Calculation of a symmetric Gnomon that approximately corrects the

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.


http://www.praezisions-sonnenuhr.de/






http://riegler.home.cern.ch/riegler/sundial/mainpage.htm

http://riegler.home.cern.ch/riegler/sundial/mainpage.htm

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


Solar Declination (Degrees)

Eq

uati

on

of

Tim

e (

Deg

rees)


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


Eq

uati

on

of

Tim

e E

rro

r (M

inu

tes)

Error from Symmetrization

Maximum 1.7 Min. around June 21


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).


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))


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:


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


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.


Z0=1mm

r(δ)

δ10Werner Riegler3/29/2009

Z0=50mm

r(δ)

δ


Z0=100mm

r(δ)


Z0=150mm

r(δ)

δ


Z0=200mm

r(δ)


Z0=216mm

r(δ)


Z0=-50mm

r(δ)


Z0=-100mm

r(δ)

δ 17Werner Riegler3/29/2009

Z0=-150mm

r(δ)


Z0=-200mm

r(δ)


Z0=-213mm

r(δ)

δ 20Werner 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.


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



Final Gnomon


Final Gnomon

-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


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


z0(mm) r0(mm) z0(mm) r0(mm) z0(mm) r0(mm) z0(mm) r0(mm)

Final Gnomon

Documents

Calculation of a symmetric Gnomon that approximately corrects the