# from observations of the distances between Venus and the center of the Sun

J.-E. Arlot, P. Rocher (IMCCE)                                                                    jeudi 13 janvier 2005

## I.Introduction

In this sheet we will give a simple formula allowing to calculate an approximation of the solar parallax (and thus the Astronomical Unit or AU) starting from the observation of the distance between Venus and the center of the Sun. Contrary to the case of the contacts for which we need two observations from distant sites, we will use only one observation with the method known as “linear” which compares the observed value of this distance with the calculated theoretical one. The difference between these two values will enable us to correct the value of the theoretical solar parallax used in calculations. It is in fact an iterative method but we will limit ourselves to the last iteration. The goal – educational- of this calculation, is to show that the value of the calculated AU depends on the quality of the measurement of distance and also of the place of the observation. It will be interesting to make several calculations starting from observations made at various moments from the same place or distant places to compare the values obtained and to average them.

Pre-calculated coefficients

To obtain this formula, we pre-calculated parameters A, B, C and D which do not depend on the position of the observers but which account for the motions of Venus and the Earth around the Sun as well as the rotational movement of the Earth around its axis. Attention, they depend on time and were tabulated consequently. The parameter dD/dt represents the instantaneous variation of the distance from the Venus center to the center of the Sun. One can thus calculate the theoretical value of the measured distance and reduce the observation (by determining a new value of the parallax - or of the AU-) by taking account of the fact that the observers, the centers of Venus, the Earth and the Sun are not in the same plan and by using the relationship between the distances of the Earth and Venus to the Sun given by the third law of Kepler. One will find these tables at the end of this sheet. Measured value  d  between Venus and the centre of the Sun

## II Observation of the distance of Venus to the centre of the Sun

You must know the following quantities :

1.      latitude and longitude of the site of observation (l, j)

2.      the distance from the centre of Venus to the centre of the Sun (cf. figure above) in minutes of degree

3.      the date of the observation in Universal Time (UTC) within one second of time

Formulae to be applied are as follows :

- transformation of the geographic latitude j of the site into geocentric latitude j‘ : where h is the elevation of the site, R the terrestrial radius (taken as 6378,140 km) and  f the flatness of the Earth (taken as 1/298.257)

and where cos u and sin u are deduced from tan u

- calculation of the approximate value Dp of the theoretical distance Dc Venus-Sun using the pre-calculated coefficients A, B, C and D provided at the end of the present sheet (F2)

where p0 =  8,79414" and where r cos j and r sin j are calculated as above

Attention: if you wish to make a more precise calculation, you need the accurate value of the theoretical distance Dc . For that purpose, you should use an ephemerides server such as the one of  http://www.imcce.fr, calculating the positions of the center of the Sun and the center of Venus at the date of the image and deducing the distance Dc from these positions.

- calculation of the new solar parallax, i.e. the new value of the AU :

In order to reduce this observation we will use the next formula: (F3)

where D0 corresponds to our measure d  given in seconds of degree and A, B, C, dD/dt given at the end of the present sheet

The solar parallax deduced from our observation becomes : p0 + dp0 i.e. 8,79414 + dp0

In order to make the calculation, please fill in the following form “distance”.

Calculating form « distance » :

Calculation of the solar  parallax thanks to the observation of the distance between Venus and the centre of the Sun

(in order to get a good result, you must average several calculations using different observations)

geographic longitude l =

geographic latitude j =

observed distance (Venus – Sun) : d =                                                         in minutes of degree

date of the observation in Universal Time (UTC) =

- transformation of the geoghraphic latitude into geocentric latitude :

(1) geographic latitude j =

(2) Calculate the cosine of j : cos (j ) =

(3) Calculate the sine of j : sin (j) =

(4) Elevation of the site of observation h =                                         in meters

(5) Calculate tan u = (1 – 1/298,257) x tan (line (1)) =

(6) Calculate the angle u (which remains between -90° and +90°) = arctan (line(5)) =

(7) Calculate the cosine of u = cos (line(6)) =

(8) Calculate the sine of u = sin (line(6)) =

(9) Calculate (r cos j’) = cos u + (h/6378140) cos j

= line (7) + [(line (4)/6378140) x line (2)] =

(10) Calculate (r sin j’) = (1 – 1/298,257) x sin u + ( h / 6378140 ) sin j

= (1 – 1/298,257) x line (8) + [(line(4) / 6378140) x line (3)]  =

- calculation of the theoretical distance « Venus-Sun » (if you prefer to use an ephemerides server, go to line 20):

(11) Calculate the cosine of l : cos l =

(12) Calculate the sine of l : sin l =

(13) Calculate (r cos jcos l) = line (9) x line (11) =

(14) Calculate (r cos j’ sin l) = line (9) x line (12) =

(15) Détermine A from the table at the end of the sheet : A =

(16) Détermine B from the table at the end of the sheet  : B =

(17) Détermine C from the table at the end of the sheet  : C =

(18) Calculate the coefficient of p0  in formula (F2) :

= line (15) x line (13) + line (16) x line (14) +line (17) x line (10) =

(19) Détermine D from the table at the end of the sheet  : D =

(20) Calculate the second member of formula (F2) i.e. the value of Dtheoretical :

= line (19) + 8.79414 x line (18) =

- calculation of the new solar parallax :

(21) Your measure of dobserved in seconds of degree =

(22) Difference « observed – calculated » = line (21) – line (20) =

(23) Correction of the parallax deduced from the observation = line (22) / line (18) =

(24) New value of the solar parallax p0 = 8.79414 + line (25) =

- calculation of the astronomical unit :

(25) Calculate the astronomical unit :

AU = line (24) x (206265.806247 / 6 378.1363) =

You may remake the same calculation using other observed distances Venus-Sun either made from the same site, or made from another site of observation, and make an average of all the results.

Remind you that the proposed method of calculation is an approximation and, in order to get a more accurate result, you must use more complete formulae (as given in sheets n°4 and 5).  Even if your observation was perfect, you will get only an approximation of the AU.

Pre-calculated coefficients to be used in the above formulae :

The two following tables provide the values of the parameters A, B, C, dD/dt, D depending on time every 5 minutes de temps during the transit. The three first parameters are without dimension, dD/dt is in seconds of degree / minute of time, D is in minutes of degree.

 T UTC A B C dD/dt "/min D ' 5h05m 2.2768 -0.1279 0.9656 -3.1409 16.6791 5h10m 2.2682 -0.0645 0.9919 -3.1085 16.4202 5h15m 2.2571 -0.0011 1.0189 -3.0746 16.1641 5h20m 2.2436 0.0622 1.0465 -3.0389 15.9108 5h25m 2.2275 0.1254 1.0749 -3.0016 15.6606 5h30m 2.2088 0.1882 1.1039 -2.9624 15.4135 5h35m 2.1874 0.2507 1.1336 -2.9214 15.1697 5h40m 2.1634 0.3128 1.1640 -2.8783 14.9295 5h45m 2.1365 0.3742 1.1951 -2.8331 14.6928 5h50m 2.1067 0.4350 1.2269 -2.7857 14.4601 5h55m 2.0741 0.4950 1.2594 -2.7359 14.2313 6h00m 2.0385 0.5540 1.2925 -2.6837 14.0068 6h05m 1.9998 0.6119 1.3263 -2.6290 13.7867 6h10m 1.9581 0.6686 1.3607 -2.5716 13.5712 6h15m 1.9133 0.7239 1.3957 -2.5115 13.3607 6h20m 1.8653 0.7776 1.4313 -2.4484 13.1552 6h25m 1.8142 0.8297 1.4674 -2.3824 12.9550 6h30m 1.7598 0.8797 1.5040 -2.3133 12.7605 6h35m 1.7022 0.9277 1.5410 -2.2411 12.5718 6h40m 1.6414 0.9734 1.5783 -2.1655 12.3892 6h45m 1.5774 1.0166 1.6159 -2.0866 12.2131 6h50m 1.5103 1.0570 1.6537 -2.0043 12.0436 6h55m 1.4400 1.0945 1.6916 -1.9185 11.8811 7h00m 1.3667 1.1288 1.7294 -1.8291 11.7258 7h05m 1.2904 1.1598 1.7671 -1.7363 11.5781 7h10m 1.2113 1.1871 1.8045 -1.6399 11.4382 7h15m 1.1295 1.2107 1.8415 -1.5401 11.3064 7h20m 1.0452 1.2302 1.8780 -1.4368 11.1831 7h25m 0.9586 1.2455 1.9138 -1.3301 11.0684 7h30m 0.8699 1.2564 1.9488 -1.2203 10.9628 7h35m 0.7793 1.2628 1.9827 -1.1073 10.8663 7h40m 0.6872 1.2644 2.0154 -0.9914 10.7794 7h45m 0.5938 1.2612 2.0469 -0.8729 10.7021 7h50m 0.4995 1.2531 2.0768 -0.7520 10.6348 7h55m 0.4046 1.2400 2.1051 -0.6289 10.5776 8h00m 0.3095 1.2220 2.1316 -0.5039 10.5306 8h05m 0.2147 1.1989 2.1563 -0.3775 10.4941 8h10m 0.1204 1.1709 2.1789 -0.2499 10.4681 8h15m 0.0271 1.1380 2.1993 -0.1216 10.4527 8h20m -0.0647 1.1004 2.2176 0.0071 10.4480 8h25m -0.1548 1.0583 2.2336 0.1357 10.4539 8h30m -0.2428 1.0118 2.2473 0.2640 10.4704 8h35m -0.3282 0.9611 2.2586 0.3914 10.4976 8h40m -0.4108 0.9065 2.2677 0.5177 10.5353 8h45m -0.4901 0.8484 2.2744 0.6424 10.5833

 T UTC A B C dD/dt "/min D ' 8h50m -0.5661 0.7869 2.2789 0.7653 10.6417 8h55m -0.6383 0.7224 2.2811 0.8860 10.7101 9h00m -0.7067 0.6552 2.2813 1.0043 10.7884 9h05m -0.7709 0.5857 2.2794 1.1198 10.8764 9h10m -0.8310 0.5141 2.2755 1.2324 10.9739 9h15m -0.8867 0.4409 2.2699 1.3419 11.0806 9h20m -0.9380 0.3663 2.2625 1.4482 11.1962 9h25m -0.9849 0.2905 2.2535 1.5511 11.3204 9h30m -1.0273 0.2140 2.2432 1.6506 11.4531 9h35m -1.0653 0.1370 2.2314 1.7466 11.5938 9h40m -1.0989 0.0598 2.2185 1.8390 11.7424 9h45m -1.1281 -0.0175 2.2045 1.9279 11.8985 9h50m -1.1531 -0.0945 2.1896 2.0133 12.0618 9h55m -1.1740 -0.1711 2.1738 2.0953 12.2320 10h00m -1.1908 -0.2471 2.1573 2.1738 12.4089 10h05m -1.2036 -0.3222 2.1402 2.2490 12.5921 10h10m -1.2127 -0.3964 2.1225 2.3209 12.7814 10h15m -1.2181 -0.4696 2.1044 2.3896 12.9766 10h20m -1.2200 -0.5414 2.0860 2.4553 13.1773 10h25m -1.2184 -0.6120 2.0673 2.5180 13.3833 10h30m -1.2137 -0.6811 2.0484 2.5778 13.5945 10h35m -1.2058 -0.7488 2.0293 2.6349 13.8104 10h40m -1.1950 -0.8148 2.0102 2.6893 14.0310 10h45m -1.1813 -0.8792 1.9911 2.7412 14.2560 10h50m -1.1649 -0.9419 1.9719 2.7907 14.4852 10h55m -1.1460 -1.0028 1.9529 2.8379 14.7183 11h00m -1.1247 -1.0620 1.9339 2.8828 14.9553 11h05m -1.1010 -1.1194 1.9151 2.9257 15.1960 11h10m -1.0752 -1.1749 1.8964 2.9665 15.4401 11h15m -1.0473 -1.2286 1.8779 3.0055 15.6875 11h20m -1.0175 -1.2804 1.8596 3.0426 15.9380 11h25m -0.9858 -1.3303 1.8415 3.0780 16.1916 11h30m -0.9524 -1.3783 1.8237 3.1118 16.4480 11h35m -0.9174 -1.4244 1.8061 3.1440 16.7072