Sandburg Center for Sky Awareness
A Fairfax County Public Schools Planetarium
Calculate and Chart the Analemma
Finding the Analemma by Computation. Given the latitude
& longitude
of the observer, the Sun's altitude
& azimuth
(A & az) at exactly 12:00 noon Standard Time,
and the height of an aperture
through which the Sun's image is cast (d), the x,y coordinates
for points along the analemma
may be calculated using the following formulas (after Waugh, Sundials:
Their Theory and Construction, pp. 21-28):
(1) VS = d × COT(A)
(2) BV = VS × COS(a)
(3) BS = BV × TAN(a)
Where:
\|/
--o--
/|\
(to Sun)
\
\
=
\
\
C
|\
| \
| \
| \
d | \
| \
| \
| \
| \
V---------\---B
\
\
\
S
C = Sun aperture (a hole ~1/4 in. diameter)
d = 48 in. (given height of Sun aperture)
V = the base, i.e., the point on the floor (or ground) vertically below aperture
BV = the meridian
CS = beam of sunlight passing through aperture
S = Sun's image ("spot" on the floor)
ÐA = CSV (Altitude of Sun)
Ða = BVS (azimuth of Sun or angle from the south)
Note: Solar altitude & azimuth data are obtained from the USNO Altitude and Azimuth
of the Sun or Moon During One Day Web page. The values for
COT(A), COS(a), and TAN(a) are obtained from Table 1.
Any point along the analemma can be located on the Cartesian Plane by an
ordered
pair of numbers (x,y), called the coordinates. The origin
represents Point V, and in this case, also represents the center of
a compass
rose. The y-axis represents the meridian
(BV): north is toward the top of the y-axis; south is toward the
bottom. East is toward the right end of the x-axis; west is toward the
left. Construct a graph of the approximate shape
of an analemma (as projected on a horizontal surface) using the values for
"x" & "y" in Table 2 (see orange columns).
The shortest and longest and distances
from the base (Point V) are inferred from the values of
BV for JUN 21 and DEC 21 respectively.
Table 1. Altitude & Azimuth of the
Sun (at 12:00 noon EST)
Washington, D.C., ~39°N (38°53'N), ~77°W
(77°02'W)
| Month |
Day |
Altitude (A) |
ÐVCS = 90 -
A |
COT(A) = TAN(ÐVCS) |
Azimuth (az) |
a = az - 180 |
COS(a) |
TAN(a) |
| JAN |
1 |
28.1 |
61.9 |
1.8728 |
177.0 |
-3.0 |
0.9986 |
-0.0524 |
|
15 |
29.9 |
60.1 |
1.739 |
175.3 |
-4.7 |
0.997 |
-0.082 |
| FEB |
1 |
33.8 |
56.2 |
1.4937 |
173.8 |
-6.2 |
0.9941 |
-0.1086 |
|
15 |
38.2 |
51.8 |
1.271 |
173.1 |
-6.9 |
0.993 |
-0.121 |
| MAR |
1 |
43.3 |
46.7 |
1.0611 |
173.0 |
-7.0 |
0.9925 |
-0.1227 |
|
15 |
48.8 |
41.2 |
0.875 |
173.5 |
-6.5 |
0.994 |
-0.114 |
|
21 |
51.2 |
38.8 |
0.804 |
173.9 |
-6.1 |
0.994 |
-0.107 |
| APR |
1 |
55.6 |
34.4 |
0.6847 |
174.7 |
-5.3 |
0.9957 |
-0.0927 |
|
15 |
60.9 |
29.1 |
0.557 |
175.8 |
-4.2 |
0.9973 |
-0.0734 |
| MAY |
1 |
66.2 |
23.8 |
0.4411 |
176.9 |
-3.1 |
0.9985 |
-0.0542 |
|
15 |
70.0 |
20.0 |
0.36397 |
176.9 |
-3.1 |
0.9985 |
-0.0542 |
| JUN |
1 |
73.1 |
16.9 |
0.3038 |
175.3 |
-4.7 |
0.9966 |
-0.0822 |
|
15 |
74.3 |
15.7 |
0.281 |
172.7 |
-7.3 |
0.992 |
-0.128 |
|
21 |
74.5 |
15.5 |
0.2773 |
171.6 |
-8.4 |
0.9892 |
-0.1476 |
| Month |
Day |
Altitude (A) |
ÐVCS = 90 -
A |
COT(A) = TAN(ÐVCS) |
Azimuth (az) |
a = az - 180 |
COS(a) |
TAN(a) |
| JUL |
1 |
74.0 |
16.0 |
0.2867 |
170.0 |
-10.0 |
0.9848 |
-0.1763 |
|
15 |
72.4 |
17.6 |
0.317 |
169.2 |
-10.8 |
0.982 |
-0.191 |
| AUG |
1 |
68.9 |
21.1 |
0.3858 |
170.4 |
-9.6 |
0.9859 |
-0.1691 |
|
15 |
65.0 |
25.0 |
0.466 |
172.7 |
-7.3 |
0.992 |
-0.128 |
| SEP |
1 |
59.3 |
30.7 |
0.5937 |
176.0 |
-4.0 |
0.9975 |
-0.0699 |
|
15 |
54.1 |
35.9 |
0.724 |
178.6 |
-1.4 |
0.9997 |
-0.024 |
|
23 |
51.0 |
39.0 |
0.810 |
179.8 |
-0.2 |
0.9999 |
-0.003 |
| OCT |
1 |
47.9 |
42.1 |
0.9035 |
180.8 |
+0.8 |
0.9999 |
0.0139 |
|
15 |
42.6 |
47.4 |
1.087 |
182.0 |
+2.0 |
0.999 |
0.035 |
| NOV |
1 |
36.6 |
53.4 |
1.3465 |
182.5 |
+2.5 |
0.9990 |
0.0436 |
|
15 |
32.6 |
57.4 |
1.564 |
182.1 |
+2.1 |
0.999 |
0.037 |
| DEC |
1 |
29.3 |
60.7 |
1.7819 |
180.8 |
+0.8 |
0.9999 |
0.0139 |
|
15 |
27.9 |
62.1 |
1.8887 |
179.2 |
-0.8 |
0.9999 |
-0.01396 |
|
21 |
27.7 |
63.3 |
1.905 |
178.4 |
-1.6 |
0.9996 |
-0.028 |
Table 2. Sample Values of the
Analemma (at 12:00 noon EST)
Washington, D.C., ~39°N (38°53'N), ~77°W
(77°02'W)
| Month |
Day |
VS (in.) |
y (in.) = BV |
x (in.) = BS |
| JAN |
1 |
89.904 |
89.814 |
-4.670 |
|
15 |
83.472 |
83.222 |
-6.824 |
| FEB |
1 |
71.712 |
71.282 |
-7.7697 |
|
15 |
61.008 |
60.581 |
-7.330 |
| MAR |
1 |
50.928 |
50.572 |
-6.220 |
|
15 |
42.000 |
41.748 |
-4.759 |
|
21 |
38.592 |
38.360 |
-4.105 |
| APR |
1 |
32.880 |
32.748 |
-3.045 |
|
15 |
26.736 |
26.664 |
-1.957 |
| MAY |
1 |
21.173 |
21.141 |
-1.146 |
|
15 |
17.471 |
17.445 |
-0.946 |
| JUN |
1 |
14.592 |
14.548 |
-1.193 |
|
15 |
13.488 |
13.380 |
-1.713 |
|
21 |
13.296 |
13.1497 |
-1.946 |
| Month |
Day |
VS (in.) |
y (in.) = BV |
x (in.) = BS |
| JUL |
1 |
13.776 |
13.569 |
-2.388 |
|
15 |
15.216 |
14.942 |
-2.854 |
| AUG |
1 |
18.528 |
18.269 |
-3.087 |
|
15 |
22.368 |
22.189 |
-2.840 |
| SEP |
1 |
28.512 |
28.455 |
-1.992 |
|
15 |
34.752 |
34.742 |
-0.834 |
|
23 |
38.880 |
38.876 |
-0.117 |
| OCT |
1 |
43.392 |
43.349 |
+0.607 |
|
15 |
52.176 |
52.124 |
+1.824 |
| NOV |
1 |
64.656 |
64.591 |
+2.842 |
|
15 |
75.072 |
74.997 |
+2.775 |
| DEC |
1 |
85.536 |
85.450 |
+1.196 |
|
15 |
90.658 |
90.649 |
-1.265 |
|
21 |
91.440 |
91.403 |
-2.559 |
Figure 1.0 Approximate shape of an analemma
projected on a horizontal surface. The analemma shows the Sun's
annual north-south and east-west migration caused by changes in the Sun's
declination
(see reference table)
and Equation of Time
(see reference table)
respectively.
The actual shape and orientation of the analemma depends upon the time of
day as well as the time difference between the location of the observer
and the Standard
Time Meridian (STM):
 See a
side-by-side comparison of the analemma for the same location at two
different times: Washington, D.C. at 10
a.m. & 12 noon.
For locations
east of the STM, the figure eight is offset to the right of the y-axis;
for locations west of the STM, the offset is to the left. For example,
Washington, D.C. (77°W) is located two degrees west of the STM for the
Eastern Time
Zone (75°W), therefore the noon analemma for D.C. (shown right) is
offset to the left of the y-axis (meridian). Contrast the shape and
orientation of the D.C. analemma with the analemma for the ETZ STM (at the same latitude and
time). See a side-by-side comparison of the analemmas for the same
latitude (39°N) at two different longitudes: 75°W and 77°W (at 12 noon).
|
Experiment with analemmas for other locations (both real-sky and
projected) as well as the Equation of Time using SunAnalemma.xls, the Microsoft® Excel workbook
by Dr. Robert L. Kellogg, Treasurer, North American Sundial Society. Download and open
the workbook. At the following external prompt, click the "Enable
Macros" button:
Note: If you launch Microsoft Excel from within Microsoft Internet
Explorer, the following prompt appears in an external dialog box:
The Sandburg Center
for Sky Awareness is a
trusted source; click the "Yes" button.] ENTER the "Longitude of [Standard
Time] Meridian" and "Longitude of Concern" (e.g., your longitude). Voila!
Related Resources
- Charting
an Indoor Analemma, honoring the work of the late Dr. James R.
Griffith, Ph.D., U.S. Naval Research
Laboratory, Washington, D.C.
- An Indoor
Analemma projection by Robert Terwilliger, Coconut Grove (Miami),
Florida (Webmaster, North American
Sundial Society)
- The
M&M Millenial Analemma, an outdoor analemma projection by Michael
Kauper, Minneapolis, Minnesota (See also Astronomy with Children: Our
Backyard Analemma Project, Sky &
Telescope magazine, March 2003, pp. 77-81.)
- Determine the exact time of day by using a
radio-controlled atomic clock, such as the ExactSet™ RM806 from Oregon
Scientific, Inc.
© Copyright 2003-2009 Walter Sanford. All rights
reserved.
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