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!