calculation of a symmetric gnomon that approximately corrects the
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Calculation of a symmetric Gnomon that approximately corrects the Equation of Time
Werner Riegler, 30.3. 2009
1Werner Riegler3/29/2009
The Gnomon of a sundial can be shaped such that the Equation of Time is automatically corrected and the Sundial shows the Civil Time.
Because the sun has the same declination twice a year, but the Equation of Time is different at these dates, the gnomon has to be changed twice a year, like it is e.g. the case for the dial of Martin Bernhardt:http://www.praezisions-sonnenuhr.de/
In order to avoid changing the gnomon twice a year I developed a ‘semi transparent’ double gnomon that incorporates both:http://riegler.home.cern.ch/riegler/sundial/mainpage.htm
The absolute value of the equation of time at a given declination is not very different, so by approximating the analemma with a symmetric curve one can arrive at a single Gnomon that doesn’t have to be changed. The error introduced by this approximation is at maximum 1.7 minutes.
The calculation of this Gnomon is described in the next pages.
2Werner Riegler3/29/2009
Days after December 21st
Days after December 21st
Days after December 21st
So
lar
Declin
ati
on
(D
eg
rees)
Eq
uati
on
of
Tim
e (
Min
ute
s)
Eq
uati
on
of
Tim
e (
Deg
rees)
Solar Declination
Equation of Time (EoT):Civil Time = Sundial Time + EoT
Equation of Time in Degrees=EOT (Minutes)/4
3Werner Riegler3/29/2009
Solar Declination (Degrees)
Eq
uati
on
of
Tim
e (
Deg
rees)
Solar Declination (Degrees)
Eq
uati
on
of
Tim
e (
Deg
rees)
EoT vs. Declination
Dec. 21 to Jun. 21Jun. 21 to Dec. 21
EoT vs. Declination
Averaged to make it symmetric
Solar Declination (Degrees)
Eq
uati
on
of
Tim
e E
rro
r (M
inu
tes)
Error from Symmetrization
Maximum 1.7 Min. around June 21
4Werner Riegler3/29/2009
Solar Declination δ (Degrees)
Sym
me
triz
ed
Eq
uati
on
of
Tim
eτ
(De
gre
es
)
Using the symmetric Approximation the Equation of Time τ is now a unique function of the Solar Declination δ .
We call the function absolute value of the approximated equation of Time τ (δ).
5Werner Riegler3/29/2009
South
12:00
x
y
P
Calculation of the Gnomon:At 12:00 + EoT the Sun is exactly in southern direction A Ray connected to Point P and pointing South with a declination of δ must touch the Gnomon.
This defines a ray for every declination δ
EoT (degrees)
R
x=-R*Sin(τ (δ))y=-R*Cos(τ (δ))
6Werner Riegler3/29/2009
South
Parametric Representation of the sunrays:
x=- R*Sin(τ (δ)) + k*0y=- R*Cos(τ (δ)) + k*1z= 0 + k*Tan(δ)
δ =-23.45 to 23.45k= 0 to 1500 mm
Magnifications:
7Werner Riegler3/29/2009
Sunrays:
x (k, δ)=- R*Sin(τ (δ)) + k*0y (k, δ) =- R*Cos(τ (δ)) + k*1z (k, δ) = 0 +k*Tan(δ)
δ =-23.45 to 23.45k= to 1500 mm
Gnomon:
The Gnomon must be the rotationally symmetric body that touches the surfaceof sun rays.
How can we calculate this body ?
To find the radius of the Gnomon at height z0 we cut the sunrays with a plane at z=z0 which gives a 1-dimensional curve.
Then we find the circle centered at zero that touches this curve. The radius r0 of this circle is the radius of the Gnomon at height z0.
The parametrization of this curve at z=z0 is:
z0= k*Tan(δ) k=z0/ Tan(δ)
X(δ)=- R*Sin(τ (δ)) y (δ) =- R*Cos(τ (δ)) + z0/ Tan(δ)δ =-23.45 to 23.45
Example z0=150mm
To find the circle, centered at zero, that touches this curve, we simply have to find the minimum distance between the point (0,0) and the curve, i.e.
r(δ)2= x(δ)2+y(δ)2 = = R2*Sin(τ (δ)) 2+[R*Cos(τ (δ)) -z0/Tan(δ)]2
Minimum r0
8Werner Riegler3/29/2009
On the following pages the touching circles at different heights z0 above the equatorial plane are shown.
The radius R of the dial is assumed to be 500mm.
9Werner Riegler3/29/2009
Above and below a certain limiting declination there is no solution to the minimization problem.
The reason is that the correction for one day is shadowing the correction of the next day, so the correcting Gnomon can only exist in a certain range of declinations.
The solar declination ranges from -23.45 to 23.45 degrees.
The rage where a solution for the correcting Gnomon can be found is -23.30 to 23.42 degrees.
This means that only ±10 days around Decmeber21st and ±2 days around June 21st the indicator isn’t precise.
The optimum shape to minimize the error during these days is to simply cut the gnomon at this height.
21Werner Riegler3/29/2009
Z0=-213mm
r(δ)
δ
r(δ)
δ
Limit at positive declination: 23.42 degrees, z0=216mm
Limit at negative declination: 23.3 degrees, z0=-213mm
The total height of the gnomon is therefore 213+216 = 429mm
22Werner Riegler3/29/2009
-213. 8.49-212. 9.98-211. 11.31-210. 12.46-209. 13.48-208. 14.44-207. 15.32-206. 16.14-205. 16.9-204. 17.6-203. 18.27-202. 18.91-201. 19.52-200. 20.1-199. 20.65-198. 21.17-197. 21.66-196. 22.14-195. 22.6-194. 23.06-193. 23.49-192. 23.91-191. 24.31-190. 24.7-189. 25.08-188. 25.44-187. 25.78-186. 26.11-185. 26.43-184. 26.73-183. 27.03-182. 27.32-181. 27.6-180. 27.87-179. 28.14-178. 28.38-177. 28.62-176. 28.85-175. 29.08-174. 29.3-173. 29.52-172. 29.73-171. 29.92-170. 30.12-169. 30.29-168. 30.47-167. 30.64-166. 30.8-165. 30.97-164. 31.11
-163. 31.26-162. 31.40-161. 31.54-160. 31.67-159. 31.8-158. 31.92-157. 32.03-156. 32.13-155. 32.24-154. 32.34-153. 32.43-152. 32.52-151. 32.61-150. 32.69-149. 32.77-148. 32.84-147. 32.91-146. 32.97-145. 33.02-144. 33.08-143. 33.12-142. 33.17-141. 33.21-140. 33.25-139. 33.28-138. 33.31-137. 33.34-136. 33.36-135. 33.38-134. 33.4-133. 33.41-132. 33.42-131. 33.43-130. 33.43-129. 33.43-128. 33.42-127. 33.42-126. 33.41-125. 33.39-124. 33.38-123. 33.36-122. 33.34-121. 33.31-120. 33.29-119. 33.25-118. 33.22-117. 33.19-116. 33.15-115. 33.1-114. 33.06
-113. 33.01-112. 32.96-111. 32.9-110. 32.84-109. 32.79-108. 32.73-107. 32.66-106. 32.6-105. 32.53-104. 32.47-103. 32.39-102. 32.32-101. 32.25-100. 32.17-99. 32.09-98. 32.-97. 31.91-96. 31.82-95. 31.73-94. 31.64-93. 31.55-92. 31.45-91. 31.35-90. 31.26-89. 31.16-88. 31.05-87. 30.94-86. 30.83-85. 30.72-84. 30.61-83. 30.49-82. 30.38-81. 30.26-80. 30.14-79. 30.02-78. 29.900-77. 29.78-76. 29.65-75. 29.52-74. 29.39-73. 29.26-72. 29.13-71. 28.99-70. 28.86-69. 28.72-68. 28.58-67. 28.44-66. 28.29-65. 28.15
-64. 28.01-63. 27.86-62. 27.71-61. 27.56-60. 27.40-59. 27.25-58. 27.09-57. 26.93-56. 26.78-55. 26.62-54. 26.46-53. 26.31-52. 26.15-51. 25.99-50. 25.82-49. 25.65-48. 25.48-47. 25.31-46. 25.14-45. 24.97-44. 24.79-43. 24.62-42. 24.44-41. 24.27-40. 24.09-39. 23.91-38. 23.73-37. 23.54-36. 23.36-35. 23.18-34. 23.-33. 22.81-32. 22.63-31. 22.45-30. 22.27-29. 22.08-28. 21.90-27. 21.71-26. 21.52-25. 21.32-24. 21.12-23. 20.93-22. 20.73-21. 20.54-20. 20.34-19. 20.15-18. 19.95-17. 19.75-16. 19.56-15. 19.36-14. 19.16-13. 18.96-12. 18.76-11. 18.56-10. 18.36-9. 18.16-8. 17.95-7. 17.75-6. 17.54-5. 17.34-4. 17.13-3. 16.93-2. 16.72-1. 16.52
25Werner Riegler3/29/2009
Final Gnomon
z0(mm) r0(mm) z0(mm) r0(mm) z0(mm) r0(mm) z0(mm) r0(mm)
1. 16.112. 15.93. 15.694. 15.485. 15.276. 15.067. 14.858. 14.659. 14.4410. 14.2411. 14.0312. 13.8213. 13.6114. 13.415. 13.1916. 12.9817. 12.7718. 12.5519. 12.3420. 12.1321. 11.9222. 11.7123. 11.524. 11.2925. 11.0826. 10.8727. 10.6628. 10.4529. 10.2430. 10.0331. 9.8232. 9.633. 9.3934. 9.1835. 8.9736. 8.7637. 8.5538. 8.3439. 8.1240. 7.9141. 7.742. 7.4943. 7.2844. 7.0645. 6.8646. 6.6547. 6.4448. 6.2349. 6.0250. 5.81
51. 5.6152. 5.453. 5.1954. 4.9855. 4.7856. 4.5757. 4.3758. 4.1659. 3.9660. 3.7561. 3.5562. 3.3563. 3.1564. 2.9565. 2.7566. 2.5467. 2.3468. 2.1469. 1.9470. 1.7571. 1.5572. 1.3573. 1.1574. 0.9675. 0.7676. 0.5777. 0.3878. 0.1979. 080. 0.1981. 0.3782. 0.5683. 0.7484. 0.9285. 1.186. 1.2987. 1.4888. 1.6689. 1.8690. 2.0591. 2.2392. 2.4193. 2.5894. 2.7695. 2.9296. 3.0997. 3.2698. 3.4499. 3.61100. 3.78
101. 3.95102. 4.12103. 4.28104. 4.44105. 4.60106. 4.76107. 4.92108. 5.08109. 5.24110. 5.39111. 5.54112. 5.7113. 5.84114. 5.99115. 6.14116. 6.28117. 6.43118. 6.57119. 6.71120. 6.85121. 6.98122. 7.12123. 7.25124. 7.39125. 7.52126. 7.64127. 7.77128. 7.9129. 8.02130. 8.14131. 8.26132. 8.38133. 8.5134. 8.62135. 8.74136. 8.85137. 8.95138. 9.06139. 9.16140. 9.26141. 9.36142. 9.45143. 9.55144. 9.64145. 9.73146. 9.82147. 9.9148. 9.99149. 10.07150. 10.15
151. 10.22152. 10.29153. 10.36154. 10.42155. 10.49156. 10.55157. 10.6158. 10.66159. 10.71160. 10.76161. 10.8162. 10.85163. 10.89164. 10.93165. 10.96166. 10.98167. 11.168. 11.02169. 11.04170. 11.05171. 11.07172. 11.07173. 11.07174. 11.07175. 11.06176. 11.05177. 11.04178. 11.02179. 10.99180. 10.96181. 10.92182. 10.87183. 10.83184. 10.77185. 10.72186. 10.65187. 10.58188. 10.5189. 10.41190. 10.32191. 10.22192. 10.11193. 9.99194. 9.87195. 9.73196. 9.6197. 9.44198. 9.27199. 9.1200. 8.92201. 8.73202. 8.53203. 8.3204. 8.05205. 7.79206. 7.51207. 7.2208. 6.87209. 6.52210. 6.12211. 5.67212. 5.18213. 4.60214. 3.94215. 3.09216. 1.86
26Werner Riegler3/29/2009
z0(mm) r0(mm) z0(mm) r0(mm) z0(mm) r0(mm) z0(mm) r0(mm)
Final Gnomon
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