Wndsn Quadrant Telemeter Tutorials
Making the most out of our graphical telemetry computers.
Like with many complex instruments, there are multiple ways to solve certain problems and to measure the required inputs. Combining the various functions leads to a multitude of advanced uses.
Using the Obliquity Arc
Astronomy
The declination of the Sun is the angle between the Sun's rays and the equatorial plane of the Earth. The declination angle is the same for anywhere on Earth on a given day.
The ecliptic and the celestial sphere. The ecliptic is the circular path on the celestial sphere that the Sun appears to follow over the course of a year. The term also refers to the plane of this path, which is coplanar with Earth's orbit around the Sun (and hence the Sun's apparent orbit around Earth).
Obliquity of the ecliptic is the term used by astronomers for the inclination of Earth's equator with respect to the ecliptic. It is approximately 23.44°. Earth's axis remains tilted in the same direction with reference to the background stars throughout a year (regardless of where it is in its orbit).
Relationship between Earth's axial tilt to the tropics and the polar circles.
During the half of the year from March equinox to September equinox, the Northern Hemisphere tilts towards the Sun, with the maximum occurring on the June solstice. For the other half of the year, the same happens in the Southern Hemisphere, with the maximum around the December solstice. The two points where the Sun is directly overhead at the equator are the equinoxes.
This is the cause of Earth's seasons. Around the March equinox, the Northern Hemisphere is experiencing spring as the hours of daylight increase, and the Southern Hemisphere is experiencing autumn as daylight hours shorten. Summer occurs in the Northern hemisphere when the north pole is directed towards the Sun and vice versa.
Traversing the Quadrant
Over the course of a year -- historically and astronomically starting on March Equinox -- the declination oscillates between 0° and 23.44° every approximately 90 days: 0° on the equinoxes, +23.44° on the day of June solstice and -23.44° on the day of December solstice.
Sun declination by day. The obliquity of 23.44° is the amplitude while 90° is 1/4 of the wavelength (of 360° which we equate with days for our approximation) forming a sine wave which we can read on the quadrant.
This means that we have to walk the quadrant back and forth; between March equinox and June solstice, we are moving from 0° to 23.44° (0-90), between June solstice and September equinox, we are moving back from 23.44° to 0° (90-0), then from 0° to -23.44° (again 0-90) until December solstice, and back to 0° (90-0) until March equinox.
In order to find out the declination of a given day, we have to know which quarter we are in, map 23.44° to 90° and determine the respective day number, where 1 day ≅ 1°.
The Calculation
The counting of the days and mapping them to the obliquity can be done in a graphic way on the quadrant. To facilitate that, an arc set at 23.44° is engraved on the quadrant (we could do it without the arc just as well though).
Construction of the obliquity arc representing the sine wave of obliquity and days. The obliquity arc's radius is the result of projecting horizontally from 23.44° on the degree scale, which means that 0° corresponds to 0° and 23.44° corresponds to 90°, which we use to count the days.
The obliquity arc is a circular arc around the vertex of the quadrant, it is a projection of the Earth's orbital obliquity (or tilt of its axis). The purpose of this arc is to determine the Sun's declination (angle above or below the equator) for a given day, in order to subsequently determine the Sun's altitude at noon for that day.
Some Quadrants show a reduced obliquity mark ε at the 24/60 posiition of the sexagesimal cosine scale.
Using the obliquity arc to compute the solar declination for a given day. The 35th day after the March equinox (APR-25) has a declination of ~13°.
The Procedure
- Count the days since March equinox and determine the sign to expect.
- Set the string on the degree scale at the day number determined.
- Project horizontally from where the string crosses the obliquity arc (by holding that point down with a fingernail or similar) back to the degree scale.
- Read the respective declination for that day.
What to do with the Declination?
Knowing the Sun's declination for a given day, we can subsequently determine a number of interesting values for that day.
If we know the latitude (by quadrant observation or from data), we can calculate (knowing the maximum and minimum declination 23.44°):
- Maximum solar altitude on equinox and solstice
- Sunrise and sunset azimuth on solstices
If we, in addition to the latitude, know the date or the declination (by quadrant calculation using the maximum noon altitude of the Sun) for the respective date, we can calculate:
- Declination or noon altitude respectively
- Sunrise and sunset azimuth
- Hour angle at sunrise
- Sun hours
- Equation of time
If we, in addition to latitude and declination, know the instantaneous Sun altitude (by quadrant observation), we can calculate:
- Hour angle
- Time to or from noon
- Sun azimuth
- Shadow length
See also the Wndsn Sun Calculator to measure local latitude and determine declination for a given date, to calculate a number of latitudinal, diurnal, and/or instantaneous solar values.