Appendix A Differentiating Potential
The gravitational acceleration is given by taking the gradient of Eq. (10) as [27]
$$\begin{aligned} {\mathbf {a}} = \frac{d U}{d {\mathbf {r}}_{{\mathcal {F}}}} = \frac{\partial U}{\partial \Vert {\mathbf {r}}\Vert }\left( \frac{\partial \Vert {\mathbf {r}}\Vert }{\partial {\mathbf {r}}_{ {\mathcal {F}}}} \right) ^{T}+\frac{\partial {U}}{\partial {\theta }}\left( \frac{\partial \theta }{\partial {\mathbf {r}}_{{\mathcal {F}}}}\right) ^{T}+\frac{\partial {U}}{\partial {\phi }}\left( \frac{\partial \phi }{\partial {\mathbf {r}}_{{\mathcal {F}}}}\right) ^{T} \end{aligned}$$
(A1a)
where the partial derivatives are given by
$$\begin{aligned}&\frac{\partial U}{\partial \Vert {\mathbf {r}}\Vert }=-\frac{\mu }{\Vert {\mathbf {r}}\Vert ^{2}}\Bigg [1+ \sum _{n=1}^{\infty }\sum _{p=0}^{n}(n+1)\Bigg (\frac{R_{M}}{\Vert {\mathbf {r}}\Vert }\Bigg )^n\bar{P}_{n,p}(\sin {\phi })(\bar{C}_{n,p}\cos {p\theta } +\dots \nonumber \\&\quad \bar{S}_{n,p} \sin {p \theta })\Bigg ] \end{aligned}$$
(A1b)
$$\begin{aligned}&\frac{\partial U}{\partial \theta }=\frac{\mu }{\Vert {\mathbf {r}}\Vert }\left[ \sum _{n=1}^{\infty }\sum _{p=0}^{n}\Bigg (\frac{R_{M}}{\Vert {\mathbf {r}}\Vert }\Bigg )^n\bar{P}_{n,p}(\sin {\phi })p(-\bar{C}_{n,p}\sin {p\theta } +\bar{S}_{n,p}\cos {p \theta })\right] \end{aligned}$$
(A1c)
$$\begin{aligned}&\frac{\partial U}{\partial \phi }= \frac{\mu }{\Vert {\mathbf {r}}\Vert }\left[ \sum _{n=1}^{\infty }\sum _{p=0}^{n}\Bigg (\frac{R_{M}}{\Vert {\mathbf {r}}\Vert }\Bigg )^n\left( \bar{P}_{n,p+1}(\sin {\phi })-\dots \right. \right. \nonumber \\&\quad \left. p\tan {\phi }{\bar{P}}_{n,p}(\sin {\phi })\right) (\bar{C}_{n,p}\cos {p\theta } +\bar{S}_{n,p} \sin {p \theta })\Bigg ] \end{aligned}$$
(A1d)
$$\begin{aligned}&\frac{\partial \Vert {\mathbf {r}}\Vert }{\partial {\mathbf {r}}_{{\mathcal {F}}}}=\frac{{\mathbf {r}}_{{\mathcal {F}}}^{T}}{\Vert {\mathbf {r}}\Vert } \end{aligned}$$
(A1e)
$$\begin{aligned}&\frac{\partial \theta }{\partial {\mathbf {r}}_{{\mathcal {F}}}}=-\frac{{\mathbf {r}}_{{\mathcal {F}}}^{T}\left( \hat{{\mathbf {e}}}_{3}^{'}\right) ^{\times } }{{\mathbf {r}}_{{\mathcal {F}}}^{T}\bar{{\mathbf {A}}}_{3}{\mathbf {r}}_{{\mathcal {F}}}} \end{aligned}$$
(A1f)
$$\begin{aligned}&\frac{\partial \phi }{\partial {\mathbf {r}}_{{\mathcal {F}}}}=\frac{1}{\sqrt{{\mathbf {r}}_{{\mathcal {F}}}^{T}\bar{{\mathbf {A}}}_{3}{\mathbf {r}}_{{\mathcal {F}}}}}\left( \left( \hat{{\mathbf {e}}}_{3}^{'}\right) ^{T}-\frac{{\mathbf {r}}_{{\mathcal {F}}}^{T}\left( \hat{{\mathbf {e}}}_{3}^{'}\right) ^{T}{\mathbf {r}}_{{\mathcal {F}}}}{\Vert {\mathbf {r}}\Vert }\right) \end{aligned}$$
(A1g)
Define the position vector from the origin to a differential mass element as \({\mathbf {r}}_{dm}={\mathbf {r}}_{{\mathcal {B}}}+\delta {\mathbf {r}}_{{\mathcal {B}}}\) where \({\mathbf {r}}_{{\mathcal {B}}}\) represents the position vector from the origin to the rigid body’s center of mass and \(\delta {\mathbf {r}}_{{\mathcal {B}}}\) is a position vector from the center of mass to the small mass element, both expressed in the \({\mathcal {B}}\) frame. Thus, employing the definitions in Eq. (13) and the definitions for \({\mathbf {A}}_{3}\) and \(\hat{{\mathbf {e}}}_{i}\) in Sect. 3, the differential force per unit mass is obtained as
$$\begin{aligned} \begin{aligned} d{\mathbf {F}}_g=&\mu \left[ -\frac{{\mathbf {r}}_{dm}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^\frac{3}{2}}\left( 1+ \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{A}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^\frac{n}{2}}\right) + \dots \right. \\&\left. \frac{ \hat{{\mathbf {e}}}_{3}^{\times }{\mathbf {r}}_{dm}}{\left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) }\left( \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{B}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^\frac{n+1}{2}}\right) +\dots \right. \\&\left. \frac{{\mathbf {r}}_{dm}^{T}{\mathbf {r}}_{dm}\hat{{\mathbf {e}}}_{3}-{\mathbf {r}}_{dm}^{T}\hat{{\mathbf {e}}}_{3}{\mathbf {r}}_{dm}}{\left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) ^{\frac{1}{2}}}\left( \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{C}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^{\frac{n+3}{2}}}\right) \right] dm\\ {}&=d{\mathbf {F}}_{gA}+d{\mathbf {F}}_{gB}+d{\mathbf {F}}_{gC} \end{aligned} \end{aligned}$$
(A2)
which is integrated over the body \({\mathbf {F}}_{g}=\int _{{\mathcal {B}}} d {\mathbf {F}}_{g}\) to obtain all orbit-attitude coupling effects. Similarly, the gravity gradient torque is obtained via \({\mathbf {L}}_{g}=\int _{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times }d{\mathbf {F}}_{g}\). Such derivations are presented in Appendix C, employing the binomial expansions in Appendix B.
Appendix B Binomial Expansions
Using binomial expansions, it can be shown that for an integer \(m\in {\mathbb {Z}}\) that
$$\begin{aligned} \begin{aligned} \left( {\mathbf {r}}_{dm}^{T}{\mathbf {r}}_{dm}\right) ^{-\frac{m+1}{2}}=&\frac{1}{\Vert {\mathbf {r}}\Vert ^{m+1}}\left( 1-\frac{\left( m+1\right) {\mathbf {r}}_{{\mathcal {B}}}^{T} \delta {\mathbf {r}}_{{\mathcal {B}}}}{\Vert {\mathbf {r}}\Vert ^{2}}-\frac{\left( m+1\right) \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} \delta {\mathbf {r}}_{{\mathcal {B}}}}{2\Vert {\mathbf {r}}\Vert ^{2}} +\dots \right. \\&\left. \frac{\left( m+1\right) \left( m+3\right) \left( {\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}}\right) ^2}{2\Vert {\mathbf {r}}\Vert ^4} \right) +{\mathcal {O}}(3) \end{aligned} \end{aligned}$$
(B1)
where \({\mathbf {r}}_{dm}={\mathbf {r}}_{{\mathcal {B}}}+\delta {\mathbf {r}}_{{\mathcal {B}}}\). Hence,
$$\begin{aligned} \begin{aligned} \left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) ^{-\frac{m+1}{2}}=&\frac{1}{\left( {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}\right) ^{\frac{m+1}{2}}}\left( 1-(m+1)\frac{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}-\dots \right. \\&\quad \left. (m+1)\frac{\delta {\mathbf {r}}_{{\mathcal {B}}} ^{T}{\mathbf {A}}_{3} \delta {\mathbf {r}}_{{\mathcal {B}}}}{2{\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}+\dots \right. \\ {}&\left. \frac{(m+1)(m+3)}{2}\frac{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}})^{2}}{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{2}} \right) +{\mathcal {O}}(3) \end{aligned} \end{aligned}$$
(B2)
Appendix C Gravitational Effect Derivation
The derivations for the gravitational effects are presented below, where the following common identities are utilized:
$$\begin{aligned}&\int _{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} \delta {\mathbf {r}}_{{\mathcal {B}}} dm=\frac{1}{2}\text {tr}({\mathbf {J}}), \quad \int _{{\mathcal {B}}}-\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times } \delta {\mathbf {r}}_{{\mathcal {B}}}^{\times }dm={\mathbf {J}},\\& \int _{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} dm=\frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}},\quad\int _{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}} dm = 0 \end{aligned}$$
and, defining \(\odot\) as the Hadamard element-wise matrix product and the vector for summation \(\varvec{\sigma }=[1\quad 1\quad 1]^{T}\),
$$\begin{aligned} \int _{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{i}\delta {\mathbf {r}}_{{\mathcal {B}}}~dm=\frac{1}{2}\left( \text {tr}({\mathbf {A}}_{i})\text {tr}({\mathbf {J}})-2\varvec{\sigma }^{T}\left( {\mathbf {A}}_{i}\odot {\mathbf {J}}\right) \varvec{\sigma }\right) \quad (i=1,2,3) \end{aligned}$$
1.1 C.1 Gravitational Force
The derivations for \({\mathbf {F}}_{gA1}\) have been presented in other sources [1], so that derivation is neglected here.
1.1.1 C.1.1 Term \({\mathbf {F}}_{gA2}\)
Term \({\mathbf {F}}_{gA2}\):
$$\begin{aligned} {\mathbf {F}}_{gA2}=-\mu \sum _{n=1}^{\infty }\sum _{p=0}^{n}\int _{{\mathcal {B}}} \frac{Q_{n,p}^{A}{\mathbf {r}}_{dm}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^\frac{n+3}{2}}dm \end{aligned}$$
Applying Equ. (B1) and rearranging the inner products for the integral evaluation gives
$$\begin{aligned} \begin{aligned} {\mathbf {F}}_{gA2}=&-\mu \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{A}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\int _{{\mathcal {B}}}\left( {\mathbf {r}}_{{\mathcal {B}}}-\frac{\left( n+3\right) \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}}}{2\Vert {\mathbf {r}}\Vert ^{2}}{\mathbf {r}}_{{\mathcal {B}}}+\dots \right. \\ {}&\left. \frac{\left( n+3\right) \left( n+5\right) {\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}} {\mathbf {r}}_{{\mathcal {B}}}}{2\Vert {\mathbf {r}}\Vert ^4} {\mathbf {r}}_{{\mathcal {B}}} -\frac{\left( n+3\right) \delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}}{\Vert {\mathbf {r}}\Vert ^{2}}{\mathbf {r}}_{{\mathcal {B}}}\right) ~dm \end{aligned} \end{aligned}$$
which, after some simplification, leads to
$$\begin{aligned} \begin{aligned} {\mathbf {F}}_{gA2}=&-\mu m\sum _{n=1}^{\infty }\sum _{p=0}^{n} \frac{Q_{n,p}^{A}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\left\{ 1+\frac{\left( n+3\right) }{m \Vert {\mathbf {r}}\Vert ^{2}}\left[ {\mathbf {J}} + \dots \right. \right. \\ {}&\left. \left. \frac{1}{2}\left( \frac{n+2}{2}\text {tr}({\mathbf {J}})-\left( n+5\right) \frac{{\mathbf {r}}_{{\mathcal {B}}}^T {\mathbf {J}} {\mathbf {r}}_{{\mathcal {B}}}}{\Vert {\mathbf {r}}\Vert ^{2}}\right) {\mathbf {I}}_3\right] \right\} {\mathbf {r}}_{{\mathcal {B}}} \end{aligned} \end{aligned}$$
(C1)
Eqs. (16c) and (16d) are obtained from Eq. (C1).
1.1.2 C.1.2 Term \({\mathbf {F}}_{gB}\)
Term \({\mathbf {F}}_{gB}\):
$$\begin{aligned} {\mathbf {F}}_{gB}=\mu \int _{{\mathcal {B}}}\frac{ \hat{{\mathbf {e}}}_{3}^{\times }{\mathbf {r}}_{dm}}{\left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) }\left( \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{B}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^\frac{n+1}{2}}\right) dm \end{aligned}$$
Switching the order of integral and series, applying the appropriate binomial expansions from Eqs. (B1) and (B2) and definition for \({\mathbf {r}}_{dm}\), distributing retaining up to second order terms in \(\delta {\mathbf {r}}_{{\mathcal {B}}}\), and rearranging inner products leads to
$$\begin{aligned} \begin{aligned} {\mathbf {F}}_{gB}=&\frac{\mu \hat{{\mathbf {e}}}_{3}^{\times }}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{B}}{\Vert {\mathbf {r}}\Vert ^{n+1}}\int _{{\mathcal {B}}}\left( {\mathbf {r}}_{{\mathcal {B}}}-\frac{(n+1)\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}}{\Vert {\mathbf {r}}\Vert ^{2}}{\mathbf {r}}_{{\mathcal {B}}}-\dots \right. \\&\left. \frac{(n+1)\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} \delta {\mathbf {r}}_{{\mathcal {B}}}}{2\Vert {\mathbf {r}}\Vert ^{2}}{\mathbf {r}}_{{\mathcal {B}}}+\frac{(n+1)(n+3){\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {r}}_{{\mathcal {B}}}}{2\Vert {\mathbf {r}}\Vert ^{4}}{\mathbf {r}}_{{\mathcal {B}}}-\dots \right. \\&\left. \frac{2\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3} {\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}r}+\frac{2(n+1){\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {r}}_{{\mathcal {B}}}}{\Vert {\mathbf {r}}\Vert ^{2}({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})}{\mathbf {r}}_{{\mathcal {B}}}-\dots \right. \\&\left. \frac{\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}{\mathbf {r}}_{{\mathcal {B}}}+\frac{4{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {A}}_{3} {\mathbf {r}}_{{\mathcal {B}}}}{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{2}}{\mathbf {r}}_{{\mathcal {B}}}\right) dm \end{aligned} \end{aligned}$$
Evaluating the integrals and rearranging gives the expression in Eq. (15c).
$$\begin{aligned} \begin{aligned} {\mathbf {F}}_{gB}=&\frac{\mu \hat{{\mathbf {e}}}_{3}^{\times }}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{B}}{\Vert {\mathbf {r}}\Vert ^{n+1}}\Bigg \{m+\dots \\ {}&\left[ -\frac{\text {tr}({\mathbf {J}})\text {tr}({\mathbf {A}}_{3})-2\varvec{\sigma }^{T}\left( {\mathbf {A}}_{3}\odot {\mathbf {J}}\right) \varvec{\sigma }}{2{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}+\dots \right. \\ {}&\left. \frac{4}{\left( {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}\right) ^{2}}{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}+\dots \right. \\&\left. \frac{2(n+1)}{\Vert {\mathbf {r}}\Vert ^{2}({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})}{\mathbf {r}}_{{\mathcal {B}}}^{T}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}-\frac{(n+1)\text {tr}({\mathbf {J}})}{4\Vert {\mathbf {r}}\Vert ^{2}}+\dots \right. \\&\left. \frac{(n+1)(n+3)}{2\Vert {\mathbf {r}}\Vert ^{4}}{\mathbf {r}}_{{\mathcal {B}}}^{T}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {r}}_{{\mathcal {B}}}\right] {\mathbf {I}}_3-\dots \\ {}&\frac{2}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {A}}_{3}-\frac{(n+1)}{\Vert {\mathbf {r}}\Vert ^{2}}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) \Bigg \}{\mathbf {r}}_{{\mathcal {B}}} \end{aligned} \end{aligned}$$
(C2)
1.1.3 C.1.3 Term \({\mathbf {F}}_{gC1}\)
Term \({\mathbf {F}}_{gC1}\):
$$\begin{aligned} {\mathbf {F}}_{gC1}=\mu \int _{{\mathcal {B}}}\frac{{\mathbf {r}}_{dm}^{T}{\mathbf {r}}_{dm}\hat{{\mathbf {e}}}_{3}}{\left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) ^{\frac{1}{2}}}\left( \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{C}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^{\frac{n+3}{2}}}\right) dm \end{aligned}$$
Rearranging and applying the binomial expansions from Eq. (B1), distributing and retaining up to second order terms in \(\delta {\mathbf {r}}_{{\mathcal {B}}}\) gives
$$\begin{aligned} \begin{aligned} {\mathbf {F}}_{gC1}=&\frac{\mu }{\left( {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}\right) ^{\frac{1}{2}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{C}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\int _{{\mathcal {B}}} \left( \Vert {\mathbf {r}}\Vert ^{2}-\frac{2(n+3){\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {r}}_{{\mathcal {B}}}}{\Vert {\mathbf {r}}\Vert ^{2}}-\dots \right. \\ {}&\left. \frac{(n+1)\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}}}{2}+\frac{\left( n+3\right) \left( n+5\right) {\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {r}}_{{\mathcal {B}}}}{2\Vert {\mathbf {r}}\Vert ^{2}}-\dots \right. \\ {}&\left. \frac{2_{{\mathcal {B}}}r^{T} \delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3} {\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}+\frac{(n+3){\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3} {\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}-\dots \right. \\ {}&\left. \frac{\Vert {\mathbf {r}}\Vert ^{2}(\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}})}{2{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}+\frac{3}{2}\frac{\Vert {\mathbf {r}}\Vert ^{2}({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})}{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{2}}\right) dm ~ \hat{{\mathbf {e}}}_{3} \end{aligned} \end{aligned}$$
Rearranging and evaluating the integrals gives Eq. (15d).
$$\begin{aligned} \begin{aligned} {\mathbf {F}}_{gC1}=&\frac{\mu }{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{\frac{1}{2}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{ Q_{n,p}^{C}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\left[ m\Vert {\mathbf {r}}\Vert ^{2}-\frac{n+1}{4}\text {tr}({\mathbf {J}})-\dots \right. \\ {}&\left. \frac{\Vert {\mathbf {r}}\Vert ^{2}}{4{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\left( \text {tr}({\mathbf {J}})\text {tr}({\mathbf {A}}_{3})-2\varvec{\sigma }^{T}\left( {\mathbf {A}}_{3}\odot {\mathbf {J}}\right) \varvec{\sigma }\right) +\dots \right. \\ {}&\left. \left( \frac{3}{2}\frac{\Vert {\mathbf {r}}\Vert ^{2}{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}}{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{2}}-\frac{2{\mathbf {r}}_{{\mathcal {B}}}^{T}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}+\dots \right. \right. \\ {}&\left. \left. \frac{(n+3){\mathbf {r}}_{{\mathcal {B}}}^{T}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\right) \left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}+\dots \right. \\&\left. \left( \frac{(n+3)(n+5)}{2\Vert {\mathbf {r}}\Vert ^{2}}-\frac{2(n+3)}{\Vert {\mathbf {r}}\Vert ^{2}}\right) {\mathbf {r}}_{{\mathcal {B}}}^{T}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {r}}_{{\mathcal {B}}} \right] \hat{{\mathbf {e}}}_{3} \end{aligned} \end{aligned}$$
(C3)
1.1.4 C.1.4 Term \({\mathbf {F}}_{gC2}\)
Term \({\mathbf {F}}_{gC2}\):
$$\begin{aligned} {\mathbf {F}}_{gC2}=-\mu \int _{{\mathcal {B}}} \frac{{\mathbf {r}}_{dm}^{T}\hat{{\mathbf {e}}}_{3}{\mathbf {r}}_{dm}}{\left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) ^{\frac{1}{2}}}\left( \sum _{n=1}^{\infty }\sum _{p=0}^{n} \frac{Q_{n,p}^{C}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^\frac{n+3}{2}}\right) dm \end{aligned}$$
Switching the integral and series terms, applying the binomial expansions in Eqs. (B1) and (B2), distributing, and rearranging gives
$$\begin{aligned} \begin{aligned} {\mathbf {F}}_{gC2}=&-\frac{\mu }{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{\frac{1}{2}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n} \frac{Q_{n,p}^{C}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\int _{{\mathcal {B}}} \left(\vphantom{\frac{3}{2}\frac{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {A}}_{3} {\mathbf {r}}_{{\mathcal {B}}}}{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{2}}(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}} (\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}+\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} \hat{{\mathbf {e}}}_{3}-\dots \right. \\ {}&\left. \frac{(n+3)\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {r}}_{{\mathcal {B}}}}{\Vert {\mathbf {r}}\Vert ^{2}}(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}})-\frac{(n+3){\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} \hat{{\mathbf {e}}}_{3}}{\Vert {\mathbf {r}}\Vert ^{2}}{\mathbf {r}}_{{\mathcal {B}}}-\dots \right. \\ {}&\left. \frac{(n+3)\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} \delta {\mathbf {r}}_{{\mathcal {B}}}}{2 \Vert {\mathbf {r}}\Vert ^{2}}(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}+\dots \right. \\ {}&\left. \frac{(n+3)(n+5)({\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {r}}_{{\mathcal {B}}})}{2\Vert {\mathbf {r}}\Vert ^{4}}(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}-\dots \right. \\ {}&\left. \frac{\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3} {\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}})-\frac{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} \hat{{\mathbf {e}}}_{3}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}} {\mathbf {r}}_{{\mathcal {B}}}+\dots \right. \\ {}&\left. \frac{(n+3){\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {r}}_{{\mathcal {B}}}}{\Vert {\mathbf {r}}\Vert ^{2}({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})}(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}-\frac{\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {A}}_{3} \delta {\mathbf {r}}_{{\mathcal {B}}}}{2{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}+\dots \right. \\ {}&\left. \frac{3}{2}\frac{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {A}}_{3} {\mathbf {r}}_{{\mathcal {B}}}}{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{2}}(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}\right) dm \end{aligned} \end{aligned}$$
Evaluating the integrals and rearranging gives Eq. (15e).
$$\begin{aligned} \begin{aligned} {\mathbf {F}}_{gC2}=&-\mu \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{C}}{\Vert {\mathbf {r}}\Vert ^{n+3}({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{\frac{1}{2}}}\left\{ \vphantom{\frac{(n+3)\left( \hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}\right) }{\Vert {\mathbf {r}}\Vert ^{2}}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right)} \left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) \hat{{\mathbf {e}}}_{3}+\dots \right. \\&\left. \left. \left. \Bigg [m(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){-\frac{\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {A}}_{3}}+\dots \right. \right. \right. \\ {}&\left. \left. \left. \Big [-\frac{(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}})(n+3)\text {tr}({\mathbf {J}})}{4\Vert {\mathbf {r}}\Vert ^{2}}-\dots \right. \right. \right. \\ {}&\left. \left. \left. \frac{\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}}{4{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\left( \text {tr}({\mathbf {J}})\text {tr}({\mathbf {A}}_{3})-2\varvec{\sigma }^{T}\left( {\mathbf {A}}_{3}\odot {\mathbf {J}}\right) \varvec{\sigma }\right) +\dots \right. \right. \right. \\ {}&\left. \left. \left. \left( \frac{(n+3)(n+5)(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}^{T}}{2\Vert {\mathbf {r}}\Vert ^{4}}-\dots \right. \right. \right. \right. \\ {}&\left. \left. \left. \left. \frac{(n+3)\hat{{\mathbf {e}}}_{3}^{T}}{\Vert {\mathbf {r}}\Vert ^{2}}\right) \left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {r}}_{{\mathcal {B}}}+\left( \frac{(n+3)(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}){\mathbf {r}}_{{\mathcal {B}}}^{T}}{\Vert {\mathbf {r}}\Vert ^{2}({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})}+\dots \right. \right. \right. \right. \\ {}&\left. \left. \left. \frac{3}{2}\frac{(\hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}})}{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{2}}{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}-\frac{\hat{{\mathbf {e}}}_{3}^{T}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\right) \left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}\right. \Big ]{\mathbf {I}}_3-\dots \right. \\&\left. \frac{(n+3)\left( \hat{{\mathbf {e}}}_{3}^{T}{\mathbf {r}}_{{\mathcal {B}}}\right) }{\Vert {\mathbf {r}}\Vert ^{2}}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) \Bigg ]{\mathbf {r}}_{{\mathcal {B}}}\right\} \end{aligned} \end{aligned}$$
(C4)
1.2 C.2 Gravity Gradient Torque
The derivations for the term \({\mathbf {L}}_{gA1}\) can be found in [27] and [1], so they are omitted here.
1.2.1 C.2.1 Term \({\mathbf {L}}_{gA2}\)
Term \({\mathbf {L}}_{gA2}\):
$$\begin{aligned} {\mathbf {L}}_{gA2}=-\mu \int _{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times } {\mathbf {r}}_{dm} \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{A}}{\left( {\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm}\right) ^\frac{n+3}{2}}dm \end{aligned}$$
If the order of the cross product is reversed, the series and integration operators switched, the binomial expansion in Eq. (B1) is applied, and the products distributed, the expression becomes
$$\begin{aligned} {\mathbf {L}}_{gA2}=-\mu \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{A}{\mathbf {r}}_{{\mathcal {B}}}^{\times }\left( n+3\right) }{\Vert {\mathbf {r}}\Vert ^{n+5}}\left[ \int _{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} dm\right] {\mathbf {r}}_{{\mathcal {B}}} \end{aligned}$$
Evaluating the integrals reveals the final from of \({\mathbf {L}}_{gA2}\) in Eq. (18b).
$$\begin{aligned} {\mathbf {L}}_{gA2}=\mu {\mathbf {r}}_{{\mathcal {B}}}^{\times }\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{A}\left( n+3\right) }{\Vert {\mathbf {r}}\Vert ^{n+5}}{\mathbf {J}} {\mathbf {r}}_{{\mathcal {B}}} \end{aligned}$$
1.2.2 C.2.2 Term \({\mathbf {L}}_{gB}\)
Term \({\mathbf {L}}_{gB}\):
$$\begin{aligned} {\mathbf {L}}_{gB}=\mu \int _{{\mathcal {B}}}\frac{\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times } \hat{{\mathbf {e}}}_{3}^{\times }{\mathbf {r}}_{dm}}{\left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) } \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{B}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^\frac{n+1}{2}} dm \end{aligned}$$
Reversing the order of the second cross product (i.e. \(\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times }\hat{{\mathbf {e}}}_{3}^{\times }{\mathbf {r}}_{dm}=-\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times }{\mathbf {r}}_{dm}^{\times }\hat{{\mathbf {e}}}_{3}\)), implementing the product of the binomial expansions (up to first order in \(\delta {\mathbf {r}}_{{\mathcal {B}}}\)) given in Eqs. (B1) and (B2), substituting for \({\mathbf {r}}_{dm}\), distributing, and applying the property \(\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times }{\mathbf {r}}_{dm}^{\times }\hat{{\mathbf {e}}}_{3}={\mathbf {r}}_{dm}\left( \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}\hat{{\mathbf {e}}}_{3}\right) -\hat{{\mathbf {e}}}_{3}\left( \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {r}}_{dm}\right)\) yields
$$\begin{aligned} \begin{aligned} {\mathbf {L}}_{gB}=&\frac{\mu }{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{B}}{\Vert {\mathbf {r}}\Vert ^{n+1}}\Bigg [\int _{{\mathcal {B}}} -\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}\hat{{\mathbf {e}}}_{3}+\hat{{\mathbf {e}}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T} \delta {\mathbf {r}}_{{\mathcal {B}}}+\dots \\ {}&\left( \frac{\left( n+1\right) {\mathbf {r}}_{{\mathcal {B}}}^{T}\delta {\mathbf {r}}_{{\mathcal {B}}}}{\Vert {\mathbf {r}}\Vert ^{2}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}+\frac{2{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}\delta {\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}\right) \left( \hat{{\mathbf {e}}}_{3}{\mathbf {r}}_{{\mathcal {B}}}-{\mathbf {r}}_{{\mathcal {B}}}\hat{{\mathbf {e}}}_{3}\right) dm\Bigg ] \end{aligned} \end{aligned}$$
Evaluating the common integrals leads to the final expression for \({\mathbf {L}}_{gB}\) in Eq. (18c).
$$\begin{aligned} \begin{aligned} {\mathbf {L}}_{gB}=&\frac{\mu }{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{B}}{\Vert {\mathbf {r}}\Vert ^{n+1}}\Bigg [{\mathbf {J}}\hat{{\mathbf {e}}}_{3}+\dots \\ {}&\left( \frac{\left( n+1\right) {\mathbf {r}}_{{\mathcal {B}}}^{T}}{\Vert {\mathbf {r}}\Vert ^{2}}+\frac{2{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\right) \left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) \left( \hat{{\mathbf {e}}}_{3}{\mathbf {r}}_{{\mathcal {B}}}-{\mathbf {r}}_{{\mathcal {B}}}\hat{{\mathbf {e}}}_{3}\right) \Bigg ] \end{aligned} \end{aligned}$$
1.2.3 C.2.3 Term \({\mathbf {L}}_{gC1}\)
Term \({\mathbf {L}}_{gC1}\):
$$\begin{aligned} {\mathbf {L}}_{gC1}=\mu \int _{{\mathcal {B}}} \frac{\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times } \left( {\mathbf {r}}_{dm}^{T}{\mathbf {r}}_{dm}\hat{{\mathbf {e}}}_{3}\right) }{\left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) ^{\frac{1}{2}}}\left[ \sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{C}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^{\frac{n+3}{2}}}\right] dm \end{aligned}$$
Switching the integral and series terms, applying the product of the binomial expansions (up to first order in \(\delta {\mathbf {r}}_{{\mathcal {B}}}\)) given in Eqs. (B1) and (B2), changing the order of the cross product, substituting for \({\mathbf {r}}_{dm}\), distributing, and rearranging yields
$$\begin{aligned} \begin{aligned} {\mathbf {L}}_{gC1}=&-\frac{\mu \hat{{\mathbf {e}}}_{3}^{\times }}{\left( {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}\right) ^{\frac{1}{2}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{C}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\int _{{\mathcal {B}}}\Bigg [2\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {r}}_{{\mathcal {B}}}-(n+3)\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {r}}_{{\mathcal {B}}} -\dots \\ {}&\frac{\Vert {\mathbf {r}}\Vert ^{2} \delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3} {\mathbf {r}}_{{\mathcal {B}}}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\Bigg ] dm \end{aligned} \end{aligned}$$
Evaluating the integrals gives the final form seen in Eq. (18d).
$$\begin{aligned} \begin{aligned} {\mathbf {L}}_{gC1}=&\frac{\mu \hat{{\mathbf {e}}}_{3}^{\times }}{\left( {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}\right) ^{\frac{1}{2}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{C}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\Bigg [\left( n+1\right) \left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) +\dots \\ {}&\frac{\Vert {\mathbf {r}}\Vert ^{2}}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) {\mathbf {A}}_{3}\Bigg ]{\mathbf {r}}_{{\mathcal {B}}} \end{aligned} \end{aligned}$$
(C5)
1.2.4 C.2.4 Term \({\mathbf {L}}_{gC2}\)
Term \({\mathbf {L}}_{gC2}\):
$$\begin{aligned} \mathbf{L}_{gC2}=-\mu \int _{{\mathcal {B}}} \frac{\delta {\mathbf {r}}_{{\mathcal {B}}}^{\times }\left( {\mathbf {r}}_{dm}^{T}\hat{{\mathbf {e}}}_{3}{\mathbf {r}}_{dm}\right) }{\left( {\mathbf {r}}_{dm}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{dm}\right) ^{\frac{1}{2}}}\left( \sum _{n=1}^{\infty }\sum _{p=0}^{n} \frac{Q_{n,p}^{C}}{({\mathbf {r}}_{dm}^{T} {\mathbf {r}}_{dm})^\frac{n+3}{2}}\right) dm \end{aligned}$$
Switching the integral and series terms, applying the product of the binomial expansions (up to first order in \(\delta {\mathbf {r}}_{{\mathcal {B}}}\)) given in Eqs. (B1) and (B2), changing the order of the cross product, substituting for \({\mathbf {r}}_{dm}\), distributing, and rearranging yields
$$\begin{aligned} \begin{aligned} {\mathbf {L}}_{gC2}=&\frac{\mu {\mathbf {r}}_{{\mathcal {B}}}^{\times }}{\left( {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}\right) ^{\frac{1}{2}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n} \frac{Q_{n,p}^{C}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\int _{{\mathcal {B}}}\Bigg [\delta {\mathbf {r}}_{{\mathcal {B}}}\delta {\mathbf {r}}_{{\mathcal {B}}}^{T}\hat{{\mathbf {e}}}_{3}-\dots \\ {}&\frac{\left( n+3\right) \left( {\mathbf {r}}_{{\mathcal {B}}}^{T}\hat{{\mathbf {e}}}_{3}\right) }{\Vert {\mathbf {r}}\Vert ^{2}}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T} {\mathbf {r}}_{{\mathcal {B}}}-\frac{\left( {\mathbf {r}}_{{\mathcal {B}}}^{T}\hat{{\mathbf {e}}}_{3}\right) }{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}\delta {\mathbf {r}}_{{\mathcal {B}}} \delta {\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}} \Bigg ]dm \end{aligned} \end{aligned}$$
Evaluating the integrals gives the final form of Eq. (18e).
$$\begin{aligned} \begin{aligned} {\mathbf {L}}_{gC2}=&\frac{\mu {\mathbf {r}}_{{\mathcal {B}}}^{\times }}{({\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}})^{\frac{1}{2}}}\sum _{n=1}^{\infty }\sum _{p=0}^{n}\frac{Q_{n,p}^{C}}{\Vert {\mathbf {r}}\Vert ^{n+3}}\Bigg [\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) \hat{{\mathbf {e}}}_{3}-\dots \\ {}&{\mathbf {r}}_{{\mathcal {B}}}^{T}\hat{{\mathbf {e}}}_{3}\left( \frac{1}{2}\text {tr}({\mathbf {J}}){\mathbf {I}}_3-{\mathbf {J}}\right) \left( \frac{(n+3)}{\Vert {\mathbf {r}}\Vert ^{2}}{\mathbf {I}}_3+\frac{1}{{\mathbf {r}}_{{\mathcal {B}}}^{T}{\mathbf {A}}_{3}{\mathbf {r}}_{{\mathcal {B}}}}{\mathbf {A}}_{3}\right) {\mathbf {r}}_{{\mathcal {B}}}\Bigg ] \end{aligned} \end{aligned}$$
