Calibration of a Horizontal Sundial Barbara Rovšek Citation: The Physics Teacher 48, 397 (2010); doi: 10.1119/1.3479720 View online: http://dx.doi.org/10.1119/1.3479720 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/48/6?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Development of a novel voltage divider for measurement of sub-nanosecond rise time high voltage pulses Rev. Sci. Instrum. 87, 024703 (2016); 10.1063/1.4941834 Construction and calibration of an impact hammer Am. J. Phys. 77, 945 (2009); 10.1119/1.3177062 Period-speed analysis of a pendulum Am. J. Phys. 76, 956 (2008); 10.1119/1.2937897 Josephson device for simultaneous time and energy detection Appl. Phys. Lett. 82, 2109 (2003); 10.1063/1.1564297 TI-TIMER: A solution to Stopwatch frustration Phys. Teach. 36, 340 (1998); 10.1119/1.880102
Calibration of a Horizontal Sundial Barbara Rovšek, Faculty of Education, University of Ljubljana, Slovenia This paper describes how a horizontal sundial can be calibrated in a classroom without using the nontrivial equations of projective geometry. If one understands how a simple equatorial sundial works, one will also understand the procedure of calibrating a horizontal (or garden, as it is also called) sundial. Two parts of any sundial are a gnomon, parallel to the Earth s rotational axis, 1 and a dial plate, where the gnomon s shadow is observed. In the equatorial sundial the dial plate lies in a dial plane, which is parallel to an equatorial plane, and a gnomon is perpendicular to it. At parts of the Earth that are away from the poles, the dial plate of the equatorial sundial is tilted with respect to the horizontal plane. In the Northern Hemisphere the tilt angle (toward the North Pole) is b = 90 o a, where a is the latitude, as shown in Fig. 1. During the day the dial plate rotates with the Earth around its rotational axis. To an observer who sits on the same carousel, it appears that the dial plate is at rest and the gnomon s shadow rotates on the dial plate around the gnomon, due to the apparent rotation of the Sun. This shadow s rotation is uniform with constant angular velocity, which is the same as the angular velocity of the Earth, 360 o /day = 15 o /h. Successive hour lines, which indicate the direction of the shadow at a particular hour of the day, are separated by an angle of 15 o. In one day the length of the gnomon s shadow in the dial plane remains almost the same, while it changes substantially during one year. On two occasions the Sun s rays are parallel to the equatorial plane on the spring and fall equinoxes and the length of the shadow diverges then. From the spring to the fall equinox the shadow is located on the northern side of the dial plate, and in the second part of the year on the southern, if the gnomon is pushed through the dial plate, as drawn in Fig. 1. On an equatorial sundial the direction of the shadow at the same hour of the day is almost the same every day throughout the year. Some minor deviations (up to 15 minutes) are caused by two reasons. The first is due to the fact that the Earth s path around the Sun is elliptical rather than Fig. 1. Geometry of the equatorial sundial (at noon). The gnomon s shadow is observed in the dial plane, which is parallel to the equatorial plane. The gnomon is parallel to the Earth s rotational axis and perpendicular to the dial plane. Fig. 2. In the horizontal sundial the dial plane is parallel to the plane of the horizon. The gnomon, being parallel to the Earth s rotational axis, is on the Northern Hemisphere tilted toward the North Pole at an angle a. circular. 2-5 The other is the tilt of the Earth s rotational axis, which causes a variation in the length of the day and leads to the appearance of an analemma. The analemma is clearly explained in Ref. 5. In this paper we shall not consider any of DOI: 10.1119/1.3479720 The Physics Teacher Vol. 48, September 2010 397
a) Fig. 3. Horizontal (left) and equatorial sundial. Both gnomons are parallel to the Earth s rotational axis. these deviations and will be content with the limited precision of the sundial. While the equatorial sundial is probably the most simple to understand, one more often sees horizontal or vertical sundials in gardens or on walls or even worn as a necklace.6 The reason is that they are easy to build. As their names suggest, they have a horizontal or vertical dial plate instead of a dial plate parallel to the equatorial plane. The gnomon remains parallel to the Earth s rotational axis, which makes it tilted with respect to the dial plate, as shown in Fig. 2, for the horizontal sundial. A gnomon s tilt angle equals the latitude a at the location where the sundial stands. On the horizontal dial plate, the gnomon s shadow rotates, but not in a trivial manner. During the day the length of the shadow changes, and even more important (and disturbing), the angular velocity of the shadow s rotation changes as well. The equations of projective geometry can be applied2-8 to obtain a relation between the angles of the hour lines qe on the equatorial sundial and qh on the horizontal sundial: qh = tan-1 [sin a tan qe ]. (1) For the noon line qh = qe = 0, and for other hour lines qe and qh indicate the angles between the hour line and the noon line on the equatorial and horizontal sundial, respectively. Equation (1) is nevertheless nontrivial and probably not easy to understand at the introductory level. Here we suggest a much simpler projecting procedure for drawing the dial for the horizontal sundial. Equatorial and horizontal sundials are closely related. We can use the regular dial of the equatorial sundial to calibrate the more complicated dial of the horizontal sundial. The only difference between these two is the orientation of the dial plate. In Fig. 3 both sundials are shown for comparison. If we put both dial plates under the same gnomon, we 398 b) c) Fig. 4. (a) The dial of the equatorial sundial with evenly distributed hour marks. (b) Side view of the horizontal and equatorial dial plates. (c) If the latitude is a, the angle between the dial plates, used in designing the supporting triangles, is b = 90 a. The Physics Teacher Vol. 48, September 2010
a) b) Fig. 5. (a) A projective tool. (b) Hour lines on the bottom, horizontal dial plate. References 1. A gnomon is usually oriented parallel to the Earth s rotational axis, but can also be oriented in a different way. For example, the Greek astronomer Eratosthenes, who was the first to calcurealize that at any instant one shadow of the gnomon may be viewed as a projection of the other. We can apply this rule and make the dial of the horizontal (or vertical as well, with minor modifications) sundial in a classroom. Construction is especially simple if one lives, as we do, close to 45 o latitude (Ljubljana s latitude is 44 o N, which is close enough). We take two sheets of hard cardboard and draw a half circle with a compass on one of them near its margin, as shown in Fig. 4(a). The circle s diameter is parallel to the margin. The half circle is divided into equal sectors of 15 o, each corresponding to one hour s rotation of the Earth (one hour rotation of the gnomon s shadow on the equatorial sundial). The mark for noon is exactly below the center of the half circle. We make a small hole at each hour mark. Then we take two plastic right-angled isosceles triangles and use them as a support and linkage between the two sheets [dial plates, see Fig. 4(b)]. If one lives away from the 45 o parallel of latitude (as most of us do), one should construct special supporting triangles with suitable angles, as shown in Fig. 4(c). We push a stick of proper length through a hole in the center of the half circle and fix it at the bottom sheet, so the stick (gnomon) is perpendicular to the tilted shadow screen, as shown in Figs. 4(b) and (c). The point where the gnomon touches the bottom sheet is the central point of the dial of the horizontal sundial. Then we use another thin, straight, and sufficiently long stick as a projecting tool. We push one end through each hour-mark hole to the bottom sheet, while the other part of the stick passes closely by the gnomon s end, as shown in Fig. 5(a). We put a label where the projecting stick touches the bottom sheet. Finally we draw the hour lines, which connect all the labels with the central point on the bottom sheet, as shown in Fig. 5(b), and obtain in this way the dial of the horizontal sundial. With a protractor we can measure the angles of the drawn hour lines and find an excellent agreement between the measured angles and those calculated from Eq. (1). For com- Table I. Hour-line angles in equatorial and horizontal sundials. hour line equatorial q e [ ] horizontal a = 45 N horizontal a = 44 N measured a = 45 N noon 0 0 0 0 13 15 10.7 10.5 10.5 14 30 22.2 21.8 21.5 15 45 35.3 34.8 35 16 60 50.8 50.3 51 17 75 69.2 68.9 68 18 90 90 90 89 parison they are written in Table I for the latitudes 44 o N and 45 o N. With the limited precision of our method, it is not possible to differentiate between both latitudes. The final check can now be done. We have to go outside, find a suitable location in the Sun, determine the north-south direction, and place the appropriately oriented dial plate with drawn hour lines on a horizontal surface. The noon line points directly to the north, the 6:00 hour line points toward west, and the 18:00 hour line points toward east. We push the gnomon through the central point of the dial and tilt it properly toward the North Pole. The horizontal sundial is now ready to use. With a reasonably simple practical procedure, we have succeeded in plotting the hour lines on the dial of the horizontal sundial, without using Eq. (1). We find this method to be very suitable for use with students at the introductory level. The Physics Teacher Vol. 48, September 2010 399
late the circumference of Earth from his measurements, used a vertical gnomon (perpendicular to the horizontal plane). 2. Malcolm M. Thomson, Sundials, Phys. Teach. 10, 117 121 (March 1972). 3. pass.maths.org.uk/issue11/features/sundials/index.html. 4. C. J. Budd and C. J. Sangwin, Mathematics Galore! (Oxford University Press, 2001). 5. www.analemma.com. 6. Michelle B. Larson, Constructing a portable sundial, Phys. Teach. 37, 113 114 (Feb. 1999). 7. home.netcom.com/~abraxas2/horz.htm. 8. www.math.nus.edu.sg/aslaksen/projects/sundials/diff_ horizontal.html. Barbara Rovšek is a teaching assistant of physics at Faculty of Education, University of Ljubljana. Faculty of Education, University of Ljubljana, Slovenia; Barbara. Rovsek@pef.uni-lj.si The OSP Collection in ComPADRE contains three models that can be used in conjunction with the preceding paper by Barbara Rovšek (see Guest Editorial, on p. 362). The models show how gnomon shadows are produced and how these shadows change throughout the day and throughout the year. 1) The Noon Shadow model shows the geometry of the shadow cast by a gnomon at noon. Users can change the orientation of the gnomon as well as its latitude. The height of the gnomon and its shadow length are displayed in Earth radius units. www.compadre.org/osp/items/detail.cfm?id=9980 2) The Eratosthenes model displays the shadows cast by two gnomons at different locations on Earth. This model shows how Eratosthenes determines the circumference of the Earth. www.compadre.org/osp/items/detail.cfm?id=9756 3) The Gnomon model shows how the gnomon shadow changes throughout the day. The simulation shows the observer s horizon plane on the spherical Earth, as well as the ecliptic and the apparent path of the Sun. The Earth view can be set to let Earth rotate or remain fixed. www.compadre.org/osp/items/detail.cfm?id=9378 These supplemental simulations for the article by Barbara Rovšek have been approved by the author and the TPT editor. Wolfgang Christian 400 The Physics Teacher Vol. 48, September 2010