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Theses and Dissertations
1912
Computation of a comet's orbit
Orley Hosmer Truman
State University of Iowa
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Truman, Orley Hosmer. "Computation of a comet's orbit." MS (Master of Science) thesis, State University of Iowa, 1912.
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THE COMPUTATION OF A COMMET'S ORBIT.
Thesis presented in part fulfillment of the requirements
for the degree of Master of Science, at the State University
of Iowa, June 12, 1912, by O. N. Truman.
-
1
-
The Computation.of a Comet’s Orbit.
The computation of a comet’s orbit is described by
both Ilewton and. Laplace as "a problem of great difficulty,"
and. one to which they had been forced to devote muc5i reflection
before reaching a solution.
But while it is true that tbedis-
covery of the methods of work is a problem which might well
have taxed their geniu3, the methods once found are not hard
to understand.
i will, therefore, give in the following pages a
course of proceedure which i think is within the comprehension
of one with no further mathematics than analytical geometry, and
no greater knowledge of astronomy than is contained in an ordinary
elementary course.
In such an explanation; of course, the building
up of the formulae from their base in the law of gravitation
must be omitted, but this omission would have to be made on account
of lack of space, anyhow.
So if the reader is willing to-assume
the foundation stones as laid, he may, by the means I shall set
forth,build up the superstructure, and compute the orbit of a
comet very satisfactoryKy.
The proceedure which I shall use is a combination of
the work of Olbers, Euler, and Lambert, which I have modified
in a few places for the sake of greater clearness and ease of
understanding, even at the expense, perhaps, of lightly more
lengthy calculation.
I think that it is well for a beginner,
especially ir he is a geometrically- and not an algebraicallyminded person, to have a method, every step of which, as far
as possible, can be represented by something in a drawing and
which he can thus preserve in terms of visual images.
-
2
As will be explained, in due course, observations
of a comet are taken with reference to movable lines, and
therefore when a second and third observation are taken,
the lines of reference will be different from what they
were at the time of the first.
So before any progress
can be made in computing an orbit a single fixed system of
reference lines must be chosen anci all the observations
reduced to it.
The description of how to do this will, therefore,
be first taken up in this paper; that accomplished, I shall
assume the nature of the paths 01 comets in space and the
lav/s of their motion
in those paths, and on that as a basis
show how to find the positions end dimensions of the orbits.
I will illustrate each step, as closely as practible,
by actual numerical computation as I go along, as it seems to
me easier for the beginner to comprehend computational
methods when thus set forth than when a great mass of for­
mulae are first deduced and finally applied to a numerical
problem.
To illustrate the computation, I will work out the
orbit of one 01 Brook’s comets. Comet, 3, 1911 as I saw this
one myself and it is thus of special interest to me.
discovered by Mr.
It was
rt. Brooks, at Geneva, Hew York, on July
20, 1911, and at the time was only of about the 10th or 11th
magnitude, and approximately opposite the sun.
It slowly
increased in brightness, traveled north until in the latter
-
3
-
part of August it was less than 4C° from the north pole, and
then swept southward towards the sun.
In the latter part of
October, when I saw it, it was a very satisfactory appearing
comet in the eastern shy, with a straight tail at least 20°
long, and a head sometimes of the 3d, sometimes of the 2d,
magnitude.
By that time it was rapidly receding from the
earth, and soon passed from the view of the northern hemi­
sphere, drifting away in its voyage out into space.
Vith these pre] iminary remarks the actual astronomical
work may be taken up.
The place in the sky of a comet or other heavenly
body (to refresh the reader's memory by a resume of the usual
elementary treatment of the subject) is established by re­
ference to two great circles on the‘surface of the celestial
sphere--the celestial equator, and the ec>iptic.
(Por the
sake of brevity "equator" in the ensuing pages will mean
"celestial equator" unless otherwise stated.)
These intersect
at two opposite points, ana that one of those points where
the sun in his annual motion comes north of the equator is
the vernal equinox.
7/e can now in our imagination pa3S through a commet
a meridian or great circle perpendicular to the equator and
therefore passing through the poles of the equator.
Then
the angular distance of the comct from the equator, measured
along this mediaian, is its declination, designated by
<T
Thus declination may be either north or south, the former
being indicated by 7. or +, the latter by 3 or -.
case it may have any value between 0° and 90°,
In either
.
_ 4
The angle between the vernal equinox ana the
meridan we have drawn, from the vernal equinox along the
equator towards the east to the meridan, is the light ascen­
sion, denoted by
running from
(«
oC
It is usually specified in time,
A
j
0 0 0 up to 24%&, but for trigonmetrical cal­
culation must be converted into angular units, e-
oaoil-g
aooor-'
i'i1 ish e& infro-■aaa^ular- vmi£ft, a thing easily accomplished
A
°
I to
by reason of the fact that 24 is equivalent to 360 , 1
15*
, I*1 to 151 , and I5 to lb".
All this seems very simple, end would be as
simple as it seems if the celestial equator and ecliptic
were actual visible great circles traced upon the surface
of the sky, and remaining fixed in their places from cen­
tury to century.
To determine the place of a newly dis­
covered comet, say, it would only be necessary either to
measure its angular distance from the equator and the ang­
ular distance of its meridian from the meridian passing
through the vernal equinox, or to measure the distance of
the comet from two stars whose position was known, or to
measure the distance and direction of the comet from one
star whose position was known.
The first oi these pro-
ceedures, which would be carried out by a transit instru­
ment, would give «c and
S
directly; and
either of the two
latter, for whose execution a micrometer would be used,
would give< and S after a little calculation in spherical
trigonometry.
- 5 -
But as a matter of fact the celestial equator
and ecliptic are neither visible nor fixed, end the deter­
mination of the data by which the rt. ascen. and dec. of
a heavenly body may be found has been, and still is, a
problem of the centuries.
All that can be said at present
is that we have approximate methods; only the appearance
or non-appearance of cumulative errors in the course of
time will show how close these approximations are.
The reasons for this state of affairs are var­
ious and to give even a passable explanation of them, ana
of the theories by which the many changes of position are
taken into account, would require more than the entire
length of this paper.
I must content myself, then,with referring the
curious reader to works on theoretical astronomy for such
inform- tion, and confine myself to the bare facts needed
in the work before me.
It has been found, then, in the theories above
hinted at, that the ecliptic and equator both have small
motions on the surface of the celestial sphere, and these
have been broken up into two classes--secular, or those
going steadily in one direction over as great a range of
time as man’s observation comprehends, and complex perioaic motions, some going through their changes in about
19 years, some in less.
The effects of the former motions
-
6
-
on the place ox a star are called precession , as are also
the motions themselves; the latter motions or their effects
are called nutatIon.
By taking both of these into account
by means of formulae which have been produced for the pur­
pose^, the position of the ecliptic and equator at any time
may be found with reference to their positions at any other
time, and as a consequence if the position of a star with
reference to the equator and ecliptic of any time are known,
its position may be found with reference to the equator
and ecliptic of any other time.
To make these facts more apparent and at the
same time lead to further statements, an illustration
may be helpful.
If a fly sat on the wheel of a wagon
which was being driven along the road, the most suitable
way to designate the position of that fly with reference
to the road would be to state the position of the wagon,
With respect to fixed objects, a mile post in the road,sa^,
and then that of the fly with respect to the wagon. The
motion of the wagon, being continuous, would correspond
to precession; that of the wheel, which would move the
fly forward-and back, and up and down, and thus would be
periodic, would correspond to ‘nutation. That part of
the motion of the fly due to the former cause along might
be celled the average oT mean motion, and
his position
at any time as it v/ould have been had no other cause acted
his mean position. If to these were combined bis motion
due to the motion of the wheel, and his change of position
- 7 -
clue to that motion, respectively,we should, obtain his
true motion and. his true position at any instant.
In just the same way in astronomy, we H a v e
the true equator and ecliptic at any date, meaning the
actual equator and ecliptic, and the mean equator and
ecliptic at that date.
r:o get from the former to the
latter we need only substract the pert of their motions
due to nutation, and to go from-the latter to the former
we must add. that part.
It has been found most suitable to calculate the
position of a comet's orbit with respect to the equator
and ecliptic at the' beginning of the !,3esselian fictici­
ous year," and this time is defined as the instant when
the "mean" longitude of the sun, referred to the mean
ecliptic and mean equinox of the instant, minus 20".47,
is equal to 260?
The time of beginning or this fictitious year
may be found from the llautical Almanac; for 1911, the
year with which we are concerned, it is given on P. 504
as Jan 07764, Washington time.
This is in astronomical
time, and means 0**.236 before noon on Jan. 1 at Washington,
ordinary civil time.
In view of all that has appeared thus far, then,
the true place ox a star at any time during the year is
its mean place at the beginning ox that Besselian year,
plus the change aue to precession in the intervening'
time, plus the correction due to nutation for the date
in question.
'But there is a third disturbing influence,
- 8 -
aberration of light, which must be remarked upon before
the complete reauction formulae can be noticed.
This phenomenon of aberration depenas upon
the fact that light aoes not travel with infinite vel­
ocity, but only with very high velocity, ana hence as the
earth rushes through space it runs to meet the light rays,
so to speak, ana they appear to us to come from a airection slightly uinerent from their true one, in a similar
way as vertically falling raindrops appear to be falling
in a slanting airection to one who is running through
them.
It has been touna by complex measurement ana cal­
culation that light rays falling upon the earth in a line
perpendicular to the line of the earth's motion appear
to us to come from a direction 20.,r4 7 (according to the
Paris Conference ) aifferent from their true one, ena of
course when they ao not iall perpenaicularly to the line
of motion, the variation, or aberration, is less, becoming
zero for rays coming irom exactly the airection towara
which, or irom which, we are going.
Thus is the value of
the 20. ''47 noticea in the remarks upon the Besselian
year
explained; not, however, the reason for its introduction
w hicii
in that connect ion
must be passed over.
This aberration of course afreets the apparent
position of every star, except, indeed, those exactly in
-
9
the line of the earth's motion, ana its influence on right
ascene/ion ana aeclinr tion can he found by spherical trig­
onometry.
The calculation, however, like that of presession
ana nutation, is too involved to he gone into here, ana
so we must content ourselves with complete and final
results alone.
•
To cotne at" once, t o t h e s e ,
then, it is possible to unite the formulae for
nutation, precession ana aberration, ana sepfrate out cer­
tain quantities which depend only upon the time, ana others
which depend only upon the position of the star in question.
ata.rs
The former quantities are thus the same for all^at any in­
stant and can he calculated end tabulated in the Nautical
Almanac for every day in each year.
These combined with the
quantities dependant upon the position of the star, which
f
we must find for ourselves, give the correction to be added
to the mean coorainates of the star at the beginning of
the Besselian year, to reduce then to those at any other
time within that year.
Conversely, of course, if the latter
coordinates are known, the former can be found by substracting the correction.
There are two systems f6 formulae in use for these
purposes, the Besselian and the Independent, ana both are
found, together with the tables for use with them, in the
llautical Almanac.
Inasmuch as the Independant star numbers
are most convenient for our purposes, these alone will be
explained.
On p. 526 of the Aautical Almanac for 1911 are
found these two formulae
-
10
for each day in the year,
-
'f is the time from the
■beginning of the Besselian year, or the time from 1911.0,
to use the designation usually employed in this connection.
The only place where
terns f
and
of a"fixed"
appears in our formulae is in the
which have to do with the proper motion
star, and we will not need these terms at all,
s o /f is of no consequence to us, and the terms containing
it will be dropped.
f, + f
The f in the first formula is
„ 0 l the tables; oc, and
the star at 1911.0, and oc
and a
are vttQ coordinates of
are its coordinates at
the date required.
The task with which we are concerned is,however,
the reverse of that contemplated in the formulae; there
e are given oc, ana
problem we have
, to find oc and S ; in the present
and cT , the observed place® of the
comet referred to the true equinox and equator of the date,
and it would seem as if the only way to obtain formulae
for e>ct and S, in terms of these would be to solve the equations
for
and
S9 . This difficult course is made unnecessary
by the fact that «c -
, and /- o9 are both very small quant­
ities, and so we may write oc for op, and / for <£ in the trig­
onometric parts of the formulae with no material error.
- 11 -
Doing this, and transposing, we find for the formulae we
wish (omitting the terms in -f, which we have no use for)
With these equations at hand we can now select
observations of the comet, referred each to the equator
and equinox of its own date, and reduce them to the mean
equator and equinox of 1911.C
^or reasons which will
appear later three observations must be taken, and the
intervals of time between the 1st and 2d and the 2d and
3d should be as nearly equal as possible. The best length
of these intervals cannot be presrcibed, but in the ex­
ample, as will be seen, it is about 5 days.
-
12
-
- 14 Certain tilings in this list require explanation.
"'irst of all, the 2d and 3d of these observations were
taken by means of a micrometer, with which the difference
of right ascensions and difference of declinations of the
comet and some nearby star were measured.
These differences
are given in the columns marked A0C and
; the quantities
in those columns aie
a
S
to be added, (algebraically) to the
coordinates of the cor. parison star in order to get 1he
coordinates of the comet.
The distinguishing number which the observer has
given to each comparison star appears in the column marked
and by looking up the same number in his list of com­
parison stars we find the mean place of that star for
1911.0, the beginning o± the Bessel!m star year.
In the
same list he gives in the column marked "Authority," the
designation of the catalogue from whence the place was
taken.
Tow returning to the observation list, we find
in the column TTHed to ap. plT! the combined
corrections
for precession, nutation, and aberration which must be
added to the mean coordinates of each star to get the
apparent coordinates at the time of observation.
Fin­
ally, these app. coordinates plus their4«C and A &
,
respectively, will give the apparent place of the comet
at the time of observation.
The work of performing the
additions ourselves is save'd us, for the observer has
already placed the results in the columns rT oC
and
" c) ejvfo "
meaning, of course, apparent ri'v '
ascension and apparent- declination.
«
- 15 -
the same as given in the respective columns.
V
,
A j these
refer to a correction of the observations made necessary
by a cause not yet touched upon, i. e., parallax.
The
observer, it will be remembered, saw the comet not from
the center of the earth, but from a point on the surface,
and it therefore appeared to him in a slightly different
position from what is would if he had seen it from the
earth’s center.
The influence of this displacement on the
two coordinates of the comet depends upon the altitude and
azimuth of the comet when observed, the position of the
observer on the earth, and the distance of the comet. All
these things but the lest are
known to the observer, and
by special tables computed by him l o r hi
ticular
observatory he can make the correction much easier than we.
Thus he he.s calculated it as far as he could, and gives
in the
column, the logarithm of the proauot of the
parallax correction multiplied by the distance of the comet,
the latter expressed in units of the radius of the earth’s
orbit.
Co if in any way, say from a previously computed
orbit.we knew the aistance at the time of an observation,
all we should have to do woula be to sub; tract its log-
-
arithm from
16
"y&jjoA
-
"
and we should have the logarithm
oi the parallax correction to be applied to eacn coorainate,
thecxl correction in seconds ox time, and 6 correction in
seconds of arc.
These corrections should be taken as -
or +, according as the letter n appears after their log­
arithms or not.
in jny computation I shall not use them,
as 1 shall compute the orbit without previous knowledge
of the distance of the comet.
The only remaining puzzling column is that
marked TTComp,r in English and Trench, or "Yerg" in Ger­
man.
Two numbers are usually contained in it; for in­
stance in the 2d, observation, there is"lo, 10'T. This
means that the observer made 15 measurements of AoC
and 10 of A
S.
,
The only usefulness of these facts is
sometimes in ellaborete discussions of the relative
value of observations, end even here they are of doubt­
ful worth.
I shall in this work have nothing more to
do with them.
Y7ith the exception of one remaining preliminary
step we are now ready to take the
"*<app!T and
B S app."
which are in each case referred to the equator snd
equinox of the date or observation, and distorted by
parallax and aberration, and use the formula heretoxore
noticed to reduce them to the places referred to the mean
equator and equinox of 1911.0.
As I have remarked, the
parallax error must be allowed to remain uncorrected.
The preliminary step mentioned is the reduction
17
of the times of observation to Grenwich time, end. to
decimal parts or a day, instead of hra., rnin.; and sec.
To cio this we must look up the longitudes of the places
from Greenwich in the nautical Almanac, pp. 520 to t>24,add them (Algebraically) to the times of
observation,
and reduce the result to decimals of a day.
Y/e hove for
the three observations:
Tables are given in some astronomical works by which the
reduction to decimals of a day is easily made.
13y anjr
means, however, we get
The necessity of making these reductions in this
place comes from the fact that they are now to be used in
interpolating for 0, H, i, g, and h, which are to be used
in making the reductions to mean places for 1911.0.
A little inconvenience still arises trom the
fact that th $
II i g h ,quant it ies, are given in the
tables for 'Washington mean midnight, not for Greenwich
-
mean noon.
18
-
" idnight, July 21 " is midnight, July 21-22.
So from each observation we must substract 0.21414 day.,
the longitude of Washington expressed in days,'and then
0.5
day, to reauce the tine to a day which begins on a
midnight instead of the previous noon.
Both sub. tractions
are of course performed at once by suhstracting C.71414
day and we have
- 19 -
- 20 -
- 21 Thus we can easily find the values of the
and
<S by substraction.
If the values ox the corrections obtained for
the 2d observation are compared, with those given by the
observer.under "Red. to Appl."» it is seen that they agree
exactly, and this of course might have been expected from
the beginning, for the corrections are small and slowly
changing quanitites, and their variation in passing over
the few minutes of cxC between the star and. comet would
be insignificant as rar as this hind of work is concerned.
This might of course be proven more definitely
by differentiation 01 the expressions lor oCe-cC-
S.-J
with respect too: and
d
and
d
, respectively, and
substituting numerical values in the resulting derivatives,
’but this is not worth the trouble.
In most cases it will
be found, as above, that the corrections given for star
positions, if they are given r.t all, are fully sufficient
for the comet, but it is well in any problem to work out
one or two corrections by the formula, as has been done
rs&l 1y
above, to make sure that the discrepancy is^insignicicant.
In this case it manitestedly is, and so for the 3d ob­
servation I shall izse the values +• 2*24
and 44,"1 with
no further question, remembering, however, that these are
to he substracted from < and
to gevoC* end £0 , not added.
-
22
-
Since in the method, of computing an orbit which
I shall use the plane of the ecliptic ana not the plane
of the equator is to hs taken as a fundamental plane of
reference, these right ascensions and declinations must
now he changed into latitudes and longitudes.
The
standard -^Oimulae "by vvhich this is accomplished are found
- 23 -
in many estronomioal works,
'.’hey depend for their demon­
stration on spherical trigonometry, and I will simply give
them without deriving them.
If o£ and / are the right ascension and dec­
lination of a star, 6
the ecliptic, and
the inclination of the equator to
X and ^3 the longitude and latitude,
respectively, which it is desired to find, we have first
to find
-
24 -
-
2
5
-
- 26 -
Having now obtained these latitudes and long­
itudes, we can go no further without some knowledge of
the nature of the paths of comets in space, and the form­
ulae by which their motions in those paths can he predicted.
In this matter again I must ask the reader to take the
formulae on faith, for while their derivation from
the lav/s of motion is most interesting, it would occupt
a prohibitive amount of space.
it is shown, then, in celestial mechanics that
if we neglect the comparatively insignificant attraction
of the planets, and pay attention only to that of the
sun, every comet moves about the sim in either an ellipse,
a parabola, or a hyperbola, having the sun at the locus,
if the first, the
comet would return in its path at re­
gular intervals indefinately; if either of the other two,
and
we should see it but onceyit would go out into space,
never to return,
^hc kina of curve depends only upon
the velocity of the comet, and the relation is expressed
in the following manner.
Let
gravitation"
be the value which thetfconstant of
has if we take the sun's mass as the unit
of mass, the second as the unit of time, and the semi­
major axis of the earth's orbit as the unit of length.
This value, which proves to be
0 ! 7 I. D
o j
is generally called the "Gaussian constant of gravi­
tation."
It is such a value that if
,and.
are two
masses expressed in units of the sun’s mass, and they
-
27
-
are separated, a distanceyi- expressed in units of the
Sun's distance from the earth, the force pulling them
f7~
A *
/- w
towards each other will be f - 'ft
--- —
-pO T Q P
s i
Since acceleration
--- -—
, if
—
mass
' were let go it .}
v;ould fall toward m, with an acceleration at first —~—i
z
units per sec, per seo„ ana if raL were let go it would
toward m
with an acceleration at first =
_
It is to he noticed that I have here defined the force
in the only way that force can be aefined--by telling what
it will do— what acceleration it will impart to a given
mass.
How let
A
be the distance of a comet from the
sun; and v the velocity with which it is moving; if
that comet is moving in a hyperbola; if
A /
its path is a parabola, and
i f 0-<A[JL±-t
<2T=
travels in an
ellipse.
It therefore appears that of an infinite
number of velocities a comet might have, only one would
cause it to move exactly in a parabola, ana so the chances
are infinity to one that no comet moves in that curve,
It is, moreover, known from observation that of all comets
whose orbits have been computea, not more than 3 or 4
appearea to be hyperbolas, ana those were not certainly
so, so that on this ground alone there is strong reason
to believe that all comets are moving in ellipses.
It
has proven to be a fact, however, that in the cases of all
F ig . 1 .
-
28
-
"but a few tlieir orbits are so nearly parabolas that
their performances oan he very well predicted by
assuming a parabola as the true form, and in the case
of a newly discovered comet this is always done.
I
shall deal, then, only with a parabolic orbit.
Let 3, figure 1, represent the sun, -
&
parabola in.which a comet is moving, and V the vertex of
that parabola.
If r is any radius making an angle v
with the line S V, we have from analytic geometry A =
y
Z
SY is in all our future work to be called q, the parihelion distance of the comet from the sun, and v is called
the anomoly of the comet.
Also let T be the time when the comet is at
Y , and t the time when it is at G.
Then the following
equation holds good
x.
>
2-
^ i.
where k is the 4jaassian constant already explained. 3o
knowing q, and T, we can find t for any anomoly v, or
conversely we could find v for any given time t.
The
latter process would necessitate the solution of a
cubic equation, and to avoid the trouble of this the
equation has been multiplied through by 7b, giving
rr
-y-
yr
°ables have then been constructed giving II or log L'
for every value of v, (by steps of only I 1 or in
some cases only 10” from 0
to 180*
and *80 when we
have v or I. the other can be quickly found.
are called barker’s Tables.
These
The most complete of
*
Fig. 2.
29
-
them is found in Oppolzer's ,rDe determination des orbits
des oometes et planete.s.TT
and goes by steps of 10", with
interpolation tables for intermediate values.
yj A- .
or to
is found to equal log™/
rT
use the usual symbol
r
If.
C0 /I?
(9.9601277-10)
'7 y
7J
so as
the complete formula for the relation between v and t
we have
Thus far, then, I have shown how to reduce the
observations of a comet to a known plane of reference, and
how to find the place of a comet in its orbit at any time.
We are now able to proceed towards finding the dimensions
and position of trie oruit in space, the ultimate end of
our work.
For this purpose we shall take the sun S as
origin, ana from it draw an x axis in the plane of the
ecliptic through the vernal equinox,
r
j
a y axis per­
pendicular to this and in the plana of the ecliptic,
and a z axis perpendicular to the plane of the ecliptic,
and let the ellipse represent the earth's orbit. 7e
thus have the beginning of tig. 2, which represents
space, of course^
a
.
a
.
.
-
30
-
Suppose now that at the 3 times of observation
the earth was at "Jf B l and E_ . The positions of these
points oan be founa from the Tautical Almanac, as will
he seen later.
Then if X y and A t are the longitude and
latitude of the comet at the first observation, let the
lire
t Cy he drawn through B. in the plane of the ec­
liptic, giving angle h, its proper value, and perpendicul­
arly above or below it, according a s ^ is -+• or -, let
S(0tbe drawn^ making angle C; E4 3/ =y3/
•
We now have
the position of one line of sight to the comet from the
earth, and in the same way we can locate the other two,
- A and E fC5 .
ITow at the instants of each observation (except
for a source of slight inaccuracy which will be taken up
later) the comet was somewhere on the corresponding line.
Suppose that its three places were c, c^ ana c3 . Sinoe a
comet moves in a plane which passes through the sun, the
first condition these noints must satisfy is that
a m IS
a U.
n7r
single plane^cutting the ecliptic on the linel/L/ and
making with the ecliptic an angle i, contains the sun
and all three of them.
IText, a comet moves in a par­
abola, with the sun as the focus, so it must be possible
to draw such a parabola through 0, 0^ 0 in the plane ^-£2. XJfc •
Let the vertex be P, which now becomes the perihelion, or
point where the comet was nearest the sun.Then P s
will
have a certain length cj_, the shortest distance from the
comet to the sun, or perihelion distance, and will make
Avl
cO ^
S
.
31
-
-
■'inally, if I is the time the comet is at P,
then if it is to obey the laws of parabolic motion the
anomoliec v
\
»
- tt
2
o/
of
Ctc^must satisfy the equations
* n ~TT
^
2-
/ i—
^ ^ +7 ^
^ - Tr
^ %,
So if we are to find, three points oo^c, which. truly represent
the places of the comet, and from them find the orbit, the
points only need satisfy all the above requirements.
It is apparent, however, that fulfillment of
these conditions would demand that the plane of the orbit be
placed in some exceptional position, and it would be hopeless
to try to locate it there by direct trial.
So the most that
we have accomplished by this excursion is the discovery of
co-j
the four unknown quantities,
position of the line
CL
;
besides the
which must be determined.
In
regard to some of these a few additional remarks are de­
manded.
In the first ^lace the comet in the drawing pierces
the plane of the ecliptic and descends below it at the point
II, so this point is called the descending noae,
IT S, or J*!.
\y~
is the line of nodes.
comet would ascend
and the line
Sometime,too,
the
above the plane of the ecliptic again,
at some point out on the line 6
called the ascending node.
-Q.
and this point would be
The lineS-Qis of especial import­
ance, inasmuch as the angle/I. which it makes with the X axis,
measured around from
position*
T
toward the east, fully determines its
This angle.whioh may be anything from o'to 360* is
'
-
always called jQ.
32
-
, the longitude of the ascending node,
and may he plaoed with i,
co , q, and T, making five
■unknowns in all.
As regards i, if we stand at the ascending node
and look towards the sun, it is reckoned upward from the
plane of the ecliptic, towards the left, in the same way
as angles are ordinarily reckoned.
0° to 180"
.
It may he anything from
If it is between 0* and 00* , it shows that
the comet is moving around the sun in the same direction
as the earth and the other planets, or that its motion
is direct;
if between 90* and 180* , the comet moves in
the opposite direction, or retrograde.
The angle60may be anything from 0* to 360
,
and is reckoned fromSH.along the plane of the orbit in
the direction of the comet's motion.
T is of coxirse the time when the comet is at
j
P, the nearest point it gets to the sun, or perihelion.
As regards q nothing more need be said.
Paving thus cleared up the details in regard
to the elements
of the orbit, we can proceed again
towards the determination of them.
As has been remarked,
U.IX-
the attempt to find them by direct trial would be successA
ful, so other means must be sought.
A suggestion as to the proper path to pursue is
gained if it ie observed that once the points c, and c s
alone
c%) were properly located on the lines E. 0,
and E,v 0.o , that is if the distances Ec,/ and E c.3 , which
I shall hereafter call A
and
, were known, all of the
- 33 elements are fixed, ana only //oula need to "be calculated.
For from the principles of analytical geometry only one
plane coula be passed through S, c/ and c5 , and in that
plane, only one parabola with 3 as the focus coula be made
,
C-j,
to go through
and c,^ . So J
a
n
d
q are all deter­
mined, and by using one of the times of observation, t f
say. in the eauation
of course being also known from the parabola, T could
easily be found, and our task would be done.
Eut I have saia "if the points c( and c3 were
properly located in the lines
processes coula be carried out.
in it the requirements;
0, and
Z
11
the above
This "properly" contains
(1) that the parabola passed
through c, ana c^ shoula also contain cz , ana (2) that
the T ae bermine a from the equation above usea snoula also
sauisiy tne equations
ana tne question arises, how we could tell without going
through the long calculation hinted at above, whether two
chosen values for
^0,
andwould
of the needful conditions.
secure the fulfillment
Fig. 3.
-
34
- -
One step towards the solution of this difficulty
may suggest itself upon the- reflection that while v.
v3 and T are obtainable from given values of
A
, vt
and
^ 3
only at the expense of exceedingly tedious calculation,
the distances of the comet from the sun at the two
ZitncS,
I
r, and r. , and the distance X of the 1st and Sd places
of the comet from each other, can he found with comparative
ease,
^or consider the 1st observation, and refer to fig. 3.
-
35 -
F i g . 4.
- 36
-
It is apparent from plane trigonometry that in
the triangle S Q/
o3
Pig. 4 ,
M L - A , 11
- 2. A , A }
fa -V jJ
In these five equations there appear seven unknown quantities
v
v3 T
q, r; , ?3 , K , ancl it ought to he possible to
entirely get rid of the four unknowns v; , r , T, and q,
whose oalciili tions from^anQL/^jinvolves so much trouble,
leaving only a definite relation which must be satisfied
by the quantities r,
It is well known, however, that comprehension
of the possibility
of eliminating unknowns between
equations, and the actual accomplishment of it, are two
entirely different matters, and in this case one must pick
his way exactly right if he is to meet with success.
I will jtherefore,
give^method of elimination
which is found in Watson's Theoretical Astronomy, without
attempting to tell how it was ever hit upon.
($ )
-
3
7
-
-
38
-
-
39
-
-
40
-
-
41
-
This extraordinary result is due to Euler, hut
always 'bears the name of "Lambert’s equation", after
Johann Heinrich Lambert, who considered similar relations
in the case of the ellipse as well as the parabola, and
jpro^uced a far shorter but less trustworthy proof of the
theorem.
Once in possession of it, it would be more within
the range Of practicability to find by trial values of /?/
a n d t o
which values for r, , r3 and X
satisfying Lam­
bert's equation would correspond, and finally determine
whether the parabola passed through c; and c5 would contain
the point
ct
.
It is said indeed,that methods based on
this principle are among the best for improving an already
known approximate orbit, but to find an orbit at first
entirely unknown they would again lead to hopelessly long
calculation.
Relief from this dilemnja is found in the xorincipfe
of Olber's and the resulting equation, which provides
means for calculating from known quantities a closely
(mate
approx^ value of the ratio
> 80 ‘clia‘fc in our formulae
for- finding r a and^which formerly contained^/^ ^ /3t can
be introduced instead.
Then Only/?, remains to be found by
trial so as to give values of
x
r
a n d * which satisfy
Lamberts eqiiation, and from there on the problem is merely
one of analytic geometry.
U
It is^ Olber's equation, then, that we mtist next
devote our attention.
This equation, takes its origin in a principle
F ig. 5.
-
4
2
-
ox orbital motion not heretofore mentioned** tfhe famous
law of Kepler,elaborated by llewton, that if a comet moves
about the sun in any of the conic sections, its radius vector
sweeps over areas proportional to the times consumed in
describing them.
Fig. 6.
- 4 3It is upon the accuracy of this "nearly' that
the Giber's method rests, and as ’’nearly” is the more or
less exact, the orbit obtained is the more or less correct.
We must, however, take the assumption for what it is worth,
and I will later discuss some factors which influence its
effectiveness.
Assuming
A--,. -A
—
. we may draw, in fig. 6.
f e/
c , ct c
a figure of a portion of a comet orbit
T in space,
with the sun at 5 , and a portion of the earth’s orbit
E IEL*
The planes of tie two orbits intersect on the line s n .
Let the places of the comet and earth at the three lines
of observation be c, ® tC.and E,E,E3 , respectively, and draw
the lines of sight E, o,
, E^ 0 v, E o , the radii S* c , 3 ot S o t
S E , , S Et ,S E 4 , and the chords c; o and E#~} intersecting
3 oxand 3 7^at q and Q, respectively.
We now make the Olber’s assumption as well of
the segments of the chord of the
earth's
as of the chord of the comet's orbit
c
orbit E,E
c
,
and so say
Fig. 7.
F i g . 8.
F i g .
9 .
- 4 4 -
Fig.
1 0 .
-45-
Fig. 11.
-46
-
Fi g. 12.
-
47
-
Pirst choose the point Q, and draw the trianles
Q o' c/ and Q G^Oj
, and the lines Q 3,
icj and Q oj in their
proper positions.
Then draw a sphere about Q, as a center,
and produce all the lines going otxt from Q till they cut it.
Then if we transfer the letters
c/ c/ 0/ q
and 3
out to
the surface or the sphere, we shall have a spherical triangle
S c 'c/ with a great circle drawn across it rrom S through
o' cutting the side o' c 7at c.
To make the subsequent opera­
tions plainer, it will he necessary to construct a figure,
figure 12, in which the system is seen from a different
angle, and the lines are in different relative positions.
In that figure Q ;S ,and q have the same significance as beiore
while c, c and c3 c orrespond to
o/ g ^o , in
the former drawing;
S I is a portion of the ecliptic, and P is its pole.
low it is
— -
that we must find, and it is
apparent that its value depends.upon the angles </, a^ ' fit.
and the sides S c, and S c3 of the spherical triangle S c/
Tie nrust now find these quantities.
Draw the arc of a great circle ?
^
be the vernal equinox on the ecliptic,
L, and let
®
..ill then
be the longitude of the sun as seen from Q at the time of
the second observation, and because Q is on the line join­
ing the sun and the earth at that observation,
will
also he the longitude of the sun as seen from the earth
at that observation, and can he found from the Uautical
Almanac.
will he the longitude of the comet at the first
observation, because while the comet
was seen from EL; in
figure 6, and not from „ , the direction of the line of
.
- 4 8 sight was preserved intact in transferring that line of
sight to the new position where it had its base at Q.
For
the sane reasons A fwill "be the latitude of the comet at
that observation.
Angle 3 I c/5 is of course equal to
O
90
, for ?c, L passes through the pole of the
'7e now know two sides
S c*C~ (^/ " &) CL^J~c, •(-
ecliptic.
f
of
the right triangle 3 I c, and can find the side 5 a f ^
ich I will aall// and the angle
SL, or
Y/ •
spherical trigonometry we have
Going now to the triangles c, Sq and ^ c :, we find
that they have a common side ~ q, and that their sides
have "become known, as likewise their angles
4
I* K - J ”)
we have
4rh'Yt* So referring again to spherical trigonometry
-49
-
-
5
0
-
51
So we must first oaloulate
Q>R,, *
:,: T & W, "by the formulae above, and. then r;\ rf and rf are
easily found for any assumed value of^<P/ .
Since the computation of the nine quantities
above is rather long, it is necessary to have a check to
guard, against mistakes.
A good, one is found in the tact
that
as is easily seen hy multiplying out these expressions.
If the values obtained in this way xor X I anu. W agree
with those previously found, it is almost certain that not
only K.Ii & W, hut the P's
s;ana P.'s are correct.
“or the computation of P.* , P_ Q./ , etc., we have
already found expressions for A
A>;'3(; B,; 3/; 0,, hut the
X*s Y's and z»s Were only roughly defined on p. 3 4 IS he in™
the coordinates of the earth at the times of observation.
As a matter of fact, for the best results they
as the
must be token .coordinates of the points on the earth from
A
whence the comet was seen, at the times of observation, and
are each made up of two parts,
of
"or example, X # is the sum
x.e. i ,
the x coordinate of the center of the earth
7
referred to the sun as origin, and
x,/;the coordinate of
the point of observation referred to the center of the
earth as origin and axis parallel to the former ones as
53
axes.
Ir the sarae way y, = zjt/ + <
jfpl, and so on.
X^i (ylt^4/,^ , ^ A j p-re easily obtained as -follows:
On p. Ill for each month in the "autical Almanac
(p. 112 for July, 1911) are found the logarithm of the
radius vector of the earth, the true longitude
or the sun,as seen from the earthy referred to the equinox
of 1911.0 fin the column A /
and the latitude ^/3h of
the sun, for Greenwich Liean iloon of every day.
If we let
he the longitude ana latitude of the earth as
seen from the sun at the time of the first observation,
say, it is easy to see that
Then if R, is the radius vector of the earth we should have
from fig. 5,
F ig . 1 3 .
-54Similarly we can find, values for
The case of *>,,
is more com­
plex, for these depend upon the distance or the place of
observation from the center of the earth, the geocentric
latitude, and the sidereal time of observation.
seen by reference to fig. 13.
This is
Here C is the center of the
earth, and x, y, and z, axes are set up as shown.with 0
as origin, paralell to the great x y and 2 axes whose origin
is at the sun.
Then x points towards the vernal equinox and z
towards the pole of the ecliptic.
How let P be a point on the earth*s surface and
r its radius from the earth's center.
Describe a sphere
the
about C with radius r.
?] n .x y plane will trace off a
A
circle x, y, -x, -y, on the sphere, and the plane of the
equator another, making an angle €
north pole of the earth.
and the arc ^3
arc /mix
^S
with it.
let p be the
Pass the meridian ffcPm through P,
will be the geocentric latitude of P.
The
will be the sidereal time of observation, at
the place of observationsxpressed in arc, since sidereal
time is defined as the Jiour angle of the vernal equinox.
Then in the rt. spherical triangle
55
To determine the quadrants of £ and <lr we only
need notice that "both of them are always "between 0? and
180*^ so that the cosine of one and the tangent of the other
fully fix their values.
nhe means are now all in hand for finding ^3/
and thence ^
, that is for locating the actual posi­
tions of the comet in space at the times of the 1st and
last observations, and we can proceed to their actual
detefmination.
Before doing this, however, it must be decided
whether it is worth while to bother with the quantities
^ < i'Xfe )
Tince, as has been stated, the lati­
tude of the sun is never more than l."0,
than a very small quantity,
is never more
moreover, the earth is but
a speck ooiupureu. wi-cn une maginitudes with which we are
dealing, and so an observer on its surface is almost at
the same position in space as though he were at its
center.
and zp
_'or rough calculation, then, x^
be entirely neglected, and then
can
y*. > X = o- '>
X= * < ,
so that considerable work is saved.
It may be said, too, that even it we do compute
and use the values of ^
jo *****
/
we will
still have only an approximate orbit, because in deriving
the formula for the value or IJ, all these were, and had to
-
be, taken equal to zero.
5 6 -
This oannot be denied, ana
though the effect in any particular case cannot be pre­
dicted, it is probable that in many instances the entire
omission of the quantities in question will have no sub­
stantial influence on the final results, but it is equally
certain that in the average case a closer approximation
will be obtained by taking them into accoxmt than by throwing
them out.
In this problem, therefore, I will use them,
leaving the reader to his own inclinations as to whether
he will do likewise in his own work.
But with log R the hourly differences change so
rapidly that the use of them alone would five slightly
erroneous results.
tion formula.
We must therefore use a n interpola­
Of these there are many, and a derivation
of them and discussion of their relative merits constitutes
a large subject in itself.
I will therefore use the one
given in Campbell's Practical Astronomy, P. 24b.
- 58 -
If we had. interpolated, hy the hourly differences
we shoula have had .0069042.
In most oases the hourly
differences will give sufficient accuracy, hut the ahove
formulae will he used, here for greater certainty.
V/e find hy its means in exactly the same manner
as for !R,,log R, =
.0064539.
59
:no find
for the two observations, we
must find for each place its. distance from the center of
the earth, r, its geocentric latitude, and the sidereal
time of observation. The former two are taken from pp.
520 to 524 of the nautical almanac. ^he r's are given
direct, while the geocentric latitudes ^
must be found
by adding a given correction to the geographical latitudes.
Latitudes to the nearest minute will be ample.
These, r's are given with the equatorial radius
of the earth as a unit, and to reduce them to parts of
the radius of the earth's orbit, they must be multiplied
s& j fT * 4 y 9 0
.
?)
j
(, L J *r /
-. C
i
f
/ oJ»
-
-
I
/
0
This .gives for our new log r's
G.Ct-f'f’
L-/*
i
C l f *r2L
putting dots over the o's to indicate that 10 is to be
substracted from the logs. 7e must next find 3, the local
sidereal time at the time of observation.
7/e can do this
with sufficient accuracy for these purposes by merely add­
ing to the hours, minutes and seconds of mean solar time
at the place of observation the sidereal time at the be-
-60-
ginning of the day, which is given on P.Ill in the column
"Sidereal time, or Eight Ascension of mean sun."
The error
made "by this method will never exceed 4 or o minutes,
and is here of no consequence.
61
62
63
64
66
6
7
68
6 9
70
71.
-72
73
7
4
Fig. 14.
75
F i g . 15.
76
77
The next thing to do is to find J 2 and
a *,whioh
)
determine the plane or the orbit, and first of all -C2 *
The equation of a plane passing through the
origin is given in cartesian coordinates by
If this plane goes through the points */ y, J , and
>r> J y ^ y
then
If we solve these equations for a and b we get
ITow it is the position of the line on which this plane cuts
the xy plane which we wish to find, so we set
^ - 0 and
get the equation of that line as
o .
+■ Jfy ~
The tangent of the angle which it makes with the x axis is
%
=
-
^
- C l .
values of a and b, we have
Substituting the
78
In order to decide which one of these values to give _ Q ,
we have .only to notioe that the comet is going in towards
the sun, since r 3 was less than r; , and that it was when
discovered in the fourth quadrant with respect to longitude,
and that it was above the plane of the ecliptic. So as it
goes arouna the sun it must pass into the second quadrant,
and there it will descend
below the plane of the ecliptic.
The quadrant in which it was when discovered is therefore
the one in which to look xor the asoer.ainr node, so we say
79
-80
-
Having now found, the plane in which the orbit
lies, we must next find the orbit; that is we must obtain
the values of CO ana q.
open.
/or this no direct way seems
It proves to be the case, however, that rormulae
(c O
-
V}J
can be derived for finding the angles
course, v
and
'
1where, of
v5 are the anomolies of the comet at the
two times of observation, and from these far, - &~3) is
quickly obtained.
/hen another formula gives v; or v
in terms of £v t — v^) and other known quantities, and the
rest is easy.
I shall first show how to find Co-flr and w -
. Referring again to figure
Id, if S I is the
perihelion line, thenw has the place indicated, and draw­
ing the radius vector Xj S of the comet,
^
.
The question now is, whether we should c a l l - i n this position
+ or -, for the signs of some of the functions in which it
appeared would be altered if its sign were changed.
Since
the majority of comets have their orbits computed before
81
they reach'. ... perihelion, we shall more often avoid
bothering with negative angles' if during the computation
of an orbit we say that before perihelion v io + , s m
after perihelion, minus.
This isS,of course, contrary to
the most natural way of taking the signs, but the con­
venience here will justify a departure from the common
us'a g e .
If v; is to be called + , then, it is obvious
that angle
and here again we can use our previously found value of
S B.to good purpose.
It is easy to see that by substitu­
tion of z for z. , S 3 for S 3. , etc. we have
The quadrants of(w>-07/ana
can be easily decided upon from
the relative positions of
y, z/ , >: y, z% , and the line S ± 2 .
because since we are finding these angles by their tangents,
there is a full 180* between the two possible choices,
and one could not easily be mistaken by that much,
vive• (o'- 2^
(Or,
nd (cu-
, their difference gives
and the first step is taken.
We must next
derive
the formula for finding v, *■
O
-
Since a parabola A. - — rlr
<Urc>
is to be so located
that it passes through the points x fy /z/ , and 7: y9 a3 , we
must have
82
These are only two equations containing three unknown
quantities, q ,v; .and v5 , hut a third equation for
determining them comes from the fact that since we know
r tf -
we Should "be able to eliminate q "by direct divi­
sion and substitute for vs in terms of ^ and v - v
giving an equation in v( and known quantities only.
Division and extraction of the square root gives
,
83
AS to the quadrant of *L,we know that v, is reckoned either
as "betweenC* and 180
always "between -90
will tell us which.
, or 0 and -13 u
and +90
.
-1
is
, and the sign ox its tangent
-84
-
It is apparent from this that #7 is not de­
termined with exactness, "but the fault is in the data,
not in the method.
The short arc of a parabola between
the 1st and 3 d observations was not great enough to
closely fix the perihelion line on purely geometrical
considerations.
It is possible, however, to introduce
another condition, not geometrical, the condition that the
times of perihelion computed from v, and v. must be the
same, and use this to build up a formula for correction
of this preliminary value of v. .
But space will not per­
mit the introduction of that subject in this paper.
85
There is a difference of 18 between these two
values, but this is so small as to warrant the belief
that it is due to the unavoidable uncertainty about v# which
I have spoken of
above. So I will take
log q =
9.673674
q =
.471709
The only remaining element to be found is T,
the time of perihelion.
This is obtained by determining
the time it would, take the comet to go from each of the
points where it was observed to its perihelion position,
according to the laws of parabolic motion, and adding these
figures to the times when it was observed.
On p.
where
0^1 -■ ?
20
we had
-h
}
and is to
be found from the value of v in a Barker's table.
Tran­
sposing the first equation, we have
_£i— .—
Jjr. f C 0 I I ? 7j
9% - f 2-- r^3 ? b7z3>7 0 / .
U
This was made upon the supposition that v/ was 4- after
pe^helion, instead of before, as we have been using it,
but the proper sign for t - T is easily picked out without
86
bothering with the signs for v» for we already know that
the comet when observed was not yet at perihelion, and so
t(- ? should, be -, and T - t„ +.
follows
7e can now calculate as
87
- 8 8
-
T'bell’s elements, taken from Astronomishe
:Tachrichten 4olO, were merely provisional figures computed
on observations taken July 21, 22 and 2 5.
Thus the
intervals between observations were only about one day,
and the great disagreement of his figures with the others
is not at all surprising.
Young’s elements, from Lick Observatory Bulletin
202, were derived from observations on July 21, Aug. 4,
and Aug. 18, so his intervals were 14 days.
rillosevich’s elements, from Astronomiche
Ilaohriohben 4b36, were c o m p u t e d from a combination of
fotir observations, extending over the interval from July
25 to Oct. 5, and are undoubtedly the best of all four sets.
It is to be noticed that Young’s elements with
intervals of 14 days closely agree with ^lillosevich.,s ;
Bbell's, with intervals of 1 day, are widely at variance,
while those derived in this paper, with intervals of 5
days, are in fairly close agreement.
It appears then,
that in this particular case better results would have
been obtained if a longer interval had been chosen.
The proper length to select, however, depends
•
upon such a variety of conditions that no rule can be
laid down.
On the one hand, there are the unavoidable
accidental errors of the observations to be considered,
and on the other the degree of approximation of that
,0,
value of M given by our formulae to the true ratio of /-— .
/*'
As to errors of observation, it is obvious that
the longer the arc swept over by the comet between
observations, the smaller the influence of these errors,
- 8 9 and with regard only to these, the ideal way to compute
an orbit would he to choose one observation when the comet
was first discovered, one about midway of its period of
visibility, and one just
before it went out of sight.
The method used here, and any other direct method of
calculating an orbit, would.howeve^ he wholly inapplicable
to such a case, so a much shorter interval must be taken.
In respect to the value of LI at least two opposing
factors influence its exactness.
The first is the geometrical
inaccuracy of the assumption made on p ,42-
that
areas are proportional to the triangle areas.
the sector
This
assumption will be the more nearly correct as the arc is
shorter.
On the other hand, the statement that "the areas
swept over by the radii vectors of bodies moving about the
sun are in proportion to the times" is, as said on p. 2 . 6 ,
only true if we neglect the attraction of all other bodies
but the sun.
These attractions are not entirely negligible
and the result is that the planets
pull
a comet slightly
away from the true parabolic path assumed for it, and the
jj.Lane os and the moon have the same efiect on the .otherwise
elliptical orbit of the earth.
Besides this, there is the
fact heretofore mentioned, that T is computed on the
assumption that the comet was seen from the center of the
earth, not from its surface, and the apparent place would
be slightly different in the two cases.
Some of these effects would dictate the choice of
long intervals between observations, some of short, and the
combined effect is so complex that nothing but.the average
90
or experiences in a great number of cases would indicate
what intervals to select.
There is one remaining source of error (alluded
to on p . 3 0 ) in the elements computed, the so called
.planetary aberration, which comes from the fact that the
comet was on the observed line of sight not at the instant
of observation but at the instant when the light started
from the comet to us.
Fig. 16.
Thus in figure 16, suppose c, and c t to be two
successive positions of a comet in space.
At a certain
tine t . let the comet send out a ray c, E,
At this instant
the earth may be at some point E . , but
light arrives at
when the ray of
, say. the earth is there to receive
it, and to an observer the comet would seem at that instant
to be at 3, , when as a matter of tact it had moved on
to c r ,
The error arising from this fact can be entirely
corrected by sub. tracting from each observed time, the inter­
val needed for light to come to us from the comet, thus
obtaining the actual times when the comet was on the lines
of sight, and using these new times for t. t^ and t5 , in
place of the times of observation.
But inasmuch as the
91
distance
of the comet is not. known- until an orbit has
heen computed, we must first calculate an orbit, neglecting
the planetary aberration, thence obtain
and
>
the.distances of the comet from the earth, from these
estimate the corrections to the three times, apply them,
ana recompute the orbit.
'i’he second computation is far less difficult than
the first, for one can use the first as a basis and simply
make small changes in the last figures of the logs* without
bothering to look.the logs.up all over again.
A recomputation of the present orbit in this
way turnishes the following elements, which are given in
parallcll columns with*, original for the sake of comparison.
It appears, then, that the difference is small,
but for an orbit
which shall represent positions of a
comet with the greatest possible accuracy it is necessary
to take it into account.
To find how nearly the above final orbit re­
presented actual positions of the comct, the position of the
latter was computed for the -morning of ITov. 1, 1911. three
months and ten days after discovery.
The methods by
which this was aone cannot be explained here, so only the
92recults will be given, as follows
It is here seen that the error in oc is nearly
1(T, or 2' *, and that in S
over
2/3 of a degree,
so
it seems that while an orbit computed on observations at
such an early aate after the discovery of a comet as was
this one will represent its general positions through the
rest of its appearance, and show what sort of a display it
will make, the orbit will not give accurate places throughout
as long an interval as three months.
Bibliography.
The work chiefly consulted. in the preparation of this
paper
s bee
n the "She ore cishc Astronorie* ox
edition of I8‘
JC
J, rna That aouii is
o Dr dtiongl
'.linkerfues,
reo/o-
cnaea
to whoever aerires ro go into This sub.i? oa ■ ■ rreater length.
The explanations
re given in as simple ana thorough c form ce
po siole, ana the desire to build everything up xirrol" and
clesrl;
irom the fundamentals is not xoregone :or rhr s ko of
a little greater deftness in rh*- demonstrations.
’.Tatson's mheoroticrl Astronomy js i w o r i n
English
on the sane subject, but. because o- iTs age ; s' well as other
reasons I cannot rco/zror.rcna it..
-
One desiring t
9 3 -
A
'iv
.m a y or orbit
computation, taking up methods that have been, !c we&las Those
th.- t
re at present ir, uc. , n;.y consul: the t'ollo' ing works:
"De Determination aec Orbits clec 3ometes et Planetes.1'
by ''ppolser.
i-
'theoria "otus Corporum Doelestiun,'T by Gauss, an
’"nglish t:. anslrti or. oi which was oubli she a by I H tlr , Brown,
Tr,
Jo. , or Dost on, in 185 7 .
Laplace’s kecaniqué Celeste, 3ook T r.
In -Tewton1s T-rincipia there is given
graphic r.cthoa
or linding the orbit or a cornet, which, so far
t]
c I know, is
ïirst successful sol ition or the -problem. Te applies it
to several 01 the oomets ox his-time., ana v it;■ rw-oa results.
In Popular Astronomy of 1909 there was published, ser­
ially
■
modern graphic
• : 4hoa,
aevisea by
'n .
0 . Penrose,
Which will furnish an interesting exercise, ana will sometimes
q
s* Its value
beginner an exact vis u 1 :an: or rl
is
cl
'
v
dporti< ns
or his erbiJ ; r.sid- trom tk.t it has little worth except es
a curiosity/.