The sundial worked before was to Declining Quadrant in this installment we look at the Azimuth sundial.
sundial in the Azimuthal Azimuthal or the gnomon is parallel to the axis of the world such as watches that are based on the equatorial quadrant, but is positioned vertically, perpendicular to the ground and the hours are defined by the projection of the shade of the same according to the height of the sun in the sky and the angle measured on the horizon and azimuth angle, ultimately by the sun's position in the sky with reference the horizontal plane.
In principle, it seems that we can make a sundial by simply placing a vertical rod and drawing on the ground grade the hours spread evenly around the vertical rod. Nothing could be further from reality, as when performing this simple design we will quickly realize that the clock does not indicate the correct time, and what is worse, the shadow on the ground is not equal in length or angle though we measure at the same time from month to month.
Unlike Equatorial sundial in azimuth time is not determined by the angle of the shadow over the southern boundary, but is determined by two variables: the length of the shadow and the angle is with respect to the meridional plane.
To understand a little what happens to the shadow, we observe the behavior of the Sun, in their movement across the sky throughout the day and year.
Although we can divide the day into 24 hours and that the Sun apparently moves to meet that schedule, we note that in the different seasons of the year the sun's position is not fixed with respect to reference objects that we have on our horizon that will slowly changing. The cause of this phenomenon is due to two reasons:
First, the axis of rotation of the earth is inclined to its orbital plane and the second by the observer's location on the globe, ie the latitude.
Being tilted the axis of rotation of the earth on its orbit, the Sun elevation changes throughout the year in northern latitudes and reaches its maximum height above the horizon at the summer solstice, and the height minimum in the winter solstice, which is our case here in Venezuela. During the equinoxes the Sun moves across the sky Ecuador.
The above figure represents the sky for two different observers, one located in Ecuador and the other above the equator. Each observer will see the sky slightly different due to its geographical position.
To the observer who is in Ecuador, the Sun reaches the zenith at noon on the equinoxes and its shadow would be right at your feet, while in the midday sun solstices is inclined with respect to the vertical an angle equal to the inclination of the earth on its orbit, which is about 23.5 degrees. In this case, the shadow of the observer will have a length and the same would be projected in the direction of one pole as the winter solstice or the summer.
For the observer located at another latitude, during the equinoxes the sun is not in the zenith distance if not the same at an angle equal to your latitude which depends on where the observer is.
If you pay attention to the trajectories of the Sun shown in the figure above, we conclude that for the latitudes they are over Ecuador and particularly in those places that are beyond the tropics (Cancer and Capricorn) the arc described by the sun during the summer solstice is higher than that reported during the winter solstice is the reason why in summer the days are longer in winter while the equinoxes, the days and nights are equal. The same is true for observers who are below the equator, only here, the longest day occurs during the winter solstice.
The system used by convention to give the position of a star in the firmament is the "celestial coordinates" whose reference points are the right ascension and declination.
The right ascension is equivalent to the meridian system used geographically with respect to a reference point for the case of the celestial coordinate system is the vernal equinox (start of the spring equinox.)
The decline is the equivalent latitude, taking as reference the celestial Ecuador Ecuador is the projection of Earth in the sky.
For the development of our watch we are only interested azimuth of the sun's decline over the months. His ascension did not take into account, but the hour angle measured from the meridian that passes through the zenith of the observer.
our clock to calculate azimuth, we have to base on the horizontal coordinate system in which the position of a celestial body is defined by its azimuth (azimuth angle is measured on the horizon taking as "zero" reference meridian of the place normally see South Pole) and the same height above the horizon. With these data we can set the length of shadow of our gnomon with its projection angle in the horizontal plane and thus trace our azimuthal quadrant.
If we have the position of a star "A" in the sky (see figure above), and draw lines linking the star "A" to the zenith "Z" of the observer and the elevated pole " P "we obtain the spherical triangle" AZPA "which secures the side" PZ "located on the meridian of the place, while the side" PA "stands on the hour circle of the sun and rotates around the pole "P", causing the height and azimuth of the sun varies over time.
This triangle "AZPA" is called "triangle of position and by means of spherical trigonometry can be deduced given other elements, thus solving all problems arising in celestial navigation and projection problems for our clock azimuth.
From the figure we can raise the hand "PZ" is the zenith distance of the pole and is the complement of latitude "l": PZ = 90 - l.
The fixed side of convenience "PA" is the polar distance of the star and is the complement of its declination "d": PA = 90 º - d.
side "ZA" is the zenith distance of the star, snap up "a": ZA = 90 ° - a.
The hour angle of the sun is equal to the time since passing the meridian of the place and that it is local solar time you start counting from that time. The times are negative numbered degrees of longitude east and west longitude positive. However, before we can determine that for equidistant hours before noon and after noon, the sun's altitude will be the same.
short, the height "a" of the sun we can determine the following findings:
Sen = sin sin d + cos lx cos lx cos dx H.
Where "H" is the hour angle measured from the meridian, this angle is 15 ° for each hour and 7 ° for half hour. Determined
height "a" of the Sun in terms of hour angle we can calculate the azimuth (Z), which is the other information we need to chart our azimuthal quadrant.
Cos Z = (lx Sen Sen - Sen d) ÷ (lx Cos Cos a).
To solve these equations need to know the declination "d" of the Sun during the months of the year. This information is readily available on the Web. However the table used development of this watch is as follows.
To obtain the required values \u200b\u200balong the solar day to make our path, we can facilitate our calculations relying on a spreadsheet. Completed
calculations, we have the position of the Sun in our sky hour by hour and month by month, which will establish the length and direction of the shadow.
The azimuth determined by this formula does not tell us if it is negative or positive (if another formula determined by the hour angle), ie if the azimuth angle is before or after Meridian. What if we determine this formula is when the sun is north or south is. This point is very important to know where it is headed in the shadow of the gnomon our determined with full accuracy for the case of zero angle "0 °", which corresponds to noon. To this point of time, the azimuth can take the value zero "0 °" or 180 degrees. If the calculated value is 0 º, our sun is to the south and if the calculation indicates 180 °, the Sun will be the North. The azimuth angle is taken from the meridian of the place counted from the South Pole.
This point is important in order to correct the path so and properly target our clock with respect to geographic north, in the same way we had to target other sundials outlined in previous articles.
To determine the length of the shadow according to the height of the sun in the sky enough to solve a simple trigonometric equation.
In the figure above we can deduce the length of the shadow produced by the gnomon.
Tangent (a) = HEIGHT GNOMON / LONG SHADOW
From equation
know the angle of the Sun's altitude (a) and height of the gnomon, with these data we can calculate the length of the shadow. With the calculated azimuth angle we can draw on our quadrant azimuth lines that indicate the time and month.
For the drawing of lines on our quadrant azimuth, we can do "old" with a ruler and square, or, like me taking advantage of technology and printing on paper with the dial 1:1 scale previously developed a program design.
Our reference is the place where is the projection of the tip of our gnomon on a horizontal plane from the place where it will be built. For each hour will place a point whose location is defined by the azimuth and length of the shadow in the direction that corresponds (North or South) as the azimuth either 0 º or 180 º. These points will unite with a continuous line being drawn is a parable that is the journey that makes the tip of the shadow on the floor as they pass the hours. Parables drawn for the months, we join the points for hours, so are drawn on the dial a series of straight lines tend to converge.
The gnomon for my watch is 100 mm high with a pyramid shape and the template to construct shown in Fig.
The following photos show the azimuthal quadrant template, trim excess paper to put on quadrant carton of 2 mm thick substrate would be the same.
time when we are placing the template on the cardboard printed azimuthal quadrant.
Quadrant stuck to the cardboard base.
trim excess cardboard.
ready azimuthal quadrant.
gnomon templates glued to cardboard 2 mm thick.
Cutting the gnomon.
Armando gnomon.
placing the gnomon on the dial.
Local Solar Time 9:00 AM.
Note that solar time is defined by the tip of the triangular shadow.
Azimuth dial clock dial to Landscape.
As I mentioned at the beginning of the post, this watch azimuth has the advantage of letting us know the month of the year and even the date and time if the azimuth plane size is large enough to allow this resolution.
The basic problem is that the trajectory of the Sun from January to July, is retraced by the same route, which prevents good to first determine the month of the year. One way to save the point is to divide the quadrant into two equal parts by the sagittal axis, so that put a half months ranging from January to June and the other half of the layout of the remaining months as shown in the picture below.
to interpret, at least we know where half of the year we are making reference to the station in question or knowing that he has passed one of the solstices.
If we were quadrant with the layout of the previous figure for the corresponding months of January through July, we would see reflected in the evenings and the rest of the year during the morning.
The following image shows the tip of the shadow falling on a parabolic lines of the azimuth quadrant, in this case (not identified in this prototype) corresponds to 1 September and True Solar Time 9:00 AM .
I should clarify that the sundial gives the True Local Time, which does not correspond to the legal or official time.
In the case shown, the sundial indicates approximately 9:00 am while mechanical clock that gives us 8:40 am. We quickly realize that there is a difference of about 20 minutes between the two "hours" for the day when the photos were taken. This difference is more pronounced in this clock with respect to the first that have been developed in this Blog by time change decreed by the Government of Venezuela.
In the next release (published in Technical Note) will expose more explicit this phenomenon to clarify any doubts that this difference of hours awake in the observers, as a rule people unconsciously compare their mechanical watch against the Solar, blaming the lack of "accuracy" between times read as a problem of design or manufacture of the sun clock
I hope this brief summary has been clear enough for those who wish to make your own sundial dial Azimuthal can build it without major problems and enjoy the fascination that these small monuments to the sun awaken our consciences.
