The SunPath User's Guide

Determination of the Sun's Position

The position of the Sun in the sky varies throughout the day and season due to the spin of the earth around its axis and to its orbiting around the Sun. Knowledge of spherical and plane trigonometry as well as some elementary notions of astronomy are required in order to understand, and not just apply, the formulas for the computation of the times and directions of the direct incoming radiation on any locality of a planet.
In order to determine the position of the celestial bodis in the sky, they are assumed to lie on a single sphere, the Celestial sphere (Clelestial Vault). The center is depending on the different conventions, the center of our Celestial Vault coincides with the position of the observer (horizontal system) that is indicated by the piramid in the center of the picture.
The apparent daily rotation of the celestial sphere about the earth's axis may be expressed in terms of the hour angle: the angular distance between the hour circle and the observer's meridian. One hour is equivalent to 360°/24 = 15° of the rotation of the celestial vault. All the values of time in solar energy computations are expressed in terms of apparent solar time (this is also known as true solar time). But in certain casses, it may be necessary to express the successive positions of the Sun relative to a fixed point on the earth's surface in term of local clock time that differs from the apparent solar day. This program is based on the apparent solar time.

The basic terms and formulas used in the SunPath applet

The latitude of a place on the surface of the Earth is its angular displacement above or below the Equator, measured from the center of the Earth. It is given between 0 ° and 90 ° N or 90 ° S.
In the applet you can change the latitude of the location by writing in the text field the latitude with the right sign, + for the Northern Hemisphere and - for the Southern Hemisphere (examples: for San Francisco, that has a latitude of 37° 37 North, you have to write 37.62 or +37.62; for Canberra, that has a latitude of 35° 17 South, you have to write -35.28). You cannot set the latitude to 90° or to -90° because at those particular points of the globe the cardinal points lose definition.

The solar time is defined as the number of hours before or after noon. Noon is defined as the time when the Sun is highest in the sky.

For mathematical analysis the solar time is converted to an angle in degrees of radians. This is known as the hour angle and is the angular displacement of the sun from its noon position.

The declination is the angular displacement of the Sun from the plane of the Earth's equator. The value of the declination will vary throughout the year between +23.45° and -23.45° because the axis of the Earth is tilted at a constant angle of its rotation about the Sun.
During a single day the declination delta can be assumed constant and equal to its value at midday. The solar declination for the Northern Hemisphere is calculated as:

delta = 23.45 * sin [(day_number - 81) * 2 * PI / 365] (deg) Eq. 1

where day_number is the day number of the year, starting from 1 on the 1th of January.

The sunrise hour angle is the hour angle at sunrise:

omega_s = arccos -(tan fi tan delta) (rad) Eq. 2

The sunset hour angle is equal at -omega_s.

The solar altitude is the angle a direct ray from the Sun makes with the horizontal at a particular place on the surface of the Earth. The Sun will be at its highest above the horizon at noon each day.
The altitude of the Sun at any time of the day is determined by:

alfa = arcsin(sin delta * sin fi + cos delta * cos fi * cos omega) (rad) Eq. 3

You can draw the altitude angle on the picture by clicking on altitude button.

The solar azimuth angle is defined as the angle a horizontal projection of a direct ray from the Sun makes with the true north-south axis. This is usually given as an angular displacement through 360° from true north, but for the purposes of SunPath it is express in a set of (-180°, +180°).
The azimuth of the Sun at any time of the day is given by:

spi = PI - arccos[(cos fi sin delta - cos delta sin fi cos omega) / cos alfa] (rad) Eq. 4

You can draw the azimuth angle on the picture by clicking on azimuth button.

SunPath applet's buttons

In the Sun's path simulation you can navigate in time and space.
You can move in time using the buttons situated on the bottom of the picture. You can also stop, play forward or play back the simulation and in a day simulation there is the possibility to play fast and play back fast too.
You can move in space using the buttons situated on the right of the simulation picture to rotate and to tilt the Earth, it is also possible zoom in and zoom out the simulation picture using the corresponding buttons.
You can set the simulation type, the month, the day and the hour by pushing the mouse button down on the correspondent choice menu, this cause a menu to appear with the current choice highlighted.