VI.
ON THE USE OF THE VECTOR METHOD IN THE DETERMINATION OF ORBITS
Letter to Dr. Hugo Buchholz, Editor of Klinkerfues' Theoretische Astronomie.^{[1]}
New Haven, October, 1898.
Dr. Hugo Buchholz,
My dear Sir,—The opinion of Fabritius^{[2]} on the comparative convenience of different methods is entitled to far more weight than mine, for I am no astronomer, and have calculated very few orbits, none, indeed, except for the trial of my own formulæ. The object of my paper was to show to astronomers, who are rather conservative (and with right, for astronomy is the oldest of the exact sciences), the advantage in the use of vector notations, which I had learned in Physics from Maxwell. This object could be best obtained, not by showing, as I might have done, that much in the classic methods could be conveniently and perspicuously represented by vector notations, but rather by showing that these notations so simplify the subject, that it is easy to construct a method for the complete solution of the problem. That the method given is the best possible, I certainly do not claim, but only that it is much better than I could have found without the use of vector notations. Some of the more obvious crudities in my paper have been corrected in that of Beebe and Phillips.^{[3]} Doubtless many more remain, even if the general method be preserved.
My first efforts, however, to solve the fundamental approximative equation were along the same lines which Fabritius has followed:—to set $r_{1}=r_{2}$ and $r_{3}=r_{2}$ in equation second of (2) of Fabritius, which will give $\rho _{2}$ and $r_{2}$, then to get $r_{1}$ from the first of (3) of Fabritius, and then $r_{3}$ either from equation second of (3) or from some other which would serve the purpose, and then to find better values of $\rho _{2},r_{2}$ by setting in equation second of (2)
$r_{1}=\left[{\frac {r_{1}}{r_{2}}}\right]r_{2},$$r_{3}=\left[{\frac {r_{3}}{r_{2}}}\right]r_{2},$


the expressions in brackets denoting numbers derived from the approximate values already found. This is similar to or identical with the method of Fabritius, except that he combines with it the principle of interpolation (for the first value in the third "hypothesis"). As I found the approximation by this method sometimes slow or failing, notably in the case of Swift's comet, 1880 V, I tried the method published in my paper. Indeed, it may be said that the method of my paper was constructed to meet the exigencies of the case of the comets 1880 V.
In ordinary cases I think that the method of Fabritius may very likely be better than that which I published. The equations are very simply and perspicuously represented in vector notations. I shall use the notations of my paper, writing ${\bar {E}},{\bar {F}},$ etc., for German letters.^{[4]} To eliminate $\rho _{1}$ and $\rho _{3}$ from equation (7) in my paper, multiply directly by ${\mathfrak {F}}_{1}\times {\mathfrak {F}}_{3}.$ This gives
$A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)({\mathfrak {E}}_{1}{\mathfrak {F}}_{1}{\mathfrak {F}}_{3})\left(1{\frac {B_{2}}{r_{2}^{3}}}\right)[({\mathfrak {E}}_{2}{\mathfrak {F}}_{1}{\mathfrak {F}}_{3})+\rho _{2}({\mathfrak {F}}_{2}{\mathfrak {F}}_{1}{\mathfrak {F}}_{3})]+A_{3}\left(1+{\frac {B_{3}}{r_{3}^{3}}}\right)({\mathfrak {E}}_{3}{\mathfrak {F}}_{1}{\mathfrak {F}}_{3})=0.$

(a)

To eliminate
$\rho _{3}$ and
$r_{3},$ multiply by
${\mathfrak {E}}_{3}\times {\mathfrak {F}}_{3}$ which gives
$A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)[({\mathfrak {E}}_{1}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})+\rho _{1}({\mathfrak {F}}_{1}{\mathfrak {E}}_{2}{\mathfrak {F}}_{3})]\left(1{\frac {B_{2}}{r_{2}^{3}}}\right)[({\mathfrak {E}}_{1}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})+\rho _{2}({\mathfrak {F}}_{2}{\mathfrak {E}}_{3}{\mathfrak {F}}_{3})]=0.$

(b)

When we have found
$\rho _{1},r_{1},\rho _{2},r_{2}$ it is not necessary to eliminate any of them, and to save labor in forming the equation for
$\rho _{3},r_{3},$ I should be inclined to take the components in (7) in the direction of one of the coordinate axes, choosing that one which is most nearly directed towards the third observed position. However, I will write
$A_{1}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right)[({\mathfrak {E}}_{1}.{\mathfrak {B}})+\rho _{1}({\mathfrak {F}}_{1}.{\mathfrak {B}})]\left(1{\frac {B_{2}}{r_{2}^{3}}}\right)[({\mathfrak {E}}_{2}.{\mathfrak {B}})+\rho _{2}({\mathfrak {F}}_{2}.{\mathfrak {B}})]+A_{3}\left(1+{\frac {B_{3}}{r_{3}^{3}}}\right)[({\mathfrak {E}}_{3}.{\mathfrak {B}})+\rho _{3}({\mathfrak {F}}_{3}.{\mathfrak {B}})]=0,$

(c)

where
${\mathfrak {B}}$ may represent an axis of coordinates, or
$({\mathfrak {E}}_{1}\times {\mathfrak {F}}_{1})$ which would give Fabritius' equation. It might be directed towards the pole
of the ecliptic, which wotdd make
$({\mathfrak {E}}_{1}.{\mathfrak {B}}),({\mathfrak {E}}_{2}.{\mathfrak {B}}),({\mathfrak {E}}_{3}.{\mathfrak {B}})$ vanish, except for exceedingly minute quantities depending on the latitude of the sun and the geocentric coordinates of the observatories, if these are included in
${\mathfrak {E}}_{1},{\mathfrak {E}}_{2},{\mathfrak {E}}_{3}.$
The equations (a), (b), (c), which are together equivalent to (7), I would solve as follows, almost in the same way as Fabritius, but relying a little more on interpolation, and less on the convergence of which he speaks, which in special cases may more or less fail.
Setting $r_{1}=r_{2}$ and $r_{3}=r_{2}$ in (a), which thus modified I shall call (a'), and solving this (a') by "trial and error," using $\rho _{2}$ as the independent variable, as soon as I have a value of $\rho _{2}$ which I think will give a residual of (a') of the same order of smallness as the effect of changing ${\frac {1}{r_{1}^{3}}}$ and ${\frac {1}{r_{3}^{3}}}$ into ${\frac {1}{r_{2}^{3}}},$ I determine from this value by (b) and (c), $r_{1}$ and $r_{3},$ and then find the residual of (a), using the values of $r_{1},r_{2},r_{3}$ derived all from the same assumed $\rho _{2}.$ Now using the last value of ${\frac {\Delta {\text{(residual)}}}{\Delta \rho _{2}}}$ in my previous calculations on (a') which indeed applies only roughly to the (a), I would get a value $\rho _{2}$ which I would use for the second "hypothesis" in (a). This will give a second residual in (a), which will enable me to make a more satisfactory interpolation. As many more interpolations may be made as shall be found necessary.
Some such method, which should perhaps be called the method of Fabritius, would, I think, in most cases probably be the best for solution of equation (7).
Of course I am quite aware that the merit of my paper, if any, lies principally in the fundamental approximation (1). I will add a few words on this subject.
The equation may be written more symmetrically
${\begin{aligned}{\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {\tau _{2}^{2}+\tau _{3}^{2}3\tau _{1}^{2}}{24r_{1}^{3}}}\right){\mathfrak {R}}_{1}&\left(1+{\frac {\tau _{3}^{2}+\tau _{1}^{2}3\tau _{2}^{2}}{24r_{2}^{3}}}\right){\mathfrak {R}}_{2}\\&{\frac {\tau _{3}}{\tau _{2}}}\left(1+{\frac {\tau _{1}^{2}+\tau _{2}^{2}3\tau _{3}^{2}}{24r_{3}^{3}}}\right){\mathfrak {R}}_{3}=0.\end{aligned}}$

I

It might be made entirely symmetrical by writing
$\tau _{2}$ for
$\tau _{2}.$ If an expression ending with
$t^{3}$ had been used, we could still have satisfied two of the conditions relating to acceleration, and should have obtained
${\frac {\tau _{1}}{\tau _{2}}}{\mathfrak {R}}_{1}\left(1+{\frac {\tau _{1}^{2}\tau _{2}^{2}}{6r_{2}^{2}}}\right){\mathfrak {R}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}\left(1+{\frac {\tau _{1}^{2}\tau _{3}^{2}}{6r_{3}^{2}}}\right){\mathfrak {R}}_{3}=0,$

IIa

or
${\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {\tau _{2}^{2}\tau _{1}^{2}}{6r_{1}^{2}}}\right){\mathfrak {R}}_{1}{\mathfrak {R}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}\left(1+{\frac {\tau _{2}^{2}\tau _{3}^{2}}{6r_{3}^{2}}}\right){\mathfrak {R}}_{3}=0,$

IIb

or
${\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {\tau _{3}^{2}\tau _{1}^{2}}{6r_{1}^{2}}}\right){\mathfrak {R}}_{1}\left(1+{\frac {\tau _{3}^{2}\tau _{2}^{2}}{6r_{2}^{2}}}\right){\mathfrak {R}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}{\mathfrak {R}}_{2}=0.$

IIc

Using an expansion ending with
$t^{2}$ we can only satisfy one condition relating to acceleration, say the second. This will give
${\frac {\tau _{1}}{\tau _{2}}}{\mathfrak {R}}_{1}\left(1{\frac {\tau _{1}\tau _{3}}{2r_{2}^{3}}}\right){\mathfrak {R}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}{\mathfrak {R}}_{3}=0.$

III

(Gauss uses virtually
${\frac {\tau _{1}}{\tau _{2}}}{\mathfrak {R}}_{1}{\frac {{\mathfrak {R}}_{2}}{1+{\frac {\tau _{1}\tau _{3}}{2r_{2}^{3}}}}}+{\frac {\tau _{3}}{\tau _{2}}}{\mathfrak {R}}_{3}=0,$


which is a little more convenient, but not, I think, generally quite so accurate.)
Writing an equation analogous to III for the earth and subtracting from (7), Mem. Nat. Acad., we have
${\frac {\tau _{1}}{\tau _{2}}}\rho _{2}{\mathfrak {F}}_{1}\left(1{\frac {\tau _{1}\tau _{3}}{2r_{2}^{3}}}\right)\rho _{2}{\mathfrak {F}}_{2}+{\frac {\tau _{3}}{\tau _{2}}}\rho _{3}{\mathfrak {F}}_{3}={\frac {\tau _{1}\tau _{3}}{2}}\left({\frac {1}{{\mathfrak {E}}_{2}^{3}}}{\frac {1}{r_{2}^{3}}}\right){\mathfrak {E}}_{2},$


which gives, on multiplication by
${\mathfrak {F}}_{1}\times {\mathfrak {F}}_{3}$ and
${\mathfrak {F}}_{2}\times {\mathfrak {E}}_{2},$ theorems of Olbers and Lambert.
It is evident that in general the error in I is of the fifth order, in IIa, IIb, IIc of the fourth, and in III of the third. But for equal intervals, the error in I is of the sixth order, and in III of the fourth. And when $\tau _{2}^{2}+\tau _{3}^{2}3\tau _{1}^{2}=0,$ IIa becomes identical with I, and its error is of the fifth order.
The same is true of IIc in the corresponding case. It follows that when the intervals are nearly as 5:8 we should use IIa or IIb instead of I. This will evidently abbreviate the solution given above as only one of the quantities $r_{1},r_{3},$ is to be used.
The formulæ IIa, IIb, IIc may also be obtained by the following method, which will show their relative accuracy.
The interpolation formula
${\frac {\tau _{1}}{\tau _{2}}}{\frac {{\mathfrak {R}}_{1}}{r_{1}^{3}}}{\frac {{\mathfrak {R}}_{2}}{r_{2}^{3}}}+{\frac {\tau _{3}}{\tau _{2}}}{\frac {{\mathfrak {R}}_{3}}{r_{3}^{3}}}=0$


has an error evidently of the second order. If we multiply by
${\frac {\tau _{2}^{2}+\tau _{3}^{2}3\tau _{1}^{2}}{24}}$ and subtract from
I, we get
IIa. So if we multiply by
${\frac {\tau _{3}^{2}+\tau _{1}^{2}3\tau _{2}^{2}}{24}}$or${\frac {\tau _{1}^{2}+\tau _{2}^{2}3\tau _{3}^{2}}{24}}$


we get
IIb or
IIc. The errors due to using one of these equations instead of
I are therefore proportional to these multipliers and very unequal.
Again, in case of equal intervals,
IIa and
IIc become identical with
III. There is, therefore, no reason for using
IIa or
IIc when the intervals are nearly equal.
IIb is in this case much less accurate than
III.
It will be observed that all the formulaæ I, IIa, IIb, IIc, III may be expressed in the general form
${\frac {\tau _{1}}{\tau _{2}}}\left(1+{\frac {B_{1}}{r_{1}^{3}}}\right){\mathfrak {R}}_{1}\left(1+{\frac {B_{2}}{r_{2}^{3}}}\right){\mathfrak {R}}_{2}+\left(1+{\frac {B_{3}}{r_{3}^{2}}}\right){\mathfrak {R}}_{3}=0,$


except that the letters
$B_{1},B_{2},B_{3}$ have different values in the different cases, some vanishing in the more simple formulæ. Moreover, if the values of
$B_{1},B_{2},B_{3}$ have been calculated for
I, the values for
IIa,
IIb, or
IIc are found simply by subtraction of one of the numbers from the three. It is evident that
IIb will hardly be useful except in special cases, as in the determination of a parabolic orbit in the failing case of Olbers' method, and then it would be a question whether it would not be better to determine the orbit from
$\rho _{2}$ and
$\rho _{3},$ or
$\rho _{2}$ and
$\rho _{1},$ using
IIb or
IIc.
Equations IIa and IIc are very appropriate for the determination of an elliptic orbit when the observed motion is nearly in the ecliptic, by means of four observations with intervals nearly in the ratio 5 : 8 : 5.
It is evident that the solution of (7) given above may be varied, in ways too numerous to mention, by the use of the simpler forms IIa, IIc, or III for II in the earlier stages of the work. This only involves changing the values of $B_{1},B_{2},B_{3}$ in (a), (b) and (c).
It is not correct to say that in my expressions for the ratios of the triangles the error is of the fifth order in general, or for equal intervals, of the sixth. If we write $p_{1},p_{2},p_{3},$ for the coefficients of ${\mathfrak {R}}_{1},{\mathfrak {R}}_{2},{\mathfrak {R}}_{3}$ in I, and ${\mathfrak {T}}$ for the error of the equation, we have exactly
$p_{1}{\mathfrak {R}}_{1}p_{2}{\mathfrak {R}}_{2}+p_{3}{\mathfrak {R}}_{3}={\mathfrak {T}},$


which gives
$p_{1}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}p_{2}{\mathfrak {R}}_{2}\times {\mathfrak {R}}_{3}={\mathfrak {T}}\times {\mathfrak {R}}_{3},$


${\frac {{\mathfrak {R}}_{3}\times {\mathfrak {R}}_{3}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{1}}{p_{2}}}{\frac {{\mathfrak {T}}\times {\mathfrak {R}}_{3}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$


Now
${\frac {p_{1}}{p_{2}}}$ is my expression for the ratio of the triangles, and
${\frac {{\mathfrak {T}}\times {\mathfrak {R}}_{3}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}$ is its error. This is of the fourth order in general (since the denominator is of the first), and for equal intervals, of the fifth. The same is true of the two other ratios. Thus we have
${\frac {{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{2}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{3}}{p_{2}}}{\frac {{\mathfrak {R}}_{1}\times {\mathfrak {T}}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$


Adding these equations and subtracting 1 [from both sides] we have
${\frac {{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{2}+{\mathfrak {R}}_{2}\times {\mathfrak {R}}_{3}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}{{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}={\frac {p_{1}+p_{3}p_{2}}{p_{2}}}+{\frac {({\mathfrak {R}}_{3}{\mathfrak {R}}_{1})\times {\mathfrak {T}}}{p_{2}{\mathfrak {R}}_{1}\times {\mathfrak {R}}_{3}}}\cdot$


Here the last term, which represents the error, is of the fifth order in general, or for equal intervals, of the sixth. But the
quantity sought is of the second order, and the relative error is of the third order in the general case, or the fourth for equal intervals. It is precisely this error which is most important in the case of elliptic orbits.
It will be observed that the accuracy of the expressions for the ratios $[r_{1}r_{2}]:[r_{2}r_{3}]:[r_{1}r_{3}]$ affords no measure of the accuracy of the formula for the determination of elliptic orbits.
I think that this hasty sketch will illustrate the convenience and perspicuity of vector notations in this subject, quite independently of any particular method which is chosen for the determination of the orbit. What is the best method? is hardly, I think, a question which admits of a definite reply. It certainly depends upon the ratio of the time intervals, their absolute value, and many other things.
P.S.—If we wish to use the curtate distances, with reference to the ecliptic or the equator, let $\rho _{1}$ be defined as the distance multiplied by cosine (lat. or dec), and ${\mathfrak {F}}_{1}$ as a vector of length secant (lat. or dec). For the most part the formulae will require no change, but the square of ${\mathfrak {F}}_{1}$ will be $\sec ^{2}$(lat. or dec.) instead of unity, so that the last terms of (8) will have this factor. (${\mathfrak {F}}_{1}{\mathfrak {F}}_{2}{\mathfrak {F}}_{3}$) will then be Gauss' (0.1.2.), whereas in my paper (${\mathfrak {F}}_{1}{\mathfrak {F}}_{2}{\mathfrak {F}}_{3}$) is Lagrange's ($C'C''C'''$ ).