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SWIFT
SWIFTsim
Commits
5b055e8a
Commit
5b055e8a
authored
Dec 25, 2017
by
Matthieu Schaller
Browse files
Improvements to the gravity derivative descriptions in the TeX document.
parent
c07c0c94
Changes
1
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theory/Multipoles/potential_derivatives.tex
View file @
5b055e8a
...
...
@@ -18,8 +18,10 @@ by constructing derivatives of the truncated potentials:
\chi
^{
(4)
}
(r, r
_
s)
&
=
\frac
{
16
}{
r
_
s
^
4
}
\left
(48
\alpha
(x)
^
5 - 120
\alpha
(x)
^
4 + 100
\alpha
(x)
^
3 -30
\alpha
(x)
^
2 + 2
\alpha
(x)
\right
)
\nonumber
\\
\chi
^{
(5)
}
(r, r
_
s)
&
=
\frac
{
32
}{
r
_
s
^
5
}
\left
(240
\alpha
(x)
^
6 - 720
\alpha
(x)
^
5 + 780
\alpha
(x)
^
4 - 360
\alpha
(x)
^
3 + 62
\alpha
(x)
^
2 - 2
\alpha
(x)
\right
)
\nonumber
\end{align}
We can now construct common quantities that appear in derivatives of
multiple orders:
In the Newtonian limit (
$
r
_
s
\rightarrow\infty
$
) the first expression
reduces to
$
\chi
(
r,r
_
s
)
=
1
$
whilst all higher-order derivatives
vanish. We can now construct common quantities that appear in
derivatives of multiple orders:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
...
...
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