scattering at altitudes up to 100 km or higher, in which case ga might grow to 20 or so. However, because of layering of the atmosphere and the general lack of vertical circulation above 15 km or so, it is unlikely that terrestrial dust would reach altitudes above 50 km except perhaps as a result of extreme vulcanism.

Using the equation given in Figure 2 and the approximate equations for 26' and Z, I find the following corresponding values, assuming hm = 50 km:

ho ga Z approximate average b
needed to scatter enough
light to obscure Venus
10 km 6.41 357 km 4 x 10-5/m sterad.
20 km 9.06 293 km 5 x 10-5
30 km 11.09 180 km 8 x 10-5
40 km 12.80 83 km 18 x 10-5
50 km 14.30 0 km "infinite"


It is clear from the preceding table that if the concentration of dust in the upper layers of the atmosphere increased, the arcus visionis would increase. However, one might well ask whether or not it makes sense to talk about layers of the atmosphere as "structures" capable of carrying (supporting) dust. The answer to this question is that the atmosphere does indeed have a layered structure, with the various layers known as the troposphere (0 to ~15 km), the stratosphere (15 to 50 km), the mesosphere (50-100 km), the ionosphere (100-500 km), and the exosphere (beyond 500 km). General properties of these "spheres" are the following: the troposphere contains most of the atmosphere and almost all weather processes; the stratosphere contains a layer of sulfate particles, most of which are droplets of concentrated water solutions of sulfuric acid, and the layer of greatest ozone concentration; some of the upper altitude weather processes occur in the lower regions of the stratosphere; the mesosphere is the layer in which most meteors burn out and thus is a source of upper atmospheric dust; the mesosphere is also the location of so-called noctilucent clouds; the ionosphere, a portion of which is often referred to as the thermosphere, contains ionized gases in a near vacuum. The ionosphere is responsible for auroras and for effects on radio transmission.

It should be emphasized that the boundaries of these regions are not sharp, but that characteristics of the atmosphere within these regions are markedly different. The main differences in characteristics result from the Earth's gravitational field and the "ideal gas" nature of the gaseous constituents of the atmosphere. The physics of an ideal gas indicate that the density and pressure should decrease as the altitude increases, which is what actually happens with the atmosphere. With regard to density, if all of the atmosphere above a point on the Earth's surface were squeezed into a volume so that the average density of the volume equalled the sea level density, and if the cross-section of the volume were kept constant while this squeezing took place, the height of the volume would be only about 8 to 10 km. Another characteristic of the atmosphere is the temperature variation with height. Starting with 0 km the average temperature drops as height increases up to the tropopause, which lies in the range 10-18 km, depending upon latitude and season of the year. The tropopause is the boundary between the tropo- and stratospheres. The temperature at the tropopause is around -56C. The temperature then increases continually with height.

Because of the extreme temperature and density differences between these "spheres", vertical mixing of the atmosphere from one layer to another takes place only at the boundaries. Of course there is considerable vertical mixing within the troposphere itself, but beyond this the stratification of the atmosphere is nearly constant. Atmospheric dust (solid particles, droplets) can be suspended within these layers, with the amount of suspended dust decreasing as the altitude increases. Most dust is within the troposphere at altitudes lower than 10 km. Under stable conditions of the atmosphere dust at higher altitudes gradually filters downward, thus keeping the atmosphere above 10 km quite "clean". This explains roughly why the arcus visionis for Venus is less than 6 under normal, clear conditions. However, light scattering experiments using searchlights and laser beams have shown that there are stable layers of dust at altitudes up to 30 km, even under normal conditions. Although the quantity of dust in these layers is normally very low, after volcanic eruptions the dust in these layers may increase significantly and remain for periods of weeks, months, or even years. After Krakatoa exploded in 1883 optical effects on sunsets, etc., which are indicative of upper atmosphere dust loading, persisted for several years. Thus I expect that measurements of the arcus visionis of Venus made within a year after the Krakatoa eruption might have yielded values somewhat larger than 6 and perhaps as large as 10.


The arcus visionis is a quantity which is related to conditions of the Earth's atmosphere. Values of the arcus visionis for Venus typically lie in the range 5-6. Sizeable increases in the arcus visionis would require conditions of extreme vulcanism on the surface of the Earth or an extraterrestrial source of dust or a combination of these. Under such conditions one might expect severe effects on the environment ranging from optical effects (very red sunsets and sunrises, sky greyish rather than blue, poor visibility of the sky just after sunset, a bluish tinge to moonlight) to thermal effects (gradual heating or cooling of the atmosphere producing extreme summers, extreme winters, and unusual amounts of generally dirty precipitation). Thus if one were to find in historical records clear evidence of sizeable increases in the arcus visionis of Venus, one might search for coincidental records of environmental and heavenly "catastrophes" and variation in the values of the arcus visionis of other heavenly bodies (e.g. Sirius).(5)


1. L.E. Rose and R.C. Vaughan, "Analysis of the Babylonian Observations of Venus," KRONOS 11:2, pp. 3 ff.
2. E.J. McCartney, Optics of the Atmosphere, John Wiley and Sons, New York, N. Y. (1976).
3. W.E.K. Middleton, Vision through the Atmosphere, University of Toronto Press Toronto, Canada (Oxford University Press), (1952).
4. S.H. Langdon, J.K. Fotheringham, and C. Schoch, The Venus Tablets of Ammizaduga, Oxford University Press, (1928).
5. I thank R.C. Vaughan and L.E. Rose for constructive comments on earlier versions of this paper.

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