The scalar functions in Eq. (LABEL:S_I_2PN) are given by (cf. Eqs. (D29),
(D31), (D33), (D35), (D37), (D39) in Zschocke_Quadrupole_1 )

$\displaystyle{\cal W}_{\left(3\right)}\left(t\right)$ $\displaystyle=$
$\displaystyle\ln\left(x_{\rm
N}-\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm N}\right),$ (172)
$\displaystyle{\cal W}_{\left(5\right)}\left(t\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\,\frac{1}{\left(d_{\sigma}\right)^{2}}\,\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}}{x_{\rm N}}\;,$ (173) $\displaystyle{\cal
W}_{\left(7\right)}\left(t\right)$ $\displaystyle=$
$\displaystyle-\frac{2}{15}\,\frac{1}{\left(d_{\sigma}\right)^{2}}\left(\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}}{x_{\rm
N}}\,\frac{1}{\left(d_{\sigma}\right)^{2}}+\frac{1}{2}\,\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}}{\left(x_{\rm N}\right)^{3}}\right),$ $\displaystyle{\cal
X}_{\left(3\right)}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\left(d_{\sigma}\right)^{2}}\left(x_{\rm
N}+\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm N}\right),$ (175)
$\displaystyle{\cal X}_{\left(5\right)}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{2}{3}\,\frac{1}{\left(d_{\sigma}\right)^{2}}\left(\frac{x_{\rm
N}+\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}}{\left(d_{\sigma}\right)^{2}}-\frac{1}{2}\,\frac{1}{x_{\rm N}}\right),$
(176) $\displaystyle{\cal X}_{\left(7\right)}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{8}{15}\frac{1}{\left(d_{\sigma}\right)^{2}}$ (177)
$\displaystyle\times\left(\frac{x_{\rm
N}+\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{\rm
N}}{\left(d_{\sigma}\right)^{4}}-\frac{1}{2}\frac{1}{x_{\rm
N}}\frac{1}{\left(d_{\sigma}\right)^{2}}-\frac{1}{8}\frac{1}{\left(x_{\rm
N}\right)^{3}}\right),$

where $\mbox{\boldmath$x$}_{\rm N}=\mbox{\boldmath$x$}_{\rm N}\left(t\right)$
and $x_{\rm N}=x_{\rm N}\left(t\right)$. One also needs the scalar functions
${\cal W}_{\left(3\right)}\left(t_{0}\right)\,,\,\dots\,,\,{\cal
X}_{\left(7\right)}\left(t_{0}\right)$ which one obtains from (172) - (177) by
replacing $\mbox{\boldmath$x$}_{\rm N}$ and $x_{\rm N}$ by
$\mbox{\boldmath$x$}_{0}$ and $x_{0}$, respectively, because $x_{\rm
N}\left(t_{0}\right)=\mbox{\boldmath$x$}_{0}$ and $x_{\rm
N}\left(t_{0}\right)=\mbox{\boldmath$x$}_{0}$; note that
$\mbox{\boldmath$d$}_{\sigma}$ is time-independent.

##  Appendix E Tensorial coefficients in (62) and (63)

It is convenient to introduce the impact vector,

$d$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}\times\left(\mbox{\boldmath$x$}\times\mbox{\boldmath$\sigma$}\right),$
(178)

where the spatial variable $x$ can either be the unperturbed light ray
$\mbox{\boldmath$x$}_{\rm N}$ in (59) or the light ray in 1PN approximation
$\mbox{\boldmath$x$}_{\rm 1PN}$ in (60); the spatial components of this impact
vector are $d^{i}$.

The tensorial coefficients of monopole-monopole term of the new representation
of light trajectory in (62) and (63) are

$\displaystyle U^{i}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{i}\;.$ (179) $\displaystyle
U^{i}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle d^{i}\;.$ (180)

The tensorial coefficients of monopole-quadrupole term of the new
representation of light trajectory in (62) and (63) are

$\displaystyle V^{i\,ab}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}\delta^{bi}\;.$ (181) $\displaystyle
V^{i\,ab}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle d^{a}\delta^{bi}\;.$ (182) $\displaystyle
V^{i\,ab}_{\left(3\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\sigma^{a}\sigma^{b}\sigma^{i}\;.$ (183) $\displaystyle
V^{i\,ab}_{\left(4\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\sigma^{a}d^{b}\sigma^{i}\;.$ (184) $\displaystyle
V^{i\,ab}_{\left(5\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle d^{a}d^{b}\sigma^{i}\;.$ (185) $\displaystyle
V^{i\,ab}_{\left(6\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle d^{a}d^{b}d^{i}\;.$ (186) $\displaystyle
V^{i\,ab}_{\left(7\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\sigma^{a}\sigma^{b}d^{i}\;.$ (187) $\displaystyle
V^{i\,ab}_{\left(8\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\sigma^{a}d^{b}d^{i}\;.$ (188)

The tensorial coefficients of quadrupole-quadrupole term of the new
representation of light trajectory in (63) are

$\displaystyle W^{i\,abcd}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}\sigma^{b}\delta^{di}\;.$ (189)
$\displaystyle W^{i\,abcd}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}d^{b}\delta^{di}\;.$ (190)
$\displaystyle W^{i\,abcd}_{\left(3\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}\sigma^{b}\sigma^{c}\delta^{di}\;.$
(191) $\displaystyle
W^{i\,abcd}_{\left(4\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\sigma^{a}\sigma^{b}d^{c}\delta^{di}\;.$ (192) $\displaystyle
W^{i\,abcd}_{\left(5\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\sigma^{a}d^{b}\sigma^{c}\delta^{di}\;.$ (193) $\displaystyle
W^{i\,abcd}_{\left(6\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\sigma^{a}d^{b}d^{c}\delta^{di}\;.$ (194) $\displaystyle
W^{i\,abcd}_{\left(7\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle d^{a}d^{b}\sigma^{c}\delta^{di}\;.$ (195) $\displaystyle
W^{i\,abcd}_{\left(8\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle d^{a}d^{b}d^{c}\delta^{di}\;.$ (196) $\displaystyle
W^{i\,abcd}_{\left(9\right)}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle\delta^{ac}\delta^{bd}\sigma^{i}\;.$ (197) $\displaystyle
W^{i\,abcd}_{\left(10\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}\sigma^{b}\sigma^{d}\sigma^{i}\;.$
(198) $\displaystyle
W^{i\,abcd}_{\left(11\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}\sigma^{b}d^{d}\sigma^{i}\;.$ (199)
$\displaystyle W^{i\,abcd}_{\left(12\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}d^{b}d^{d}\sigma^{i}\;.$ (200)
$\displaystyle W^{i\,abcd}_{\left(13\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\sigma^{a}\sigma^{b}\sigma^{c}\sigma^{d}\sigma^{i}\;.$ (201)
$\displaystyle W^{i\,abcd}_{\left(14\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\sigma^{a}\sigma^{b}\sigma^{c}d^{d}\sigma^{i}\;.$ (202)
$\displaystyle W^{i\,abcd}_{\left(15\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}\sigma^{b}d^{c}d^{d}\sigma^{i}\;.$
(203) $\displaystyle
W^{i\,abcd}_{\left(16\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}d^{b}\sigma^{c}d^{d}\sigma^{i}\;.$
(204) $\displaystyle
W^{i\,abcd}_{\left(17\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}d^{b}d^{c}d^{d}\sigma^{i}\;.$ (205)
$\displaystyle W^{i\,abcd}_{\left(18\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle d^{a}d^{b}d^{c}d^{d}\sigma^{i}\;.$ (206)
$\displaystyle W^{i\,abcd}_{\left(19\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}\delta^{bd}d^{i}\;.$ (207)
$\displaystyle W^{i\,abcd}_{\left(20\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}\sigma^{b}\sigma^{d}d^{i}\;.$ (208)
$\displaystyle W^{i\,abcd}_{\left(21\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}\sigma^{b}d^{d}d^{i}\;.$ (209)
$\displaystyle W^{i\,abcd}_{\left(22\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\delta^{ac}d^{b}d^{d}d^{i}\;.$ (210)
$\displaystyle W^{i\,abcd}_{\left(23\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\sigma^{a}\sigma^{b}\sigma^{c}\sigma^{d}d^{i}\;.$ (211)
$\displaystyle W^{i\,abcd}_{\left(24\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}\sigma^{b}\sigma^{c}d^{d}d^{i}\;.$
(212) $\displaystyle
W^{i\,abcd}_{\left(25\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}\sigma^{b}d^{c}d^{d}d^{i}\;.$ (213)
$\displaystyle W^{i\,abcd}_{\left(26\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}d^{b}\sigma^{c}d^{d}d^{i}\;.$ (214)
$\displaystyle W^{i\,abcd}_{\left(27\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\sigma^{a}d^{b}d^{c}d^{d}d^{i}\;.$ (215)
$\displaystyle W^{i\,abcd}_{\left(28\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle d^{a}d^{b}d^{c}d^{d}d^{i}\;.$ (216)

##  Appendix F Scalar functions in (62) and (63)

To simplify the notation, it is appropriate to introduce the following scalar
functions,

$\displaystyle a_{\left(n\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\left(x+\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}\right)^{n}\;,$
(217) $\displaystyle b_{\left(n\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{\left(x\right)^{n}}\;,$ (218)
$\displaystyle c_{\left(n\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}}{\left(x\right)^{n}}\;,$
(219) $\displaystyle d_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\ln\left(x-\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}\right),$
(220) $\displaystyle d_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\arctan\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}}{d}+\frac{\pi}{2}\;,$
(221) $\displaystyle d_{\left(3\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\arctan\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}}{d}\;,$
(222) $\displaystyle d_{\left(4\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}}{d}\left(\arctan\frac{\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}}{d}+\frac{\pi}{2}\right).$
(223)

Then, the scalar functions in the new representation in (62) and (63) can be
expressed in terms of these functions (217) - (223).

The scalar functions of the monopole term of the new representation in (62)
are given by

$\displaystyle{F}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle+2\,d_{\left(1\right)}\,.$ (224)
$\displaystyle{F}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-2\,\frac{a_{\left(1\right)}}{\left(d\right)^{2}}\,.$ (225)

The scalar functions of the quadrupole term of the new representation in (62)
are given by

$\displaystyle{G}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-2\,\frac{c_{\left(1\right)}}{\left(d\right)^{2}}\,.$ (226)
$\displaystyle{G}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+4\,\frac{a_{\left(1\right)}}{\left(d\right)^{4}}-2\,\frac{b_{\left(1\right)}}{\left(d\right)^{2}}\,.$
(227) $\displaystyle{G}_{\left(3\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+\frac{c_{\left(1\right)}}{\left(d\right)^{2}}+c_{\left(3\right)}\;.$
(228) $\displaystyle{G}_{\left(4\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-4\,\frac{a_{\left(1\right)}}{\left(d\right)^{4}}+2\,\frac{b_{\left(1\right)}}{\left(d\right)^{2}}+2\,b_{\left(3\right)}\;.$
(229)

$\displaystyle{G}_{\left(5\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-2\,\frac{c_{\left(1\right)}}{\left(d\right)^{4}}-\frac{c_{\left(3\right)}}{\left(d\right)^{2}}\,.$
(230) $\displaystyle{G}_{\left(6\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{8}{\left(d\right)^{6}}a_{\left(1\right)}+4\,\frac{b_{\left(1\right)}}{\left(d\right)^{4}}+\frac{b_{\left(3\right)}}{\left(d\right)^{2}}\,.$
(231) $\displaystyle{G}_{\left(7\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{2}{\left(d\right)^{4}}a_{\left(1\right)}+\frac{b_{\left(1\right)}}{\left(d\right)^{2}}-b_{\left(3\right)}\,.$
(232) $\displaystyle{G}_{\left(8\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+4\,\frac{c_{\left(1\right)}}{\left(d\right)^{4}}+2\,\frac{c_{\left(3\right)}}{\left(d\right)^{2}}\,.$
(233)

The scalar functions of the monopole-monopole term of the new representation
in (63) are given by

$\displaystyle{X}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+4\,\frac{a_{\left(1\right)}}{\left(d_{\sigma}\right)^{2}}+\frac{c_{\left(2\right)}}{4}-\frac{15}{4}\,\frac{d_{\left(3\right)}}{d}\,.$
(234) $\displaystyle{X}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+4\,\frac{a_{\left(2\right)}}{\left(d_{\sigma}\right)^{4}}+\frac{b_{\left(2\right)}}{4}-\frac{15}{4}\frac{d_{\left(4\right)}}{\left(d\right)^{2}}\,.$
(235)

These functions in combination with the coefficients (179) and (180) are in
agreement with Eq. (51) in Article_Zschocke1 .

The scalar functions of the monopole-quadrupole term of new representation in
(63) are given by

$\displaystyle{Y}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+12\,\frac{a_{\left(1\right)}}{\left(d\right)^{4}}-4\,\frac{b_{\left(1\right)}}{\left(d\right)^{2}}-\frac{93}{32}\,\frac{c_{\left(2\right)}}{\left(d\right)^{2}}-\frac{7}{16}\,c_{\left(4\right)}$
(236)
$\displaystyle-\frac{285}{32}\,\frac{d_{\left(3\right)}}{\left(d\right)^{3}}\,.$
$\displaystyle{Y}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-16\,\frac{a_{\left(2\right)}}{\left(d\right)^{6}}-\frac{91}{32}\,\frac{b_{\left(2\right)}}{\left(d\right)^{2}}-\frac{7}{16}\,b_{\left(4\right)}+4\,\frac{c_{\left(1\right)}}{\left(d\right)^{4}}$
(237)
$\displaystyle+\frac{465}{32}\,\frac{d_{\left(4\right)}}{\left(d\right)^{4}}\,.$
$\displaystyle{Y}_{\left(3\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-8\,\frac{a_{\left(1\right)}}{\left(d\right)^{4}}+2\,\frac{b_{\left(1\right)}}{\left(d\right)^{2}}+2\,b_{\left(3\right)}+\frac{29}{64}\,\frac{c_{\left(2\right)}}{\left(d\right)^{2}}$
(238)
$\displaystyle+\frac{111}{32}\,c_{\left(4\right)}-\frac{5}{8}\left(d\right)^{2}c_{\left(6\right)}+\frac{285}{64}\,\frac{d_{\left(3\right)}}{\left(d\right)^{3}}\,.$
$\displaystyle{Y}_{\left(4\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+16\,\frac{a_{\left(2\right)}}{\left(d\right)^{6}}+\frac{155}{32}\,\frac{b_{\left(2\right)}}{\left(d\right)^{2}}+\frac{199}{16}\,b_{\left(4\right)}-\frac{5}{4}\,\left(d\right)^{2}b_{\left(6\right)}$
(239)
$\displaystyle-8\,\frac{c_{\left(1\right)}}{\left(d\right)^{4}}-4\,\frac{c_{\left(3\right)}}{\left(d\right)^{2}}-\frac{465}{32}\,\frac{d_{\left(4\right)}}{\left(d\right)^{4}}\,.$
$\displaystyle{Y}_{\left(5\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+8\,\frac{a_{\left(1\right)}}{\left(d\right)^{6}}-4\,\frac{b_{\left(1\right)}}{\left(d\right)^{4}}-2\,\frac{b_{\left(3\right)}}{\left(d\right)^{2}}-\frac{209}{64}\,\frac{c_{\left(2\right)}}{\left(d\right)^{4}}$
(240)
$\displaystyle-\frac{91}{32}\,\frac{c_{\left(4\right)}}{\left(d\right)^{2}}+\frac{5}{8}\,c_{\left(6\right)}-\frac{465}{64}\,\frac{d_{\left(3\right)}}{\left(d\right)^{5}}\,.$
$\displaystyle{Y}_{\left(6\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+48\,\frac{a_{\left(2\right)}}{\left(d\right)^{8}}+\frac{263}{64}\,\frac{b_{\left(2\right)}}{\left(d\right)^{4}}+\frac{883}{32}\,\frac{b_{\left(4\right)}}{\left(d\right)^{2}}+\frac{5}{8}\,b_{\left(6\right)}$
(241)
$\displaystyle-16\,\frac{c_{\left(1\right)}}{\left(d\right)^{6}}\,-4\,\frac{c_{\left(3\right)}}{\left(d\right)^{4}}-\frac{2325}{64}\,\frac{d_{\left(4\right)}}{\left(d\right)^{6}}\,.$
$\displaystyle{Y}_{\left(7\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+16\,\frac{a_{\left(2\right)}}{\left(d\right)^{6}}+\frac{235}{64}\,\frac{b_{\left(2\right)}}{\left(d\right)^{2}}-\frac{71}{32}\,b_{\left(4\right)}-\frac{5}{8}\left(d\right)^{2}b_{\left(6\right)}$
(242)
$\displaystyle+4\,\frac{c_{\left(3\right)}}{\left(d\right)^{2}}-\frac{855}{64}\,\frac{d_{\left(4\right)}}{\left(d\right)^{4}}\,.$
$\displaystyle{Y}_{\left(8\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-32\,\frac{a_{\left(1\right)}}{\left(d\right)^{6}}+12\,\frac{b_{\left(1\right)}}{\left(d\right)^{4}}+8\,\frac{b_{\left(3\right)}}{\left(d\right)^{2}}+\frac{81}{32}\,\frac{c_{\left(2\right)}}{\left(d\right)^{4}}$
(243)
$\displaystyle+\frac{91}{16}\,\frac{c_{\left(4\right)}}{\left(d\right)^{2}}+\frac{5}{4}\,c_{\left(6\right)}+\frac{465}{32}\,\frac{d_{\left(3\right)}}{\left(d\right)^{5}}\,.$

The scalar functions of the quadrupole-quadrupole term of the new
representation in (63) are given by

$\displaystyle{Z}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+8\,\frac{a_{\left(1\right)}}{\left(d\right)^{6}}-8\,\frac{b_{\left(1\right)}}{\left(d\right)^{4}}-\frac{327}{128}\,\frac{c_{\left(2\right)}}{\left(d\right)^{4}}-\frac{7}{192}\,\frac{c_{\left(4\right)}}{\left(d\right)^{2}}+\frac{13}{48}\,c_{\left(6\right)}+\frac{185}{128}\frac{d_{\left(3\right)}}{\left(d\right)^{5}}\,.$
(244) $\displaystyle{Z}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-16\,\frac{a_{\left(2\right)}}{\left(d\right)^{8}}-\frac{985}{384}\,\frac{b_{\left(2\right)}}{\left(d\right)^{4}}-\frac{5}{192}\,\frac{b_{\left(4\right)}}{\left(d\right)^{2}}+\frac{13}{48}\,b_{\left(6\right)}+8\,\frac{c_{\left(1\right)}}{\left(d\right)^{6}}+\frac{985}{128}\,\frac{d_{\left(4\right)}}{\left(d\right)^{6}}\,.$
(245) $\displaystyle{Z}_{\left(3\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+4\,\frac{a_{\left(1\right)}}{\left(d\right)^{6}}+4\,\frac{b_{\left(1\right)}}{\left(d\right)^{4}}-\frac{2103}{512}\,\frac{c_{\left(2\right)}}{\left(d\right)^{4}}+\frac{451}{256}\,\frac{c_{\left(4\right)}}{\left(d\right)^{2}}+\frac{23}{64}\,c_{\left(6\right)}+\frac{9}{32}\left(d\right)^{2}c_{\left(8\right)}-\frac{5175}{512}\,\frac{d_{\left(3\right)}}{\left(d\right)^{5}}\,.$
(246) $\displaystyle{Z}_{\left(4\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-16\,\frac{a_{\left(2\right)}}{\left(d\right)^{8}}-\frac{27019}{1536}\,\frac{b_{\left(2\right)}}{\left(d\right)^{4}}+\frac{1585}{768}\,\frac{b_{\left(4\right)}}{\left(d\right)^{2}}+\frac{5}{96}\,b_{\left(6\right)}+\frac{9}{32}\left(d\right)^{2}b_{\left(8\right)}+20\,\frac{c_{\left(1\right)}}{\left(d\right)^{6}}-8\,\frac{c_{\left(3\right)}}{\left(d\right)^{4}}+\frac{5515}{512}\,\frac{d_{\left(4\right)}}{\left(d\right)^{6}}\,.$
(247) $\displaystyle{Z}_{\left(5\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+16\,\frac{a_{\left(2\right)}}{\left(d\right)^{8}}-\frac{3859}{768}\,\frac{b_{\left(2\right)}}{\left(d\right)^{4}}+\frac{1609}{384}\,\frac{b_{\left(4\right)}}{\left(d\right)^{2}}+\frac{79}{96}\,b_{\left(6\right)}+\frac{9}{16}\,\left(d\right)^{2}b_{\left(8\right)}-8\,\frac{c_{\left(1\right)}}{\left(d\right)^{6}}-\frac{2285}{256}\,\frac{d_{\left(4\right)}}{\left(d\right)^{6}}\,.$
(248) $\displaystyle{Z}_{\left(6\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-16\,\frac{a_{\left(1\right)}}{\left(d\right)^{8}}+24\,\frac{b_{\left(1\right)}}{\left(d\right)^{6}}-16\,\frac{b_{\left(3\right)}}{\left(d\right)^{4}}+\frac{6381}{256}\,\frac{c_{\left(2\right)}}{\left(d\right)^{6}}-\frac{2323}{384}\,\frac{c_{\left(4\right)}}{\left(d\right)^{4}}-\frac{119}{96}\,\frac{c_{\left(6\right)}}{\left(d\right)^{2}}-\frac{9}{16}\,c_{\left(8\right)}+\frac{2285}{256}\,\frac{d_{\left(3\right)}}{\left(d\right)^{7}}\,.$
(249) $\displaystyle{Z}_{\left(7\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+16\,\frac{a_{\left(1\right)}}{\left(d\right)^{8}}+16\,\frac{b_{\left(1\right)}}{\left(d\right)^{6}}-\frac{1419}{512}\,\frac{c_{\left(2\right)}}{\left(d\right)^{6}}-\frac{2443}{768}\,\frac{c_{\left(4\right)}}{\left(d\right)^{4}}-\frac{143}{192}\,\frac{c_{\left(6\right)}}{\left(d\right)^{2}}-\frac{9}{32}\,c_{\left(8\right)}-\frac{5515}{512}\,\frac{d_{\left(3\right)}}{\left(d\right)^{7}}\,.$
(250) $\displaystyle{Z}_{\left(8\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+\frac{4831}{512}\,\frac{b_{\left(2\right)}}{\left(d\right)^{6}}-\frac{877}{256}\,\frac{b_{\left(4\right)}}{\left(d\right)^{4}}-\frac{43}{64}\,\frac{b_{\left(6\right)}}{\left(d\right)^{2}}-\frac{9}{32}\,b_{\left(8\right)}+8\,\frac{c_{\left(3\right)}}{\left(d\right)^{6}}-\frac{2205}{512}\,\frac{d_{\left(4\right)}}{\left(d\right)^{8}}\,.$
(251) $\displaystyle{Z}_{\left(9\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+\frac{1}{128}\,\frac{c_{\left(2\right)}}{\left(d\right)^{4}}+\frac{1}{192}\,\frac{c_{\left(4\right)}}{\left(d\right)^{2}}+\frac{5}{48}\,c_{\left(6\right)}+\frac{1}{128}\,\frac{d_{\left(3\right)}}{\left(d\right)^{5}}\,.$
(252) $\displaystyle{Z}_{\left(10\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-8\,\frac{a_{\left(1\right)}}{\left(d\right)^{6}}+8\,\frac{b_{\left(1\right)}}{\left(d\right)^{4}}+\frac{839}{256}\,\frac{c_{\left(2\right)}}{\left(d\right)^{4}}+\frac{199}{384}\,\frac{c_{\left(4\right)}}{\left(d\right)^{2}}-\frac{85}{96}\,c_{\left(6\right)}+\frac{15}{16}\left(d\right)^{2}c_{\left(8\right)}-\frac{185}{256}\,\frac{d_{\left(3\right)}}{\left(d\right)^{5}}\,.$
(253) $\displaystyle{Z}_{\left(11\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+16\,\frac{a_{\left(2\right)}}{\left(d\right)^{8}}+\frac{2521}{384}\,\frac{b_{\left(2\right)}}{\left(d\right)^{4}}+\frac{197}{192}\,\frac{b_{\left(4\right)}}{\left(d\right)^{2}}-\frac{85}{48}\,b_{\left(6\right)}+\frac{15}{8}\left(d\right)^{2}b_{\left(8\right)}-8\,\frac{c_{\left(1\right)}}{\left(d\right)^{6}}-\frac{985}{128}\,\frac{d_{\left(4\right)}}{\left(d\right)^{6}}\,.$
(254) $\displaystyle{Z}_{\left(12\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{985}{256}\,\frac{c_{\left(2\right)}}{\left(d\right)^{6}}-\frac{217}{384}\,\frac{c_{\left(4\right)}}{\left(d\right)^{4}}-\frac{5}{96}\,\frac{c_{\left(6\right)}}{\left(d\right)^{2}}-\frac{15}{16}\,c_{\left(8\right)}-\frac{985}{256}\,\frac{d_{\left(3\right)}}{\left(d\right)^{7}}\,.$
(255) $\displaystyle{Z}_{\left(13\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-4\,\frac{a_{\left(1\right)}}{\left(d\right)^{6}}-4\,\frac{b_{\left(1\right)}}{\left(d\right)^{4}}+14\,\frac{b_{\left(3\right)}}{\left(d\right)^{2}}+\frac{3237}{2048}\,\frac{c_{\left(2\right)}}{\left(d\right)^{4}}-\frac{969}{1024}\,\frac{c_{\left(4\right)}}{\left(d\right)^{2}}+\frac{395}{256}\,c_{\left(6\right)}-\frac{369}{128}\left(d\right)^{2}c_{\left(8\right)}+\frac{15}{16}\left(d\right)^{4}c_{\left(10\right)}+\frac{15525}{2048}\,\frac{d_{\left(3\right)}}{\left(d\right)^{5}}\,.$
$\displaystyle{Z}_{\left(14\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+\frac{8507}{512}\,\frac{b_{\left(2\right)}}{\left(d\right)^{4}}-\frac{1217}{256}\,\frac{b_{\left(4\right)}}{\left(d\right)^{2}}+\frac{393}{64}\,b_{\left(6\right)}-\frac{369}{32}\left(d\right)^{2}b_{\left(8\right)}+\frac{15}{4}\left(d\right)^{4}b_{\left(10\right)}-12\,\frac{c_{\left(1\right)}}{\left(d\right)^{6}}+8\,\frac{c_{\left(3\right)}}{\left(d\right)^{4}}-\frac{945}{512}\,\frac{d_{\left(4\right)}}{\left(d\right)^{6}}\,.$
(257) $\displaystyle{Z}_{\left(15\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-16\,\frac{a_{\left(1\right)}}{\left(d\right)^{8}}-\frac{2677}{1024}\,\frac{c_{\left(2\right)}}{\left(d\right)^{6}}+\frac{5515}{1536}\,\frac{c_{\left(4\right)}}{\left(d\right)^{4}}+\frac{335}{384}\,\frac{c_{\left(6\right)}}{\left(d\right)^{2}}+\frac{249}{64}\,c_{\left(8\right)}-\frac{15}{8}\left(d\right)^{2}c_{\left(10\right)}+\frac{5515}{1024}\,\frac{d_{\left(3\right)}}{\left(d\right)^{7}}\,.$
(258) $\displaystyle{Z}_{\left(16\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+16\,\frac{a_{\left(1\right)}}{\left(d\right)^{8}}-24\,\frac{b_{\left(1\right)}}{\left(d\right)^{6}}+16\,\frac{b_{\left(3\right)}}{\left(d\right)^{4}}-\frac{10477}{512}\,\frac{c_{\left(2\right)}}{\left(d\right)^{6}}+\frac{5395}{768}\,\frac{c_{\left(4\right)}}{\left(d\right)^{4}}+\frac{311}{192}\,\frac{c_{\left(6\right)}}{\left(d\right)^{2}}+\frac{249}{32}\,c_{\left(8\right)}-\frac{15}{4}\left(d\right)^{2}c_{\left(10\right)}-\frac{2285}{512}\,\frac{d_{\left(3\right)}}{\left(d\right)^{7}}\,.$

$\displaystyle{Z}_{\left(17\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{7667}{512}\,\frac{b_{\left(2\right)}}{\left(d\right)^{6}}+\frac{2153}{256}\,\frac{b_{\left(4\right)}}{\left(d\right)^{4}}+\frac{143}{64}\,\frac{b_{\left(6\right)}}{\left(d\right)^{2}}+\frac{261}{32}\,b_{\left(8\right)}-\frac{15}{4}\left(d\right)^{2}b_{\left(10\right)}-8\,\frac{c_{\left(3\right)}}{\left(d\right)^{6}}+\frac{2205}{512}\,\frac{d_{\left(4\right)}}{\left(d\right)^{8}}\,.$
(260) $\displaystyle{Z}_{\left(18\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+\frac{2205}{2048}\,\frac{c_{\left(2\right)}}{\left(d\right)^{8}}-\frac{3361}{1024}\,\frac{c_{\left(4\right)}}{\left(d\right)^{6}}-\frac{365}{256}\,\frac{c_{\left(6\right)}}{\left(d\right)^{4}}-\frac{129}{128}\,\frac{c_{\left(8\right)}}{\left(d\right)^{2}}+\frac{15}{16}\,c_{\left(10\right)}+\frac{2205}{2048}\,\frac{d_{\left(3\right)}}{\left(d\right)^{9}}\,.$
(261) $\displaystyle{Z}_{\left(19\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{5}{384}\,\frac{b_{\left(2\right)}}{\left(d\right)^{4}}-\frac{1}{192}\,\frac{b_{\left(4\right)}}{\left(d\right)^{2}}+\frac{5}{48}\,b_{\left(6\right)}+\frac{5}{128}\,\frac{d_{\left(4\right)}}{\left(d\right)^{6}}\,.$
(262) $\displaystyle{Z}_{\left(20\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{3997}{768}\,\frac{b_{\left(2\right)}}{\left(d\right)^{4}}-\frac{569}{384}\,\frac{b_{\left(4\right)}}{\left(d\right)^{2}}-\frac{95}{96}\,b_{\left(6\right)}+\frac{15}{16}\left(d\right)^{2}b_{\left(8\right)}+\frac{925}{256}\,\frac{d_{\left(4\right)}}{\left(d\right)^{6}}\,.$
(263) $\displaystyle{Z}_{\left(21\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-48\,\frac{a_{\left(1\right)}}{\left(d\right)^{8}}+40\,\frac{b_{\left(1\right)}}{\left(d\right)^{6}}+8\,\frac{b_{\left(3\right)}}{\left(d\right)^{4}}+\frac{985}{128}\,\frac{c_{\left(2\right)}}{\left(d\right)^{6}}+\frac{601}{192}\,\frac{c_{\left(4\right)}}{\left(d\right)^{4}}+\frac{5}{48}\,\frac{c_{\left(6\right)}}{\left(d\right)^{2}}-\frac{15}{8}c_{\left(8\right)}+\frac{985}{128}\,\frac{d_{\left(3\right)}}{\left(d\right)^{7}}\,.$
(264) $\displaystyle{Z}_{\left(22\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+64\,\frac{a_{\left(2\right)}}{\left(d\right)^{10}}+\frac{13039}{768}\,\frac{b_{\left(2\right)}}{\left(d\right)^{6}}+\frac{611}{384}\,\frac{b_{\left(4\right)}}{\left(d\right)^{4}}+\frac{5}{96}\,\frac{b_{\left(6\right)}}{\left(d\right)^{2}}-\frac{15}{16}\,b_{\left(8\right)}-\frac{48}{\left(d\right)^{8}}\,c_{\left(1\right)}-\frac{6895}{256}\,\frac{d_{\left(4\right)}}{\left(d\right)^{8}}\,.$
(265) $\displaystyle{Z}_{\left(23\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+12\frac{a_{\left(2\right)}}{\left(d\right)^{8}}+\frac{31153}{2048}\frac{b_{\left(2\right)}}{\left(d\right)^{4}}-\frac{2371}{1024}\frac{b_{\left(4\right)}}{\left(d\right)^{2}}-\frac{37}{256}b_{\left(6\right)}-\frac{111}{128}\left(d\right)^{2}b_{\left(8\right)}+\frac{15}{16}\left(d\right)^{4}b_{\left(10\right)}-4\frac{c_{\left(1\right)}}{\left(d\right)^{6}}+8\frac{c_{\left(3\right)}}{\left(d\right)^{4}}-\frac{25875}{2048}\frac{d_{\left(4\right)}}{\left(d\right)^{6}}.$
$\displaystyle{Z}_{\left(24\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+24\,\frac{a_{\left(1\right)}}{\left(d\right)^{8}}-36\,\frac{b_{\left(1\right)}}{\left(d\right)^{6}}+16\,\frac{b_{\left(3\right)}}{\left(d\right)^{4}}-\frac{11343}{512}\,\frac{c_{\left(2\right)}}{\left(d\right)^{6}}+\frac{59}{256}\,\frac{c_{\left(4\right)}}{\left(d\right)^{4}}-\frac{65}{64}\,\frac{c_{\left(6\right)}}{\left(d\right)^{2}}-\frac{9}{32}\,c_{\left(8\right)}-\frac{15}{4}\left(d\right)^{2}c_{\left(10\right)}+\frac{945}{512}\,\frac{d_{\left(3\right)}}{\left(d\right)^{7}}\,.$
$\displaystyle{Z}_{\left(25\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+64\frac{a_{\left(2\right)}}{\left(d\right)^{10}}+\frac{93133}{3072}\frac{b_{\left(2\right)}}{\left(d\right)^{6}}-\frac{11479}{1536}\frac{b_{\left(4\right)}}{\left(d\right)^{4}}-\frac{433}{384}\frac{b_{\left(6\right)}}{\left(d\right)^{2}}-\frac{9}{64}b_{\left(8\right)}-\frac{15}{8}\left(d\right)^{2}b_{\left(10\right)}-48\frac{c_{\left(1\right)}}{\left(d\right)^{8}}+16\frac{c_{\left(3\right)}}{\left(d\right)^{6}}-\frac{19405}{1024}\frac{d_{\left(4\right)}}{\left(d\right)^{8}}.$
$\displaystyle{Z}_{\left(26\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-64\,\frac{a_{\left(2\right)}}{\left(d\right)^{10}}+\frac{5893}{1536}\,\frac{b_{\left(2\right)}}{\left(d\right)^{6}}-\frac{5887}{768}\,\frac{b_{\left(4\right)}}{\left(d\right)^{4}}-\frac{457}{192}\,\frac{b_{\left(6\right)}}{\left(d\right)^{2}}-\frac{9}{32}\,b_{\left(8\right)}-\frac{15}{4}\left(d\right)^{2}b_{\left(10\right)}+48\,\frac{c_{\left(1\right)}}{\left(d\right)^{8}}+\frac{6395}{512}\,\frac{d_{\left(4\right)}}{\left(d\right)^{8}}\,.$
(269) $\displaystyle{Z}_{\left(27\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-48\,\frac{b_{\left(1\right)}}{\left(d\right)^{8}}+\frac{48}{\left(d\right)^{6}}\,b_{\left(3\right)}-\frac{26781}{512}\,\frac{c_{\left(2\right)}}{\left(d\right)^{8}}+\frac{7457}{256}\,\frac{c_{\left(4\right)}}{\left(d\right)^{6}}+\frac{493}{64}\,\frac{c_{\left(6\right)}}{\left(d\right)^{4}}+\frac{129}{32}\,\frac{c_{\left(8\right)}}{\left(d\right)^{2}}+\frac{15}{4}\,c_{\left(10\right)}-\frac{2205}{512}\,\frac{d_{\left(3\right)}}{\left(d\right)^{9}}\,.$
(270) $\displaystyle{Z}_{\left(28\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{47575}{2048}\,\frac{b_{\left(2\right)}}{\left(d\right)^{8}}+\frac{10965}{1024}\,\frac{b_{\left(4\right)}}{\left(d\right)^{6}}-\frac{445}{256}\,\frac{b_{\left(6\right)}}{\left(d\right)^{4}}+\frac{129}{128}\,\frac{b_{\left(8\right)}}{\left(d\right)^{2}}+\frac{15}{16}\,b_{\left(10\right)}-24\,\frac{c_{\left(3\right)}}{\left(d\right)^{8}}+\frac{19845}{2048}\,\frac{d_{\left(4\right)}}{\left(d\right)^{10}}\,.$
(271)

##  Appendix G Agreement of (55)-(LABEL:S_I_2PN) and (62)-(63)

In this Appendix some basic ideas are presented about how to get from the old
representation (54) with (55) and (LABEL:S_I_2PN), to the new representation
(61) with (62) and (63). For that demonstration one needs the following
relations which are valid up to terms of the order ${\cal
O}\left(c^{-4}\right)$:

$\displaystyle\mbox{\boldmath$x$}_{\rm 1PN}\left(t\right)$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$x$}_{\rm
N}\left(t\right)+\Delta\mbox{\boldmath$x$}_{\rm 1PN}\left(t,t_{0}\right)\;,$
(272) $\displaystyle x_{\rm 1PN}\left(t\right)$ $\displaystyle=$
$\displaystyle x_{\rm N}\left(t\right)+\frac{\mbox{\boldmath$x$}_{\rm
N}\left(t\right)\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(t,t_{0}\right)}{x_{\rm N}\left(t\right)}\;,$ (273)
$\displaystyle\frac{1}{\left(x_{\rm 1PN}\left(t\right)\right)^{n}}$
$\displaystyle=$ $\displaystyle\frac{1}{\left(x_{\rm
N}\left(t\right)\right)^{n}}-\frac{n}{\left(x_{\rm
N}\left(t\right)\right)^{n}}\,\frac{\mbox{\boldmath$x$}_{\rm
N}\left(t\right)\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(t,t_{0}\right)}{\left(x_{\rm N}\left(t\right)\right)^{2}}\;.$

Let us notice here that

$\displaystyle\mbox{\boldmath$x$}_{0}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$x$}_{\rm
N}\left(t_{0}\right)=\mbox{\boldmath$x$}_{\rm 1PN}\left(t_{0}\right)$ (275)

which follow from (5) and (13). Furthermore, one encounters the following
impact vector

$\displaystyle\hat{\mbox{\boldmath$d$}_{\sigma}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}\times\left(\mbox{\boldmath$x$}_{\rm
1PN}\left(t\right)\times\mbox{\boldmath$\sigma$}\right)$ (276)

and its absolute value
$\hat{d}_{\sigma}=|\hat{\mbox{\boldmath$d$}_{\sigma}}|$. This impact vector
$\hat{\mbox{\boldmath$d$}_{\sigma}}$ in (276) is related to the impact vector
$\mbox{\boldmath$d$}_{\sigma}$ in (58) as follows (up to terms of the order
${\cal O}\left(c^{-4}\right)$):

$\displaystyle\hat{\mbox{\boldmath$d$}_{\sigma}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$d$}_{\sigma}+\mbox{\boldmath$\sigma$}\times\left(\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(t,t_{0}\right)\times\mbox{\boldmath$\sigma$}\right),$ (277)
$\displaystyle\hat{d}_{\sigma}$ $\displaystyle=$ $\displaystyle
d_{\sigma}+\frac{\mbox{\boldmath$d$}_{\sigma}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(t,t_{0}\right)}{d_{\sigma}}\;,$ (278)
$\displaystyle\frac{1}{(\hat{d}_{\sigma})^{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(d_{\sigma}\right)^{n}}-\frac{n}{\left(d_{\sigma}\right)^{n}}\,\frac{\mbox{\boldmath$d$}_{\sigma}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(t,t_{0}\right)}{\left(d_{\sigma}\right)^{2}}\;.$ (279)

In relations (272) - (LABEL:Appendix_x_C) as well as (277) - (279) one needs
the light ray perturbation in 1PN approximation,
$\Delta\mbox{\boldmath$x$}_{\rm 1PN}\left(t,t_{0}\right)$, where it is
advantageous to take Eqs. (282) and (283).

The entire procedure is separated into four steps:

First step: The 1PN terms in Eq. (55) contain $6$ tensorial coefficients given
in (165) - (170):

$\displaystyle{\cal A}^{i}_{\left(3\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right)\;,\;{\cal B}^{i}_{\left(3\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right)\;,\;{\cal C}^{i\,ab}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right)\;,\;{\cal D}^{i\,ab}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right).$ (280)

These tensorial coefficients (280) consist of $10$ different tensors as given
by (179) - (188) with the argument
$\mbox{\boldmath$x$}=\mbox{\boldmath$x$}_{\rm N}$:

$\displaystyle U^{i}_{\left(1\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right)\;,\;U^{i}_{\left(2\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right)\;,\;V^{i\,ab}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right).$ (281)

Accordingly, one may rewrite the 1PN terms in Eq. (55) in terms of these $10$
individual tensors:

$\displaystyle\Delta{x}^{i}_{\rm 1PN}\left(t,t_{0}\right)$ $\displaystyle=$
$\displaystyle\Delta{x}^{i}_{\rm 1PN}\left(t\right)-\Delta{x}^{i}_{\rm
1PN}\left(t_{0}\right)$ (282)

with

$\displaystyle\Delta{x}^{i}_{\rm 1PN}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{GM}{c^{2}}\sum\limits_{n=1}^{2}\left({U}^{i}_{\left(n\right)}\,{F}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)$ (283)
$\displaystyle+\frac{G\hat{M}_{ab}}{c^{2}}\sum\limits_{n=1}^{8}\left({V}^{i\,ab}_{\left(n\right)}\,{G}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right),$

where the scalar functions ${F}_{\left(n\right)}$ and ${G}_{\left(n\right)}$
are given by Eqs. (224) - (225) and Eqs. (226) - (233), respectively, where
the argument $\mbox{\boldmath$x$}=\mbox{\boldmath$x$}_{\rm N}$. One may easily
show that (282) with (283) is identical with (55).

Second step: Similarly, the 2PN terms in Eq. (LABEL:S_I_2PN) contain $51$
tensorial coefficients given by Eqs. (E28) - (E39) and Eqs. (E41) - (E65) as
well as Eqs. (E67) - (E87) in Zschocke_Quadrupole_1 :

$\displaystyle{\cal E}^{i}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right),{\cal F}^{i}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right),{\cal G}^{i}_{\left(5\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right),{\cal H}^{i}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right),$ $\displaystyle{\cal
K}^{i\,ab}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm N}\right),{\cal
L}^{i\,ab}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm N}\right),{\cal
M}^{i\,ab}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm N}\right),{\cal
N}^{i\,ab}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm N}\right),$
$\displaystyle{\cal
P}^{i\,abcd}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm N}\right),{\cal
Q}^{i\,abcd}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm N}\right).$ (284)

These $51$ tensorial coefficients (284) consist of $38$ different tensors
given by (179) - (188) and (189) - (216):

$\displaystyle U^{i}_{\left(1\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right),\;U^{i}_{\left(2\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right),\;V^{i\,ab}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right),\;W^{i\,abcd}_{\left(n\right)}\left(\mbox{\boldmath$x$}_{\rm
N}\right).$ (285)

Accordingly, one may rewrite the 2PN terms in Eq. (LABEL:S_I_2PN) in terms of
these $38$ individual tensors:

$\displaystyle\Delta{x}^{i}_{\rm 2PN}\left(t,t_{0}\right)$ $\displaystyle=$
$\displaystyle\Delta{x}^{i}_{\rm 2PN}\left(t\right)-\Delta{x}^{i}_{\rm
2PN}\left(t_{0}\right)$ (286)

with

$\displaystyle\Delta{x}^{i}_{\rm 2PN}\left(t\right)$ $\displaystyle=$
$\displaystyle+\frac{G^{2}M^{2}}{c^{4}}\sum\limits_{n=1}^{2}\left(U^{i}_{\left(n\right)}\,{\tilde{X}}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)$ (287)
$\displaystyle+\frac{G^{2}M\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{8}\left(V^{i\,ab}_{\left(n\right)}\,{\tilde{Y}}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)$
$\displaystyle+\frac{G^{2}\hat{M}_{ab}\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{28}\left(W^{i\,abcd}_{\left(n\right)}\,{\tilde{Z}}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right).$

The scalar functions in (287) can be deduced just by inserting these $51$
tensorial coefficients (284) into (LABEL:S_I_2PN) and then combining all those
scalar terms belonging to one and the same tensorial coefficient in (285).
However, these scalar functions are an intermediate step and will not be given
in their explicit form here, in order to simplify the representation. It is
noticed again that (286) and (287) is identical with (LABEL:S_I_2PN).

The form of (286) and (287) resembles already the structure of (62) and (63),
respectively. However, the arguments in (286) are the unperturbed light rays,
$\mbox{\boldmath$x$}_{\rm N}$, while in (62) the arguments are the light rays
in 1PN approximation, $\mbox{\boldmath$x$}_{\rm 1PN}$. Furthermore, the scalar
functions ${\tilde{X}}_{\left(n\right)}$, ${\tilde{Y}}_{\left(n\right)}$,
${\tilde{Z}}_{\left(n\right)}$ in (287) are not identical with the scalar
functions ${X}_{\left(n\right)}$, ${Y}_{\left(n\right)}$,
${Z}_{\left(n\right)}$ in (63). In order to arrive at (62) and (63) two
further steps are necessary.

Third step: In order to arrive at (62) and (63) the argument in the tensorial
coefficients as well as in the scalar functions in (283) have to be replaced
by the light ray in 1PN approximation. Then one obtains:

$\displaystyle\Delta{x}^{i}_{\rm 1PN}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{GM}{c^{2}}\sum\limits_{n=1}^{2}\left({U}^{i}_{\left(n\right)}\,{F}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
1PN}\right)$ (288)
$\displaystyle+\frac{G\hat{M}_{ab}}{c^{2}}\sum\limits_{n=1}^{8}\left({V}^{i\,ab}_{\left(n\right)}\,{G}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
1PN}\right)+\delta{x}^{i}_{\rm 2PN}\,,$

where $\delta{x}^{i}_{\rm 2PN}$ is just the difference (283) minus (288):

$\displaystyle\delta{x}^{i}_{\rm 2PN}=$
$\displaystyle+\frac{GM}{c^{2}}\sum\limits_{n=1}^{2}\bigg{[}\left({U}^{i}_{\left(n\right)}\,{F}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)-\left({U}^{i}_{\left(n\right)}\,{F}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
1PN}\right)\bigg{]}$
$\displaystyle+\frac{G\hat{M}_{ab}}{c^{2}}\bigg{[}\sum\limits_{n=1}^{8}\left({V}^{i\,ab}_{\left(n\right)}\,{G}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)-\left({V}^{i\,ab}_{\left(n\right)}\,{G}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
1PN}\right)\bigg{]}.$ (289)

Eq. (288) is identical with (283).

Fourth step: In order to determine the expression in (289), one has to perform
a series expansion of those terms in (289) having as argument the light ray in
1PN approximation. For that calculation one needs the same relations as given
previously by Eqs. (272) - (LABEL:Appendix_x_C) and Eqs. (277) - (279).

The determination of $\delta{x}^{i}_{\rm 2PN}$ in (289) has been assisted by
the computer algebra system Maple Maple . One finally arrives at the following
form,

$\displaystyle\delta{x}^{i}_{\rm 2PN}$ $\displaystyle=$
$\displaystyle+\frac{G^{2}M^{2}}{c^{4}}\sum\limits_{n=1}^{2}\left(U^{i}_{\left(n\right)}\,{\hat{X}}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)$ (290)
$\displaystyle+\frac{G^{2}M\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{8}\left(V^{i\,ab}_{\left(n\right)}\,{\hat{Y}}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)$
$\displaystyle+\frac{G^{2}\hat{M}_{ab}\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{28}\left(W^{i\,abcd}_{\left(n\right)}\,{\hat{Z}}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right),$

which is separated into three terms proportional to monopole-monopole,
monopole-quadrupole and quadrupole-quadrupole. The tensorial coefficients are
defined by (179) - (180), (181) - (188), and (189) - (216), respectively. The
scalar functions in (290) are an intermediate step and will not be given in
their explicit form here, in favor of a clear representation.

The term $\delta{x}^{i}_{\rm 2PN}$, defined by Eq. (289) and determined by Eq.
(290), is obviously of second post-Newtonian order and should, therefore, be
added to (287) rather than (288). Accordingly, the sum of (287) and (288) can
be written in the form

$\displaystyle\mbox{\boldmath$x$}_{\rm 2PN}\left(t\right)$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$x$}_{0}+c\left(t-t_{0}\right)\mbox{\boldmath$\sigma$}+\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(t\right)-\Delta\mbox{\boldmath$x$}_{\rm 1PN}\left(t_{0}\right)$
(291) $\displaystyle+\,\Delta\mbox{\boldmath$x$}_{\rm
2PN}\left(t\right)-\Delta\mbox{\boldmath$x$}_{\rm 2PN}\left(t_{0}\right),$
$\displaystyle\Delta{x}^{i}_{\rm 1PN}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{GM}{c^{2}}\sum\limits_{n=1}^{2}\left({U}^{i}_{\left(n\right)}\,{F}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
1PN}\right)$ (292)
$\displaystyle+\frac{G\hat{M}_{ab}}{c^{2}}\sum\limits_{n=1}^{8}\left({V}^{i\,ab}_{\left(n\right)}\,{G}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
1PN}\right),$ $\displaystyle\Delta{x}^{i}_{\rm 2PN}\left(t\right)$
$\displaystyle=$
$\displaystyle\frac{G^{2}M^{2}}{c^{4}}\sum\limits_{n=1}^{2}\left(U^{i}_{\left(n\right)}\,{X}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)$ (293)
$\displaystyle+\frac{G^{2}M\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{8}\left(V^{i\,ab}_{\left(n\right)}\,{Y}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right)$
$\displaystyle+\frac{G^{2}\hat{M}_{ab}\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{28}\left(W^{i\,abcd}_{\left(n\right)}\,{Z}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}_{\rm
N}\right),$

where, by taking account of (287) and(290), the new scalar functions

$\displaystyle{X}_{\left(n\right)}$ $\displaystyle=$
$\displaystyle{\tilde{X}}_{\left(n\right)}+{\hat{X}}_{\left(n\right)}\;,$
(294) $\displaystyle{Y}_{\left(n\right)}$ $\displaystyle=$
$\displaystyle{\tilde{Y}}_{\left(n\right)}+{\hat{Y}}_{\left(n\right)}\;,$
(295) $\displaystyle{Z}_{\left(n\right)}$ $\displaystyle=$
$\displaystyle{\tilde{Z}}_{\left(n\right)}+{\hat{Z}}_{\left(n\right)}\;,$
(296)

have been introduced. The solution (291) with (292) and (293) agrees with
expression (61) with (62) and (63), where the scalar functions (294) - (296)
are given by Eqs. (234) - (271) in their explicit form.

##  Appendix H Calculation of
$\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm 2PN}$ in terms of $k$

In this Appendix we consider the term

$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
2PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
2PN}}\left(\mbox{\boldmath$x$}_{1}\right)-\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
2PN}}\left(\mbox{\boldmath$x$}_{0}\right)$

in Eq. (77) which needs fully to be expressed in terms of vector $k$. The
expression of $\Delta\mbox{\boldmath$x$}_{{\rm
2PN}}\left(\mbox{\boldmath$x$}\right)$ is given by Eq. (63), hence one obtains

$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
2PN}}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle+\frac{G^{2}M^{2}}{c^{4}}\sum\limits_{n=1}^{2}\\!\left(k^{i}U^{i}_{\left(n\right)}\,{X}_{\left(n\right)}\\!\right)\left(\mbox{\boldmath$x$}\right)$
(298)
$\displaystyle+\frac{G^{2}M\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{8}\\!\left(k^{i}V^{i\,ab}_{\left(n\right)}{Y}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}\right)$
$\displaystyle+\frac{G^{2}\hat{M}_{ab}\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{28}\\!\left(k^{i}W^{i\,abcd}_{\left(n\right)}{Z}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}\right).$

The tensorial coefficients in (179) - (216) as well as the scalar functions in
(234) - (271) are given in terms of vector $\sigma$ rather than vector $k$.
But in view of relation (73) we have
$\mbox{\boldmath$\sigma$}=\mbox{\boldmath$k$}+{\cal O}\left(c^{-2}\right)$.
Thus, a replacement $\sigma$ by $k$ in the tensorial coefficients as well as
in the scalar functions in (298) would cause an error of the order ${\cal
O}\left(c^{-6}\right)$ in line with the 2PN approximation. The tensorial
coefficients in (298) are contracted with $k^{i}$. For instance one obtains up
to terms of the order ${\cal O}\left(c^{-2}\right)$:
$k^{i}U^{i}_{\left(1\right)}=1$, $k^{i}U^{i}_{\left(2\right)}=0$,
$k^{i}V^{i\,ab}_{\left(1\right)}=k^{a}\,k^{b}$ , $\dots$,
$k^{i}W^{i\,abcd}_{\left(28\right)}=0$. After performing these contractions
one may distinguish the following tensors:

$\displaystyle S_{\left(1\right)}^{ab}$ $\displaystyle=$ $\displaystyle
k^{a}\,k^{b},\;S_{\left(2\right)}^{ab}=k^{a}\,d_{k}^{b},\;S_{\left(3\right)}^{ab}=d_{k}^{a}\,d_{k}^{b},$
(299) $\displaystyle T_{\left(1\right)}^{abcd}$ $\displaystyle=$
$\displaystyle\delta^{ac}\,k^{b}\,k^{d},\;T_{\left(2\right)}^{abcd}=\delta^{ac}\,k^{b}\,d_{k}^{d},\;T_{\left(3\right)}^{abcd}=k^{a}\,k^{b}\,k^{c}\,k^{d},$
$\displaystyle T_{\left(4\right)}^{abcd}$ $\displaystyle=$ $\displaystyle
k^{a}\,k^{b}\,k^{c}\,d_{k}^{d},\;T_{\left(5\right)}^{abcd}=k^{a}\,d_{k}^{b}\,k^{c}\,d_{k}^{d},$
$\displaystyle T_{\left(6\right)}^{abcd}$ $\displaystyle=$ $\displaystyle
k^{a}\,k^{b}\,d_{k}^{c}\,d_{k}^{d},\;T_{\left(7\right)}^{abcd}=k^{a}\,d_{k}^{b}\,d_{k}^{c}\,d_{k}^{d},\;$
$\displaystyle T_{\left(8\right)}^{abcd}$ $\displaystyle=$
$\displaystyle\delta^{ac}\,\delta^{bd},\;T_{\left(9\right)}^{abcd}=\delta^{ac}\,d_{k}^{b}\,d_{k}^{d},\;T_{\left(10\right)}^{abcd}=d_{k}^{a}\,d_{k}^{b}\,d_{k}^{c}\,d_{k}^{d},$

where the symmetries $a\leftrightarrow b$ and $c\leftrightarrow d$ as well as
$a\leftrightarrow c\land b\leftrightarrow d$ and $a\leftrightarrow d\land
b\leftrightarrow c$ have been taken into account, according to the
corresponding symmetries of the quadrupole tensors in front of the individual
terms in (298). As mentioned, in the scalar functions (234) - (271) one may
replace $\sigma$ by $k$. Then, one obtains the following expression:

$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
2PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle\frac{G^{2}M^{2}}{c^{4}}\;u_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
(301)
$\displaystyle+\frac{G^{2}M\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{3}S_{\left(n\right)}^{ab}\;v_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle+\frac{G^{2}\hat{M}_{ab}\,\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{10}T_{\left(n\right)}^{abcd}\;w_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$

where the scalar functions are given by

$\displaystyle
u_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{2}}\,e_{\left(1\right)}+\frac{g_{\left(2\right)}}{4}-\frac{15}{4}\frac{h_{\left(1\right)}}{d_{k}}\,.$
(302) $\displaystyle
v_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{4}}\,e_{\left(1\right)}-\frac{2}{\left(d_{k}\right)^{2}}\,f_{\left(1\right)}+2\,f_{\left(3\right)}-\frac{157}{64}\,\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{2}}+\frac{97}{32}\,g_{\left(4\right)}-\frac{5}{8}\left(d_{k}\right)^{2}g_{\left(6\right)}-\frac{285}{64}\,\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{3}}\,.$
(303) $\displaystyle
v_{\left(2\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{2}{\left(d_{k}\right)^{2}}\,f_{\left(2\right)}+12\,f_{\left(4\right)}-\frac{5}{4}\left(d_{k}\right)^{2}\,f_{\left(6\right)}-\frac{4}{\left(d_{k}\right)^{4}}\,g_{\left(1\right)}-\frac{4}{\left(d_{k}\right)^{2}}\,g_{\left(3\right)}\;.$
(304) $\displaystyle
v_{\left(3\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{8}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}-\frac{4}{\left(d_{k}\right)^{4}}\,f_{\left(1\right)}-\frac{2}{\left(d_{k}\right)^{2}}\,f_{\left(3\right)}-\frac{209}{64}\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{4}}-\frac{91}{32}\frac{g_{\left(4\right)}}{\left(d_{k}\right)^{2}}+\frac{5}{8}\,g_{\left(6\right)}-\frac{465}{64}\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{5}}\,.$
(305) $\displaystyle
w_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{185}{256}\,\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{4}}+\frac{185}{384}\,\frac{g_{\left(4\right)}}{\left(d_{k}\right)^{2}}-\frac{59}{96}\,g_{\left(6\right)}+\frac{15}{16}\left(d_{k}\right)^{2}g_{\left(8\right)}+\frac{185}{256}\,\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{5}}\,.$
(306) $\displaystyle
w_{\left(2\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{4}}\,f_{\left(2\right)}+\frac{f_{\left(4\right)}}{\left(d_{k}\right)^{2}}-\frac{3}{2}\,f_{\left(6\right)}+\frac{15}{8}\left(d_{k}\right)^{2}f_{\left(8\right)}\,.$
(307) $\displaystyle
w_{\left(3\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{14}{\left(d_{k}\right)^{2}}\,f_{\left(3\right)}-\frac{5175}{2048}\,\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{4}}+\frac{835}{1024}\,\frac{g_{\left(4\right)}}{\left(d_{k}\right)^{4}}+\frac{487}{256}\,g_{\left(6\right)}-\frac{333}{128}\left(d_{k}\right)^{2}g_{\left(8\right)}+\frac{15}{16}\left(d_{k}\right)^{4}g_{\left(10\right)}-\frac{5175}{2048}\,\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{5}}\,.$
$\displaystyle
w_{\left(4\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{6}{\left(d_{k}\right)^{4}}\,f_{\left(2\right)}+\frac{3}{2}\,\frac{f_{\left(4\right)}}{\left(d_{k}\right)^{2}}+\frac{449}{64}\,f_{\left(6\right)}-\frac{171}{16}\left(d_{k}\right)^{2}f_{\left(8\right)}+\frac{15}{4}\left(d_{k}\right)^{4}f_{\left(10\right)}\,.$
(309) $\displaystyle
w_{\left(5\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{2285}{512}\,\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{6}}+\frac{749}{768}\,\frac{g_{\left(4\right)}}{\left(d_{k}\right)^{4}}+\frac{73}{192}\,\frac{g_{\left(6\right)}}{\left(d_{k}\right)^{2}}+\frac{231}{32}\,g_{\left(8\right)}-\frac{15}{4}\left(d_{k}\right)^{2}g_{\left(10\right)}+\frac{2285}{512}\,\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{7}}\,.$
(310) $\displaystyle
w_{\left(6\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{16}{\left(d_{k}\right)^{6}}\,f_{\left(1\right)}-\frac{5515}{1024}\,\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{6}}+\frac{629}{1536}\,\frac{g_{\left(4\right)}}{\left(d_{k}\right)^{4}}+\frac{49}{384}\,\frac{g_{\left(6\right)}}{\left(d_{k}\right)^{2}}+\frac{231}{64}\,g_{\left(8\right)}-\frac{15}{8}\left(d_{k}\right)^{2}g_{\left(10\right)}-\frac{5515}{1024}\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{7}}\,.$
(311) $\displaystyle
w_{\left(7\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{709}{128}\,\frac{f_{\left(2\right)}}{\left(d_{k}\right)^{6}}+\frac{319}{64}\,\frac{f_{\left(4\right)}}{\left(d_{k}\right)^{4}}+\frac{25}{16}\,\frac{f_{\left(6\right)}}{\left(d_{k}\right)^{2}}+\frac{63}{8}\,f_{\left(8\right)}-\frac{15}{4}\left(d_{k}\right)^{2}f_{\left(10\right)}\,.$
(312) $\displaystyle
w_{\left(8\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{1}{128}\,\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{4}}+\frac{1}{192}\,\frac{g_{\left(4\right)}}{\left(d_{k}\right)^{2}}+\frac{5}{48}\,g_{\left(6\right)}+\frac{1}{128}\,\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{5}}\,.$
(313) $\displaystyle
w_{\left(9\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{985}{256}\,\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{6}}-\frac{217}{384}\,\frac{g_{\left(4\right)}}{\left(d_{k}\right)^{4}}-\frac{5}{96}\,\frac{g_{\left(6\right)}}{\left(d_{k}\right)^{2}}-\frac{15}{16}\,g_{\left(8\right)}-\frac{985}{256}\,\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{7}}\,.$
(314) $\displaystyle
w_{\left(10\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{2205}{2048}\,\frac{g_{\left(2\right)}}{\left(d_{k}\right)^{8}}-\frac{3361}{1024}\,\frac{g_{\left(4\right)}}{\left(d_{k}\right)^{6}}-\frac{365}{256}\,\frac{g_{\left(6\right)}}{\left(d_{k}\right)^{4}}-\frac{129}{128}\,\frac{g_{\left(8\right)}}{\left(d_{k}\right)^{2}}+\frac{15}{16}\,g_{\left(10\right)}+\frac{2205}{2048}\,\frac{h_{\left(1\right)}}{\left(d_{k}\right)^{9}}\,,$

where the abbreviations

$\displaystyle
e_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle\left(x_{1}+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}\right)^{n}-\left(x_{0}+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}\right)^{n}\;,$
(316) $\displaystyle
f_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(x_{1}\right)^{n}}-\frac{1}{\left(x_{0}\right)^{n}}\;,$
(317) $\displaystyle
g_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{\left(x_{1}\right)^{n}}-\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}{\left(x_{0}\right)^{n}}\;,$
(318) $\displaystyle
h_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle\arctan\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{d_{k}}-\arctan\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}{d_{k}}\;,$
(319) $\displaystyle
h_{\left(2\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{d_{k}}\left(\arctan\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{d_{k}}+\frac{\pi}{2}\right)$
(320)
$\displaystyle-\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}{d_{k}}\left(\arctan\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}{d_{k}}+\frac{\pi}{2}\right),$

have been introduced.

##  Appendix I Calculation of
$\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm 1PN}$ in terms of
vector $k$

In this Appendix we consider the term

$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1}\right)-\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{0}\right)$

in Eq. (77) which needs fully to be expressed in terms of vector $k$. The
expression of $\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}\right)$ is given by Eq. (62). One obtains

$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle+\frac{GM}{c^{2}}\sum\limits_{n=1}^{2}\left(k^{i}{U}^{i}_{\left(n\right)}\,{F}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}\right)$
(322)
$\displaystyle+\frac{G\hat{M}_{ab}}{c^{2}}\sum\limits_{n=1}^{8}\left(k^{i}{V}^{i\,ab}_{\left(n\right)}\,{G}_{\left(n\right)}\right)\left(\mbox{\boldmath$x$}\right)$

where the spatial variable $x$ can either be $\mbox{\boldmath$x$}_{1}$ or
$\mbox{\boldmath$x$}_{0}$. The tensorial coefficients in Eqs. (179) - (188)
and the scalar functions in Eqs. (224) - (233) are given in terms of vector
$\sigma$ and need to be expressed in terms of vector $k$.

The boundary value problem is defined by Eqs. (64) and (65), that means the
spatial position of the source, $\mbox{\boldmath$x$}_{0}$, and the spatial
position of the observer, $\mbox{\boldmath$x$}_{1}$. Hence, in
(LABEL:Term_Shapiro_A_5) - (322) one naturally encounters both impact vectors

$\displaystyle\mbox{\boldmath$d$}_{\sigma}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}\times\left(\mbox{\boldmath$x$}_{0}\times\mbox{\boldmath$\sigma$}\right),$
(323) $\displaystyle\hat{\mbox{\boldmath$d$}_{\sigma}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\sigma$}\times\left(\mbox{\boldmath$x$}_{1}\times\mbox{\boldmath$\sigma$}\right).$
(324)

For the treatment of the boundary value problem a further impact vector in
terms of $k$ is needed, defined by

$\displaystyle\mbox{\boldmath$d$}_{k}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$k$}\times\left(\mbox{\boldmath$x$}_{0}\times\mbox{\boldmath$k$}\right)=\mbox{\boldmath$k$}\times\left(\mbox{\boldmath$x$}_{1}\times\mbox{\boldmath$k$}\right).$
(325)

In order to rewrite (LABEL:Term_Shapiro_A_5) fully in terms of vector $k$ one
needs a relation between the impact vector (323) and (325) and between the
impact vector (324) and (325). These relations can be obtained by inserting
(LABEL:k_sigma_1PN) into Eqs. (323) and (324):

$\displaystyle\mbox{\boldmath$d$}_{\sigma}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$d$}_{k}+\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}\,\mbox{\boldmath$k$}+\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}{R}\,\mbox{\boldmath$k$}\times\left(\Delta\mbox{\boldmath$x$}_{\rm
1PN}\times\mbox{\boldmath$k$}\right),$
$\displaystyle\hat{\mbox{\boldmath$d$}_{\sigma}}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$d$}_{k}+\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}\,\mbox{\boldmath$k$}+\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{R}\,\mbox{\boldmath$k$}\times\left(\Delta\mbox{\boldmath$x$}_{\rm
1PN}\times\mbox{\boldmath$k$}\right),$

where $\Delta\mbox{\boldmath$x$}_{\rm 1PN}=\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$. These
relations are valid up to terms of the order ${\cal O}\left(c^{-4}\right)$.
The subsequent relations will be applied which are valid up to terms of the
order ${\cal O}\left(c^{-4}\right)$:

$\displaystyle\mbox{\boldmath$k$}\cdot\mbox{\boldmath$\sigma$}$
$\displaystyle=$ $\displaystyle 1\,,$ (328)
$\displaystyle\mbox{\boldmath$k$}\cdot\mbox{\boldmath$d$}_{\sigma}$
$\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}\,,$ (329)
$\displaystyle\mbox{\boldmath$k$}\cdot\hat{\mbox{\boldmath$d$}_{\sigma}}$
$\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}\,,$ (330) $\displaystyle\frac{1}{\left(d_{\sigma}\right)^{n}}$
$\displaystyle=$
$\displaystyle\frac{1}{\left(d_{k}\right)^{n}}-\frac{n}{R}\,\frac{\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}\right)\,\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)}{\left(d_{k}\right)^{n+2}}\,,$ (331)
$\displaystyle\frac{1}{(\hat{d}_{\sigma})^{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(d_{k}\right)^{n}}-\frac{n}{R}\,\frac{\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}\right)\,\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)}{\left(d_{k}\right)^{n+2}}\,,$ (332)
$\displaystyle\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{0}$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}-\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}\,,$ (333)
$\displaystyle\mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$x$}_{1}$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}-\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}\,,$ (334)

where $\Delta\mbox{\boldmath$x$}_{\rm 1PN}=\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$. These
relations follow from (LABEL:k_sigma_1PN) and (LABEL:Term_Shapiro_15A) and
(LABEL:Term_Shapiro_15B). Here it useful to notice that
$\left(\mbox{\boldmath$k$}\times\mbox{\boldmath$x$}\right)\cdot\left(\mbox{\boldmath$k$}\times\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)=\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm 1PN}$.

Using (LABEL:k_sigma_1PN) and (LABEL:Term_Shapiro_15A) - (330) one obtains for
the tensorial coefficients in (322) when expressed in terms of vector $k$ the
following expressions, which are valid up to terms of the order ${\cal
O}\left(c^{-4}\right)$:

$\displaystyle k^{i}{U}^{i}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle 1\,,$ (335) $\displaystyle
k^{i}{U}^{i}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{R}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right),$ (336) $\displaystyle
k^{i}{V}^{i\,ab}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle k^{a}\,k^{b}-\frac{k^{b}}{R}\,\Delta
x^{a}_{\rm
1PN}+\frac{k^{a}k^{b}}{R}\,\left(\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right),$ (337) $\displaystyle
k^{i}{V}^{i\,ab}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle
d_{k}^{a}\,k^{b}+\frac{1}{R}\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)\Delta
x^{a}_{\rm 1PN}\,k^{b}$
$\displaystyle+\frac{1}{R}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)k^{a}\,k^{b}-\frac{1}{R}\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)\left(\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)k^{a}\,k^{b}\,,$ $\displaystyle
k^{i}{V}^{i\,ab}_{\left(3\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle k^{a}\,k^{b}-\frac{2}{R}\,k^{(a}\,\Delta
x^{b)}_{\rm
1PN}+2\,\frac{k^{a}k^{b}}{R}\,\left(\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right),$ $\displaystyle
k^{i}{V}^{i\,ab}_{\left(4\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle
k^{a}\,d_{k}^{b}+\frac{1}{R}\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)k^{a}\,\Delta
x^{b}_{\rm 1PN}$ (340)
$\displaystyle+\frac{1}{R}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)k^{a}\,k^{b}-\frac{1}{R}\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)\left(\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)k^{a}\,k^{b}$ $\displaystyle-\frac{1}{R}\,\Delta x^{a}_{\rm
1PN}\,d_{k}^{b}+\frac{1}{R}\,\left(\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)k^{a}\,d_{k}^{b}\,,$ $\displaystyle
k^{i}{V}^{i\,ab}_{\left(5\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$ $\displaystyle
d_{k}^{a}\,d_{k}^{b}+\frac{2}{R}\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)d_{k}^{(a}\,\Delta
x^{b)}_{\rm 1PN}$
$\displaystyle+\frac{2}{R}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)d_{k}^{(a}\,k^{b)}-\frac{2}{R}\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)\left(\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)d_{k}^{(a}\,k^{b)}\,,$ $\displaystyle
k^{i}{V}^{i\,ab}_{\left(6\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\frac{d_{k}^{a}\,d_{k}^{b}}{R}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right),$ (342) $\displaystyle
k^{i}{V}^{i\,ab}_{\left(7\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\frac{k^{a}\,k^{b}}{R}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right),$ (343) $\displaystyle
k^{i}{V}^{i\,ab}_{\left(8\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle\frac{k^{a}\,d_{k}^{b}}{R}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right),$ (344)

where in (336) - (344) the abbreviation $\Delta\mbox{\boldmath$x$}_{\rm
1PN}=\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$ is used, and
$A^{(a}\,B^{b)}=\left(A^{a}\,B^{b}+A^{b}\,B^{a}\right)/2$ denotes
symmetrization. Similarly, using (331) - (334) one obtains for the scalar
functions in (322) when expressed in terms of vector $k$ the following
expressions:

$\displaystyle{F}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+2\,\ln\left(x-\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)+\frac{2}{\left(d_{k}\right)^{2}}\left(x+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}+{\cal O}\left(c^{-4}\right),$ (345)
$\displaystyle{F}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{2}{\left(d_{k}\right)^{2}}\left(x+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)+{\cal
O}\left(c^{-2}\right),$ (346)
$\displaystyle{G}_{\left(1\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{2}{\left(d_{k}\right)^{2}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}+\frac{4}{\left(d_{k}\right)^{4}}\,\frac{\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)^{2}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)}{R\,x}+\frac{2}{\left(d_{k}\right)^{2}}\,\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R\,x}+{\cal O}\left(c^{-4}\right),$ (347)
$\displaystyle{G}_{\left(2\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{4}}\left(x+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)-\frac{2}{\left(d_{k}\right)^{2}}\,\frac{1}{x}-\frac{4}{\left(d_{k}\right)^{4}}\,\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}\left(1-\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}+4\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{\left(d_{k}\right)^{2}}\left(x+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)\right)+{\cal
O}\left(c^{-4}\right),$ (348)
$\displaystyle{G}_{\left(3\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+\frac{1}{\left(d_{k}\right)^{2}}\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}+\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{\left(x\right)^{3}}-\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R\,x}\left(\frac{1}{\left(d_{k}\right)^{2}}+\frac{1}{\left(x\right)^{2}}+2\,\frac{\left(\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)^{2}}{\left(d_{k}\right)^{4}}\right)+{\cal
O}\left(c^{-4}\right),$ (349)
$\displaystyle{G}_{\left(4\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{4}{\left(d_{k}\right)^{4}}\left(x+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)+\frac{2}{\left(d_{k}\right)^{2}}\,\frac{1}{x}+\frac{2}{\left(x\right)^{3}}+\frac{4}{\left(d_{k}\right)^{4}}\,\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R}\left(1-\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}+4\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{\left(d_{k}\right)^{2}}\left(x+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)\right)+{\cal
O}\left(c^{-4}\right),$
$\displaystyle{G}_{\left(5\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{2}{\left(d_{k}\right)^{4}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}-\frac{1}{\left(d_{k}\right)^{2}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{\left(x\right)^{3}}-\frac{1}{\left(d_{k}\right)^{2}}\,\frac{\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}}{R\,x}\left(\frac{4}{\left(d_{k}\right)^{2}}+\frac{1}{\left(x\right)^{2}}-8\,\frac{\left(x\right)^{2}}{\left(d_{k}\right)^{4}}\right)+{\cal
O}\left(c^{-4}\right),$ (351)
$\displaystyle{G}_{\left(6\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{8}{\left(d_{k}\right)^{6}}\left(x+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)+\frac{4}{\left(d_{k}\right)^{4}}\,\frac{1}{x}+\frac{1}{\left(d_{k}\right)^{2}}\,\frac{1}{\left(x\right)^{3}}+{\cal
O}\left(c^{-2}\right),$ (352)
$\displaystyle{G}_{\left(7\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle-\frac{2}{\left(d_{k}\right)^{4}}\left(x+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)+\frac{1}{\left(d_{k}\right)^{2}}\,\frac{1}{x}-\frac{1}{\left(x\right)^{3}}+{\cal
O}\left(c^{-2}\right),$ (353)
$\displaystyle{G}_{\left(8\right)}\left(\mbox{\boldmath$x$}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{4}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}+\frac{2}{\left(d_{k}\right)^{2}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{\left(x\right)^{3}}+{\cal
O}\left(c^{-2}\right),$ (354)

where the functions in (346) and (352) - (354) need to be calculated up to
terms of the order ${\cal O}\left(c^{-2}\right)$ because their corresponding
tensorial coefficients in (336) and (342) - (344) contain only terms of the
order ${\cal O}\left(c^{-2}\right)$. By inserting (335) - (344) and (345) -
(354) into (322) one obtains:

$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}\right)$ $\displaystyle=$
$\displaystyle+\frac{2\,G\,M}{c^{2}}\,\ln\left(x-\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}\right)$
$\displaystyle-\frac{2\,G\hat{M}_{ab}}{c^{2}}\left[\frac{1}{\left(d_{k}\right)^{4}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}\,d_{k}^{a}\,d_{k}^{b}+\frac{1}{2}\,\frac{1}{\left(d_{k}\right)^{2}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{\left(x\right)^{3}}\,d_{k}^{a}\,d_{k}^{b}+\frac{1}{2}\,\frac{1}{\left(d_{k}\right)^{2}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}\,k^{a}k^{b}-\frac{1}{2}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{\left(x\right)^{3}}\,k^{a}k^{b}-\frac{1}{\left(x\right)^{3}}\,d_{k}^{a}\,k^{b}\right]$
$\displaystyle-\frac{2\,G\hat{M}_{ab}}{c^{2}}\,\frac{1}{\left(d_{k}\right)^{4}}\,\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}}{x}\left[\frac{4}{\left(d_{k}\right)^{2}}\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)d_{k}^{a}\,d_{k}^{b}+\left(\mbox{\boldmath$d$}_{k}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)k^{a}\,k^{b}+2\left(\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right)d_{k}^{a}\,k^{b}-2\,d_{k}^{a}\,\Delta x^{b}_{\rm 1PN}\right]+{\cal
O}\left(c^{-6}\right),$

where $\Delta\mbox{\boldmath$x$}_{\rm 1PN}=\Delta\mbox{\boldmath$x$}_{\rm
1PN}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$ and the
spatial argument $x$ in (LABEL:Appendix_Term_Shapiro_A) can either be
$\mbox{\boldmath$x$}_{1}$ or $\mbox{\boldmath$x$}_{0}$. By inserting (70) with
(62) into (LABEL:Appendix_Term_Shapiro_A) and taking account of
(LABEL:Term_Shapiro_A_5), one finally arrives at

$\displaystyle\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{G\,M}{c^{2}}\,P_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)+\frac{G\hat{M}_{ab}}{c^{2}}\sum\limits_{n=1}^{3}S^{ab}_{\left(n\right)}\,Q_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
(356)
$\displaystyle+\frac{G^{2}M^{2}}{c^{4}}\,r_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)+\frac{G^{2}M\,\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{3}S^{ab}_{\left(n\right)}\,s_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)+\frac{G^{2}\hat{M}_{ab}\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{10}T^{abcd}_{\left(n\right)}\,t_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)+{\cal
O}\left(c^{-6}\right),$

where the tensors have been defined by Eqs. (299) - (LABEL:Tensor_T) and the
scalar functions are

$\displaystyle
P_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+2\,\ln\frac{x_{1}-\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{x_{0}-\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}=-2\,\ln\frac{x_{1}+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{x_{0}+\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}\;.$
$\displaystyle
Q_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{g_{\left(1\right)}}{\left(d_{k}\right)^{2}}+g_{\left(3\right)}\;.$
(358) $\displaystyle
Q_{\left(2\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$ $\displaystyle+2\,f_{\left(3\right)}\;.$ (359) $\displaystyle
Q_{\left(3\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{2}{\left(d_{k}\right)^{4}}\,g_{\left(1\right)}-\frac{g_{\left(3\right)}}{\left(d_{k}\right)^{2}}\;.$
(360) $\displaystyle
r_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$ $\displaystyle 0\;.$ (361) $\displaystyle
s_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{4}}\,e_{\left(1\right)}\,g_{\left(1\right)}\;.$
(362) $\displaystyle
s_{\left(2\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$ $\displaystyle 0\;.$ (363) $\displaystyle
s_{\left(3\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{8}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,g_{\left(1\right)}\;.$
(364) $\displaystyle
t_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$ $\displaystyle 0\;.$ (365) $\displaystyle
t_{\left(2\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{8}{\left(d_{k}\right)^{6}}\,g_{\left(1\right)}\,g_{\left(1\right)}\;.$
(366) $\displaystyle
t_{\left(3\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,g_{\left(1\right)}-\frac{2}{\left(d_{k}\right)^{4}}\,f_{\left(1\right)}\,g_{\left(1\right)}$
(367)
$\displaystyle+\frac{2}{\left(d_{k}\right)^{2}}\,f_{\left(3\right)}\,g_{\left(1\right)}\;.$
$\displaystyle
t_{\left(4\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{6}}\,g_{\left(1\right)}\,g_{\left(1\right)}-\frac{4}{\left(d_{k}\right)^{4}}\,g_{\left(1\right)}\,g_{\left(3\right)}\;.$
(368) $\displaystyle
t_{\left(5\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{16}{\left(d_{k}\right)^{8}}\,e_{\left(1\right)}\,g_{\left(1\right)}+\frac{8}{\left(d_{k}\right)^{6}}\,f_{\left(1\right)}\,g_{\left(1\right)}\;.$
(369) $\displaystyle
t_{\left(6\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{16}{\left(d_{k}\right)^{8}}\,e_{\left(1\right)}\,g_{\left(1\right)}-\frac{8}{\left(d_{k}\right)^{6}}\,f_{\left(1\right)}\,g_{\left(1\right)}$
(370)
$\displaystyle-\frac{4}{\left(d_{k}\right)^{4}}\,f_{\left(3\right)}\,g_{\left(1\right)}\;.$
$\displaystyle
t_{\left(7\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{8}{\left(d_{k}\right)^{6}}\,g_{\left(1\right)}\,g_{\left(3\right)}\;.$
(371) $\displaystyle
t_{\left(8\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$ $\displaystyle 0\;.$ (372) $\displaystyle
t_{\left(9\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{16}{\left(d_{k}\right)^{8}}\,e_{\left(1\right)}\,g_{\left(1\right)}-\frac{8}{\left(d_{k}\right)^{6}}\,f_{\left(1\right)}\,g_{\left(1\right)}\;.$
(373) $\displaystyle
t_{\left(10\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{4}{\left(d_{k}\right)^{6}}\,f_{\left(3\right)}\,g_{\left(1\right)}\;.$
(374)

The scalar functions $e_{\left(n\right)}$, $f_{\left(n\right)}$,
$g_{\left(n\right)}$, $h_{\left(n\right)}$ were introduced by Eqs. (316) -
(320).

##  Appendix J Calculation of
$\left|\mbox{\boldmath$k$}\times\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right|^{2}$

In this Appendix we consider the term

$\displaystyle\left|\mbox{\boldmath$k$}\times\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)\right|^{2}$
$\displaystyle=$ $\displaystyle\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$ (375)
$\displaystyle-\left(\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)\right)^{2}$

in Eq. (77). The calculation of (375) can considerably be simplified by
omitting all terms proportional to vector $k$ in
$\Delta\mbox{\boldmath$x$}_{{\rm 1PN}}$. Then, by inspection of (62) one
obtains

$\displaystyle\left|\mbox{\boldmath$k$}\times\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)\right|^{2}$
$\displaystyle=$
$\displaystyle\frac{G^{2}M^{2}}{c^{4}}\,x_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
(376)
$\displaystyle+\frac{G^{2}M\,\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{3}S^{ab}_{\left(n\right)}\,y_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle+\frac{G^{2}\hat{M}_{ab}\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{10}T^{abcd}_{\left(n\right)}\,z_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$

where the tensors have been defined by Eqs. (299) - (LABEL:Tensor_T) and
scalar functions are

$\displaystyle
x_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{2}}\,e_{\left(1\right)}\,e_{\left(1\right)}\;.$
(377) $\displaystyle
y_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{8}{\left(d_{k}\right)^{4}}e_{\left(1\right)}e_{\left(1\right)}-\frac{4}{\left(d_{k}\right)^{2}}e_{\left(1\right)}f_{\left(1\right)}+4e_{\left(1\right)}f_{\left(3\right)}\,.$
(378) $\displaystyle
y_{\left(2\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{8}{\left(d_{k}\right)^{4}}\,e_{\left(1\right)}\,g_{\left(1\right)}-\frac{8}{\left(d_{k}\right)^{2}}\,e_{\left(1\right)}\,g_{\left(3\right)}\;.$
(379) $\displaystyle
y_{\left(3\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{16}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,e_{\left(1\right)}-\frac{8}{\left(d_{k}\right)^{4}}\,e_{\left(1\right)}\,f_{\left(1\right)}-\frac{4}{\left(d_{k}\right)^{2}}\,e_{\left(1\right)}\,f_{\left(3\right)}\;.$
(380) $\displaystyle
z_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{4}}\,g_{\left(1\right)}\,g_{\left(1\right)}\;.$
(381) $\displaystyle
z_{\left(2\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{16}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,g_{\left(1\right)}+\frac{8}{\left(d_{k}\right)^{4}}\,f_{\left(1\right)}\,g_{\left(1\right)}\;.$
(382) $\displaystyle
z_{\left(3\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{4}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,e_{\left(1\right)}-\frac{4}{\left(d_{k}\right)^{4}}\,g_{\left(1\right)}\,g_{\left(1\right)}-\frac{4}{\left(d_{k}\right)^{4}}\,e_{\left(1\right)}\,f_{\left(1\right)}+\frac{1}{\left(d_{k}\right)^{2}}\,f_{\left(1\right)}\,f_{\left(1\right)}+\frac{4}{\left(d_{k}\right)^{2}}\,e_{\left(1\right)}\,f_{\left(3\right)}-2\,f_{\left(1\right)}\,f_{\left(3\right)}$
(383)
$\displaystyle+\left(d_{k}\right)^{2}f_{\left(3\right)}\,f_{\left(3\right)}\;.$
$\displaystyle
z_{\left(4\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{8}{\left(d_{k}\right)^{6}}e_{\left(1\right)}\,g_{\left(1\right)}-\frac{4}{\left(d_{k}\right)^{4}}f_{\left(1\right)}\,g_{\left(1\right)}-\frac{8}{\left(d_{k}\right)^{4}}e_{\left(1\right)}\,g_{\left(3\right)}-\frac{4}{\left(d_{k}\right)^{2}}f_{\left(3\right)}\,g_{\left(1\right)}+\frac{4}{\left(d_{k}\right)^{2}}f_{\left(1\right)}\,g_{\left(3\right)}-4\,f_{\left(3\right)}\,g_{\left(3\right)}.$
(384) $\displaystyle
z_{\left(5\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{16}{\left(d_{k}\right)^{8}}\,e_{\left(1\right)}\,e_{\left(1\right)}+\frac{16}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,f_{\left(1\right)}+\frac{8}{\left(d_{k}\right)^{4}}\,g_{\left(1\right)}\,g_{\left(3\right)}-\frac{4}{\left(d_{k}\right)^{4}}\,f_{\left(1\right)}\,f_{\left(1\right)}+\frac{4}{\left(d_{k}\right)^{2}}\,g_{\left(3\right)}\,g_{\left(3\right)}\;.$
(385) $\displaystyle
z_{\left(6\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{16}{\left(d_{k}\right)^{8}}e_{\left(1\right)}\,e_{\left(1\right)}-\frac{16}{\left(d_{k}\right)^{6}}e_{\left(1\right)}\,f_{\left(1\right)}+\frac{4}{\left(d_{k}\right)^{4}}e_{\left(1\right)}\,f_{\left(3\right)}+\frac{4}{\left(d_{k}\right)^{4}}f_{\left(1\right)}\,f_{\left(1\right)}-\frac{2}{\left(d_{k}\right)^{2}}f_{\left(1\right)}\,f_{\left(3\right)}-2\,f_{\left(3\right)}\,f_{\left(3\right)}.$
(386) $\displaystyle
z_{\left(7\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{16}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,g_{\left(3\right)}-\frac{8}{\left(d_{k}\right)^{4}}\,f_{\left(3\right)}\,g_{\left(1\right)}+\frac{8}{\left(d_{k}\right)^{4}}\,f_{\left(1\right)}\,g_{\left(3\right)}\;.$
(387) $\displaystyle
z_{\left(8\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$ $\displaystyle 0\;.$ (388) $\displaystyle
z_{\left(9\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle+\frac{16}{\left(d_{k}\right)^{8}}\,e_{\left(1\right)}\,e_{\left(1\right)}-\frac{16}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,f_{\left(1\right)}+\frac{4}{\left(d_{k}\right)^{4}}\,f_{\left(1\right)}\,f_{\left(1\right)}\;.$
(389) $\displaystyle
z_{\left(10\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$
$\displaystyle=$
$\displaystyle-\frac{8}{\left(d_{k}\right)^{6}}\,e_{\left(1\right)}\,f_{\left(3\right)}+\frac{4}{\left(d_{k}\right)^{4}}\,f_{\left(1\right)}\,f_{\left(3\right)}+\frac{1}{\left(d_{k}\right)^{2}}\,f_{\left(3\right)}\,f_{\left(3\right)}\;.$
(390)

The scalar functions $e_{\left(n\right)}$, $f_{\left(n\right)}$,
$g_{\left(n\right)}$, $h_{\left(n\right)}$ are given by Eqs. (316) - (320).

##  Appendix K Estimation of Shapiro time-delay

###  K.1 The expression of the Shapiro time-delay

According to Eq. (77) the time delay of a light signal in the field of one
body at rest, where its monopole and quadrupole structure is taken into
account, is given by

$\displaystyle c\left(t_{1}-t_{0}\right)$ $\displaystyle=$ $\displaystyle
R-\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)-\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{{\rm
2PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)$ (391)
$\displaystyle+\,\frac{1}{2\,R}\left|\mbox{\boldmath$k$}\times\Delta\mbox{\boldmath$x$}_{{\rm
1PN}}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right)\right|^{2}+{\cal
O}\left(c^{-6}\right).$

The term $\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm 2PN}$ has
been given by Eq. (301) in Appendix H. The term
$\mbox{\boldmath$k$}\cdot\Delta\mbox{\boldmath$x$}_{\rm 1PN}$ has been given
by Eq. (356) in Appendix I. The term
$\left|\mbox{\boldmath$k$}\times\Delta\mbox{\boldmath$x$}_{\rm
1PN}\right|^{2}$ has been given by Eq. (376) in Appendix J. According to these
results the Shapiro time delay in 2PN approximation in the gravitational field
of one body at rest with monopole and quadrupole structure is given as follows
(cf. Eq. (LABEL:Shapiro_2PN_Final)):

$\displaystyle c\left(t_{1}-t_{0}\right)$ $\displaystyle=$ $\displaystyle
R+\Delta c\tau^{M}_{\rm 1PN}+\Delta c\tau^{M_{ab}}_{\rm 1PN}$ (392)
$\displaystyle+\,\Delta c\tau^{M\times M}_{\rm 2PN}+\Delta c\tau^{M\times
M_{ab}}_{\rm 2PN}+\Delta c\tau^{M_{ab}\times M_{cd}}_{\rm 2PN}\,,$

up to terms of the order ${\cal O}\left(c^{-6}\right)$ and where the
individual terms are

$\displaystyle\Delta c\tau^{M}_{\rm 1PN}$ $\displaystyle=$
$\displaystyle-\frac{G\,M}{c^{2}}\,P_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right),$
(393) $\displaystyle\Delta c\tau^{M_{ab}}_{\rm 1PN}$ $\displaystyle=$
$\displaystyle-\frac{G\hat{M}_{ab}}{c^{2}}\sum\limits_{n=1}^{3}S^{ab}_{\left(n\right)}\,Q_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right),$
(394)

$\displaystyle\Delta c\tau^{M\times M}_{\rm 2PN}$ $\displaystyle=$
$\displaystyle+\frac{G^{2}M^{2}}{c^{4}}\,R_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right),$
(395) $\displaystyle\Delta c\tau^{M\times M_{ab}}_{\rm 2PN}$ $\displaystyle=$
$\displaystyle+\frac{G^{2}M\,\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{3}S^{ab}_{\left(n\right)}\,S_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right),$
(396) $\displaystyle\Delta c\tau^{M_{ab}\times M_{cd}}_{\rm 2PN}$
$\displaystyle=$
$\displaystyle+\frac{G^{2}\hat{M}_{ab}\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{10}T^{abcd}_{\left(n\right)}\,T_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right).$

The tensors $S^{ab}_{\left(n\right)}$ and $T_{\left(n\right)}^{abcd}$ are
defined by Eqs. (299) and (LABEL:Tensor_T) and the scalar functions are
introduced:

$\displaystyle R_{\left(1\right)}$ $\displaystyle=$ $\displaystyle-
r_{\left(1\right)}-u_{\left(1\right)}+\frac{1}{2\,R}\,x_{\left(1\right)}\;,$
(398) $\displaystyle S_{\left(n\right)}$ $\displaystyle=$ $\displaystyle-
s_{\left(n\right)}-v_{\left(n\right)}+\frac{1}{2\,R}\,y_{\left(n\right)}\;,$
(399) $\displaystyle T_{\left(n\right)}$ $\displaystyle=$ $\displaystyle-
t_{\left(n\right)}-w_{\left(n\right)}+\frac{1}{2\,R}\,z_{\left(n\right)}\;.$
(400)

The functions in (398) are defined by Eqs. (361) and (302) and (377). The
functions in (399) are defined by Eqs. (362) - (364) and (303) - (305) and
(378) - (380). The functions in (400) are defined by Eqs. (365) - (374) and
(306) - (LABEL:w_10) and (381) - (390). In these functions the abbreviations
as given by Eqs. (316) - (320) have been used.

In this Appendix we will determine the upper limits of the individual terms in
Shapiro time delay formula (392). One may distinguish two scenarios of Shapiro
time delay measurements: one-way and two-way scenario. In the one-way scenario
a signal is emitted from the celestial object (e.g. spacecraft, pulsar) and
received by the observer. In the two-way scenario a signal is emitted from the
observer, then reflected off the celestial object (e.g. planet or spacecraft),
and finally received back by the observer. If one assumes that the gravitating
body as well as observer and celestial object are at rest, then both these
scenarios just differ by a factor $2$. Here the upper limits are given for the
one-way Shapiro effect.

###  K.2 Estimation of 2PN monopole-monopole term

The 2PN monopole-monopole term in (392) reads

$\displaystyle\Delta c\tau_{\rm 2PN}^{M\times M}$ $\displaystyle=$
$\displaystyle\frac{G^{2}M^{2}}{c^{4}}\,R_{\left(1\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right),$
(401)

where the scalar function $R_{\left(1\right)}$ has been defined by Eq. (398).
Eq. (401) agrees with the 2PN term in Eq. (3.2.51) in Brumberg1991  as well as
Eq. (69) in Article_Zschocke1  (for PPN parameter the values of GR,
$\gamma=1$, must be chosen); note that
$R\,d_{k}=\left|\mbox{\boldmath$x$}_{0}\times\mbox{\boldmath$x$}_{1}\right|$.
Inserting the abbreviations (361) and (302) and (377) into (398) one obtains
for the function $R_{\left(1\right)}$:

$\displaystyle R_{\left(1\right)}$ $\displaystyle=$
$\displaystyle+\frac{2}{\left(d_{k}\right)^{2}}\frac{\left(x_{1}-x_{0}\right)^{2}-R^{2}}{R}-\frac{1}{4}\left(\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{\left(x_{1}\right)^{2}}-\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}{\left(x_{0}\right)^{2}}\right)$
(402)
$\displaystyle+\,\frac{15}{4}\,\frac{1}{d_{k}}\left(\arctan\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{d_{k}}-\arctan\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}{d_{k}}\right).$

In order to determine the upper limit of (402), the relations for the angle
$\beta_{0}=\delta\left(\mbox{\boldmath$k$},\mbox{\boldmath$x$}_{0}\right)$ and
$\beta_{1}=\delta\left(\mbox{\boldmath$k$},\mbox{\boldmath$x$}_{1}\right)$ are
very useful:

$\displaystyle\cos\beta_{0}$ $\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{0}}{x_{0}}=\frac{\left(x_{1}\right)^{2}-\left(x_{0}\right)^{2}-R^{2}}{2Rx_{0}}\;,$
(403) $\displaystyle\cos\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{\mbox{\boldmath$k$}\cdot\mbox{\boldmath$x$}_{1}}{x_{1}}=\frac{\left(x_{1}\right)^{2}-\left(x_{0}\right)^{2}+R^{2}}{2Rx_{1}}\;.$
(404)

These relations are exactly valid and can be shown by using (72). The impact
parameters are $d_{k}=x_{0}\,\sin\beta_{0}=x_{1}\,\sin\beta_{1}$. Then, the
expression in (402) can be rewritten in terms of variable $z$ in (162) as well
as angle
$\alpha=\delta\left(\mbox{\boldmath$x$}_{0},\mbox{\boldmath$x$}_{1}\right)$ in
(163). By using the computer algebra system Maple Maple  one obtains for the
upper of (401)

$\displaystyle\left|\Delta c\tau_{\rm 2PN}^{M\times M}\right|$
$\displaystyle\leq$
$\displaystyle\frac{8}{\left(d_{k}\right)^{2}}\,x_{1}\,\frac{G^{2}M^{2}}{c^{4}}\;.$
(405)

Numerical values of (405) are presented in Table 3 for the Sun and giant
planets. If one implements the inequality

$\displaystyle R\,\frac{x_{1}\,x_{0}}{\left(x_{1}+x_{0}\right)^{2}}$
$\displaystyle\leq$ $\displaystyle x_{1}$ (406)

into the first term on the r.h.s. of Eq. (70) in Article_Zschocke1 , one
verifies that the estimation in (405) is in agreement with that estimation in
our article Article_Zschocke1 . Here we note that the term which was estimated
by Eq. (71) in Article_Zschocke1  has been absorbed in our upper limit given
in (405).

###  K.3 Estimation of 2PN monopole-quadrupole term

The 2PN monopole-quadrupole term in (392) reads

$\displaystyle\Delta c\tau_{\rm 2PN}^{M\times
M_{ab}}=\frac{G^{2}M\,\hat{M}_{ab}}{c^{4}}\sum\limits_{n=1}^{3}S^{ab}_{\left(n\right)}\,S_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right),$
(407)

where the tensorial coefficients $S^{ab}_{\left(n\right)}$ are given by Eqs.
(299) and the scalar functions $S_{\left(n\right)}$ have been defined by Eq.
(399). Their explicit form is obtained by inserting the abbreviations (316) -
(320) into the scalar functions $s_{\left(n\right)}$ (given by Eqs. (362) -
(364)) and $v_{\left(n\right)}$ (given by Eqs. (303) - (305)) and
$y_{\left(n\right)}$ (given by Eqs. (378) - (380)) into (399). In order to
estimate the upper limit of the individual terms in (407) the assumption is
adopted that to a good approximation the giant planets can be considered as
axially symmetric bodies, that means the STF quadrupole tensor in the
following form is used (cf. Eq. (127))

$\displaystyle\hat{M}_{ab}=M\,J_{2}\,P^{2}\left(\frac{1}{3}\,\delta_{ab}-\delta_{a3}\,\delta_{b3}\right),$
(408)

where it is assumed that the $x^{3}$-axis of the coordinate system is aligned
with the symmetry axis $\mbox{\boldmath$e$}_{3}$ of the massive body. The
parameter in (408), that means $M$ (mass of the body) $J_{2}$ (actual second
zonal harmonic coefficient), $P$ (equatorial radius of the body) are given in
Table 1 for the Sun and giant planets of the solar system. It is advisable to
apply relations (403) - (404) as well as the parameter (162) - (163), which
considerably simplify the expressions in (407). Then, the estimation proceeds
in very similar way as for (401) and one finds, by means of the computer
algebra system Maple Maple  the following upper limit:

$\displaystyle\left|\Delta c\tau_{\rm 2PN}^{M\times M_{ab}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{12}{\left(d_{k}\right)^{2}}\,x_{1}\frac{G^{2}M^{2}}{c^{4}}\,\frac{P^{2}}{\left(d_{k}\right)^{2}}\left|J_{2}\right|.$
(409)

Numerical values of (409) are presented in Table 3 for the Sun and giant
planets. In order to get correct upper limits one has to take into account
that $k$ and $\mbox{\boldmath$d$}_{k}$ are perpendicular to each other, which
restricts their possible values and angles with $\mbox{\boldmath$e$}_{3}$ (see
also endnote [99] in Zschocke_Quadrupole_1 ).

###  K.4 Estimation of 2PN quadrupole-quadrupole term

The 2PN quadrupole-quadrupole term in (392) reads

$\displaystyle\Delta c\tau_{\rm 2PN}^{M_{ab}\times M_{cd}}$ $\displaystyle=$
$\displaystyle\frac{G^{2}\hat{M}_{ab}\hat{M}_{cd}}{c^{4}}\sum\limits_{n=1}^{10}T^{abcd}_{\left(n\right)}\,T_{\left(n\right)}\left(\mbox{\boldmath$x$}_{1},\mbox{\boldmath$x$}_{0}\right),$

where the tensorial coefficients $T^{abcd}_{\left(n\right)}$ are given by Eqs.
(LABEL:Tensor_T) and the scalar functions $T_{\left(n\right)}$ have been
defined by Eq. (400). In order to estimate the upper limit of the individual
terms in (LABEL:Estimation_Shapiro_2PN_Q_Q_5) the assumption is adopted that
to a good approximation the giant planets can be considered as axially
symmetric bodies, that means for the product of two mass-quadrupole tensors
(cf. Eq. (127)) the following expression is used

$\displaystyle\hat{M}_{ab}\,\hat{M}_{cd}$ $\displaystyle=$ $\displaystyle
M^{2}\,\left|J_{2}\right|^{2}\,P^{4}$
$\displaystyle\times\left(\frac{1}{9}\delta_{ab}\delta_{cd}-\frac{1}{3}\delta_{ab}\delta_{c3}\delta_{d3}-\frac{1}{3}\delta_{a3}\delta_{b3}\delta_{cd}+\delta_{a3}\delta_{b3}\delta_{c3}\delta_{d3}\right),$

where it is assumed that the $x^{3}$-axis of coordinate system is aligned with
the symmetry axis $\mbox{\boldmath$e$}_{3}$ of the massive body. It is
advisable to introduce the parameter (162) - 163) as well as relations 403) -
(404), which considerably simplify the expressions in
(LABEL:Estimation_Shapiro_2PN_Q_Q_5). Then, the estimation proceeds in very
similar way as for (401) and one finds, by means of the computer algebra
system Maple Maple  the following upper limit:

$\displaystyle\left|\Delta c\tau_{\rm 2PN}^{M_{ab}\times M_{cd}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{8}{\left(d_{k}\right)^{2}}\,x_{1}\frac{G^{2}M^{2}}{c^{4}}\,\frac{P^{4}}{\left(d_{k}\right)^{4}}\left|J_{2}\right|^{2}.$
(412)

Numerical values of (412) are presented in Table 3 for the Sun and giant
planets. In order to get correct upper limits one has to take into account
that $k$ and $\mbox{\boldmath$d$}_{k}$ are perpendicular to each other, which
restricts their possible values and angles with $\mbox{\boldmath$e$}_{3}$ (see
also endnote [99] in Zschocke_Quadrupole_1 ).

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