CHARTING THE analemma.
Analemma Mechanics, analytical development, its layout and its interpretation.
while investigating solar clocks arises time and again analemma word as a device to "correct" Solar Time with respect to the legal time, what is sought is the correlation between clock time "mechanics" with the time indicated by clocks
Sun
already seen in Article Legal Time, Time, which is the Solar source of the discrepancy between Legal Time and time given by the Sundials
The Analemma not a natural phenomenon such as is revealed only by comparing the position of the sun every day at the same time, Legal time that is not natural but an invention of necessity of man.
The Analemma is a geometric shape that resembles the number eight (8) and is generated by the combination of two oscillatory movements perpendicular to each other, a product of the Equation of Time takes place in the East-West and the other by the decline of the sun that occurs in the north-south axis, the result is a Lissajous figure.
A quick way to visualize graphically the function Analemma is taking advantage of the EXCEL spreadsheet for graphing, as values \u200b\u200bof "X" and "Y" put those indicated in the table of the Equation of Time and those shown in the table of the declination of the Sun, both in degrees, but we can make a combination degree-minutes. As Excel automatically adjusts the scale for the axes, the graph is showing developing a fairly distorted analemma stylish domed and not as it really is. This distortion of Analemma not allow us to superimpose sundial. However, it is possible to explore the changes undergone by the figure by adjusting the scales of the axes to obtain an approximate picture we would see in the sky when the sun's position fotografiáramos several days throughout the year. The following image shows the Analemma developed in EXCEL with automatic scaling.
On this figure we can put some important points or interest, as the line of the equinoxes and the solstices, the meridian line, the days when the equation of time is zero and when is the highest among other things.
Analemma For the layout of the sundial have to start from the position of the sun in the sky and the projection of the shadow of the gnomon on the sundial.
The Analemma can be made for any time, but the present one it will draw only noon in order to determine the delay or advance with respect to the True Sun Sun noon Legal Middle right, ie Equation time.
Analemma To chart the need to establish two important references:
1 - Length of the shadow.
2 - Azimuth of the shadow.
These parameters are actually polar coordinates that establish the position of the points on the sundial that will allow us to draw the analemma.
length of the shadow.
The length of the shadow is a function of the height of the gnomon that we use and the height of the sun above the horizon.
The height of the Sun is itself a trigonometric function (spherical trigonometry) the decline of the sun, the latitude of the place where the gnomon and hour angle as discussed in the article in the Azimuthal Sundial .
formula allows us to determine the height "a" King star in the firmament is:
Sen (a) = Sen (l) x Sen (d) + Cos (l) x Cos (d) x Cos (H)
Where: a =
Sun's altitude above the horizon in degrees.l = latitude of location in degrees.
d = Declination of the Sun in degrees.
H = hour angle measured relative to the meridian of the place in degrees.
The decline of the Sun will be available from the following table.
Keep in mind that the height of the sun given by the formula is measured facing the South Pole or North Pole. The sun reaches its zenith when the decline of that takes the value of the latitude. From this point you should consider the following:
For those who are inside the Tropic of Cancer, when the sun's declination is negative, the height is taken looking south, this orientation is maintained until the Sun reaches Peak. For consecutive days, the height is measured facing north.
The hour angle is a function of the Legal Time, hour by difference in length between the meridian of the place where analemma and delineate the chosen time zone in the country, and the Equation of Time.
In our case, the hour angle is the angular displacement of the Sun with respect to the meridian the place and this defines the Solar Time (Hs), Solar Time can be calculated using the following formula:
ET Where: Hs =Hs = HL + DL +
Solar Time. Legal
HL = Time taken referenced to UTC. For Venezuela is -4:30. Hour-by
DL = longitude difference.
ET = equation of time.
The difference of longitude "DL" is established as the angular difference, expressed in time between the meridian of the place "ML" and the UTC.
DL = ML - UTC
When the difference in length is negative, the meridian of the place is behind with respect to UTC and if the length difference is positive, indicates that the meridian of the place is advanced to UTC, so the sun reaches the meridian of the place first before coming to UTC.
In the case of Puerto Ordaz, where the meridian is the analemma is -62 °: 27 'and the UTC for the country is -4:30, which is expressed in degrees -67 °: 30'. The difference gives 5 °: 3 ', which corresponds to time 20': 12. " As the value is positive, telling me that Puerto Ordaz has advanced Solar Time with respect to the Legal Time in 20 ': 12. "
Remember that today technological resources at hand than a couple of decades ago were reserved for scientific institutes and associations and / or technology, some of these resources are computers and the Internet. With this pair of "tools" we know our latitude and longitude of our very precise and just go to Google Earth and place in the world where you place our watches Sun With the help of this technology is that I have given my geographic location.
The equation of time is defined by:
ET = Sm - Sv
where "Sm" is the ascension of the sun Middle "Sv" is the True Sun, the equation of time "ET" takes values positive or negative depending on the ascension of the sun is more or less true that the ascension of the sun means, ie, the True Sun "running ahead" to the Middle Sol or "delayed", reflecting a delay or an advance of the Hour Solar with respect to the mean solar time. For example, if by October 1 Equation of Time has a value of +10: 04 ", the true Sun is behind (lower right ascension) to the sun and Middle Eastern Daylight Time effect, this value" ET "corresponds to a preview of Solar Time with respect to the Legal Time in that amount with its own sign, so for the meridian UTC, at noon on the clock Sun is ahead in 10 ': 04 "to the Official Time.
Hence the equation of time can also be expressed as follows:
ET = Hs - Where
HL "HL" is the Official Time and "Hs" Solar Time.
For our calculations, which will be legal for half days, the angle at 12 PM law is zero, HL = 0, so that the equation to determine the hour angle of the sun at 12 PM is legal reduced to:
The following table gives the Equation of Time for each day of the year expressed in degrees.
Hs = DL + ET
The length of the shadow is defined by the following expression of plane trigonometry:
AG ÷ Ls = Tg (a)
Where:
Ls = Length the shade.
AG = Height of the gnomon.
a = height of the Sun in degrees.
the gnomon height define it according to the size of the projection we want, for the purposes of this project, the prefecture shall height 100 mm.
AZIMUTH OF THE SHADOW.
The azimuth angle is a function of height Sun, the hour angle of declination of the Sun and the latitude of the observation and can be calculated by:
Cos (Z) = (Sen (l) x Sen (a) - Sen (d) ) ÷ (Cos (l) x Cos (a))
Where:
Z = azimuth angle measured from the meridian and positive face South East.
l = length of the place.
d = Declination of the sun for the day in question.
a = height of the sun for the day and time in question.
Determining the length of the shadow and the azimuth angle for example, for every afternoon of 1, 12, 21 and 30 of each month, we get the table next.
The results of the length of the shadow got to a height of Nomon (AG) of 100 mm. It is worth mentioning that the calculated azimuth angle is the Sun's position in the sky and we are working with the projection of the shadow of the gnomon, so that the calculated azimuth we must add 180 ° to give us direction and meaning the shade. These are the azimuth values \u200b\u200bshown by the previous spreadsheet.
the analemma.
For the layout of Analemma already have the polar coordinates of the points for days 1, 12, 21 and 30 of each month where the length of the shadow is the radius vector and the azimuth angle. With the help of a ruler and protractor or better yet using a computer aided design software with a computer we can put points on paper according to polar coordinates found. Obviously, the curve will be better defined and accurate to the extent they are used more points (days) for their layout.
The figure below shows the layout of the shadow of the gnomon for the months of January and December on the horizontal plane.
Once the layout of all the "shadows" on the paper, join their free ends in the same order lines were placed representing the shadow of the gnomon and it got the figure of Analemma.
The following image shows the envelope Analemma as the lines of "shadows."
The following figure is the same image without the lines of "shadows", the result is a fantastic and evocative image, simple and beautiful it seems to keep a secret.
The dotted line "NS" represents the meridian of the place (North-South) which traced the Analemma and can be easily seen that the curve is shifted "E" (East) because the city of Puerto Ordaz (Ciudad Guayana) is ahead (east) over the reference meridian -4:30 opted for the country to set the Time Legal. The shift shaft Analemma over the line "NS" corresponds to the 20 ': 12''advance that Puerto Ordaz with respect to UTC, as can be seen, not Analemma intersects the southern boundary "NS" This means that Solar Time and Time Legal ever match for this location.
On this curve we can put some information of interest as mentioned at the beginning of the article. The solstices, equinoxes, aphelion and perihelion, and values \u200b\u200bthe advancement of solar time with respect to the Legal Time in minutes and other information.
The figure shows the Analemma with some additional information.
In the picture you can see that the closest point between Solar Time and Time Legal is the month of February, around day 12 with a difference of 5.9 minutes and the time maximum disparity in early November with 36.7 minutes in advance between Solar Time and Legal.
The following image shows the Analemma overlapped Azimuth sundial clock.
The number of subsequent frames show us for a Nomon Analemma 100 mm height of pyramid shape.
The first picture shows the moment to "align" the North-South route Analemma. In a future entry will be shown a simple and practical method for the orientation of the Sundials
the gnomon on the dial placed horizontally. The shadow indicates that solar noon is over as shown in the picture below.
Time it is 12:00 noon Legal.
photography gives us a detail of the shadow of the gnomon on Analemma, note that almost the tip of the shadow of the gnomon is on the point that represents the day 21, the Summer Solstice (the zodiac sign is Cancer.) The photos were made on the eve of the solstice, on June 20.
The picture below shows the figure of Analemma on Azimuth sundial clock.
In this picture you can see the legal time indicated by the cell clock and time given by the clock Solar Azimuth. Legal noon reads the Analemma just as the shadow of the gnomon makes contact with the figure in the relevant month You can clearly see the difference between the two hours, with advance the sundial. The advance is about 18 minutes to the time of photography, data that indicated no defaults on Analemma drawn on the sundial.
With this article I have tried to clarify what the Analemma, its origin, its path on a sundial and how to read to tell the difference between Solar Time and Time Legal.
June 2010.
Note to typos mode: In pictures and photos of finished Analemma is casting a "misprint." This error is the improper placement of the symbols that mark the solstices, so that the symbol of Cancer is at the place that corresponds to Capricorn and vice versa.
