The following outlines the data elements presented on the Main Screen of the Dialist's Companion. All values can be viewed simultaneously, with dynamic elements updated every second.

Sunrise and Sunset are the standard times when the sun's center is on the horizon. This time is somewhat later than almanac predictions of sunrise time and earlier for sunset, since it ignores 1) refraction of the sun's rays when it is near the horizon and 2) the sun's apparent diameter. The time given here is approximately 3-4 minutes later/earlier than the sun's actual first/last appearance above the horizon.

Altitude is the current altitude of the sun above the horizon, measured in degrees. The altitude is at its maximum for any given day at noon solar time (dial time - for a dial that does not incorporate longitude corrections), when its value is 90° plus the solar declination minus the latitude.

Given the latitude L, solar hour arc T and solar declination D, the altitude A is determined as follows:

sin A = sin L sin D + cos L cos D cos T

Azimuth is the current azimuth of the sun, measured in degrees from the meridian (i.e. from the South in the Northern hemisphere and from the North in the Southern hemisphere). The azimuth is the angle on the horizon between its intersections with 1) the meridian and 2) the arc through the zenith and the sun's current location.

Given the latitude L, solar hour arc T and solar declination D, the azimuth Z is determined as follows:

cot Z = (sin L cos T - cos L tan D) / sin T

Declination is the sun's angle above or below the celestial equator, measured on the circle passing through the sun and the celestial poles. Currently, the declination ranges from approximately 23.45° at the June solstice to -23.45° at the December solstice. Equation of Time The Equation of Time is the difference between local mean and solar times.

A frequent practice among dialists is the assumption of a constant Equation of Time during the course of a single day. Actually, of course, the value undergoes continuous change. In a full 24 hour period, the maximum variation in the Equation of Time is in the neighborhood of 30 seconds.

The Dialist's Companion provides an exact calculation of the Equation of Time for the system date and location; the value is updated every second.

If we define a solar day as the time between successive crossings of our local meridian by the sun, then solar days are not all equal in length. In January the sun moves faster across the sky than it does in June.

If we consider a fictional sun which moves along the celestial equator (instead of the ecliptic) at a constant speed equal to the average speed throughout the year of the true sun, then this fictional mean sun would match our clocks, and all mean solar days would be equal to each other.

While the difference in the length of a true solar day from one day to the next is small, the variations add up over weeks and months to make the sun time recorded by a dial vary from a regular, uniformly running clock by as much as 16 minutes. On some days the true sun is ahead of the clock and the mean sun; on others, it is behind.

The equation of time is determined by three factors. The first two are:

1) the obliquity of the ecliptic ( i.e. the earth's axis of rotation is not perpendicular to the plane of its orbit - this is also the primary factor in determining our seasons); and

2) the eccentricity of the Earth's orbit (i.e. rather than moving at a uniform rate in a perfect circle, we move at varying velocities in an elliptical orbit).

The third factor, which has an impact only over very long periods of time, since it is itself very slow to change, is :

3) the distance between the perihelion (or point of closest approach to the Sun of the Earth's orbit) and the vernal equinox point.

This field identifies the sign of the Zodiac where the sun is currently found. Each sign represents a 30° interval of longitude in the sun's apparent path along the ecliptic circle.

The vernal equinox occurs when the sun's true longitude is 0°; note that this is only roughly approximated by the moment when the sun's declination is 0°. The difference results from the fact that the sun does have a small non-zero latitude.

Using the moments when the sun's declination is 0° to represent the equinoxes is an approximation; similarly, using the extremes of declination to represent the solstices is also an approximation. The season changes actually occur when the true sun moves from one sign to the next: (in the Northern hemisphere we have the following)

Spring - from Pisces to Aries

Summer - from Gemini to Cancer

Autumn - from Virgo to Libra

Winter - from Sagittarius to Capricorn

The moment of transition from one sign to the next is calculated only approximately in the Dialist's Companion. The program will generally produce transition times accurate to within 10 minutes.

Standard Time (or Daylight Saving Time if the program has been so configured) is the time currently indicated by the user's system clock.

Greenwich Mean Time is the mean time at the zero meridian in Greenwich, England.

Dial Time is the time that would be indicated on a sundial under the user defined program conditions.

Sidereal Time, which is often used in astronomy, is a measure of the hour arc of the mean vernal equinox point. The sidereal time given here is for Greenwich, England and is therefore Greenwich Mean Sidereal Time.

A mean solar day is the average time between crossings of the meridian by the sun; a mean sidereal day is the average time between crossings of the meridian by a fixed point on the ecliptic. Because the sun appears to move throughout the year with respect to the background stars (from west to east), the mean sidereal day is always slightly shorter (by approximately 3 minutes, 57 seconds) than the mean solar day. Each sidereal hour is about 9.86 seconds shorter than a standard hour.

In fact, a tropical or solar year of 365.2422 solar days equals 366.2422 sidereal days. The difference between a solar day and a sidereal day adds up over the course of a year to one full day - corresponding to one revolution of the earth around its orbit.

Babylonian and Italian hours are from two of the earliest systems of recording time to use equal hours. The Babylonian system divides the day into 24 hours beginning at sunrise and ending with the following sunrise.

The Italian system also divides the day into 24 hours but begins counting at sunset and ends with the following sunset, in a manner similar to the Jewish tradition still followed today.

To determine these hours, given latitude L, solar hour arc T (-180° <= T <= 180°), and solar declination D, let the angle S (0° <= S <= 180° ) be defined by

cos S = - tan L tan D .

Then the Babylonian hour B and Italian hour I are calculated as follows:

B = (24 + (T + S) / 15) mod 24

I = (24 + (T - S) / 15) mod 24

Temporal hours are part of a system of recording time that was prevalent before the 15th century. The system divides the day into 12 daylight hours between sunrise and sunset, and 12 night hours between sunset and sunrise.

Near the equinoxes the length of these temporal hours is close to the standard equal hour we use today. However, near the summer solstice, the temporal daylight hours are longer, while the night hours are shorter. Near the winter solstice, the reverse is true: daylight hours are shorter and night hours are longer.

Within any one daylight or night period, the temporal hours are roughly equal to each other, differing only because of the slow but continuous change in the sun's declination

The difference in the length of a temporal hour from season to season can be fairly large. At latitude 40°N, we have the following:

On the main screen display, the daylight temporal hours are followed by a sun symbol to distinguish between day and night. Note that solar noon is always 06:00 temporal time, regardless of the day of the year. Sunrise and sunset are both 00:00.

To determine the temporal hour, given latitude L, solar hour arc T (-180° <= T <= 180°), and solar declination D, let the angle S (0° <= S <= 180°) be defined by

cos S = - tan L tan D.

Then the temporal hour H is calculated as follows:

If T < -S : H = 6 * (360° + T - S) / (180° - S)

If T >= S : H = 6 * (T - S) / (180° - S)

Otherwise H = 6 * (T + S) / S

Longitude Correction is the difference between the Standard Time for the specified Time Zone and local mean time. The correction is + 4 minutes for every degree of longitude that the specified location is East of the Time Zone's central meridian, and - 4 for every degree West of the central meridian.

Refraction correction is an adjustment to account for the effect of atmospheric refraction on the apparent altitude of the sun; refraction has no impact on the sun's apparent azimuth.

The Dialist's Companion lists sunrise and sunset as the times when the sun's center is on the horizon (i.e. has altitude zero); but note that refraction will cause the sun to appear to be above the horizon at these times. A sundial will be fast in the morning, and slow in the evening - in each case corresponding to an apparent altitude that is higher than expected.

Actually, the effect of refraction is usually ignored when designing or reading a sundial. This disregard for the phenomenon arises from a number of very good reasons. The effect is always fairly small and therefore easily overlooked. Whatever adjustment might be called for actually varies in the course of a day, ranging from zero at noon to a maximum when the sun is just below the horizon (but still visible, due to the effect of refraction). The adjustment is not easy to calculate, and any calculation will only provide an approximation of the average effect.

For example, suppose we look at a traditional horizontal sundial located at 40° N 75° W on December 25, 1995. At 4 p.m. EST the solar time is 4:00:05 (accounting for the equation of time); however, the effect of refraction will be on average to increase the sun's apparent altitude by 9 arcminutes and to decrease its apparent hour angle so that the time recorded (assuming sufficient precision) will be 3:59:39. The sun appears to "slow down" as it approaches the horizon.

The actual impact on the sun's apparent altitude, and therefore on the dial reading, will vary with such things as barometric pressure, temperature, and height above sea level.

The value calculated here assumes that the dial is at sea level in an environment where the barometric pressure is 1010 millibars and the temperature is 50° F. Higher barometric pressure increases the adjustment; higher temperature decreases it.

For a discussion of the formulas involved in approximating refraction and converting a change in altitude into a new hour arc, see Fred Sawyer's article "Atmospheric Refraction" in NASS Compendium 2:4 (December 1995).

The Total Correction to be applied to the dial in question, consisting of the user selectable factors (Longitude and Refraction) and the Equation of Time. It represents the adjustment that needs to be made to the current standard time to obtain the time shown on the dial.

Astronomers assign a unique sequential number - the Julian Day number - to each day, beginning at Noon GMT on Jan. 1, 4713 B.C.

If, for example, you would like to determine how many days old you are: find the Julian Day number for the date of your birth and subtract it from today's number.

Examples of Julian Days:

Despite similarity of name, the Julian calendar and the Julian Day number are not directly related. The calendar is named for Julius Caesar; the day number recalls Julius Scaliger.

Solar Noon is the standard time on the program date at which the sun is on the local meridian - corresponding to local noon.

Shadow is the length of a shadow cast by an object of unit height under the current program conditions.

Given the sun's altitude A, the length of the shadow of a vertical gnomon with height equal to 1 is cot A.