Solar Shadows and Analemmatic Sundials
Donald L. Snyder[1]
(
Abstract
Methods for predicting the geometry of shadows cast by sunlit objects are well known. Some of these are reviewed and then applied to the design of a particular kind of interactive solar clock known as an analemmatic sundial.
Contents
Examples of Analemmatic Sundials
1. Earth Centered Coordinate Systems
2. Place Centered Coordinate Systems
IV. Analemmatic Sundial Design
A. Determining the true North-South Direction
D. Summary of Analemmatic Sundial Design
E. Example: A Design for St. Louis, MO
VI. Construction Materials and Methods
VII. Discussion and Conclusions
I. Introduction
Shadows cast by objects exposed to sunlight change with time and date as the sun's apparent position varies during the day due to the earth's rotation and during the seasons due to the earth's orbital motion around the sun. Understanding how to predict the behavior of solar shadows is important in a number of areas, including architectural design and the placement of building structures, the design and placement of solar-energy collectors, the design and placement of flower and vegetable gardens, and the design and placement of time and date indicators.
This discussion will be about solar shadows with application to sundials. At first thought, it may seem that sundials for indicating time are of little practical value in these days of high technology. After all, modern clocks are much more accurate than sundials, they work at night as well as day, they work on rainy days as well as sunny ones, and they are cheap and readily available. The study of sundials is worthwhile nonetheless. Understanding how sundials work helps to understand the heating and cooling requirements within buildings exposed to sunlight, the energy produced by solar panels, the growth of flowers and vegetables in gardens, and effects in other applications where sunlight plays a role. Moreover, shadows cast by sundials are useful in unexpected, modern ways. An example is the sundial affixed to the roving vehicle that landed on Mars in 2004, shown in Figure 1.[2] Also, and importantly, the study of sundials has educational merit. It helps students learn about the world around them, and it shows them how some of the abstract mathematics they learn in school, particularly trigonometry, provide very practical tools. Moreover, designing and making one’s own personal sundials can be fun!
The first topic to be studied is solar shadows, identifying where the sun is relative to any place of interest and where the shadows of objects fall that result from its light. The ideas are then used for the design of analemmatic sundials. These are interactive sundials, and many examples of them exist.
Figure 1 (left) The sundial is installed on the deck
of the solar array on the
Mars Rover. (center) The sundial on
the Mars Rover. (right) A picture
taken of the sundial while on Mars, showing the shadow.
Examples of Analemmatic Sundials
The shadow casting object in an analemmatic sundial, called the gnomon, is usually a vertical rod or a person. In contrast to most sundials, the gnomon is not in a single, fixed position but, rather, must be placed in a varying position that depends on the date in order for the sundial to indicate the correct time for that date. A person serving as the gnomon must stand on a date-dependent place to read the time with any accuracy. Analemmatic sundials have been built at many locations around the world and in many different styles. Some examples are in the pictures shown in Figure 2; these pictures were derived from a search of the internet for “analemmatic sundials.” The bottom picture on the left is located at the Brooklyn Children’s Museum. It was made by Robert Adzema using hundreds of one-inch tiles. The gnomon is not always a person, as seen in the table top dial of Figure 3, where a thin rod is used as the shadow caster. This dial was made by John Carmichael (see http://www.sundialsculptures.com/).
It can be seen in these pictures that an analemmatic sundial consists of two main parts. One part has some marks along a curve that indicate the hours of time; this curve is an ellipse, as discussed below. The other part of the dial is a platform containing the date marks where the person or gnomon stands to cast the shadow towards the hour marks. The design of an analemmatic sundial therefore consists of locating the hour marks along the ellipse and the date marks on the platform. Once these numerical aspects of the design are completed, there is then much leeway in completing the artistic features of the design to give the dial its unique style and character. A description of solar shadows is needed to locate the hour and date marks. This is developed in the next section.
There are various approaches for designing an analemmatic sundial. Purely graphical methods can be used; see, for example, Rohr [4]. Empirical methods can also be used by making daily and yearly observations of the shadow of a vertical rod. An ellipse is first laid out at the intended site of the sundial, with the minor axis oriented in the north-south direction. Time marks can be placed on the ellipse at selected times during a day of choice, such as hourly, by noting where the shadow of a vertical rod located on the minor axis intersects the ellipse. A date mark can be placed on the minor axis of the ellipse for that day. Observations over the course of a year will be needed to locate other date marks empirically along the minor axis. This experimental method of construction is protracted over time and tedious. Fortunately, there is a very nice analytical way of doing it, which is developed as our discussion proceeds. While this analytic method for designing analemmatic sundials can also be used for designing other types of sundials and solar calendars, including sundials with fixed shadow casters and henges, we will not explore these extensions here.
II. Solar Shadows
To a very good approximation, the earth rotates around the sun in an elliptical orbit. The orbit is not quite an ellipse because of the gravitational pull of other planets and distant stars, but the deviation from an ellipse is so small that it can be disregarded for the purpose of designing solar clocks.
Figure 2 Pictures of analemmatic sundials obtained
by searching the internet for sundials of this type
Figure 3 Table top analemmatic sundial
made by John Carmichael
A. A Brief Review of Ellipses
A standard ellipse is a curve in a plane, as shown in Figure 4. It has major and minor axes that can be aligned with the axes of a Cartesian coordinate system, with its center, C ,
Figure 4 A standard ellipse
located at the origin (0,0) of the coordinate system and its short and long axes aligned with the x-axis and y-axis, respectively, of the coordinate system.[3] The equation for points (x, y) that lie on the elliptical curve is
(1)
Where M and m are parameters that determine the size of the major and minor axes. If M and m are equal, say to r, then the ellipse is a circle of radius r. If M > m, then the major axis of the ellipse lies along the y axis of the coordinate system, as shown in Figure 4. Another way to think of the ellipse is in terms of the angle θ shown in the figure. A point P located in the plane at
is on the ellipse because
.
As θ varies from 0º to 360º, the point P moves from (x, y) = (m, 0) counterclockwise around the ellipse and back. The center C of the standard ellipse lies at the origin (0,0) of the coordinate
system. A point P on the ellipse is at a distance from the center C of the ellipse; this distance varies with θ and, hence, the location of P on the ellipse. The points (0,
f ) and (0,+ f ) are called the foci
of the ellipse if
. The distance d1 from the focal point at (0,
f ) to a point P at
on the ellipse is
(3)
Similarly, the distance d2 from the focal point at (0, f ) to P is . This yields the important property that for
any point P on the ellipse, the sum
of the distances from the two foci to that point equals the length, 2M, of the major axis,
. (4)
The eccentricity, e, of an ellipse is defined by
(5)
The eccentricity has a value between 0 and 1, 0 ≤ e ≤ 1. An ellipse having an eccentricity e = 0 is a circle. As illustrated in Figure 5, the ellipse departs more and more from a circle as e increases, becoming simply a line when e = 1.
Figure 5 Illustration of the effect of
eccentricity on the shape of an ellipse
The equation for an ellipse that is not centered at the origin (0,0) but rather at a point (x0, y0) is
An ellipse centered at (0,
f ) is shown in Figure 6. The focal
point at (0, f ) in Figure 4 is now at the origin in Figure 6,
and (6)
becomes
(7)
Figure 6 Ellipse with a focal point at the
origin
A point P on this ellipse has coordinates ,
where r is the distance from the
focal point at the origin to P, and
is the angle that the line connecting the
origin to P makes with the positive x axis. The point P at
on the ellipse is closest to the focal point
at the origin, and
is the most distant point.
B. Earth Motions and Seasons
The earth orbits the sun periodically, with a period of 365.25 days, following an elliptical path with the sun at one focal point, so we may think of the point P in Figure 6 as representing the center of the earth and the origin as representing the center of the sun. The orbital path defines a plane called the ecliptic. The center of the earth, throughout the course of its annual orbit, and the center of the sun lie in the ecliptic. While the earth flies through space around the sun, it also rotates periodically about its own axis, with a period of 24 hours.
The
closest approach of the earth to the sun, called the perihelion, occurs around January 2 each year. At perihelion, in Figure
6, and r is approximately
meters (or about 91 million miles). The point when the earth is most distant from
the sun, called aphelion, occurs around
July 3. At aphelion,
in Figure
6, and r is approximately
meters (or about 94 million miles). The eccentricity e of the earth’s orbit is
approximately 0.0167, so the orbit is very nearly a circle. This eccentricity is too small to account for
the annual seasons. The earth rotates
about its polar axis as it orbits the sun.
This axis is not perpendicular to the ecliptic but, rather, tilts about
23.45 degrees from that. The tilt of the
axis, along with the orbital motion, cause the angle of incidence of the sun’s
rays at any place on the earth to vary between
23.45º and +23.45º, resulting in the seasons. Figure
7 shows some of the critical positions of the earth in
its elliptical orbit around the sun.
Consider a plane that is perpendicular to the ecliptic and which
contains the centers of
Figure 7 Some critical positions of the
earth in its elliptical orbit around the sun
the sun and earth. I will call this the Milankovitch plane after the Serbian scientist Milankovitch who in the 1920s studied the influence of the solar cycle on climate (see http://aa.usno.navy.mil/faq/docs/seasons_orbit.html). Over the course of a year, the Milankovitch plane rotates around the sun, moving with the earth in its orbit. The axis of rotation of the earth lies in this plane only two times during the year. These are the winter and summer solstices. At a solstice, the greatest number of hours of sunshine for any day of the year occurs in the hemisphere tilted towards the sun, and this is in the summer for that hemisphere; the fewest number of hours occurs for the opposite hemisphere, which is in the winter. The winter and summer solstices indicated in Figure 7 are for the northern hemisphere. At other days of the year, the axis of rotation of the earth is tilted out of the Milankovitch plane. The greatest tilt occurs twice a year when it reaches ±23.45 degrees. These are the equinoctial days when the number of hours in the day and night are equal.
The orbits of all of the other planets of our solar system also lie in the ecliptic plane. Shown in Figure 8 is a photograph taken shortly after sunset by Jimmy Westlake on June 19, 2005. Three planets can be seen above the Colorado Rocky Mountains in the foreground. These are Saturn, Venus, and Mercury, all lying in the ecliptic. Also, see Figure 16 for an annotated version of this picture.
Figure 8
Photograph taken by Jimmy Westlake of
C. Coordinates
The earth, approximated as a
sphere, is represented in Figure
9. The two
points where the spin axis of the earth meets the surface of the earth are
called the poles. One end of the spin axis points towards the
star Polaris. The pole closest to Polaris
is called the North Pole, and the other is the South Pole. The equator
is the great circle[4] that is
perpendicular to the spin axis and midway between the poles. Great circles that pass through the poles are
called meridians. By a long standing tradition, the meridian
passing through
Identifying the location of an object on the earth’s surface or in space requires that a coordinate system be specified. There are many possible coordinate systems that can be adopted. Some are earth centered, and some are place centered.
1. Earth Centered Coordinate Systems
The equatorial coordinate system is earth centered. It is defined in terms of the equatorial plane. This is a plane of infinite extent that passes through the center of the earth and contains the earth’s equator. In the equatorial coordinate system, the location of any point in space can be specified by the values of its Cartesian coordinates (x, y, z) shown in Figure 9.
Figure 9 The equatorial coordinate system
The origin of this coordinate
system is at the center of the earth. It
is a right-handed coordinate system, with the x axis oriented towards the intersection of the prime meridian and
the equator, and the z axis is oriented towards the North Pole. A point in the equatorial plane has Cartesian
coordinates (x, y, 0). A point P on the surface of the earth will have
coordinates (x, y, z) that satisfy ,
where r is the earth’s (mean) radius. Alternatively, and commonly, the location of
the point can be specified in a three-dimensional polar coordinate system,
having two angular coordinates and one radial coordinate. These are defined in Figure 10. The longitude of P is the angle θ measured along the equator between the prime meridian
and the ”local” meridian passing through P. This angle is positive if measured
counterclockwise from the x axis in
the x,y
plane (that is, towards the east from the
prime meridian in the equatorial plane).
Otherwise, it is negative. for
example, the longitude of St. Louis, MO, is 90.3 degrees west of the prime meridian,
so θ =
90.3º
or, alternatively, θ = 269.7º. The latitude
of P is the angle
in Figure
10 measured along the
Figure 10 Longitude and latitude
coordinates
local meridian that passes
through P and
the equator. This angle is the same as
that between the equator and the point at the intersection of the parallel
containing P with the prime meridian when measured along
the prime meridian. It is positive if
measured in the counterclockwise in the x,z plane (that is, towards the northern
hemisphere from the x-axis). It is therefore positive for locations in the
northern hemisphere and negative for those in the southern hemisphere.
The
location of a place P on the surface
of the earth can be specified by its latitude and longitude angles, and
,
and its distance, r, from the earth’s
center. The distance from earth’s center
is usually omitted explicitly when a spherical earth model is assumed, but the
elevation above or below sea level is given when deviations from a sphere are
of interest. For sundial computations,
this small variation is usually disregarded. Alternatively, the location of P
can be given in terms of its (x,
y, z) coordinates. These are
related to the polar-coordinate representation in the following way:
(8)
.
Finally, note that the location of any object, whether on the surface of the earth or not, can be specified within this coordinate system. All that is required is to consider a line drawn from the center of the earth to the object’s center. The point where this line penetrates the surface of the earth specifies the latitude and longitude of the object, and its radial distance completes the specification. This includes the sun, an object in which we are very interested.
Another
earth-centered coordinate system is one in which the longitudinal reference is
not the prime meridian but, rather, the local meridian of a place P of interest.
This system is illustrated in Figure
11 for specifying the position of the sun. The Cartesian coordinate system is obtained from that of Figure 10 by rotating the coordinates
Figure 11 Hour line and declination
coordinates
about the z
axis through an angle θ
equal to the longitude of P.
A point located at the coordinates
in Figure
9 is located at the coordinates
in Figure
10, according to:
(9)
A polar coordinate representation
in the frame is important and used routinely in navigation,
sundial design, and other areas involving solar effects. There are two angles, which are analogous to
the longitude and latitude angles of Figure
10. The longitude
angle relative to the local meridian of P
is labeled τ in Figure 11. It is common
to specify this angle in the units of time rather than degrees. This is achieved taking into account that the
earth rotates through 360 degrees in each 24 hour day, so it rotates through 15
degrees per hour and 1/4 degree each minute.
This scaling is used in specifying τ in time units. The sun passes through the plane of the local
meridian of P at 12:00
hours represents the meridian separated towards
the east by 22.5 degrees of longitude from the local meridian of P.
marks the meridian 22.5 degrees of longitude
towards the west from that of P, and
degrees.
The latitude angle of the point S in Figure
11 is labeled δ. This latitude angle, measured from the
equatorial plane, is called the declination when the point S is determined by the position of the sun, as it is in Figure 11. In this
earth-centered coordinate system, the position of the sun is specified by its
hour angle, τ, its declination, δ, and the distance,
,
of its center from the center of the earth.
The sun’s declination varies between
and
over the course of the sun in its annual
orbit. The location the sun is given in
terms of its
coordinates or its polar coordinates
. These are related in the following way.
2. Place Centered Coordinate Systems
Coordinate systems that are
centered at a place of interest on the surface of the earth, P, are also commonly used. An example is shown in Figure 12. Here, lie in the plane that
Figure 12 Horizon coordinates
is tangent to the earth’s surface
at P, which is called the horizon plane, with lying in the meridian plane of P.
The
coordinate is perpendicular to this plane at P.
The coordinate system
of Figure
12 is obtained from the coordinate system
of Figure
11 by translating the origin from the center of the
earth to the place P on the surface
and a rotation about the
axis by
degrees, where
is the colatitude of P, defined by
. Thus,
(11)
Where is the earth’s radius. Hence,
Since lies in the meridian plane passing through the
place P, it points towards the south;
points towards the east; and,
points towards the zenith at P.
A
place-centered polar coordinate-system is natural and of interest for
specifying the location of objects in space, such as the sun. This is the polar coordinate system shown in Figure 13.
The Cartesian coordinates are those shown in Figure 12. The location
Figure 13 Polar horizon coordinates
of an object, such as the sun, is
specified by two angles, a and e, and its radial distance R
from the local point P on the
earth’s surface. Here, a is the azimuth angle of the object measured from the axis, and e
is the elevation angle of the object
from the horizon plane. Positive values
of the azimuth angle are measured counterclockwise from the positive
axis; that is, positive from south towards the
east. The Cartesian and polar
representations are related by:
Suppose
that the sun is the object in Figure
13, so . By equating (12) to
(13)
and replacing
by their values in (10),
there results
(14)
The following three equations are then obtained:
and
where the ratio is neglected in the last term of (17) because of its small
size. These are well known equations for
determining the position of the sun at a given place for a specified date and
time. The place provides the latitude,
;
the date provides the sun’s declination, 
,
either by calculation from known formulas or from an ephemeris table; and the
time provides the hour angle. Given
these parameters, the sun’s elevation and azimuth can be determined. Here is a summary for later reference.
__________________________________________________________________
From (17), the solar elevation is:
This elevation angle is measured positive from the horizon plane towards the zenith.
From the ratio of equations (15) and (16), the azimuth angle is:
This azimuth angle is measured
positive counterclockwise from south.
For azimuth angles measured positive clockwise from the north, use 180 a degrees.
Care is needed in using (19) to insure that the angle
is in the correct quadrant.[5]
__________________________________________________________________
The declination of the Sun is zero at the vernal equinox, about
March 21, and the autumnal equinox, about September 21. It’s maximum and minimum values are and
,
respectively. In the Northern
hemisphere, the maximum occurs on the Summer solstice, about June 21, and the
minimum on the Winter solstice, about December 21. The declination varies approximately as a sinusoid
over the course of a year. Thus, the
solar declination can be determined approximately by using the expression
Where N is the day number starting with N = 1
on January 1, N = 2 on January
2, ,
N = 81 on March 21, etc. The graph of this expression in Figure 14 shows, approximately, the variation of the solar
declination during a year. Of course,
the declination of the sun varies continuously as a function of the date and
time and not simply the day number. J.
Meeus [3, Ch. 25] gives an accurate expression for the continuously
varying solar declination in terms of the date and time.
Figure 14 Solar Declination Versus Day Number
Example: Where
is the sun at 12:00 . To determine the solar declination, use the
approximating formula (20) to get
degrees.
The hour angle from local south is
degrees.
Then, from (18),
the elevation of the sun is
degrees, and from (19),
its azimuth angle is 0 degrees (due south).
The length and direction of the shadow cast by a thin, 1 m vertical rod is
about 2.2 m towards true north. The
length and direction at other times are given in the table below and illustrated
in Figure 15.
|
solar time |
τ (degrees) |
elevation angle above horizon |
azimuth angle from south |
shadow angle (degrees) a+180º |
shadow length (m) ctan(e) |
|
9:00 |
+45 |
11.7 |
+42.4 |
222.4 |
4.8 |
|
10:00 |
+30 |
18.5 |
+32.9 |
212.9 |
3.0 |
|
11:00 |
+15 |
22.9 |
+18.5 |
198.5 |
2.4 |
|
12:00 |
0 |
24.4 |
0 |
180 |
2.2 |
|
13:00 |
-15 |
22.9 |
-18.5 |
161.5 |
2.4 |
|
14:00 |
-30 |
18.5 |
-32.9 |
147.1 |
3.0 |
|
15:00 |
-45 |
11.7 |
-42.4 |
137.6 |
4.8 |
Figure 15 Shadow lines on January 1 in
Example: Jimmy Westlake, who took the photograph
displayed in Figure 8, teaches at makes with the equatorial plane is 90º
,
so at Steamboat Springs, this angle is 90º
40.5º = 49.5º.
Since the equatorial plane is tilted at an angle of about 23.4º to the
ecliptic plane, we expect the angle between the equatorial plane and the
horizon plane at Steamboat Springs to be about 49.5º
23.4º = 26.1º degrees. An estimate of this angle is seen in Figure 16.
III . Time
A. Varieties of Time
A quick answer to the question
“What time is it?” is usually obtained by glancing at a wall clock or a wrist
watch. But being quick does not
necessarily mean being right. The complication
is that there are many versions of time, so one should first inquire about the
kind of time that is sought before answering.
A 24 hour time scale starts at
Figure 16
Annotated version of Fig. 6 showing Saturn, Venus, and Mercury in the
ecliptic plane tilted about 26.1 degrees to the horizon plane
360º/year around the sun at the center of the circle. Solar time within this approximation is called mean solar-time. Sundials are routinely designed for mean solar-time. Solar time is then obtained from mean solar-time by a correction known as the equation of time, which is discussed below.
Local
noon mean-solar-time, used in the discussion above, occurs when the sun’s center
lies in the plane of the local meridian, at which instant the sun is at its
highest elevation that day for the local place.
Local mean-solar-time is then measured by the sun’s position relative to
the local meridian. 13:00 local mean-solar-time
occurs one hour after the sun reaches its highest elevation at the location and
the earth has rotated through 15 degrees.
For most purposes, this is not a convenient way to measure time because
it is different for every place having a different longitude. For this reason, other time measures have
been adopted by governments so that a given time instant, such as noon, occurs
for locations over a wide range of contiguous longitudes. Coordinated Universal Time (UTC), known also
as Greenwich Mean Time (
longitude of in Figure
17. Times within the eastern time zone are
labeled Eastern Standard Time (EST). EST
equals
where is the longitude correction,
(22)
and
(23)
is the “equation of time,” which is discussed more below.
There is additional complication that makes answering the question “What time is it?” even more difficult. The measurement of time by fundamental physical processes has in recent years resulted in the atomic clock. With it, time is measured by natural oscillations of atoms. Important devices rely on this method of measuring time, such as: global positioning systems now found in many cars, navigation, and military systems; watches and clocks automatically updated by time-synchronized radio transmissions; and, time on the internet. While a year measured by atomic time and solar time differ by about a “leap second” that might seem insignificant, the discrepancy is far from negligible. There is much discussion currently about how to modify universal coordinated time (UTC) to correct the problem. Arguments between engineers and scientists who rely on atomic time and astronomers and sundialists who rely on solar time are intense about this one second annual variance. See:
http://www.ucolick.org/~sla/leapsecs/,
http://www.cl.cam.ac.uk/~mgk25/time/metrologia-leapsecond.pdf ,
and
http://www.ucolick.org/~sla/leapsecs/onlinebib.html
for a survey and references on leap seconds. See
http://www.mail-archive.com/leapsecs@rom.usno.navy.mil/msg00476.html
for a discussion by J. Meeus of the impact on astronomy and sundials of switching UTC to an atomic time measure.
B. The Equation of Time
The Equation of Time ( ) is the difference between solar time and mean
solar time. This correction depends on
the declination of the sun and so is date and time dependent. Mean solar time is determined under the
approximation that the earth’s orbit is circular and in the equatorial
plane. Solar time and mean solar time
differ because the earth’s orbit is elliptical, not circular, and the ecliptic is
tilted away from the equatorial plane.
Robert Urschel gives an excellent tutorial discussion of the
, (24)
in minutes, where is the day number (
=1
on Jan. 1, =2
on Jan. 2, etc.) , and the coefficients, and
,
are given in the following table.
|
k |
Ak |
Bk |
|
0 |
1.252 |
|
|
1 |
5.572 |
-7.337 |
|
2 |
-3.135 |
-9.419 |
|
3 |
-7.846 |
-3.096 |
|
4 |
-1.312 |
-1.790 |
|
5 |
-9.060 |
-1.408 |
Figure 17 World map of time zones
(from http://aa.usno.navy.mil/faq/docs/world_tzones.html)
J. Meeus [3, Ch. 28] gives an accurate expression for the ,
as a parameter along the curve; in Figure
19, the day numbers 1, 32, 60, etc. are indicated and
labeled with their corresponding dates Jan. 1, Feb. 1, Mar. 1, etc.
Figure 18 Equation of Time
Figure 19 Solar Declination and the Equation
of TIme
The curve in Figure 19 is called the solar analemma. It models the sun’s apparent position at a given time instant each day over the span of a year. For example, Bjarne H. Madsen made the composite picture of the sun in Figure 20. Using a permanently mounted camera, he acquired photographic exposures of the sun at the same time instant on 44 days between November 2000 and October 2001. These were combined with one foreground photograph to produce the composite picture. See his website at
http://home.worldonline.dk/bhm/analemma.htm.
Additional pictures of the solar analemma and some further explanations can be seen at the following websites:
http://www.perseus.gr/Astro-Solar-Analemma.htm,
http://www.analemma.de/english/analem.html,
and
http://antwrp.gsfc.nasa.gov/apod/ap020709.html.
Figure 20 Solar Analemma (photo by B. H. Madsen)
Either Figure 18 or Figure 19 is used to determine the value of the equation-of-time for a particular date of interest. Using (21), this value is subtracted from the time indicated by the sundial in order to obtain clock time.
IV. Analemmatic Sundial Design
An analemmatic sundial consists of two parts. One is an ellipse along which there are time marks. The ellipse is oriented so that its minor axis is in the true, not magnetic, north-south direction for the location of the sundial, and the major axis is perpendicular to that in the east-west direction. The center of the ellipse, at the crossing of the axes, is placed so that the completed sundial is in the desired location. The other part of the analemmatic sundial is a line along which are date marks. This line is in the north-south direction on the minor axis of the ellipse. A shadow caster, typically a person, is placed or stands on a date mark. The shadow of the caster at a certain time on that date then falls towards the appropriate time mark.
Several steps are required in the design of the sundial.
- First, the desired location of the sundial must be identified and made flat and horizontal. A design is still possible if the location is not made to lie in a horizontal plane but will require additional considerations.
- Next, the true north-south direction must be determined, which can be accomplished in a variety of ways, some of which are outlined below. Draw a north-south line once the direction is determined. Then, determine and draw an east-west line passing through the desired center location of the sundial.
- The ellipse defining the analemmatic sundial can now be drawn and the time marks placed on it, as discussed below.
- Finally, the locations of the date marks are determined and the marks placed along the north-south line.
These steps complete the mathematical portion of the design. There are still two more important steps.
- One is the design of the artistic or esthetic aspects of the sundial. Here, there is great freedom and opportunity for creative imagination. The sundial can be quite simple if desired, or it can be elaborate.
- Lastly, the sundial must be constructed, so the appropriate materials need to be selected and construction methods adopted to use those materials. It can be constructed as a temporary project, or it can be one intended to be in place for an extended time.
Finally, when the construction is complete, it’s time to try the sundial on a sunny day, take some pictures, and have some fun with the new time piece.
A. Determining the true North-South Direction
R. Rohr [4] and A. Waugh [7] describe several methods for determining the true
north-south direction at a given location.
A compass can be used, but it will indicate magnetic north. A correction factor is needed to determine
true north from magnetic north. Such
factors are available on some maps.
Another approach is to observe and mark the direction of the pole star,
Polaris. Perhaps the easiest method
begins with drawing a circle around the center location where the analemmatic
sundial is to be placed. Then, erect a
narrow rod vertically at the center point of the circle. Put two marks on the circle, one where the
tip of the shadow of the vertical rod crosses the circle before
B. Design of the Time Marks
Shown in Figure 21 is an ellipse with its axes aligned to the north-south and east-west directions of the place where the analemmatic sundial is to be placed. The semi-major and semi-minor axes have sizes M and m, respectively. Suppose that at a certain time, the azimuth
Figure 21 Design of the Time Marks
angle of the sun is a
degrees. A vertical rod placed at
the origin will cast a shadow at the angle θ = 90º + a in the opposite direction from the sun. From (2),
the location of all points on the ellipse satisfy for any choice of θ. In particular, choosing θ = 180º + a
gives the location of the point on the ellipse that is intersected by
the line of the shadow cast by the vertical rod placed at the origin when the
azimuth angle of the sun is aº. This intercept is at
.
The
time marks on the ellipse are located on an equinoctial day. On such a day, the declination of the sun, ,
is zero. From (19),
the azimuth angle of the sun then satisfies:
where τ is the solar hour angle
from the local meridian, and is the latitude of the location where the dial
is to be placed. We wish to mark the ellipse with the time τ when the azimuth
of the sun is at this hour angle. If the
dimensions of the ellipse are selected so that the minor and major axis
dimensions satisfy
,
the intercept is located on the ellipse at
and (25) is satisfied. The intercept point on the ellipse in Figure 21 can then be marked as the solar time τ hours. Various times can be marked by making various choices for τ.
1. Sunrise
There is obviously no need to place time marks on the ellipse for times between sunset and sunrise. Equation (18) can be used to determine the times of sunrise and sunset. These can then be used to limit the ellipse of an analemmatic sundial to times of sunshine. Since the elevation angle of the sun is 0º at sunrise and sunset, setting e = 0 in (18) yields
(27)
where is the time of sunset or sunrise. Thus, the times of sunset and sunrise
satisfy:
degrees, or
hours.
Let be the azimuth angle of the sun at sunset or
sunrise. Then, from (15),
with e = 0º, and (28)
(30)
Thus, the azimuth angles of the sun at sunset and sunrise satisfy:
Note that the times of sunrise and sunset occur at an equal number of hours before and after solar noon, respectively. Similarly, the azimuth angles of sunrise and sunset equal in magnitude from south.
For
example, on an equinoctial day, when the solar declination is zero, ,
equation (28)
indicates that sunrise and sunset occur at an hour angle of ±90º or, from (29),
±6 hours. Similarly, (31)
indicates that the azimuth angles of the rising and setting sun are ±90º. Thus, the sun rises at 6 hours before local
solar
90º
west of south (i.e., west). If the only
consideration were the equinoctial day, the only portion of the ellipse in Figure 21 needed for daylight hours is that above the east-west
axis. However, on other days daylight
hours may extend beyond the 12 hour day from
hours, so sunrise occurs at
2 Size of the sundial
The parameter 
defining the semi-major axis controls the
overall size of the analemmatic sundial.
The average height of people who will cast the shadow is one
consideration in selecting this sizing parameter. As seen in Figure
15, the length of the shadow on any given day is shortest
at noon, when the solar elevation is greatest.
The denote the maximum solar elevation at the
location where the sundial will be placed.
From Eq. (18),
(32)
The shortest shadow length, ,
is then
(33)
.
where h is the height of the person casting the shadow. One possible choice for sizing the sundial is
to select M so that the minor axis of the sundial, ,
is equal to
,
where is the latitude of the place where the sundial
is placed. An average height,
,
will need to be used in (34) because people with
various heights will be using the sundial.
3. Time Corrections
Time marks placed on the ellipse
of Figure
21 according to (26)
indicate mean solar-time at the meridian where the sundial is to be placed. Corrections must be applied to convert this
time into the legal time indicated by clocks.
The necessary corrections are indicated in (21). These are: a correction to account for the
difference in longitude between the standard meridian of the time zone in which
the sundial is to be located and the meridian of the sundial itself; the
equation-of-time correction, which converts mean solar-time into actual
solar-time; and, a one hour correction
for daylight savings time if that is in effect.
These corrections can be incorporated into the design of the sundial in
various ways. The longitude correction
can be incorporated into the design by replacing the hour-angle, τ, in (26)
by ,
where
, (35)
in which
. (36)
The resulting time marks are still labeled for the hour angle of τ. The shadow cast by a vertical rod will then be at the 12:00 mark (i.e., τ = 0º) when the sun passes through the plane of the standard meridian. If desired, the correction for daylight savings time can be incorporated into the design by labeling time marks with two times, one for standard time and the other for daylight time. Reading time on the sundial then requires that the user know whether or not daylight savings time is in effect and which of the two labels to use. The equation-of-time correction can be incorporated, if desired, by either displaying a graph of the type in Figure 18 at the dial location or an analemma of the type in Figure 19 along the north-south axis of the sundial at the date marks. The user of the sundial must then incorporate this correction by adding (or subtracting) the correction read from the graph or analemma to the time indicated by the time mark at the shadow line.
C. Design of the Date Marks
Suppose that and that the time marks are placed on the
ellipse of Figure
21 as described in the preceding section. On an equinoctial day, the shadow of a
vertical rod placed at the origin of the ellipse will then fall on each time
mark at the time indicated by that mark.
This will not be so on other days.
However, there is a date-dependent position along the north-south axis
where the rod can be placed so that the shadow does intersect the time marks at
the appropriate instant on a given date.
This position is found by consideration of Figure 22. A vertical
rod placed along the north-south axis at x = d casts a shadow towards
Figure 22
Vertical rod at date mark (x,y)=(d,0) casts a shadow towards point P on the ellipse when the sun is at
azimuth angle a
the time mark, τ, at point P on the ellipse with semi-major axis M and semi-minor axis . The equation for the shadow line is
where the second equality is
obtained using (19). Now consider the dashed line in Figure 22 from the origin to the point P. This would be the shadow
line of the rod if it were placed at the origin on an equinoctial day when . The equation for this line is
From (25). The location of P on the ellipse can be determined by substituting this expression for x into the equation of the ellipse
(39)
to get
(40)
Therefore, the y-coordinate of P is
and from (38), the x-coordinate is
The choice of d so that the shadow line of the rod at (d, 0) passes through P
is obtained from (37) by substituting and using (41)
and (42)
to get:
(43)
which yields the following expression for the date mark:
The date determines the solar
declination, δ, and the location of
the dial determines the latitude . The location determined by (44)
is then labeled with the date. Labels
for any dates selected can be placed, but the common choices are the first day
of each month. If desired, the date of a
special occasion, such as a birthday, wedding anniversary, or holiday, could be
selected.
D. Summary of Analemmatic Sundial Design
The steps to design an analemmatic sundial to be located on
a flat, horizontal surface at a place with latitude are:
- Determine and lay out north-south and east-west oriented lines, with their intersection at the center of the site where the sundial is to be placed. See Sec. IV.A for determining the true north-south direction.
- Select a scale M for the sundial. See Sec. IV.B.2 for guidelines on doing this.
- Determine and lay out the ellipse of the sundial and time marks along the ellipse. The property (4) of an ellipse provides one way to lay out the ellipse, as discussed below in Sec. VI. Equation (26) is used for locating the time marks. The extent of the ellipse and time marks need only include daylight hours. See Sec. IV.B.1 for determining the times of sunrise and sunset and hence the span of daylight hours. The time marks can be placed to include a time correction for the longitude of the location where the sundial is to be placed. This correction is discussed in Sec. IV.B.3.
- Determine and lay out the date marks along the north-south axis of the ellipse. The location of a date mark is determined by first determining the declination of the sun for the date, for example by using equation (20), and then using equation (44) of Sec. IV.C.
- Choose the artistic aspects of the sundial. Consider including sundial decorations; see Sec. V below for some discussion of this.
- Choose the materials and methods for constructing the sundial. See Sec. VI for some discussion of this.
E. Example: A Design for St. Louis , MO
As an example, consider the
design of an analemmatic sundial for and is aligned with the north-south direction. The ellipse is drawn in Figure 23.
Hourly
time marks 5:00, 6:00, 
,
11:00, 12:00, 13:00, ,
18:00, and 19:00 are placed on the ellipse at coordinates =
for the corresponding hour angles τ equal to
,
,
,
,
,
,
,
,
and 
,
respectively.
These coordinates are given in the following
table, and the time marks are indicated on the ellipse in Figure 23. They were determined using (26)
with M = 1.
|
time, τ |
time mark coordinates |
||
|
hours |
degrees |
x |
y |
|
5:00 |
-105 |
+0.162 |
+0.966 |
|
6:00 |
-90 |
0.000 |
+1.000 |
|
7:00 |
-75 |
-0.162 |
+0.966 |
|
8:00 |
-60 |
0.312 |
+0.866 |
|
9:00 |
-45 |
-0.441 |
+0.707 |
|
10:00 |
-30 |
-0.540 |
+0.500 |
|
11:00 |
-15 |
-0.603 |
+0.259 |
|
12:00 |
0 |
-0.624 |
0.000 |
|
13:00 |
15 |
-0.603 |
-0.259 |
|
14:00 |
30 |
-0.540 |
-0.500 |
|
15:00 |
45 |
-0.441 |
-0.707 |
|
16:00 |
60 |
-0.312 |
-0.866 |
|
17:00 |
75 |
-0.162 |
-0.966 |
|
18:00 |
90 |
-0.000 |
-1.000 |
|
19:00 |
105 |
+0.162 |
-0.966 |
Fourteen date marks are placed along the north-south axis, indicating the first day of each month, the winter solstice December 21, and the summer solstice June 21. The coordinates for these marks were determined using (20) and (44). They are given in the following table, and the date marks are indicated in Figure 23.
|
date |
day number, N |
date mark, d |
|
January 1 |
1 |
+0.332 |
|
February 1 |
32 |
+0.247 |
|
March 1 |
60 |
+0.114 |
|
April 1 |
91 |
-0.055 |
|
May 1 |
121 |
-0.208 |
|
June 1 |
152 |
-0.316 |
|
June 21 |
172 |
-0.339 |
|
July 1 |
182 |
-0.334 |
|
August 1 |
213 |
-0.253 |
|
September 1 |
244 |
-0.107 |
|
October 1 |
274 |
+0.057 |
|
November 1 |
305 |
+0.214 |
|
December 1 |
335 |
+0.317 |
|
December 21 |
355 |
+0.339 |
The ellipse of the sundial in Figure 23 is limited to the hours between 5:00 and 19:00. This range of hours was selected after examining the solar azimuth angles using (31) at sunrise and sunset.
Figure 23
Analemmatic sundial for
June 21 and Dec. 21 are indicated by the red marks on the date line.
Shown in Figure 24 are some shadow lines for the solstice and equinoctial
days. The lengths of the shadows of a
one-meter rod placed at the appropriate date marks are indicated. For a one meter rod, the length of the shadow
is ,
where e is the elevation angle of the sun, as determined
by (18)
for the date of interest. The direction
of the shadow is a + 180º,
where a is the azimuth angle of the
sun, as determined by (19) on the date of interest.
Figure 24
Shadow line lengths for a one-meter rod placed at the winter solstice
(Jan. 21, blue),
summer solstice (Jun. 21, green), and equinoxes (Mar. 21, Sept. 21, red). The shadow lines at
hours are shown for each date, and the end points of the shadow lines for each
date are shown
for all hours.
This
sundial indicates local mean-solar-time.
Its design does not account for the difference in the longitude of hours or +1.2 minutes. For example, when the sundial indicates that
the local mean-solar-time is 12:00 noon, the mean-solar-time at the standard
meridian of the time zone is 12:01:12. If desired, the dial can be redesigned to
indicate mean-solar-time at the standard meridian of the time zone by using
in place of
in (26). The coordinates for the longitudinally
corrected time marks are given in the following table, and the resulting
sundial is shown in Figure
25. There is only
a minor difference between the sundials with and without longitude correction
because the longitudes of
|
time |
time mark coordinates |
||
|
|
|
x |
y |
|
5:00 |
-104.7 |
+0.158 |
+0.967 |
|
6:00 |
-89.7 |
-0.003 |
+1.000 |
|
7:00 |
-74.7 |
-0.165 |
+0.965 |
|
8:00 |
-59.7 |
0.315 |
+0.863 |
|
9:00 |
-44.7 |
-0.444 |
+0.703 |
|
10:00 |
-29.7 |
-0.542 |
+0.496 |
|
11:00 |
-14.7 |
-0.604 |
+0.254 |
|
12:00 |
0.3 |
-0.624 |
-0.005 |
|
13:00 |
15.3 |
-0.603 |
-0.264 |
|
14:00 |
30.3 |
-0.539 |
-0.505 |
|
15:00 |
45.3 |
-0.439 |
-0.711 |
|
16:00 |
60.3 |
-0.309 |
-0.869 |
|
17:00 |
75.3 |
-0.158 |
-0.967 |
|
18:00 |
90.3 |
+0.003 |
-1.000 |
|
19:00 |
105.3 |
+0.165 |
-0.965 |
Figure 25
Analemmatic sundial for
V. Sundial Decorations
Various decorations, also called furniture, are commonly included on a
sundial. These include a graph of the
equation-of-time or an analemma so that clock time can be determined from the
time indicated by the sundial. The graph
of the
Other forms of furniture include mottos, emblems, date lines, and solar-event lines. Mottos often refer to time in some way. There are many examples, such as:
Make every hour count
Waste not time
Time waits for no one
Sunny hours are happy hours
Lists of mottos that have been used are readily available; see, for example, http://www.sundials.co.uk/mottoes.htm. An emblem may be included on a sundial to indicate its location or a special event it commemorates. Figure 24 displays lines for the equinoctial and solstice days that indicate the end point of the shadow of a 1 meter rod throughout the course of these days of special solar events. Such a line can be included for any date. For example, the line for a holiday related to the commemoration of the sundial could be included, as could someone’s birthday or wedding day. However, such lines may not be desirable for an analemmatic sundial because of the varying heights of its users.
VI. Construction Materials and Methods
There are many options in constructing an analemmatic sundial once its ellipse, time marks, and date marks are designed. There is a wide choice of materials that may be used. The character of the sundial can be greatly affected by inventive, artistic choices in its implementation. The examples in Figure 2 show a small sampling of the variety of possibilities of construction.
All
implementations will require the determination of the direction of true north,
laying out the axes of the sundial, drawing the ellipse, and placing the time
and date marks. A compass alone cannot
be used to determine the direction of true north. The deviation between magnetic and true north
varies with location. This deviation is
indicated on some maps, and a compass can be used if this deviation information
is available as well. The direction of
true north is easy to measure. One
method is to draw a circle with a vertical rod placed at its center. The tip of the shadow of the rod will cross
the circle once before ,
can then be marked. Also, the two foci
of the ellipse,
,
can be marked. One approach for laying
out the ellipse itself is to use the property of an ellipse in (4). The endpoints of a rope of length 2M
are secured at the foci of the ellipse.
A point P is the ellipse when
the rope is stretched taut, as in Figure
4. The ellipse
can be marked out by moving the point P to various locations while keeping the rope
taut. Time and date marks can be added
by using a ruler to measure their coordinates along the north-south and
east-west axes.
VII. Discussion and Conclusions
The design of analemmatic
sundials is based on predicting solar positions and the properties of
ellipses. The properties of ellipses are
reviewed in Section II-A. Earth-centered
and place-centered coordinate systems are used in predicting solar positions as
the earth rotates about its axis while orbiting the sun. These coordinate systems are reviewed in
Section II-C. Sundials indicate
mean-solar-time, so corrections must be applied to obtain clock time from a
sundial reading. These corrections are
discussed in Section
VIII. Bibliography
The following is a short list from among many publications that deal with predicting solar positions and the theory of sundials.
1. C.
J. Budd and C. J. Sangwin, “Analemmatic sundials: how to build one and why they
work,” 1997-2004 Millennium Mathematics Project,
3.
J. Meeus, Astronomical
Algorithms, Second Edition, Willmann-Bell, Inc., 1999.
4.
R. R. J. Rohr, Sundials:
History, Theory and Practice,
5. F.
W. Sawyer
6.
W. B. Stine and R. W. Harrigan, Power From The Sun,
7.
A. E. Waugh, Sundials:
Their Theory and Construction,
related websites
“Keppel Henge,” www.steveirvine.com/sundial.html. (A discussion of constructing an analemmatic sundial is included.)
“Painting a Sunclock or Human Sundial,” http://www.sunclocks.com/index.htm.
“The tree: analemmatic anthropic sundial,” http://www.nonvedolora.it/english/albero_en.htm.
Y. Masse, “Central projection analemmatic sundials,” http://perso.wanadoo.fr/ymasse/gnomon/cpaprc.htm.
“The sun’s position,” www.powerfromthesun.net/chapter3/Chapter3Word.htm.
“Earth fact sheet,” http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html.
“The seasons and the earth’s orbit Milankovitch cycles,” http://aa.usno.navy.mil/faq/docs/seasons_orbit.html.
“Ellipse,” http://mathworld.wolfram.com/Ellipse.html. (Ellipses and their properties.)
“Time zones and ‘Z’ time (universal time),” http://www.aos.wisc.edu/~hopkins/wx-doc/z-time.html.
North American Sundial Society, http://sundials.org. (Source of information about all things related to sundials.)
