Solar Observations in Ancient Mexico: Malinalco

Supposing uniform motion, we define an azimuthal angular velocity ω_Az = (180° — A)/τ where A is the azimuth of the rising sun and τ half the time during which the sun stays above the astronomical horizon (this time can be easily obtained by knowing the times of sunrise and sunset). After the time τ the sun attains its meridian altitude β = 90° — ϕ + where ϕ represents the geographic latitude and is the declination of the sun. We also define an altitudinal angular velocity ω_Alt = β/τ. The azimuthal angle 180° — A — α, where a is the angular deviation of the shrine axis from astronomical south, is therefore swept by the sun at the time t₀ = (180° — A — α)/ω_Az after sunrise. On the other hand, during this time t₀ the sun attains the altitude β* = t_oω_Alt, or β* = [1 — α(180° — A)⁻¹]β which, because of the symmetry of the solar path with respect to the local meridian, precisely equals the altitude of the sun's passage across the alignment line. Thus, such a passage of the sun takes place at the time t* = t₀ + (2α/ω_Az), that is, t* = [1 + α(180° — A)⁻¹]τ, after sunrise. In the case of the winter solstice at the Monolithic Temple we have = −23°26′, ϕ = 18°57′10“N and α = 35′. The azimuth A is given by A = arc cos(sin /cosϕ) = 114°51′52” (see for example Woolard and Clemence, 1966) and from the Malinalco times t_rise = 7^h 07^m, t_set = 18^b 06^m (Flores, personal communication, 1987) it follows that τ = 1/2(t_set — t_rise) = 5 hours 29.5 minutes. Finally, we obtain β = 47°36′.8, β* = 47°11′ and t* = 5 hours 32.4 minutes; i.e. at t_cagle = t_rise + t* = 12^h 39^m.4 the sunlight will illuminate the head of central eagle. However this statement can be true only if the height of the door has a minimal value given by H = d tanβ* where d denotes the light's minimal reach, that is, the distance between the eagle's head and the exterior edge of the door. Taking into account the measured value d = 2.70m, we obtain H = 2.91m.

We have used sinδ = sinϕ cosz + cosϕ sinz cos A, where δ, ϕ, z and A denote the declination, the geographic latitude, the zenith distance (z = 90° — h, where h is the altitude) and the azimuth respectively (see for example Woolard and Clemence, 1966). We took into account a horizon altitude of h = 3° 15′ and an azimuth of A = 105°31′ (Iwaniszewski, personal communication, 1987) for the vertex of the right-angle defined by the mountain cut. Furthermore the altitude of the horizon was corrected for the effect of atmospheric refraction by subtracting 20′ from it to get the true altitude of the rising sun (Aveni, 1980, 105). Taking ϕ = 18°57′10“N and z = 90°–2°55′ = 87°.0833 we find the solar declination for the date on which the sunrise event occurs is = — 13°39′. Consulting the ephemeris data (Flores, 1988) we obtain the dates 12 February and 29 October.

Our exploration of the cut consisted basically of visual and tactile examination of the stone wall. Whereas some portions of the wall clearly have a natural appearance, other portions are composed of relatively smooth surfaces. However, we consider that a solid-matter specialist is required to give a definitive verdict. Access to the cave is difficult on account of the steepness of the wall and the overgrown vegetation. Indeed, only one member of our survey team was able to reach the cave and enter it! The cave is rather a narrow, shallow fissure in the rock. No surface material was visually detected inside or near the entrance, nor did our archaeologist find any surface material at the footing of the wall. Of course, a more thorough study of the wall and the cave needs to be carried out.

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