HOW WARM IS THE SUMMER SUN!
A note by Eamon Henry; 27 July 2009
Foreword: This brief note compares the radiation intensity of the midsummer Sun with that of the midwinter Sun. We need first to define that we mean, and next describe how to do it. Results are then given, first a mid-day (noon) comparison, and then an average per hour comparison. A technical appendix gives the geometric (trigonometric) measurement background.
The General Background: We consider Dublin as being about 53.5 degrees (Latitude) north of the Equator. Some facts (without proof) are as follows. At the Spring and Autumn equinoxes, at Dublin the Sun at noon has about 36.5 degrees elevation above the south horizon, given by (90 less 53.5) degrees. At midwinter the Sun will be only about 13 degrees above the south horizon at noon, midway through a day about 7.5 hours long. At midsummer, the Sun at noon will be about 60 degrees above the south horizon, midway through a day about 17 hours long.
These differences in the Sun’s mid-day elevation are due to the fact that on its orbit around the Sun, the Earth’s axis points the northern hemisphere 23.5 degrees lower (than at the Equinox) at midwinter, and 23.5 degrees higher at midsummer, as measured against the plane of the Earth’s orbit.
EXPERIMENT DESCRIPTION:
We imagine a clear sky, and a square window of edge 1 metre long, and thus of area 1 square metre, always pointing directly at the Sun, and thus tilting back and rotating as the Sun rises and gains height. Directly beneath the lower edge of the window we imagine a horizontal floor on which the Sun shines through the window. As the Sun climbs, the sunlit floor area gets gradually smaller, implying more intense radiation. This sunlit area, during time, and of dimension area multiplied by time, is our measure of the Sun’s intensity. Since the window is 1 metre wide, this sunlit rectangle is measured by the length of its edge on the floor.
Because the area is very large for sunrise and shortly after, I omit effects from sunrise to 1 degree elevation of the Sun above the horizon. In treating a day, I assume symmetry, in the sense that effects from sunrise to noon are mirrored (in reverse) from noon to sunset. I also assume that the Sun’s radiation power as such is the same on both days, namely midwinter and midsummer.
Because the area is very large for sunrise and shortly after, I omit effects from sunrise to 1 degree elevation of the Sun above the horizon. In treating a day, I assume symmetry, in the sense that effects from sunrise to noon are mirrored (in reverse) from noon to sunset. I also assume that the Sun’s radiation power as such is the same on both days, namely midwinter and midsummer.
RESULTS:The easiest result is to compare the floor area sunlit at noon on midwinter day (21 December) with that on midsummer day (21 June). The midwinter area is 4.445 square metres, as compared with midsummer area 1.155 square metres, giving a ratio 3.85. This indicates that this midsummer radiation intensity is 3.85 times that of corresponding midwinter intensity.
The average per hour areas for the same two days are as follows: midwinter 11.35 square metres , as against midsummer 3.39 square metres, giving a ratio 3.35.
In other words,t he noon intensity at midsummer is 3.85 times that of midwinter, while the per hour average intensity at midsummer is 3.35 times that of midwinter.
TECHNICAL APPENDIX:The vertical cross-section form of the supposed experiment is a right-angled triangle of hypotenuse the length of the sunlit floor, with one side of the 90 degree angle the vertical cross-section of the window pointing directly at the Sun, of length 1 metre, and the other side the line from the window top to the far edge of the sunlit floor. This latter side makes with the floor-line the Sun’s angle of elevation, say A degrees. The sunlit floor length is 1/sine(A), same as cosecant (A). For a constant width of 1 metre, cosec (A) also measures the sunlit floor area.
So, we need to measure cosec(A) multiplied by time, assuming for midwinter a steady increase of elevation from 0 to 13 degrees during 3.75 hours, while omitting cosec(A) values for A between 0 and 1 degrees. However, the noon comparison as such does not involve time , but is simply cosec (13 degrees)/ cosec (60 degrees). For those who know Calculus (Differential and Integral), the integral function of cosec(A) is log (cosec(A) – cotan (A)). The logarithm is to base e, of value 2.718282 to 6 decimal places. For midwinter day, the integral function will be used as the definite integral for A in the range 1-13 degrees. Please note that this assumed same length of time for the Sun elevating through each degree of the stated range is only approximate, implying a straight-line movement, whereas the apparent path of the Sun is a bending curve.
For Midwinter Day:The definite integral value for 1-13 degrees is 2.5744, needing scaling up by 57.2987 (the value of cosec(1 degree)), to give result 147.5098. This has an average midwinter time of 0.28846 hours per degree, obtained by 3.75/13. The area X time product is 42.55, which divided by 3.75 hours gives the average of 11.35 square metres per hour.
For Midsummer Day:I take three separate measures for the 8.5 hours period from sunrise to noon:1. Sunrise to due East during 2.5 hours, for elevation 0 to about 23 degrees:2. Due East to SouthEast 23 to 47 degrees, during 3 hours;3. SouthEast to South 47 to 60 degrees during a further 3 hours to noon.
These approximate angles have been calculated by me, by using a vertical foot-ruler and its shadow length on the floor, to give a tangent value of angle A, at local times 6.00 and 9.00 hours, some 25 minutes later than GMT, for Dublin taken as 6.25 degrees West longitude. The related three separate integrals for cosec (A) give a better estimate than a single average (per degree elevation) calculation.
These three scaled-up integrals (scaling factor being same 57.2987) have values 180.594 for 1-23 degrees, with per degree multiplier 0.1087 hours; 43.533for 23-47 degrees with per degree multiplier 0.125 hours; and 16.238 for 47-60 degrees, with multiplier 0.2308 hours. The combined area X time result is 28.82 for 8.5 hours, giving an average 3.39 square metres per hour.
Without Calculus, the cosecant-based areas can be calculated by using detailed tables such as those for cosecant shown on pages 12-13 of F. Castle’s book “Five Figure Logarithmic and other tables” (Macmillan,1969). For values shown at 0.1 degree intervals, the sum of 11 cosecant values for range 1 to 2 degrees is 440.552, giving an average 40.047. Similarly, the average cosecant value for range 2 to 3 degrees is 23.303, and so on. The sum of these averages for the range 1 to 13 degrees is 147.62, which agrees closely with the Integral value 147.5098 quoted above.
