07.02 Sunpaths

Categories: Lighting

Introduction
There are many different ways of representing the position of the sun in the sky and numerous methods of investigating sunlighting in design. Only one or two of the different methods will be considered in order to demonstrate in principle how they may be used.

The basic astronomical facts will be reviewed but a detailed knowledge of them is not essential for an appreciation of sunlighting.
 

The Earth
The Earth is effectively a spherical globe which rotates about a North-South axis approximately every 24 hours, image. A globe may be partitioned in various ways which are useful in describing positions on the globe, image. If a globe is divided into two equal parts to produce two hemispheres, then the dividing line between the two parts will be a great circle. The axis about which the Earth rotates is known as the Polar Axis and this axis intersects the globe at the North Pole and the South Pole. A great circle passing through both poles is known as a Meridian. A great circle which is equidistant from the North and South Poles is known as the Equator.

Any position on the Earth may be specified in relation to, image;

• a primary meridian,
• the equator.

The Longitude describes the position of the appropriate meridian in relation to the primary meridian - it may be East or West of the primary meridian.

The Latitude describes the angle from the equator towards a Pole along a particular meridian.
 

The Earth’s orbit around the Sun
The Earth orbits the Sun approximately once every 365 days. Its orbit lies in the same plane as the Sun and is elliptical in shape with the Sun positioned at one of the ellipse’s foci, image.

One consequence of the elliptical orbit is that the earth speeds up and slows down as it moves around the sun and this means that the length of the day, measured from noon to noon, changes throughout the year. In order to simplify timekeeping the average length of day is used rather than changing the length of day from one day to the next. This leads to the familiar time convention in the UK of Greenwich Mean Time, which is based upon a mean or average length of day.

The difference between solar time, which is based upon the time from true noon each day, and mean time is called the equation of time. When it is required to know the position of the sun very accurately then a correction should be applied for the equation of time. The maximum cumulative difference between solar time and mean time is in the order of between +15 minutes and -15 minutes throughout the course of a year.
 

The tilt between the Earth’s axis and the orbital plane
A most important feature of the Earth’s circumstance is that the Polar Axis is tilted in relation to the orbital plane, image. Within the time spans considered in architecture the direction of the Polar axis relative to the orbital plane remains constant. At the present time the Tilt is at an angle of 23.4º.

One consequence of the tilted axis is that seasons of the year are experienced by those parts of the globe closer to the poles. The closer a region is to a pole then the more seasonal is the climate experienced by that region.
 

Arctic circles and tropics
The tilt also gives rise to the division of the globe into various parts, image.

The Arctic Circles divide the regions of the Earth into those that will at some time in the year experience a 24 hour day and a 24 hour night, and those regions which always experience a day and a night.

The Tropics divide the regions of the Earth into those where the sun will be directly overhead at some time in the year and those where the sun will never reach the Zenith.
 

Declination
The tilt of the earth’s axis results in a change in the relative position of the sun as the earth moves around its orbit. This change in the relative position of the sun is reflected in the change that occurs in the angle the sun’s rays make with the equatorial plane. This angle is known as the declination, image.

The declination will vary from a maximum of 23·4º at the Summer Solstice to a minimum of -23·4º at the Winter Solstice. Twice in the year the declination will be zero and this occurs at the Spring and Autumn Equinox.
 

Altitude of the Sun at Noon
At noon each day the sun will lie in the meridianal plane and be at its highest position in the sky that day, image. If the sun lies in the meridianal plane then it will be seen to be due south and shadows will be cast due north.

The highest altitude may be found quite easily if the declination is known, and for the northern latitudes is given by,

g_noon  =  90^o - Latitude + Declination

At latitudes above the arctic circle, image, there will be times when the sun is above the horizon at midnight, and for the northern latitudes the altitude of the sun at midnight will be given by,

g_midnight  =  Latitude + Declination - 90^o
 

Hemisphere of sky
If a position on the earth is considered then the sky seen from that point will occupy a hemisphere, image. The ground plane will stretch to the horizon, North will lie in the direction of the meridian, as will South, but clearly in the opposite direction. East and West will lie along the latitude line in opposite directions. The Zenith will be normal to the ground plane, pointing directly up to the sky.

The position of the sun in the hemisphere of the sky will be given by two quantities,

i) the Altitude of the sun above the horizon plane,

ii) The Azimuth of the sun on the ground plane.

It should be noted that in these notes the Azimuth will be measured from the South, but in many books the azimuth is given as the clockwise angle from North, image. The reason for choosing to measure the angle from south is that it makes the calculations of angles somewhat simpler.
 

Gnomic projections
The Gnomon is a point in space through which the rays of the sun pass to later shine upon some surface. The shadow on the ground cast by a flagpole will depend upon the sun’s altitude and azimuth, and if the topmost tip of the flagpole is considered, then it will sweep out a path on the ground as the sun moves across the sky, image. The topmost tip of the flagpole may be considered as a gnomon.

It may occur to you that a simple perspective is also constructed as a gnomic projection.
 

Sundials
Plotting the paths of the tip of the shadow for times throughout the year will produce a sun dial, image. If the height of the gnomon is known then the sundial can be used to construct the shadows created by buildings at different times of year and the sunpatches created by sunlight shining through windows.
 

Sunpath Diagrams
The sundial is the projection of the sun’s direction through a gnomon onto a simple plane. It is perhaps the easiest sunpath projection to understand and is very simple to use. However, the sundials themselves are not simply constructed and designers usually rely upon others to provide their sundials.

A slightly more contrived type of projection is the sunpath diagram and these have the advantage that some of them can be sketched very simply indeed. The designer is therefore able to construct without too much difficulty a sunpath diagram for anywhere on the globe.

If the hemisphere of sky described earlier is considered, image, then the two angles defining the position of the sun, the azimuth and the altitude may respectively be represented by the direction away from a central point and the distance from the central point, image.

The zenith is represented by the centre of the diagram and the horizon is represented by an outer circle.
 

Stereographic projection
Of the various projections possible with this type of diagram, the stereographic projection is used because the sunpaths are most easily drawn, image.

The basis of the diagram is that the radius representing an altitude is found using the formula,

r_g  = R₀ tan [ ( 90^o - g ) / 2 ]