196 
The N.Z. Journal of Science and Technology. 
[May 
Notes on the Calculation of an Ephemeris of a Comet. 
These notes refer to the calculation of an ephemeris of Halley’s Comet, 
and were written in March, 1910, and they are published here as practical 
experience has shown that the method described leads to a considerable 
reduction in labour over the usual procedure. The method is a general 
one, and may be applied to the calculation of an ephemeris of a comet 
and an asteroid. 
The usual method in calculating the positions of a comet in its orbit 
is to do so for equal intervals of time—say, for every day, or for every 
four or eight days — but this necessitates the solution of Kepler’s equation 
for the eccentric anomaly, which is a troublesome operation. Kepler’s 
equation is u° — c° sin u = nt, where u° is the eccentric anomaly in 
degrees, c° = the ellipticity of the orbit in degrees, n — the mean daily 
sidereal motion of the comet in degrees, t = the time in days from the 
perihelion passage ; and it will be seen that the equation is transcendental 
in u. 
In many cases it is not necessary to form the ephemeris at equal 
intervals of time. In these cases it is much simpler to invert the pro¬ 
cedure and take equal intervals of u —say, u = 1°, 2°, 3°, &c.—then nt 
is readily determined from Kepler’s equation. 
Example. —Halley’s Comet : This comet describes an ellipse round the 
Sun, with the Sun in one focus of the ellipse. The elements of the orbit 
given in the Monthly Notices * of the Royal Astronomical Society are— 
Time of perihelion passage 
Perihelion minus node 
Longitude of node . . 
Inclination 
Eccentricity 
Semi-axis major 
Perihelion distance . . 
T 1910, April 19-65, G.M.T. 
co 111° 42' 16" | 
Q> 57° 16' 12" j- 1910-0 
i 162° 12' 42") 
e 0-967281 
log. a = 1-2539958 a = 17-94716 
log. q = 9-7687858 q = 0-5871996 
From these elements it is required to determine t and the polar co¬ 
ordinates r and v of the comet for given values of the eccentric anomaly u. 
Since c°-ex 57°-2958 = 55°-421 
^ h 3548-18761 
and n = — = n qiF ~ = 46 ' 6673 
a* 76-0315 
Kepler’s equation becomes 
a° — 55°-421 sin u = nt 
r 
77-142 
t 
77°-142 
The radius vector, r, of the comet is obtained from 
r = a (1 — e cos u) 
.= q -\- ae vers u 
— 0-5872 -J- 17-3599 vers u. 
The comet’s true anomaly, v, is obtained from 
tan J v — 
tan \u 
= 7-75413 tan \u. 
* Vol. 70, p. 3, 1909-10. 
