Note 67 — PART II. MOTION OF COMETS THROUGH THE SYSTEM.
Reading edition
It is unnecessary to detail the whole calculations in these preliminary processes. It will be sufficient to show the computations for the first observation.
h ′ ″ h ′ ″
Sidereal time at Paramatta, = 4 19 19 ☉ M. AR at app. noon, 10 51 15·5
Longitude of Paramatta, = 10 4 14·5 Reduct. to sidereal time, 1 1.4
Sidereal time at Greenwich, = 18 15 4·5 ☉ M. AR at mean noon, = 10 52 16·7
10 52 16·9 h ′ ″
7 22 47·6 for 7 subtract 1 8·8
1 12·5 22′ .......... 3·6
Mean Solar time at Greenwich, = 7 21 35.1 47″ .......... 1
1 12·5
To compute the longitude and latitude from the right ascension and declination, we have the following formulæ given by Gauss, Theoria Motuum Cœlestium (64.); where is the declination, a the right ascension, e the obliquity of the ecliptic, the longitude, and the latitude.
° ′ ″ ° ′ ″
log tan δ = -8 49 0 = 9·1906287 - log tan (ξ - e) = -32 19 18 = 9·8012000 -
log sin a = 84 13 47 = 9·9977938 log sin λ = 83 15 42 = 9·9969896
tan ξ = -8 51 39 = 9·1928349 - tan β = -32 8 33 = 9·7981896 -
-e = -23 27 39
cos (ξ - e) = -32 19 18 = 9·9268874 log cos δ = -8 49 0 = 9·9948377
log tan a = 84 13 47 = 10·9954541 log cos a = 84 13 47 = 9·0023397
Numerator,........ 10·9223415 Numerator,........ 8·9971774
log cos ξ = -8 51 39 = 9·9947857 log cos λ = 83 15 42 = 9·0694339
tan λ = 83 15 42 = 10·9275558 + cos β = -32 8 33 = 9·9277435
To find now the Sun’s longitude, we must take it from the ephemeris for apparent time, by adding the equation of time to the mean time at Greenwich, which makes it, Sept. 4. at 7h 22′ 42″:
° ′ ″
☉ longitude on 4th Sept. = 161 22 8
For 7h 22′ 42″ add 17 54
161 40 2
As the radius vector given in the Nautical Almanack is not always correct, we take it from Schumacher’s Hülfstafeln, in which the computations are made for mean time, and at the meridian of Altona, 39′ 47″ East of Greenwich. To find the radius vector for the first observation, we must therefore take it from the ephemeris on Sept. 4·33427. On the 4th, we find the log. 10·0033772, on the following day 10·0032654. The difference for one day on the 4th being thus ·0001118, we find the difference for the interval to be ·00000373, which being subtracted from the logarithm on the 4th, gives 10·0033398 for the log. of radius vector at the first observation.