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 ca