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Module 2 - Faade Engineering

Predicting the Suns position

North Pole

Polar axis

Introduction

There are many different ways of representing the position of the sunin the sky and most of these may be usefully used to help investigate

sunlighting in design. In order to limit explanations, only one or twoof the methods will be considered in any detail, but once the principles

are understood, one should find it easy to apply those other methodsnot described in detail.

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

Figure 1 Earth rotates about Polar Axis

The Earth

The Earth is effectively a spherical globe that rotates eastwards abouta North-South axis approximately once every 24 hours, as shown inFigure1.

Polar axis Polar axis

hemisphere

Great circle

Meridians Equator

A globe may be partitioned in various ways as shown in Figure 2.These prove useful in describing parts of the Earth and locating

accurately different places on the Earth. If a globe is divided into twoequal parts to produce two hemispheres, then the dividing linebetween the two parts will be a Great Cir cle. The axis about whichthe Earth rotates is known as the Polar Axis and this axis intersects theglobe at the North Pole and the South Pole. A great circle passingthrough both poles is known as a Meridian. A great circle that is

equidistant from the North and South Poles is known as the Equator. Figure 2 Divisions of a sphere

As shown in Figure 3, any place on the Earth may be specified inrelation to;

Zenith

Primary Meridian

Polar axis

Equator

Circle of

LONG.

LAT.

Meridian of co

i) a primary meridian,nstant Longitudeii) the equator.

The Longitudedescribes the position of the appropriate meridian in

relation to the primary meridian - it may be either East or West of theprimary meridian.

constant Latitude

The Latitudedescribes the angle from the equator towards a Pole

along a particular meridian it may be either North or South of theequator.

Figure 3 Latitude and Longitude

The Earths orbit around the Sun

Tilt =23.4

Ecliptic plane

NN

side view

plan view

The Earth orbits the Sun approximately once every 365 days. Its orbitlies in the same plane as the Sun and is elliptical in shape with the Sunpositioned at one of the ellipses foci as shown in Figure 4. This plane

is known as the Ecliptic Planebecause when the moon moves into theplane there is the possibility of an eclipse.

One consequence of the elliptical orbit is that the earth speeds up andslows down as it moves around the sun and this means that the lengthof the day, measured from noon to noon, changes throughout the year.

The changing length of the Solar Dayrather complicates time keepingand it is simpler to assume a constant length of day and use the

Figure 4 Earths elliptical orbit around sun

d:\courses\facade\2002_03\sunlight_01.doc Martin Wilkinson

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Building Environmental Engineering

average length of day throughout the year. This leads to the familiartime convention in the UK of Greenwich Mean Time, which is basedupon the mean or average length of day over the whole year.

North

Zenith

Meridian

Equator

Meridian Plane

To the Sun

The difference between solar time and local mean time is called theequation of time. A correction should be applied for the equation oftime when it is required to know the position of the sun veryaccurately. However, the maximum cumulative difference betweensolar time and mean time is in the order of between +15 minutes and -

15 minutes and for many architectural purposes it may be ignored.

Solar noon occurs when the sun lies in the Meridian plane as shown inFigure 5. At noon the sun appears to be due South at higher northernlatitudes and due North at higher southern latitudes. Solar time istherefore dependent upon the particular Longitude of a location, and it

is clearly rather awkward if clocks need to be changed as one movesfrom one locale to another.

Figure 5 Sun in meridian plane at noon

Winter Solstice

N

Summer Solstice

Autumn Equinox

S

N

S

N

S

N

S

Spring Equinox

The tilt between the Earths axis and the orbital plane

A most important feature of the Earths circumstance is that the PolarAxis is tilted in relation to the orbital plane as shown in Figure 4 andFigure 6. Within the time spans considered in architecture the

direction of the Polar axis relative to the orbital plane remainsconstant. At the present time the Tilt is at an angle of 23.4.

One consequence of the tilted axis is that seasons of the year areexperienced by those parts of the globe closer to the poles. The closer

a region is to one of the poles, then the more seasonal is the climateexperienced by that region.

Figure 6 Constant tilt of polar axis as Earth orbits sun

Equator

tilt

Perpendicular to ecliptic plane

Arctic circle

declination

declination

Equator

Ecliptic plane

Tropic of Cancer

Arctic circleTropic of Capricorn

December Solstice June Solstice

Arctic circles and tropicsPolar axis Polar axis

The tilt also gives rise to the division of the globe into various parts.

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 sunwill be directly overhead at some time in the year and those where thesun will never reach the Zenith.

Figure 7 Defining the tropics and arctics

Declination

North

Sun's raysdeclination

(90-declination)

Equatorial plane

The tilt of the earths axis results in a change in the relative position ofthe sun as the earth moves in its orbit. This change in the relativeposition of the sun is reflected in the change that occurs in the anglethe suns rays make with the equatorial plane. This angle is known asthe declination.

The declination will vary from a maximum of 234 at the Summer

Solstice to a minimum of -234 at the Winter Solstice. Twice in the

year the declination will be zero and this occurs at the Spring andAutumn Equinox.

Figure 8 Declination of the sun

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There are thus 4 times in the year when the declination takes particularvalues that are especially significant:

Astronomical occurance D Calendar date

Winter Solstice -23.4 23rd.

December

Vernal (Spring) Equinox 0 21st.

March

Summer Solstice +23.4 23rd.

June

Autumnal Equinox 0 23

rd.

September

The Winter Solstice is that time of year when the declination is aminimum. Because the direction of Tilt is not exactly in line withmajor axis, it is not the case that this coincides with the shortest day of

the year.

The Summer Solstice is that time of year when the declination is amaximum. Similarly to the other solstice, the longest day does notnecessarily coincide with the summer solstice.

The Equinoxes are those times of year when the day and night are of

equal time. Thus the sun will rise at 6am and set at 6pm.( )

radians3sin001480.03cos002697.0

2sin000907.02cos006758.0

sin070257.0cos399912.0006918.0

nDeclinatioSolar

radians365

12angleDay

s

s

dd

dd

dd

d

N

+

+

+=

=

==

At other times in the year the Declination may be evaluated by theapproximate equation:

( )

+=365

284360sin4.23

NnDeclinatio degrees

Where N is the day number of the date for which the declination isbeing calculated. January 1

st.being day number 1.

North

South

West

East

Zenith

Z

Meridian

Altitude ring

Ground plane

Altitude

Azimuth

A more accurate formulation for declination is given in the margin

and this may be used for computer generated diagrams.

Positioning the sun in the sky

The position of the sun in the sky is given by two angles that areshown in Figure 9,

- the altitude of the sun above the ground or horizon planez the compass direction of the sun on the ground plane.

Figure 9 Hemisphere of sky

The Azimuth may be given in two ways; either East or West of South,or clockwise from North. In general, the azimuth is most often given

in terms of the angle from North, but in these notes the angle will begiven as an angle East or West of South.

Length of shadow = Htan

SOUTH

GNOMON

Altitude

Azimuth Vertical Pole

in ground

HFigure 10 shows a shadow cast by a vertical pole and how the lengthand position of the shadow are affected by the altitude and azimuth ofthe sun.

Figure 10 Shadow cast by a vertical pole

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The suns position may be determined from the following equations:

L

Symbol Variable Definition

D Declination The angle of the sun's rays to

the equatorial plane, positive

in summer.

Latitude The angle from the equator to

a position on Earth's surface.

H Hour angle The angle the Earth needs to

rotate to bring the meridian to

noon. Each hour of time is

equivalent to 15 deg.

N Day Number The day number, January 1st

is 1.

LDHLD sinsincoscoscossin += ,

cos

cossinsincoscoscos

LDLHDz

= ,

cos

cossinsin

DHz

= ,

DLHL

Hz

tancoscossin

sintan

= .

Application of the above formulae are not necessarily the best way toappreciate the various effects of the suns position and a more

graphical approach may usefully be adopted. However the aboveformulae may be usefully used where more precise information isneeded.

SOUTH

Winter

Equinox

Summer

Noon

10am

8am

2pm 4pm Sundials

The Gnomon is a point in space through which the rays of the sun passto later shine upon some surface. The shadow on the ground cast by aflagpole will depend upon the suns altitude and azimuth, and if the

topmost tip of the flagpole is considered, then it will sweep out a pathon the ground as the sun moves across the sky. The topmost tip of the

flagpole may be considered as a gnomon.Figure 11 Sun dial

Winter Noon

Plan of BuildingLB

HB

LS

HS

S

S

B

B

B

B

S

S

LH

HL

L

H

L

H

=

=

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

Plotting the paths of the tip of the shadow for different times

throughout the year will produce a sun dial as shown in Figure 11.

Using the horizontal sundial

If the height of the gnomon is known then the sundial can be used toconstruct the shadows created by buildings at different times of year.This is done simply by measuring the length of the shadow on the

sundial and increasing the length in proportion to how much greater isthe height of the building to the height of the gnomon. The directionof the shadow will be the same as that on the sundial. This is showndiagrammatically in Figure 12.

Figure 12 Sundial used to draw shadows

AltitudeH

L

Window

Sun Patch

SolarWallAzimuth

Sun patches created by sunlight shining through windows can beconstructed in a similar manner to show the effect of sunlight through

different types of window at different times of year as in Figure 13.

The sundial may also be used in what is sometimes called the aviatorsmethod. If the sundial is placed on a model and is viewed so that thegnomon is lined up with a particular time of year, then the eye ispositioned in the direction of the sun at that time of year as in Figure

14. All that the eye will see on the model will therefore be exposed tosunlight at that time of year. Clearly, as the eye is very much nearerFigure 13 Drawing sun patches

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Suns position in the sky

The sunpath loci are arcs of circles on the stereographic projection andthe arc of a circle can be sketched quite easily if three positions on thearc are identified.

Three positions in the sky are used to sketch a Sunpath for a given

day:i) the azimuth of the sun rise,ii) the altitude of the sun at solar noon,iii) the azimuth of the sun set.

The days course of the sun across the sky is caused by the Earthsrotation about its own axis, and as the declination changes only but alittle during the course of a day, the sun rise and the sun set may be

assumed to be symmetrically located on either side of south.

Ecliptic plane

Direction of sun

Zenith

Polar axis

LAT.

Declination

Equatorial plane

Horizon plane Whensketchinga sunpath for a particular location, it is really onlynecessary to consider four times in the year,

i) the Winter Solstice,ii) the Summer Solstice,iii) the Vernal and Autumnal Equinoxes.

The two Equinoxes have the same sun path loci and therefore it israrely the case that more than three sun path loci need be plotted onthe sun path diagram. The suns altitude at noon can be derived from

Figure 17 and is given by,

DLNoon += 90 .Figure 17 Altitude of Sun at Noon

During the summer months, above the arctic circle the sun will not set,

and therefore the sun path locus will be a circle on a stereographicprojection. All that is needed to draw a circle is its diameter. One endof the diameter will be given by the position of the sun a noon, and the

other by the suns position at midnight, as shown in Figure 18. If thealtitude is measured from the southern direction it will be given by theformula,

Equatorial plane

Declination

Horizon plane

Direction of sun

LAT.

Ecliptic plane

Zenith

North

South

south

DLMidnight = 270

and if the altitude is measured from the northern direction it will begiven by,

+==

90

)270(180

DL

DLMidnight

The azimuth of the sun at sun rise and sun set may be found using therelationship given below.

Figure 18 Altitude of sun at Midnight

L

Dz

cos

sincos 0

== .

It is worthwhile noting that, where the sun does not cross the horizon,

the above relation does not hold. Therefore, there will be no solution

when the latitude is >66 for either the summer, or the winter solstice.

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Sketching a Sunpath

0

20

40

60

S

West East

Sun at local Noon

Sun at sunset

Sun at sunriseSketching sunpaths for the City of Bath which is at a Latitude of 513North.

Noting that at solar noon the sun is due South and at the maximumaltitude given by the relation:

DLNoon += 90 .

For the equinoxes, March 21st.

and September 23rd.

, when thedeclination is 0.

=

=

7.38

3.5190Noon

Figure 19 Times needed to sketch sun paths

0

20

40

60

S

West East

Sun at local Noon

Sun at sunset

Sun at sunriseNoting also that at the equinoxes, the sun rises due East and sets dueWest, there are three known positions on the Sunpath locus for the

equinox, and these are shown on Figure 19. These may be used tosketch the first of the Sunpath loci, as is shown in Figure 20.

For the summer solstice, June 21st., when the declination is 234,

=

+=

1.62

4.237.38Noon

Figure 20 Sun path at EquinoxesUsing the relation for the azimuth when the sun rises and sets,

0

20

40

60

S

West East

129.4129.4

L

Dz cos

sincos 0

==

The declination is 23.4 and therefore,

==

==

=

=

=

4.129)635.0(cos

635.0625.0

397.0

3.51cos

4.23sincos

1

0

0

z

z

Therefore the azimuth of sunrise and sunset are respectively1294

East and West of South, as is shown in Figure 21.

Figure 21 Times at summer solsticeThere are then three positions of the sun that can be connectedtogether by an arc of a circle that denotes its Sunpath, as shown inFigure 4.

20

40

60

S

West

0

For the winter solstice, December 23rd.

, when declination is 234,

=

=

3.15

4.237.38Noon

East

The declination is 234 and therefore using the relation for azimuth

at sunrise and sunset,

Figure 22 Sun path at Summer Solstice

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==

==+

=

=

=

=

6.50)635.0(cos

635.0625.0

397.0

3.51cos

4.23sin

3.51cos

)4.23sin(cos

1

0

0

z

z

0

20

40

60

S

West East

50.650.6

129.4129.4

It is worthwhile noting at this point that,

zz

zzz

cos)180cos(

sin180sincos180cos)180cos(

=+=

and this is confirmed by the observation that,Figure 23 Times at Winter Solstice

1294+506=180.

0

20

40

60

S

West East

50.650.6

129.4129.4

Therefore, there is no need to go through the calculation of theazimuth twice. It is simpler to use the fact that the sunrise and sunset

for the two Solstices are symmetrically positioned about the East-West axis. This is shown diagrammatically in Figure 23 where the

known sun positions for the Winter Solstice are plotted.

Figure 24 shows the three points connected by the arc of a circle togive the Sunpath locus for the Winter Solstice.

The three Sunpath loci can then be collected together on the samediagram to give the range of sun positions throughout the year, as isshown in Figure 25.

Figure 24 Sunpath at Winter Solstice

0

20

40

60

S

West East

Sketching solar time lines

The hour lines on the stereographic projection are also arcs of circles.These hour lines always cross the sunpaths at 90 and this helps inconstructing them.

The easiest hour line is that of noon, which is a straight line towardsthe South. At the Equinoxes the 6am and 6pm hour lines pass throughthe horizon line due East and due West respectively as shown inFigure 26.

0

20

40

60

S

West East

0

20

40

60

S

West East

6am

8am

10am

6pm

4pm 2pm

Noon

Although not exactly correct, for the purposes of sketching, theintermediate hour lines may be positioned on the basis of spacingthem equally between the noon and 6 clock hour lines, as in

Figure 27.

Figure 25 Sun path diagram

Figure 26 Construction of Hour Lines Figure 27 Complete Sunpath diagram

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For Latitudes beyond the Arctic circle

At latitudes greater than 666N, the sun will not rise above thehorizon at the Winter Solstice. Therefore only two Sunpath loci are

required to show the extreme ranges of the suns position in the sky,that at the Equinoxes and that of the Summer Solstice.

At the Summer Solstice the sun will be above the horizon for thewhole day and therefore its Sunpath is sketched using the position ofthe sun at noon and midnight. Considering the Figure 18 used to

obtain the relation for the altitude of the sun at midnight, it should benoted that the sun will appear to be due North.

0

20

40

60

S

West East

Considering the Sunpath for a Latitude of 70 North:

At the Equinoxes the Declination is zero and the maximum altitude atnoon will be,

== 207090Noon .

The sunrise and sunset are respectively due East and West.

At the Summer Solstice the Declination is +23.4 and therefore the

maximum altitude at noon will be,

=+=+= 4.434.23204.237090Noon .Figure 28 Sunpaths for 70 North

And the minimum altitude of the sun at midnight will be,

Direction of sun

Polar axis

LAT.

Declination

Equatorial planeHorizon plane

Zenith

Ecliptic plane

=== 6.1764.2370270270 DLMidnight .

measured from South, and

= 4.36.176180

measured from North.

These are then used to plot the sunpaths as is shown in Figure 28 .

For Latitudes within the Tropics

Figure 29 Section of Earth at noonFor Latitudes within the tropics the sun will pass overhead through thezenith at some time of year. Therefore, it is important to realise that atthe summer solstice the sun may be to the North in Northern latitudesand to the south in southern latitudes. This should be apparent fromthe cross section through the earth at noon shown in Figure 29. 0

20

40

60

S

West East

As an example, sketching the sunpaths for the Latitude of 10 N:

At the Equinoxes, the Declination = 0, and the max altitude is,

=+=+= 800109090 DLNoon

At the Summer solstice, the declination is 23.4, and the altitude is,

Norththeto6.76103.4-180

souththeto4.1034.23109090

=

=+=+= DLNoon

Figure 30 Sunpaths for 10 NorthThe azimuth at sunrise and sunset is,

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The Stereographic projection

Baseline

Inclined Shading

planes

20

0

80

60

40

Vertical shadingplanes

The stereographic projection displays the whole sky on a flat plane.In order to fully utilise the diagram, it is necessary to relate it to thethree dimensional reality, and to only consider the diagram as a twodimensional drawing will lead to misunderstandings.

Particularly useful in conjunction with the projection itself is ashading protractor such as is shown in Figure xx. This aid divides thehemisphere of sky into a series of inclined plane and vertical planes.

A single inclined plane is shown in Figure xx, and it can be seen that itdivides the hemisphere of sky into two parts; that above the inclinedplane and that below the inclined plane.

VSA

GL Ground Line

Sky Vault

Zenith

This inclined plane is positioned within the hemisphere of sky by twoquantities:

i) The Vertical Shading Angle VSAii) The orientation of the Ground Line GL

Figure xx shows the single inclined plane of Figure xx plotted on astereographic projection. Some aspects should be noted about thelocus of the line positioning of the inclined plane on the diagram:

i) the altitude of the inclined plane ( VSA), and the altitude circle of

VSA coincide on a line normal to the Ground Line,Orientation of

Ground line

Inclined plane

VSA

ii) the altitude of the inclined plane is zero where the Ground

Line meets the horizon line,

iii) the locus is an arc of a circle on the stereographic

projection.

A special case of the inclined plane is when the VSA is 90. Such avertical plane is shown in Figure xx, and Figure xx shows this planeplotted on a stereographic projection. Being simply a vertical planepassing through the centre of the projection, it has a constant azimuth

and will just be a radial line emanating from the centre of theprojection.

Zenith

Ground Line

Orientation of plane

A protractor will normally have a series of radial lines representingvertical planes on the opposite side of the ground line, as is shown inthe example of a protractor in Figure xx.

Viewing a room

The stereographic projection may be used to display a room as seenfrom some point. Consider the 10m square room shown in Figure xx.Assume that the projection is centred on the centre of the room at aheight of 1m.

Ground

Line

Vertical plane

Orientation of plane

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Plan

Section45

5

5

1

2

1

z1

z2

1 2

z2

Each of the walls intersection with the ceiling will lie in an inclinedplane emanating from the rooms centre. The inclined planes, each of

a vertical shading angle 1

, will therefore represent the junction of theceiling and walls.

In this particular example;

== 315

3tan 11

These are plotted on a stereographic projection with the shadingprotractor and using the angle of VSA=31.

The walls intersect in a vertical plane with a constant azimuth and

therefore are shown by radial lines. These may be constructed either

by using the angle z2 , or drawn as a radial lines emanating from thecorners of the ceiling.

z12

The vertical sides of the window subtend an azimuth of z1 and the

head of the window lies in an inclined plane of angle 2 where;

== 8.215

2tan 12

and

==

7.385

4tan

1

1z

Thus Figure xx shows a stereographic projection of the room seenfrom its centre. This projection may then be superimposed over aSunpath diagram as is shown in Figure xx, and the sunpaths seen

through the aperture of the window will be seen by the point at thecentre of the room.

Clearly the room projection should be correctly orientated withrespect to the Sunpath diagram.

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