DETERMINATION OF ELLIPTIC ORBITS
123
This gives[ 1]
Δ
q
1
=
−
(
S
S
″
S
‴
)
(
S
′
S
″
S
‴
)
{\displaystyle \Delta q_{1}=-{\frac {({\mathfrak {SS}}''{\mathfrak {S}}''')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}}}
Δ
q
2
=
−
(
S
S
‴
S
′
)
(
S
′
S
″
S
‴
)
{\displaystyle \Delta q_{2}=-{\frac {({\mathfrak {SS}}'''{\mathfrak {S}}')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}}}
Δ
q
3
=
−
(
S
S
′
S
″
)
(
S
′
S
″
S
‴
)
⋅
{\displaystyle \Delta q_{3}=-{\frac {({\mathfrak {S}}{\mathfrak {S}}'{\mathfrak {S}}'')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}}\cdot }
(20)
From the corrected values of
q
1
,
q
2
,
q
3
{\displaystyle q_{1},q_{2},q_{3}}
we may calculate a new residual
S
,
{\displaystyle {\mathfrak {S}},}
and from that determine another correction for each of the quantities
q
1
,
q
2
,
q
3
.
{\displaystyle q_{1},q_{2},q_{3}.}
It will sometimes be worth while to use formulæ a little less simple for the sake of a more rapid approximation. Instead of equation (19) we may write, with a higher degree of accuracy,
−
S
=
S
′
Δ
q
1
+
S
″
Δ
q
2
+
S
‴
Δ
q
3
+
1
2
T
′
(
Δ
q
1
)
2
+
1
2
T
″
(
Δ
q
2
)
2
+
1
2
T
‴
(
Δ
q
3
)
2
,
{\displaystyle -{\mathfrak {S}}={\mathfrak {S}}'\Delta q_{1}+{\mathfrak {S}}''\Delta q_{2}+{\mathfrak {S}}'''\Delta q_{3}+{\tfrac {1}{2}}{\mathfrak {T}}'(\Delta q_{1})^{2}+{\tfrac {1}{2}}{\mathfrak {T}}''(\Delta q_{2})^{2}+{\tfrac {1}{2}}{\mathfrak {T}}'''(\Delta q_{3})^{2},}
(21)
where
T
′
=
{\displaystyle {\mathfrak {T}}'=}
d
2
S
1
d
q
1
2
=
{\displaystyle {\frac {d^{2}{\mathfrak {S}}_{1}}{dq_{1}^{2}}}=}
2
A
1
B
1
d
(
r
1
−
3
)
d
q
1
F
1
+
{\displaystyle 2A_{1}B_{1}{\frac {d(r_{1}^{-3})}{dq_{1}}}{\mathfrak {F}}_{1}+}
B
1
1
+
B
1
r
1
−
3
d
2
(
r
1
−
3
)
d
q
1
2
S
1
{\displaystyle {\frac {B_{1}}{1+B_{1}r_{1}^{-3}}}{\frac {d^{2}(r_{1}^{-3})}{dq_{1}^{2}}}{\mathfrak {S}}_{1}}
}
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}
(22)
T
″
=
{\displaystyle {\mathfrak {T}}''=}
d
2
S
2
d
q
2
2
=
{\displaystyle {\frac {d^{2}{\mathfrak {S}}_{2}}{dq_{2}^{2}}}=}
2
B
2
d
(
r
2
−
3
)
d
q
2
F
2
−
{\displaystyle 2B_{2}{\frac {d(r_{2}^{-3})}{dq_{2}}}{\mathfrak {F}}_{2}-}
B
2
1
−
B
2
r
2
−
3
d
2
(
r
2
−
3
)
d
q
2
2
S
2
{\displaystyle {\frac {B_{2}}{1-B_{2}r_{2}^{-3}}}{\frac {d^{2}(r_{2}^{-3})}{dq_{2}^{2}}}{\mathfrak {S}}_{2}}
T
‴
=
{\displaystyle {\mathfrak {T}}'''=}
d
2
S
3
d
q
3
2
=
{\displaystyle {\frac {d^{2}{\mathfrak {S}}_{3}}{dq_{3}^{2}}}=}
2
A
3
B
3
d
(
r
3
−
3
)
d
q
3
F
3
+
{\displaystyle 2A_{3}B_{3}{\frac {d(r_{3}^{-3})}{dq_{3}}}{\mathfrak {F}}_{3}+}
B
3
1
+
B
3
r
3
−
3
d
2
(
r
3
−
3
)
d
q
3
2
S
3
{\displaystyle {\frac {B_{3}}{1+B_{3}r_{3}^{-3}}}{\frac {d^{2}(r_{3}^{-3})}{dq_{3}^{2}}}{\mathfrak {S}}_{3}}
It is evident that
T
″
{\displaystyle {\mathfrak {T}}''}
is generally many times greater than
T
′
{\displaystyle {\mathfrak {T}}'}
or
T
‴
,
{\displaystyle {\mathfrak {T}}''',}
the factor
B
2
,
{\displaystyle B_{2},}
in the case of equal intervals, being exactly ten times as great as
A
1
B
1
{\displaystyle A_{1}B_{1}}
or
A
3
B
3
.
{\displaystyle A_{3}B_{3}.}
This shows, in the first place, that the accurate determination of
Δ
q
2
{\displaystyle \Delta q_{2}}
is of the most importance for the subsequent approximations. It also shows that we may attain nearly the same accuracy in writing
−
S
=
S
′
Δ
q
1
+
S
″
Δ
q
2
+
S
‴
Δ
q
3
+
1
2
T
″
Δ
q
2
2
.
{\displaystyle -{\mathfrak {S}}={\mathfrak {S}}'\Delta q_{1}+{\mathfrak {S}}''\Delta q_{2}+{\mathfrak {S}}'''\Delta q_{3}+{\tfrac {1}{2}}{\mathfrak {T}}''\Delta q_{2}^{2}.}
(23)
We may, however, often do a little better than this without using a more complicated equation. For
T
′
+
T
‴
{\displaystyle {\mathfrak {T}}'+{\mathfrak {T}}'''}
may be estimated very roughly as equal to
1
2
T
″
.
{\displaystyle {\tfrac {1}{2}}{\mathfrak {T}}''.}
Whenever, therefore,
Δ
q
1
{\displaystyle \Delta q_{1}}
and
Δ
q
3
{\displaystyle \Delta q_{3}}
are about as large as
Δ
q
2
,
{\displaystyle \Delta q_{2},}
as is often the case, it may be a little better to use the coefficient
6
10
{\displaystyle {\tfrac {6}{10}}}
instead of
1
2
{\displaystyle {\tfrac {1}{2}}}
in the last term.
For
Δ
q
2
,
{\displaystyle \Delta q_{2},}
then, we have the equation
−
(
S
S
‴
S
′
)
=
(
S
′
S
″
S
‴
)
Δ
q
2
+
6
10
(
T
″
S
‴
S
′
)
Δ
q
2
2
.
{\displaystyle -({\mathfrak {S}}{\mathfrak {S}}'''{\mathfrak {S}}')=({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')\Delta q_{2}+{\tfrac {6}{10}}({\mathfrak {T}}''{\mathfrak {S}}'''{\mathfrak {S}}')\Delta q_{2}^{2}.}
(24)
(
T
″
S
‴
S
′
)
{\displaystyle ({\mathfrak {T}}''{\mathfrak {S}}'''{\mathfrak {S}}')}
is easily computed from the formula
(
T
″
S
‴
S
′
)
=
1
q
2
(
1
−
5
q
2
2
r
2
2
)
(
(
S
′
S
″
S
‴
)
+
(
F
2
S
‴
S
′
)
)
{\displaystyle ({\mathfrak {T}}''{\mathfrak {S}}'''{\mathfrak {S}}')={\frac {1}{q_{2}}}\left(1-5{\frac {q_{2}^{2}}{r_{2}^{2}}}\right)\left(({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')+({\mathfrak {F}}_{2}{\mathfrak {S}}'''{\mathfrak {S}}')\right)}
}
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}}
(25)
−
B
2
q
2
r
2
3
(
1
+
q
2
2
r
2
2
)
(
F
2
S
‴
S
′
)
,
{\displaystyle -{\frac {B_{2}}{q_{2}r_{2}^{3}}}\left(1+{\frac {q_{2}^{2}}{r_{2}^{2}}}\right)({\mathfrak {F}}_{2}{\mathfrak {S}}'''{\mathfrak {S}}'),}
which may be derived from equations (18) and (22).
↑ These equations are obtained by taking the direct products of both members of the preceding equation with
S
″
×
S
‴
,
{\displaystyle {\mathfrak {S}}''\times {\mathfrak {S}}''',}
S
‴
×
S
′
,
{\displaystyle {\mathfrak {S}}'''\times {\mathfrak {S}}',}
and
S
′
×
S
″
,
{\displaystyle {\mathfrak {S}}'\times {\mathfrak {S}}'',}
respectively. See footnote on page 119.