Документ взят из кэша поисковой машины. Адрес оригинального документа : http://star.arm.ac.uk/orrery/make_orrery_2011jul04.pdf Дата изменения: Tue Jul 5 18:47:29 2011 Дата индексирования: Tue Oct 2 01:36:23 2012 Кодировка: Поисковые слова: п п п п п п

A Guide To Make a Human Orrery

Erin Higgins, Thornhill College, Derry Ryan OHare, St. Louis Grammar School, Kilkeel David ONeill, St. Louis Grammar School, Ballymena 1st July 2011

Introduction
This guide provides a step-by-step description of how to make a simple, but precise Human Orrery. This will enable predictions to be made about the positions of planets at a certain date and time. The main aims of this Human Orrery are to show the elliptical orbit for each planet around the Sun, and using a time-step of 16 days to show the positions of the planets at different dates. Each ellipse is marked out using the string and pin method; that is using a circular template with the Sun at the centre and with a pre-drilled hole to represent the second focus of the elliptical orbit for each planet. The Sun and the predrilled holes create the two focus points required for each planets orbit and allow an ellipse to be drawn by looping a string of length L, found in Table 1, around the two focus points (or pins), and a piece of chalk to draw the ellipse.

Method
The basic template will be 1.2m in diameter, and can be made from wood or a perspex plastic. This can be used both as a large protractor and as a means to define the orientation of each planetary ellipse. Having chalked out the shape of each objects orbit, then paint can be used to identify the location of the planets at certain dates. To make this orrery, a tape-measure, large protractor, string, chalk and paint will be needed to map out the ellipses. The comets orbits and that of Saturn will be drawn using pins beyond the main template, since their orbits are too large to draw using the template. A table of the points for Halley, Encke, Phaethon and Tempel-Tuttle is given to provide points to draw their respective orbits. The orbits are placed in order from the Sun and with regular intervals to determine dates and to make predictions of where the planets will be at a particular date. This can be done by dividing the number of time-steps or multiples of 16 days from the start-date of the orrery by the number of points in each planetary orbit to leave a remainder. This remainder will give the location of the planet at that date.


Table 1. The string and pin method for drawing the ellipses for the planets and other objects shown on the Human Orrery. The Sun is located at the centre of the model and represents a common focus for every object. The scale of the model is such that 1m on the ground represents 1AU in space.

Object

Length of string, S (m) 0.93 1.46 2.03 3.33 5.97 10.91 20.10 70.59 8.19 39.40 4.81

Distance between foci, 2ae (m) 0.159 0.010 0.033 0.285 0.442 0.503 1.027 34.71 3.76 18.72 2.26

Direction of the second focus in heliocentric elliptic coordinates, measured from the first point of Aries (degrees) 257.3 311.6 282.9 156.1 333.4 194.7 272.6 126.0 341.1 242.4 49.5

Mercury Venus Earth Mars (1) Ceres Jupiter Saturn 1P/Halley 2P/Encke 55P/Tempel-Tuttle (3200) Phaethon


The Template
When designing the template for drawing these orbits, a few design constraints were noticed. The template must be small enough and light enough to make it portable and accessible for various places. The template must be large enough to hold the two focus points for most of the objects. Pre-drilled holes must be a feature of the template so that they can be easily located on the ground without the need to calculate angles and distances from the centre. The template will have the Sun at the centre as a focus for all objects, with another hole for each objects second focus. If, as suggested, it is chosen to be large enough to encompass the two foci of Jupiters orbit, it may also be convenient to provide pre-drilled holes for the eleven points that together define two revolutions of Mercurys orbit.


Curriculum
The Human Orrery, a dynamic model of the Solar System in which people play the role of the moving planets, can be used to encourage children to have a better understanding of the Solar System at Key Stages 1 and 2. At these stages, children learn basic details about the Solar System. The model can be aligned to the curriculum, for example, where children learn about number ratios and multiplications of ten. This is useful when working with powers of ten in the model, and can provide an understanding of the scale of the Solar System. Children in Key Stages 1 and 2 also learn how to divide leaving a remainder, and this is useful when working out the dates in each division of the planetary orbit which will give the position of the planet at the desired date. In these stages of development, children are encouraged to imagine a real-life situation, where they have a problem to solve based on a given situation. The model of the Human Orrery will allow children to imagine the proportion and size of the Solar System showing them the elliptical orbits in a simplified model. The Human Orrery, will enable children to use their knowledge of estimations and help them to solve a real-life problem to predict the positions of the planets at any given time.

Conclusion
This guide is designed to simplify the complications of making a Human Orrery by providing the basic astronomical information and describing equipment needed accurately to lay out the planetary orbit. The Human Orrery is designed to show the positions, orbits and relative distances of the planets around the Sun. This project allows anyone to make a Human Orrery. A key feature of the model is that it develops understanding of the motion of celestial bodies around the Sun and the Earths changing position in space.

Acknowledgements
This project was done as part of our work-experience during the dates 27th June 2011 and 1st July 2011. We would like to thank the Armagh Observatory for their facilities, help and enthusiasm. We especially appreciate the help of Professor Mark Bailey who guided us through our project and gave us an insight into the aspects of an Astronomers career. We would also like to thank our schools for their support in setting up work-experience at the Armagh Observatory.

Appendix:
These ta point of distance, positions bles show the orbital periods, distances from the Sun and angles from the first Aries, for various objects shown on the Human Orrery at various times. The r, and heliocentric ecliptic longitude, L, can be used to make the planetary at the given time.


Mercury

11 Tiles at 16-day intervals (2 complete revolutions). True period (days): 87.969 Add a step ( one tile) every 520 revolutions
L 190.107 240.364 285.008 339.812 63.985 156.761 217.097 262.510 310.105 17.531 113.997 r 0.402 0.461 0.453 0.381 0.309 0.357 0.440 0.466 0.424 0.336 0.318 f Separation (cm) 112.438 37.1 162.991 34.7 207.741 38.9 262.202 46.6 346.633 48.3 79.168 40.7 139.550 35.0 185.245 36.1 232.701 42.8 299.878 48.8 36.693

JD Tile No. Year Month Day 2453371.5 0 2005 1 1 2453387.5 1 2005 1 17 2453403.5 2 2005 2 2 2453419.5 3 2005 2 18 2453435.5 4 2005 3 6 2453451.5 5 2005 3 22 2453467.5 6 2005 4 7 2453483.5 7 2005 4 23 2453499.5 8 2005 5 9 2453515.5 9 2005 5 25 2453531.5 10 2005 6 10

L DD MM ss 190 6 25 240 21 50 285 0 29 339 48 43 63 59 6 156 45 40 217 5 49 262 30 36 310 6 18 17 31 52 113 59 49

r Theta=360-L Theta=36 (metres) Degrees DD MM 0.402 169.893 169 53 0.461 119.636 119 38 0.453 74.992 74 59 0.381 20.188 20 11 0.309 296.015 296 0 0.357 203.239 203 14 0.440 142.903 142 54 0.466 97.490 97 29 0.424 49.895 49 53 0.336 342.469 342 28 0.318 246.003 246 0

0-L ss 35 10 31 17 54 20 11 24 42 8 11

L-f 77.669 77.373 77.267 77.610 -282.648 77.593 77.547 77.265 77.404 -282.347 77.304


JD

Tile No. Year Month Day

L

r

f

Separation (cm)

DD MM ss

(metres)

Degrees

DD MM ss

Venus

14 Tiles at 16-day intervals. True period (days): 224.701 Miss a step ( one tile) every
L 229.086 254.566 279.916 305.213 330.530 355.916 21.394 46.981 72.687 98.521 124.466 150.469 176.443 202.295 r 0.724 0.726 0.727 0.728 0.728 0.727 0.725 0.723 0.721 0.719 0.718 0.719 0.720 0.722

JD Tile No. Year Month Day 2453371.5 0 2005 1 1 2453387.5 1 2005 1 17 2453403.5 2 2005 2 2 2453419.5 3 2005 2 18 2453435.5 4 2005 3 6 2453451.5 5 2005 3 22 2453467.5 6 2005 4 7 2453483.5 7 2005 4 23 2453499.5 8 2005 5 9 2453515.5 9 2005 5 25 2453531.5 10 2005 6 10 2453547.5 11 2005 6 26 2453563.5 12 2005 7 12 2453579.5 13 2005 7 28

23 revolutions f Separation (cm) 97.264 32.0 122.803 31.9 148.208 31.9 173.528 31.9 198.828 32.0 224.172 32.0 249.617 32.1 275.194 32.1 300.917 32.2 326.775 32.3 352.718 32.3 18.682 32.3 44.601 32.2 70.424

L DD MM ss 229 5 10 254 33 58 279 54 58 305 12 47 330 31 48 355 54 58 21 23 38 46 58 52 72 41 13 98 31 16 124 27 58 150 28 8 176 26 35 202 17 42

r Theta=360-L Theta=36 (metres) Degrees DD MM 0.724 130.914 130 54 0.726 105.434 105 26 0.727 80.084 80 5 0.728 54.787 54 47 0.728 29.470 29 28 0.727 4.084 4 5 0.725 338.606 338 36 0.723 313.019 313 1 0.721 287.313 287 18 0.719 261.479 261 28 0.718 235.534 235 32 0.719 209.531 209 31 0.720 183.557 183 33 0.722 157.705 157 42

0-L ss 50 2 2 13 12 2 22 8 47 44 2 52 25 18

L-f 131.822 131.763 131.708 131.685 131.702 131.744 -228.223 -228.213 -228.230 -228.254 -228.252 131.787 131.842 131.871


JD

Tile No. Year Month Day

L

r

f

Separation (cm)

DD MM ss

(metres)

Degrees

DD MM ss

Earth

23 Tiles at 16-day intervals. True period (days): 365.256 Add a step ( one tile) every
L 100.605 116.908 133.174 149.365 165.447 181.399 197.208 212.874 228.406 243.824 259.155 274.430 289.685 304.957 320.280 335.688 351.207 6.858 22.652 38.589 54.659 70.839 87.099 r 0.983 0.984 0.986 0.988 0.992 0.996 1.001 1.005 1.010 1.013 1.015 1.017 1.017 1.015 1.013 1.010 1.006 1.001 0.997 0.993 0.989 0.986 0.984

JD Tile No. Year Month Day 2453371.5 0 2005 1 1 2453387.5 1 2005 1 17 2453403.5 2 2005 2 2 2453419.5 3 2005 2 18 2453435.5 4 2005 3 6 2453451.5 5 2005 3 22 2453467.5 6 2005 4 7 2453483.5 7 2005 4 23 2453499.5 8 2005 5 9 2453515.5 9 2005 5 25 2453531.5 10 2005 6 10 2453547.5 11 2005 6 26 2453563.5 12 2005 7 12 2453579.5 13 2005 7 28 2453595.5 14 2005 8 13 2453611.5 15 2005 8 29 2453627.5 16 2005 9 14 2453643.5 17 2005 9 30 2453659.5 18 2005 10 16 2453675.5 19 2005 11 1 2453691.5 20 2005 11 17 2453707.5 21 2005 12 3 2453723.5 22 2005 12 19

6 revolutions f Separation (cm) 357.741 27.9 14.039 27.9 30.300 27.8 46.484 27.7 62.556 27.6 78.498 27.5 94.305 27.3 109.976 27.2 125.518 27.1 140.946 27.1 156.283 27.0 171.563 27.0 186.826 27.0 202.108 27.0 217.446 27.1 232.871 27.2 248.406 27.3 264.072 27.5 279.876 27.6 295.815 27.7 311.877 27.8 328.039 27.9 344.272

L DD MM ss 100 36 18 116 54 29 133 10 26 149 21 54 165 26 49 181 23 56 197 12 29 212 52 26 228 24 22 243 49 26 259 9 18 274 25 48 289 41 6 304 57 25 320 16 48 335 41 17 351 12 25 6 51 29 22 39 7 38 35 20 54 39 32 70 50 20 87 5 56

r Theta=360-L Theta=36 (metres) Degrees DD MM 0.983 259.395 259 23 0.984 243.092 243 5 0.986 226.826 226 49 0.988 210.635 210 38 0.992 194.553 194 33 0.996 178.601 178 36 1.001 162.792 162 47 1.005 147.126 147 7 1.010 131.594 131 35 1.013 116.176 116 10 1.015 100.845 100 50 1.017 85.570 85 34 1.017 70.315 70 18 1.015 55.043 55 2 1.013 39.720 39 43 1.010 24.312 24 18 1.006 8.793 8 47 1.001 353.142 353 8 0.997 337.348 337 20 0.993 321.411 321 24 0.989 305.341 305 20 0.986 289.161 289 9 0.984 272.901 272 54

0-L ss 42 31 34 6 11 4 31 34 38 34 42 12 54 35 12 43 35 31 53 40 28 40 4

L-f -257.136 102.869 102.874 102.881 102.891 102.901 102.903 102.898 102.888 102.878 102.872 102.867 102.859 102.849 102.834 102.817 102.801 -257.214 -257.224 -257.226 -257.218 -257.200 -257.173


JD

Tile No. Year Month Day

L

r

f

Separation (cm)

DD MM ss

(metres)

Degrees

DD MM ss

Mars

43 Tiles at 16-day intervals. True period (days): 686.980 Add a step ( one tile) every
L 222.507 230.473 238.649 247.049 255.686 264.567 273.692 283.054 292.640 302.423 312.370 322.439 332.579 342.738 352.861 2.895 12.794 22.516 32.033 41.323 50.373 59.179 67.745 76.079 84.193 92.103 99.827 107.386 114.799 122.089 129.275 136.379 143.423 150.428 157.415 164.406 171.421 178.483 185.612 192.830 200.158 207.619 215.232 r 1.569 1.550 1.529 1.508 1.487 1.467 1.448 1.430 1.415 1.401 1.391 1.385 1.382 1.382 1.386 1.394 1.405 1.419 1.435 1.453 1.473 1.493 1.514 1.535 1.555 1.575 1.593 1.610 1.624 1.637 1.648 1.656 1.662 1.665 1.666 1.664 1.660 1.653 1.644 1.633 1.619 1.604 1.586

JD Tile No. Year Month Day 2453371.5 0 2005 1 1 2453387.5 1 2005 1 17 2453403.5 2 2005 2 2 2453419.5 3 2005 2 18 2453435.5 4 2005 3 6 2453451.5 5 2005 3 22 2453467.5 6 2005 4 7 2453483.5 7 2005 4 23 2453499.5 8 2005 5 9 2453515.5 9 2005 5 25 2453531.5 10 2005 6 10 2453547.5 11 2005 6 26 2453563.5 12 2005 7 12 2453579.5 13 2005 7 28 2453595.5 14 2005 8 13 2453611.5 15 2005 8 29 2453627.5 16 2005 9 14 2453643.5 17 2005 9 30 2453659.5 18 2005 10 16 2453675.5 19 2005 11 1 2453691.5 20 2005 11 17 2453707.5 21 2005 12 3 2453723.5 22 2005 12 19 2453739.5 23 2006 1 4 2453755.5 24 2006 1 20 2453771.5 25 2006 2 5 2453787.5 26 2006 2 21 2453803.5 27 2006 3 9 2453819.5 28 2006 3 25 2453835.5 29 2006 4 10 2453851.5 30 2006 4 26 2453867.5 31 2006 5 12 2453883.5 32 2006 5 28 2453899.5 33 2006 6 13 2453915.5 34 2006 6 29 2453931.5 35 2006 7 15 2453947.5 36 2006 7 31 2453963.5 37 2006 8 16 2453979.5 38 2006 9 1 2453995.5 39 2006 9 17 2454011.5 40 2006 10 3 2454027.5 41 2006 10 19 2454043.5 42 2006 11 4

16 revolutions f Separation (cm) 246.369 21.8 254.332 22.0 262.506 22.3 270.904 22.7 279.539 23.0 288.419 23.3 297.542 23.6 306.903 23.8 316.486 24.0 326.266 24.2 336.209 24.4 346.274 24.4 356.411 24.5 6.566 24.4 16.686 24.3 26.719 24.2 36.617 24.0 46.342 23.7 55.862 23.5 65.155 23.2 74.210 22.9 83.021 22.6 91.593 22.3 99.933 22.0 108.053 21.7 115.969 21.4 123.699 21.2 131.263 21.0 138.680 20.8 145.970 20.6 153.156 20.5 160.259 20.4 167.301 20.3 174.303 20.3 181.288 20.3 188.278 20.3 195.293 20.4 202.356 20.5 209.489 20.7 216.713 20.8 224.049 21.0 231.517 21.2 239.139

L DD MM ss 222 30 25 230 28 23 238 38 56 247 2 56 255 41 10 264 34 1 273 41 31 283 3 14 292 38 24 302 25 23 312 22 12 322 26 20 332 34 44 342 44 17 352 51 40 2 53 42 12 47 38 22 30 58 32 1 59 41 19 23 50 22 23 59 10 44 67 44 42 76 4 44 84 11 35 92 6 11 99 49 37 107 23 10 114 47 56 122 5 20 129 16 30 136 22 44 143 25 23 150 25 41 157 24 54 164 24 22 171 25 16 178 28 59 185 36 43 192 49 48 200 9 29 207 37 8 215 13 55

r Theta=360-L Theta=36 (metres) Degrees DD MM 1.569 137.493 137 29 1.550 129.527 129 31 1.529 121.351 121 21 1.508 112.951 112 57 1.487 104.314 104 18 1.467 95.433 95 25 1.448 86.308 86 18 1.430 76.946 76 56 1.415 67.360 67 21 1.401 57.577 57 34 1.391 47.630 47 37 1.385 37.561 37 33 1.382 27.421 27 25 1.382 17.262 17 15 1.386 7.139 7 8 1.394 357.105 357 6 1.405 347.206 347 12 1.419 337.484 337 29 1.435 327.967 327 58 1.453 318.677 318 40 1.473 309.627 309 37 1.493 300.821 300 49 1.514 292.255 292 15 1.535 283.921 283 55 1.555 275.807 275 48 1.575 267.897 267 53 1.593 260.173 260 10 1.610 252.614 252 36 1.624 245.201 245 12 1.637 237.911 237 54 1.648 230.725 230 43 1.656 223.621 223 37 1.662 216.577 216 34 1.665 209.572 209 34 1.666 202.585 202 35 1.664 195.594 195 35 1.660 188.579 188 34 1.653 181.517 181 31 1.644 174.388 174 23 1.633 167.170 167 10 1.619 159.842 159 50 1.604 152.381 152 22 1.586 144.768 144 46

0-L ss 35 37 4 4 50 59 29 46 36 37 48 40 16 43 20 18 22 2 1 37 37 16 18 16 25 49 23 50 4 40 30 16 37 19 6 38 44 1 17 12 31 52 5

L-f -23.862 -23.859 -23.857 -23.855 -23.853 -23.852 -23.850 -23.849 -23.846 -23.843 -23.839 -23.835 -23.832 336.172 336.175 -23.824 -23.823 -23.826 -23.829 -23.832 -23.837 -23.842 -23.848 -23.854 -23.860 -23.866 -23.872 -23.877 -23.881 -23.881 -23.881 -23.880 -23.878 -23.875 -23.873 -23.872 -23.872 -23.873 -23.877 -23.883 -23.891 -23.898 -23.907


JD

Tile No. Year Month Day

L

r

f

Separation (cm)

DD MM ss

(metres)

Degrees

DD MM ss

Jupiter

27 Tiles at 160-day intervals. True period (days): 4332.589 Miss a step ( one tile) every
L 186.903 198.981 211.077 223.247 235.542 248.014 260.707 273.660 286.903 300.449 314.299 328.433 342.809 357.366 12.025 26.699 41.299 55.742 69.962 83.909 97.558 110.901 123.949 136.726 149.269 161.619 173.826 r 5.454 5.456 5.445 5.423 5.389 5.346 5.295 5.239 5.180 5.122 5.067 5.020 4.983 4.959 4.949 4.954 4.973 5.006 5.051 5.103 5.160 5.219 5.277 5.330 5.376 5.413 5.439

JD Tile No. Year Month Day 2453371.5 0 2005 1 1 2453531.5 10 2005 6 10 2453691.5 20 2005 11 17 2453851.5 30 2006 4 26 2454011.5 40 2006 10 3 2454171.5 50 2007 3 12 2454331.5 60 2007 8 19 2454491.5 70 2008 1 26 2454651.5 80 2008 7 4 2454811.5 90 2008 12 11 2454971.5 100 2009 5 20 2455131.5 110 2009 10 27 2455291.5 120 2010 4 5 2455451.5 130 2010 9 12 2455611.5 140 2011 2 19 2455771.5 150 2011 7 29 2455931.5 160 2012 1 5 2456091.5 170 2012 6 13 2456251.5 180 2012 11 20 2456411.5 190 2013 4 29 2456571.5 200 2013 10 6 2456731.5 210 2014 3 15 2456891.5 220 2014 8 22 2457051.5 230 2015 1 29 2457211.5 240 2015 7 8 2457371.5 250 2015 12 15 2457531.5 260 2016 5 23

13 revolutions f Separation (cm) 171.988 114.8 184.322 114.9 196.656 115.2 209.092 115.8 221.586 116.7 234.236 117.7 247.107 118.9 260.135 120.3 273.432 121.6 286.949 123.0 300.745 124.2 314.702 125.2 328.876 126.0 343.213 126.4 357.589 126.5 11.993 126.2 26.329 125.5 40.589 124.6 54.603 123.4 68.456 122.1 82.077 120.7 95.424 119.4 108.551 118.1 121.447 117.0 134.189 116.1 146.704 115.4 159.158

L DD MM ss 186 54 11 198 58 52 211 4 37 223 14 49 235 32 31 248 0 50 260 42 25 273 39 36 286 54 11 300 26 56 314 17 56 328 25 59 342 48 32 357 21 58 12 1 30 26 41 56 41 17 56 55 44 31 69 57 43 83 54 32 97 33 29 110 54 4 123 56 56 136 43 34 149 16 8 161 37 8 173 49 34

r Theta=360-L Theta=36 (metres) Degrees DD MM 5.454 173.097 173 5 5.456 161.019 161 1 5.445 148.923 148 55 5.423 136.753 136 45 5.389 124.458 124 27 5.346 111.986 111 59 5.295 99.293 99 17 5.239 86.340 86 20 5.180 73.097 73 5 5.122 59.551 59 33 5.067 45.701 45 42 5.020 31.567 31 34 4.983 17.191 17 11 4.959 2.634 2 38 4.949 347.975 347 58 4.954 333.301 333 18 4.973 318.701 318 42 5.006 304.258 304 15 5.051 290.038 290 2 5.103 276.091 276 5 5.160 262.442 262 26 5.219 249.099 249 5 5.277 236.051 236 3 5.330 223.274 223 16 5.376 210.731 210 43 5.413 198.381 198 22 5.439 186.174 186 10

0-L ss 49 8 23 11 29 10 35 24 49 4 4 1 28 2 30 4 4 29 17 28 31 56 4 26 52 52 26

L-f 14.915 14.659 14.421 14.155 13.956 13.778 13.600 13.525 13.471 13.500 13.554 13.731 13.933 14.153 -345.564 14.706 14.970 15.153 15.359 15.453 15.481 15.477 15.398 15.279 15.080 14.915 14.668


JD

Tile No. Year Month Day

L

r

f

Separation (cm)

DD MM ss

(metres)

Degrees

DD MM ss

Saturn

67 Tiles at 160-day intervals. True period (days): 10759.227 Miss a step ( one tile) every
L 113.327 119.247 125.138 130.995 136.815 142.591 148.321 154.002 159.632 165.208 170.729 176.196 181.607 186.963 192.266 197.517 202.718 207.871 212.978 218.041 223.065 228.052 233.004 237.926 242.822 247.693 252.544 257.380 262.202 267.016 271.825 276.633 281.443 286.260 291.088 295.930 300.790 305.674 310.584 315.525 320.500 325.513 330.567 335.664 340.808 346.000 351.243 356.539 1.889 7.294 12.754 18.270 23.841 29.466 35.143 40.870 46.644 52.460 58.316 64.206 70.125 76.068 82.028 87.999 93.975 99.949 105.915 r 9.058 9.077 9.100 9.127 9.159 9.194 9.232 9.273 9.316 9.361 9.408 9.455 9.503 9.552 9.600 9.647 9.693 9.738 9.781 9.822 9.861 9.897 9.930 9.959 9.986 10.009 10.028 10.043 10.055 10.062 10.065 10.065 10.060 10.051 10.038 10.021 10.000 9.975 9.946 9.913 9.878 9.839 9.798 9.754 9.709 9.662 9.614 9.565 9.515 9.466 9.417 9.369 9.323 9.278 9.236 9.196 9.159 9.126 9.096 9.071 9.050 9.034 9.022 9.016 9.015 9.019 9.028

JD Tile No. Year Month Day 2453371.5 0 2005 1 1 2453531.5 10 2005 6 10 2453691.5 20 2005 11 17 2453851.5 30 2006 4 26 2454011.5 40 2006 10 3 2454171.5 50 2007 3 12 2454331.5 60 2007 8 19 2454491.5 70 2008 1 26 2454651.5 80 2008 7 4 2454811.5 90 2008 12 11 2454971.5 100 2009 5 20 2455131.5 110 2009 10 27 2455291.5 120 2010 4 5 2455451.5 130 2010 9 12 2455611.5 140 2011 2 19 2455771.5 150 2011 7 29 2455931.5 160 2012 1 5 2456091.5 170 2012 6 13 2456251.5 180 2012 11 20 2456411.5 190 2013 4 29 2456571.5 200 2013 10 6 2456731.5 210 2014 3 15 2456891.5 220 2014 8 22 2457051.5 230 2015 1 29 2457211.5 240 2015 7 8 2457371.5 250 2015 12 15 2457531.5 260 2016 5 23 2457691.5 270 2016 10 30 2457851.5 280 2017 4 8 2458011.5 290 2017 9 15 2458171.5 300 2018 2 22 2458331.5 310 2018 8 1 2458491.5 320 2019 1 8 2458651.5 330 2019 6 17 2458811.5 340 2019 11 24 2458971.5 350 2020 5 2 2459131.5 360 2020 10 9 2459291.5 370 2021 3 18 2459451.5 380 2021 8 25 2459611.5 390 2022 2 1 2459771.5 400 2022 7 11 2459931.5 410 2022 12 18 2460091.5 420 2023 5 27 2460251.5 430 2023 11 3 2460411.5 440 2024 4 11 2460571.5 450 2024 9 18 2460731.5 460 2025 2 25 2460891.5 470 2025 8 4 2461051.5 480 2026 1 11 2461211.5 490 2026 6 20 2461371.5 500 2026 11 27 2461531.5 510 2027 5 6 2461691.5 520 2027 10 13 2461851.5 530 2028 3 21 2462011.5 540 2028 8 28 2462171.5 550 2029 2 4 2462331.5 560 2029 7 14 2462491.5 570 2029 12 21 2462651.5 580 2030 5 30 2462811.5 590 2030 11 6 2462971.5 600 2031 4 15 2463131.5 610 2031 9 22 2463291.5 620 2032 2 29 2463451.5 630 2032 8 7 2463611.5 640 2033 1 14 2463771.5 650 2033 6 23 2463931.5 660 2033 11 30

4 revolutions f Separation (cm) 18.904 93.7 24.832 93.4 30.843 93.2 36.870 92.9 42.966 92.5 49.082 92.2 55.274 91.8 61.363 91.4 67.496 91.0 73.571 90.5 79.504 90.1 85.337 89.6 91.003 89.2 96.548 88.7 101.797 88.3 106.959 87.9 111.928 87.5 116.711 87.1 121.425 86.7 126.039 86.4 130.627 86.0 135.118 85.7 139.707 85.5 144.287 85.2 148.893 85.0 153.583 84.8 158.342 84.7 163.198 84.6 168.091 84.5 173.143 84.4 178.248 84.4 183.455 84.5 188.724 84.5 194.106 84.6 199.538 84.7 204.995 84.9 210.535 85.1 216.042 85.4 221.514 85.7 226.811 86.0 232.058 86.3 237.090 86.7 241.966 87.0 246.786 87.5 251.503 87.9 256.202 88.3 260.846 88.7 265.635 89.2 270.401 89.6 275.261 90.1 280.282 90.5 285.374 91.0 290.614 91.4 295.946 91.8 301.495 92.2 307.083 92.5 312.830 92.8 318.736 93.1 324.702 93.4 330.779 93.6 336.913 93.8 343.155 93.9 349.357 94.0 355.586 94.0 1.772 94.0 7.898 93.9 13.899

L DD MM ss 113 19 37 119 14 49 125 8 17 130 59 42 136 48 54 142 35 28 148 19 16 154 0 7 159 37 55 165 12 29 170 43 44 176 11 46 181 36 25 186 57 47 192 15 58 197 31 1 202 43 5 207 52 16 212 58 41 218 2 28 223 3 54 228 3 7 233 0 14 237 55 34 242 49 19 247 41 35 252 32 38 257 22 48 262 12 7 267 0 58 271 49 30 276 37 59 281 26 35 286 15 36 291 5 17 295 55 48 300 47 24 305 40 26 310 35 2 315 31 30 320 30 0 325 30 47 330 34 1 335 39 50 340 48 29 346 0 0 351 14 35 356 32 20 1 53 20 7 17 38 12 45 14 18 16 12 23 50 28 29 27 58 35 8 35 40 52 12 46 38 38 52 27 36 58 18 58 64 12 22 70 7 30 76 4 5 82 1 41 87 59 56 93 58 30 99 56 56 105 54 54

r Theta=360-L Theta=36 (metres) Degrees DD MM 9.058 246.673 246 40 9.077 240.753 240 45 9.100 234.862 234 51 9.127 229.005 229 0 9.159 223.185 223 11 9.194 217.409 217 24 9.232 211.679 211 40 9.273 205.998 205 59 9.316 200.368 200 22 9.361 194.792 194 47 9.408 189.271 189 16 9.455 183.804 183 48 9.503 178.393 178 23 9.552 173.037 173 2 9.600 167.734 167 44 9.647 162.483 162 28 9.693 157.282 157 16 9.738 152.129 152 7 9.781 147.022 147 1 9.822 141.959 141 57 9.861 136.935 136 56 9.897 131.948 131 56 9.930 126.996 126 59 9.959 122.074 122 4 9.986 117.178 117 10 10.009 112.307 112 18 10.028 107.456 107 27 10.043 102.620 102 37 10.055 97.798 97 47 10.062 92.984 92 59 10.065 88.175 88 10 10.065 83.367 83 22 10.060 78.557 78 33 10.051 73.740 73 44 10.038 68.912 68 54 10.021 64.070 64 4 10.000 59.210 59 12 9.975 54.326 54 19 9.946 49.416 49 24 9.913 44.475 44 28 9.878 39.500 39 30 9.839 34.487 34 29 9.798 29.433 29 25 9.754 24.336 24 20 9.709 19.192 19 11 9.662 14.000 14 0 9.614 8.757 8 45 9.565 3.461 3 27 9.515 358.111 358 6 9.466 352.706 352 42 9.417 347.246 347 14 9.369 341.730 341 43 9.323 336.159 336 9 9.278 330.534 330 32 9.236 324.857 324 51 9.196 319.130 319 7 9.159 313.356 313 21 9.126 307.540 307 32 9.096 301.684 301 41 9.071 295.794 295 47 9.050 289.875 289 52 9.034 283.932 283 55 9.022 277.972 277 58 9.016 272.001 272 0 9.015 266.025 266 1 9.019 260.051 260 3 9.028 254.085 254 5

0-L ss 23 11 43 18 6 32 44 53 5 31 16 14 35 13 2 59 55 44 19 32 6 53 46 26 41 25 22 12 53 2 30 1 25 24 43 12 36 34 58 30 0 13 59 10 31 0 25 40 40 22 46 48 32 2 25 48 22 24 2 38 30 55 19 4 30 4 6

L-f 94.423 94.415 94.295 94.125 93.849 93.509 93.047 92.639 92.136 91.637 91.225 90.859 90.604 90.415 90.469 90.558 90.790 91.160 91.553 92.002 92.438 92.934 93.297 93.639 93.929 94.110 94.202 94.182 94.111 93.873 93.577 93.178 92.719 92.154 91.550 90.935 90.255 89.632 89.070 88.714 88.442 88.423 88.601 88.878 89.305 89.798 90.397 90.904 -268.512 -267.967 -267.528 -267.104 -266.773 -266.480 -266.352 -266.213 -266.186 -266.276 -266.386 -266.573 -266.788 -267.087 -267.329 -267.587 92.203 92.051 92.016


JD

Tile No. Year Month Day

L

r

f

Separation (cm)

DD MM ss

(metres)

Degrees

DD MM ss

Ceres

21 Tiles at 80-day intervals. True period (days): 1681.592 Miss a step ( one tile) every
L 198.192 217.281 235.360 252.462 268.737 284.381 299.594 314.555 329.419 344.314 359.349 14.625 30.240 46.307 62.948 80.302 98.487 117.556 137.421 157.791 178.199 r 2.598 2.650 2.713 2.778 2.842 2.897 2.941 2.971 2.985 2.982 2.963 2.928 2.879 2.820 2.756 2.690 2.631 2.583 2.554 2.546 2.562

JD Tile No. Year Month Day 2453371.5 0 2005 1 1 2453451.5 5 2005 3 22 2453531.5 10 2005 6 10 2453611.5 15 2005 8 29 2453691.5 20 2005 11 17 2453771.5 25 2006 2 5 2453851.5 30 2006 4 26 2453931.5 35 2006 7 15 2454011.5 40 2006 10 3 2454091.5 45 2006 12 22 2454171.5 50 2007 3 12 2454251.5 55 2007 5 31 2454331.5 60 2007 8 19 2454411.5 65 2007 11 7 2454491.5 70 2008 1 26 2454571.5 75 2008 4 15 2454651.5 80 2008 7 4 2454731.5 85 2008 9 22 2454811.5 90 2008 12 11 2454891.5 95 2009 3 1 2454971.5 100 2009 5 20

50 revolutions f Separation (cm) 43.242 87.2 62.243 84.5 80.433 81.9 97.777 79.8 114.330 78.3 130.199 77.4 145.528 77.0 160.473 77.1 175.199 77.3 189.870 77.8 204.653 78.4 219.712 79.0 235.210 79.9 251.300 80.9 268.116 82.4 285.751 84.3 304.230 86.5 323.482 88.7 343.321 90.2 3.452 90.5 23.521

L DD MM ss 198 11 31 217 16 52 235 21 36 252 27 43 268 44 13 284 22 52 299 35 38 314 33 18 329 25 8 344 18 50 359 20 56 14 37 30 30 14 24 46 18 25 62 56 53 80 18 7 98 29 13 117 33 22 137 25 16 157 47 28 178 11 56

r Theta=360-L Theta=36 (metres) Degrees DD MM 2.598 161.808 161 48 2.650 142.719 142 43 2.713 124.640 124 38 2.778 107.538 107 32 2.842 91.263 91 15 2.897 75.619 75 37 2.941 60.406 60 24 2.971 45.445 45 26 2.985 30.581 30 34 2.982 15.686 15 41 2.963 0.651 0 39 2.928 345.375 345 22 2.879 329.760 329 45 2.820 313.693 313 41 2.756 297.052 297 3 2.690 279.698 279 41 2.631 261.513 261 30 2.583 242.444 242 26 2.554 222.579 222 34 2.546 202.209 202 12 2.562 181.801 181 48

0-L ss 29 8 24 17 47 8 22 42 52 10 4 30 36 35 7 53 47 38 44 32 4

L-f 154.950 155.038 154.927 154.685 154.407 154.182 154.066 154.082 154.220 154.444 154.696 -205.087 -204.970 -204.993 -205.168 -205.449 -205.743 -205.926 -205.900 154.339 154.678


JD

Tile No. Year Month Day

L

r

f

Separation (cm)

DD MM ss

(metres)

Degrees

DD MM ss

1P/Halley

A total of 14+13 = 27 Tiles at 80-day intervals (The true total would be 1723 Tiles at 16-day intervals. True period (days): 27563.247 Add a step ( one tile) every 3 revolutions
L 306.011 225.818 200.952 189.113 181.885 176.871 173.116 170.159 167.745 165.721 163.988 162.481 161.151 159.966 158.898 157.930 157.046 156.233 155.482 154.784 154.134 153.526 152.955 152.417 151.909 151.429 150.972 r 0.587 1.015 1.625 2.190 2.709 3.191 3.642 4.069 4.476 4.865 5.238 5.599 5.947 6.284 6.612 6.930 7.241 7.544 7.839 8.129 8.412 8.689 8.961 9.227 9.489 9.747 9.999 f Separation (cm) 0.000 81.805 107.406 68.6 119.255 60.3 126.375 54.6 131.267 50.4 134.909 47.1 137.765 44.5 140.090 42.2 142.034 40.4 143.696 38.7 145.140 37.3 146.412 36.0 147.546 34.9 148.565 33.9 149.490 32.9 150.334 32.0 151.109 31.2 151.826 30.5 152.491 29.8 153.110 29.2 153.690 28.6 154.234 28.0 154.746 27.5 155.230 27.0 155.688 26.5 156.123

JD Tile No. Year Month Day 2446471.0 0.0 1986 2 9 2446511.0 2.5 1986 3 21 2446551.0 5.0 1986 4 30 2446591.0 7.5 1986 6 9 2446631.0 10.0 1986 7 19 2446671.0 12.5 1986 8 28 2446711.0 15.0 1986 10 7 2446751.0 17.5 1986 11 16 2446791.0 20.0 1986 12 26 2446831.0 22.5 1987 2 4 2446871.0 25.0 1987 3 16 2446911.0 27.5 1987 4 25 2446951.0 30.0 1987 6 4 2446991.0 32.5 1987 7 14 2447031.0 35.0 1987 8 23 2447071.0 37.5 1987 10 2 2447111.0 40.0 1987 11 11 2447151.0 42.5 1987 12 21 2447191.0 45.0 1988 1 30 2447231.0 47.5 1988 3 10 2447271.0 50.0 1988 4 19 2447311.0 52.5 1988 5 29 2447351.0 55.0 1988 7 8 2447391.0 57.5 1988 8 17 2447431.0 60.0 1988 9 26 2447471.0 62.5 1988 11 5 2447511.0 65.0 1988 12 15

L DD MM ss 306 0 40 225 49 5 200 57 7 189 6 47 181 53 6 176 52 16 173 6 58 170 9 32 167 44 42 165 43 16 163 59 17 162 28 52 161 9 4 159 57 58 158 53 53 157 55 48 157 2 46 156 13 59 155 28 55 154 47 2 154 8 2 153 31 34 152 57 18 152 25 1 151 54 32 151 25 44 150 58 19

r Theta=360-L Theta=36 (metres) Degrees DD MM 0.587 53.989 53 59 1.015 134.182 134 10 1.625 159.048 159 2 2.190 170.887 170 53 2.709 178.115 178 6 3.191 183.129 183 7 3.642 186.884 186 53 4.069 189.841 189 50 4.476 192.255 192 15 4.865 194.279 194 16 5.238 196.012 196 0 5.599 197.519 197 31 5.947 198.849 198 50 6.284 200.034 200 2 6.612 201.102 201 6 6.930 202.070 202 4 7.241 202.954 202 57 7.544 203.767 203 46 7.839 204.518 204 31 8.129 205.216 205 12 8.412 205.866 205 51 8.689 206.474 206 28 8.961 207.045 207 2 9.227 207.583 207 34 9.489 208.091 208 5 9.747 208.571 208 34 9.999 209.028 209 1

0-L ss 20 55 53 13 54 44 2 28 18 44 43 8 56 2 7 12 14 1 5 58 58 26 42 59 28 16 41

L-f 306.011 144.013 93.546 69.858 55.510 45.604 38.207 32.394 27.655 23.687 20.292 17.341 14.739 12.420 10.333 8.440 6.712 5.124 3.656 2.293 1.024 -0.164 -1.279 -2.329 -3.321 -4.259 -5.151

1P/Halley
JD 2472994.2 2473034.2 2473074.2 2473114.2 2473154.2 2473194.2 2473234.2 2473274.2 2473314.2 2473354.2 2473394.2 2473434.2 2473474.2 2473514.2 2473554.2 2473594.2 2473634.2 2473674.2 2473714.2 2473754.2 2473794.2 2473834.2 2473874.2 2473914.2 2473954.2 2473994.2 2474034.2

13 Tiles at 80-day intervals. The 14th tile (perihelion) coincides with the zeroth one above.
Tile No. 1658.0 1660.5 1663.0 1665.5 1668.0 1670.5 1673.0 1675.5 1678.0 1680.5 1683.0 1685.5 1688.0 1690.5 1693.0 1695.5 1698.0 1700.5 1703.0 1705.5 1708.0 1710.5 1713.0 1715.5 1718.0 1720.5 1723.0 Year Month Day 2058 9 23 2058 11 2 2058 12 12 2059 1 21 2059 3 2 2059 4 11 2059 5 21 2059 6 30 2059 8 9 2059 9 18 2059 10 28 2059 12 7 2060 1 16 2060 2 25 2060 4 5 2060 5 15 2060 6 24 2060 8 3 2060 9 12 2060 10 22 2060 12 1 2061 1 10 2061 2 19 2061 3 31 2061 5 10 2061 6 19 2061 7 29 L 101.935 101.499 101.039 100.554 100.041 99.497 98.917 98.299 97.636 96.922 96.151 95.312 94.395 93.385 92.265 91.010 89.588 87.955 86.050 83.779 80.999 77.467 72.742 65.898 54.543 29.807 306.318 r 9.985 9.733 9.476 9.215 8.949 8.677 8.400 8.118 7.829 7.534 7.231 6.921 6.603 6.276 5.939 5.591 5.231 4.858 4.470 4.064 3.637 3.186 2.705 2.187 1.623 1.016 0.593 f Separation (cm) 203.965 204.403 204.864 205.352 205.868 206.416 207.000 207.624 208.294 209.015 209.796 210.647 211.578 212.605 213.746 215.028 216.483 218.157 220.116 222.458 225.335 229.004 233.935 241.109 253.046 278.788 0.000 L DD MM ss 101 56 6 101 29 56 101 2 20 100 33 14 100 2 28 99 29 49 98 55 1 98 17 56 97 38 10 96 55 19 96 9 4 95 18 43 94 23 42 93 23 6 92 15 54 91 0 36 89 35 17 87 57 18 86 2 60 83 46 44 80 59 56 77 28 1 72 44 31 65 53 53 54 32 35 29 48 25 306 19 5 r Theta=360-L Theta=36 (metres) Degrees DD MM 9.985 258.065 258 3 9.733 258.501 258 30 9.476 258.961 258 57 9.215 259.446 259 26 8.949 259.959 259 57 8.677 260.503 260 30 8.400 261.083 261 4 8.118 261.701 261 42 7.829 262.364 262 21 7.534 263.078 263 4 7.231 263.849 263 50 6.921 264.688 264 41 6.603 265.605 265 36 6.276 266.615 266 36 5.939 267.735 267 44 5.591 268.990 268 59 5.231 270.412 270 24 4.858 272.045 272 2 4.470 273.950 273 56 4.064 276.221 276 13 3.637 279.001 279 0 3.186 282.533 282 31 2.705 287.258 287 15 2.187 294.102 294 6 1.623 305.457 305 27 1.016 330.193 330 11 0.593 53.682 53 40 0-L ss 54 4 40 46 32 11 59 4 50 41 56 17 18 54 6 24 43 42 60 16 4 59 29 7 25 35 55 L-f -102.030 -102.904 -103.825 -104.798 -105.827 -106.919 -108.083 -109.325 -110.658 -112.093 -113.645 -115.335 -117.183 -119.220 -121.481 -124.018 -126.895 -130.202 -134.066 -138.679 -144.336 -151.537 -161.193 -175.211 -198.503 -248.981 306.318


JD

Tile No. Year Month Day

L

r

f

Separation (cm)

DD MM ss

(metres)

Degrees

DD MM ss

Encke

15 Tiles at 80-day intervals. True period (days): 1205.764 Miss a step ( one tile) every
L 160.951 239.058 268.022 281.967 290.521 296.510 r 0.338 0.532 0.823 1.092 1.334 1.553

JD Tile No. Year Month Day 2453003.377 0 2003 12 30 2453019.377 1 2004 1 15 2453035.377 2 2004 1 31 2453051.377 3 2004 2 16 2453067.377 4 2004 3 3 2453083.377 5 2004 3 19 6 7 8 9 2453163.377 10 2004 6 7 11 12 13 14 2453243.377 15 2004 8 26 16 17 18 19 2453323.377 20 2004 11 14 21 22 23 24 2453403.377 25 2005 2 2 26 27 28 29 2453483.377 30 2005 4 23 31 32 33 34 2453563.377 35 2005 7 12 36 37 38 39 2453643.377 40 2005 9 30 41 42 43 44 2453723.377 45 2005 12 19 46 47 48 49 2453803.377 50 2006 3 9 51 52 53 54 2453883.377 55 2006 5 28 56 57 58 59 2453963.377 60 2006 8 16 61 62 63 64 2454043.377 65 2006 11 4 66 67 68 69 2454123.377 70 2007 1 23 2454139.377 2454155.377 2454171.377 2454187.377 2454203.377 71 72 73 74 75 2007 2007 2007 2007 2007 2 2 3 3 4 8 24 12 28 13

14 revolutions f Separation (cm) 0.000 181.0 78.088 106.495 120.295 128.826 134.832 101.7

L DD MM ss 160 57 4 239 3 29 268 1 19 281 58 1 290 31 16 296 30 36

r Theta=360-L The (metres) Degrees DD 0.338 199.049 199 0.532 120.942 120 0.823 91.978 91 1.092 78.033 78 1.334 69.479 69 1.553 63.490 63

ta=36 MM 2 56 58 1 28 29

0-L ss 56 31 41 59 44 24

L-f 160.951 160.970 161.527 161.672 161.695 161.678

312.513

2.415

151.000

72.4

312

30

47

2.415

47.487

47

29

13

161.513

320.802

3.025

159.432

55.1

320

48

7

3.025

39.198

39

11

53

161.370

326.586

3.468

165.330

42.9

326

35

10

3.468

33.414

33

24

50

161.256

331.218

3.781

170.059

33.9

331

13

5

3.781

28.782

28

46

55

161.159

335.254

3.982

174.182

28.2

335

15

14

3.982

24.746

24

44

46

161.072

338.995

4.082

178.003

26.0

338

59

42

4.082

21.005

21

0

18

160.992

342.642

4.085

181.727

27.9

342

38

31

4.085

17.358

17

21

29

160.915

346.374

3.992

185.534

33.4

346

22

26

3.992

13.626

13

37

34

160.840

350.391

3.799

189.627

42.1

350

23

28

3.799

9.609

9

36

32

160.764

354.983

3.495

194.298

54.2

354

58

59

3.495

5.017

5

1

1

160.685

0.684

3.062

200.082

71.1

0

41

2

3.062

359.316

359

18

58

-199.398

8.761

2.466

208.243

99.1

8

45

40

2.466

351.239

351

14

20

-199.482

23.871 29.301 36.803 48.345 69.828 124.760

1.627 1.415 1.182 0.924 0.638 0.373

223.385 228.787 236.216 247.583 268.632 323.146

23 29 36 48 69 124

52 18 48 20 49 45

16 4 11 42 41 36

1.627 1.415 1.182 0.924 0.638 0.373

336.129 330.699 323.197 311.655 290.172 235.240

336 330 323 311 290 235

7 41 11 39 10 14

44 56 49 18 19 24

-199.514 -1 -1 -1 -1 -1 9 9 9 9 9 9 9 9 8 8 .4 .4 .2 .8 .3 8 1 3 0 8 6 3 8 4 6