A new methodology to compute the gravitational contribution of a spherical tesseroid based on the analytical solution of a sector of a spherical zonal band

Abstract

A new methodology for computing the gravitational effect of a spherical tesseroid has been devised and implemented. The methodology is based on the rotation from the global Earth-Centred Rotational reference frame to the local Earth-Centred P-Rotational reference frame, referred to the computation point P, and it requires knowledge of the height and the angular extension of each topographic column. After rotation, the gravitational effect of the tesseroid is computed via the effect of a sector of the spherical zonal band. In this respect, two possible procedures for handling the rotated tesseroids have been proposed and tested. The results obtained with the devised methodology are in good agreement with those derived by applying other existing methodologies.

Appendices

Appendix 1

Transformation from the global ECR reference frame to the local ECP reference frame.

Let point O and base $\vec {e}_i$ define the classical ECR (Earth-Centred Rotational) reference frame, where the origin O is coincident with the Earth’s centre of mass, the Z-axis, aligned with the Earth rotational axis, and the X-axis, extending from O to the intersection of the equator and the prime meridian (Fig. 15, panel a). The Y-axis completes the right-handed equatorial system. Let the same origin O and base $\vec {n}_i$ define the ECP (Earth-Centred P-Rotational) reference frame, where the z-axis is directed along $\vec {n}_3$, parallel to the line connecting the centre of the Earth O and point P, from O outwards, and axes x and y are parallel to the spherical directions evaluated in P, i.e. $\vec {n}_1$ and $\vec {n}_2$, respectively (Fig. 15, panels a$_{1}$ and a$_{2})$.

If $\lambda _{P}$ and $\theta _{P}$ indicate the longitude and the colatitude of point P, respectively, in the ECR reference frame, the transformation from ECR to ECP can be performed as

$$\begin{aligned} \begin{array}{l} \vec {n}_1 =\cos \theta _p \cos \lambda _p \vec {e}_1 +\cos \theta _p \sin \lambda _p \vec {e}_2 -\sin \theta _p \vec {e}_3 \\ \vec {n}_2 =-\sin \lambda _p \vec {e}_1 +\cos \lambda _p \vec {e}_2 \\ \vec {n}_3 =\sin \theta _p \cos \lambda _p \vec {e}_1 +\sin \theta _p \sin \lambda _p \vec {e}_2 +\cos \theta _p \vec {e}_3 \\ \end{array} \end{aligned}$$

or

$$\begin{aligned} \vec {n}_i =R_{RP} \vec {e}_j \end{aligned}$$

(4)

with $R_{RP} =\left[ {{\begin{array}{lll} {\cos \theta \cos \lambda }&{} {\cos \theta \sin \lambda }&{} {-\sin \theta } \\ {-\sin \lambda }&{} {\cos \lambda }&{} 0 \\ {\sin \theta \cos \lambda }&{} {\sin \theta \sin \lambda }&{} {\cos \theta } \\ \end{array} }} \right] _{P}$ evaluated at P.

Conversely, the transformation from ECP to ECR can be performed as

$$\begin{aligned} \begin{array}{l} \vec {e}_1 =\cos \theta _P \cos \lambda _P \vec {n}_1 -\sin \lambda _P \vec {n}_2 +\sin \theta _P \cos \lambda _P \vec {n}_3 \\ \vec {e}_2 =\cos \theta _P \sin \lambda _P \vec {n}_1 +\cos \lambda _P \vec {n}_2 +\sin \theta _P \sin \lambda _P \vec {n}_3 \\ \vec {e}_3 =-\sin \theta _P \vec {n}_1 +\cos \theta _P \vec {n}_3 \\ \end{array} \end{aligned}$$

or

$$\begin{aligned} \vec {e}_i =R_{PR} \vec {n}_j \end{aligned}$$

(5)

with $R_{PR} =\left[ {{\begin{array}{lll} {\cos \theta \cos \lambda }&{} {-\sin \lambda }&{} {\sin \theta \cos \lambda } \\ {\cos \theta \sin \lambda }&{} {\cos \lambda }&{} {\sin \theta \sin \lambda } \\ {-\sin \theta }&{} 0&{} {\cos \theta } \\ \end{array} }} \right] _P$,

where $R_{PR} =\left( {R_{RP} } \right) ^{-1}=R^{\mathrm{T}}_{RP}$.

If X, Y and Z are the coordinates of a generic point Q given in the ECR reference frame and x, y and z are the coordinates of the same point given in the ECP reference frame, it follows that

$$\begin{aligned} \left[ {x,y,z} \right] \left[ {\begin{array}{l} \vec {n}_1 \\ \vec {n}_2 \\ \vec {n}_3 \\ \end{array}} \right] =\left[ {X,Y,Z} \right] \left[ {\begin{array}{l} \vec {e}_1 \\ \vec {e}_2 \\ \vec {e}_3 \\ \end{array}} \right] \end{aligned}$$

and, from A$_{2}$,

$$\begin{aligned} \left[ {x,y,z} \right] \left[ {\begin{array}{l} \vec {n}_1 \\ \vec {n}_2 \\ \vec {n}_3 \\ \end{array}} \right]= & {} \left[ {X,Y,Z} \right] \left( {R_{PR} \left[ {\begin{array}{l} \vec {n}_1 \\ \vec {n}_2 \\ \vec {n}_3 \\ \end{array}} \right] } \right) \\= & {} ( {{\begin{array}{l} \\ {\left[ {X,Y,Z} \right] R_{PR} } \\ \\ \end{array} }} )\left[ {\begin{array}{l} \vec {n}_1 \\ \vec {n}_2 \\ \vec {n}_3 \\ \end{array}} \right] \end{aligned}$$

Finally, neglecting base $\vec {n}_i$ and transposing both terms, we obtain

$$\begin{aligned}&\left[ {x,y,z} \right] ^{\mathrm{T}}=( {{\begin{array}{l} \\ {\left[ {X,Y,Z} \right] R_{PR} } \\ \\ \end{array} }} )^{\mathrm{T}}\\&\left[ {\begin{array}{l} x \\ y \\ z \\ \end{array}} \right] =R^{\mathrm{T}}_{PR} \left[ {\begin{array}{l} X \\ Y \\ Z \\ \end{array}} \right] \end{aligned}$$

or, explicitly,

$$\begin{aligned} \left[ {\begin{array}{lll} x \\ y \\ z \\ \end{array}} \right]= & {} \left[ {{\begin{array}{l@{\quad }l@{\quad }l} {\cos \theta _p \cos \lambda _p }&{} {\cos \theta _p \sin \lambda _p }&{} {-\sin \theta _p } \\ {-\sin \lambda _p }&{} {\cos \lambda _p }&{} 0 \\ {\sin \theta _p \cos \lambda _p }&{} {\sin \theta _p \sin \lambda _p }&{} {\cos \theta _p } \\ \end{array} }} \right] \left[ {\begin{array}{lll} {X} \\ Y \\ Z \\ \end{array}} \right] \\ x= & {} \cos \theta _p \cos \lambda _p X+\cos \theta _p \sin \lambda _p Y-\sin \theta _p Z \\ y= & {} -\sin \lambda _p X+\cos \lambda _p Y \\ z= & {} \sin \theta _p \cos \lambda _p X+\sin \theta _p \sin \lambda _p Y+\cos \theta _p Z \\ \end{aligned}$$

The longitude and the colatitude of point Q in the ECP reference frame, $\lambda {'}$ and $\theta {'}$, respectively, can finally be determined as

$$\begin{aligned} \begin{array}{l} \lambda '=\hbox {arctg}\left( {\frac{y}{x}} \right) \\ \theta '=\hbox {arctg}\sqrt{\frac{x^{2}+y^{2}}{z^{2}}} \\ \end{array} \end{aligned}$$

or

$$\begin{aligned} \lambda '= & {} \hbox {arctg}\left( {\frac{-\sin \lambda _p \cdot X+\cos \lambda _p \cdot Y}{\cos \theta _p \cos \lambda _p \cdot X+\cos \theta _p \sin \lambda _p \cdot Y-\sin \theta _p \cdot Z}} \right) \\ \theta '= & {} arctg\sqrt{\frac{\left( {\cos \theta _p \cos \lambda _p \cdot X+\cos \theta _p \sin \lambda _p \cdot Y-\sin \theta _p \cdot Z} \right) ^{2}+\left( {-\sin \lambda _p \cdot X+\cos \lambda _p \cdot Y} \right) ^{2}}{\left( {\sin \theta _p \cos \lambda _p \cdot X+\sin \theta _p \sin \lambda _p \cdot Y+\cos \theta _p \cdot Z} \right) ^{2}}} \end{aligned}$$

If $\lambda$ and $\theta$ are the longitude and colatitude of point Q, respectively, given in the ECR reference frame, then

Panel (a$_{2})$ of Fig. 15 shows the definitions of $\lambda {'}$ and $\theta {'}$.

Appendix 2

$$\begin{aligned} \Delta g_{_{P}} =G\rho \Delta \lambda {'}\mathop {\int }\limits _R^{R+h} {\mathop {\int }\limits _{\theta _1^{'} }^{\theta _2^{'} } {\frac{r{'}^{2}\sin \theta {'} \cos \alpha }{\ell ^{2}}\mathrm{d}\theta {'} \mathrm{d}r{'}} } \end{aligned}$$

To carry out the integration over $r{'}$ and $\theta {'}$, we follow the procedure used by Turcotte and Schubert (1982) to calculate the gravitational acceleration due to the whole spherical Earth. Thus, for the law of planar cosines,

$$\begin{aligned} \cos \alpha= & {} \left( {\frac{r^{2}+\ell ^{2}-r{'}^{2}}{2r\ell }} \right) \\ \ell ^{2}= & {} r^{2}+r{'}^{2}-2rr{'}\cos \theta {'} \end{aligned}$$

and by differentiating the second expression with r and $r{'}$ held constant,

$$\begin{aligned} \sin \theta {'}\mathrm{d}\theta {'}=\frac{\ell }{rr{'}}\hbox {d}\ell . \end{aligned}$$

The integral gives

$$\begin{aligned} \Delta g_{P}= & {} G\rho \Delta \lambda {'}\mathop {\int }\limits _R^{R+h} {\mathrm{d}r{'}\mathop {\int }\limits _{\ell _1 }^{\ell _2 } {\frac{r{'}^{2} \frac{r^{2}+\ell ^{2}-r{'}^{2}}{2r\ell } }{\ell ^{2}}\frac{\ell }{rr{'}}\mathrm{d}\ell } } \\= & {} \frac{1}{2}\frac{G\rho }{r^{2}}\Delta \lambda {'}\mathop {\int }\limits _R^{R+h} {r{'} \mathrm{d}r{'}\mathop {\int }\limits _{\ell _1 }^{\ell _2 } {\left( {\frac{r^{2}-r{'}^{2}}{\ell ^{2}}+1} \right) \mathrm{d}\ell } } \end{aligned}$$

where $\left\{ {\begin{array}{l} \ell _1 =\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 } \\ \ell _2 =\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 } \\ \end{array}} \right.$

The integration over $\ell$ gives

$$\begin{aligned} \Delta g_{_{P}} =\frac{1}{2}\frac{G\rho }{r^{2}}\Delta \lambda {'}\mathop {\int }\limits _R^{R+h} {r{'} \left[ {\ell -\frac{r^{2}-r{'}^{2}}{\ell }} \right] _{\ell _1 }^{\ell _2 } } \mathrm{d}r{'} \end{aligned}$$

or

$$\begin{aligned} \Delta g_{_{P}}= & {} \frac{1}{2}\frac{G\rho }{r^{2}}\Delta \lambda {'}\mathop {\int }\limits _R^{R+h} {r{'} \left[ \sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 }-\frac{r^{2}-r{'}^{2}}{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 }}\right. }\\&{\left. -\,\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 }+\frac{r^{2}-r{'}^{2}}{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 }} \right] \mathrm{d}r{'}}\\ \Delta g_{_P}= & {} \frac{1}{2}\frac{G\rho }{r^{2}}\Delta \lambda {'}\mathop {\int }\limits _R^{R+h} r{'} \left[ \frac{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 -r^{2}+r{'}^{2}}{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 }}\right. \\&\left. -\frac{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 +r^{2}-r{'}^{2}}{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 }} \right] \mathrm{d}r{'} \\= & {} \frac{1}{2}\frac{G\rho }{r^{2}}\Delta \lambda {'}\mathop {\int }\limits _R^{R+h} {r{'} \left[ {\frac{2r{'}^{2}-2rr{'}\cos \theta {'}_2 }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 }}+\frac{-2r{'}^{2}+2rr{'}\cos \theta {'}_1 }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 }}} \right] \mathrm{d}r{'}} \\= & {} \frac{1}{2}\frac{G\rho }{r^{2}}\Delta \lambda {'}\mathop {\int }\limits _R^{R+h} {r{'} \left[ {\frac{2r{'}\left( {r{'}-r\cos \theta {'}_2 } \right) }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 }}+\frac{-2r{'}\left( {r{'}-r\cos \theta {'}_1 } \right) }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 }}} \right] \mathrm{d}r{'}} \\= & {} \frac{G\rho }{r^{2}}\Delta \lambda {'}\mathop {\int }\limits _R^{R+h} { \left[ {\frac{r{'}^{ 2}\left( {r{'}-r\cos \theta {'}_2 } \right) }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 }}-\frac{r{'}^{ 2}\left( {r{'}-r\cos \theta {'}_1 } \right) }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 }}} \right] \mathrm{d}r{'}} \end{aligned}$$

Let $I_{r{'}}$ be the indefinite integral, defined as

$$\begin{aligned} I_{r{'}} ={\int }\left[ {\frac{r{'}^{ 2}\left( {r{'}-r\cos \theta {'}_2 } \right) }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 }}-\frac{r{'}^{ 2}\left( {r{'}-r\cos \theta {'}_1 } \right) }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 }}} \right] \mathrm{d}r{'} \end{aligned}$$

The integration over $r{'}$ gives the following partial integrals

$$\begin{aligned}&{\int }{ \left[ {\frac{r{'}^{ 2}\left( {r{'}-r\cos \theta {'}_1 } \right) }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_1 }}} \right] \mathrm{d}r{'}} \\&\quad =\frac{1}{3}\left[ {r{'}^{ 2}+r^{ 2}\left( {3\cos ^{2}\theta {'}_1 -2} \right) +r r{'}\cos \theta {'}_1 } \right] \\&\qquad \times \,\sqrt{r{'}^{ 2}+r^{ 2}-2r r{'}\cos \theta {'}_1 } + \cos \theta {'}_1 \left( {\cos ^{2}\theta {'}_1 -1} \right) r^{3}\\&\qquad \times \,\ln \left[ {2\left( {\sqrt{r{'}^{ 2}+r^{ 2}-2r r{'}\cos \theta {'}_1 }+r{'}-r\cos \theta {'}_1 } \right) } \right] \\&{\int }{ \left[ {\frac{r{'}^{ 2}\left( {r{'}-r\cos \theta {'}_2 } \right) }{\sqrt{r^{2}+r{'}^{2}-2rr{'}\cos \theta {'}_2 }}} \right] \mathrm{d}r{'}}\\&\quad =\frac{1}{3}\left[ {r{'}^{ 2}+r^{ 2}\left( {3\cos ^{2}\theta {'}_2 -2} \right) +r r{'}\cos \theta {'}_2 } \right] \\&\qquad \times \,\sqrt{r{'}^{ 2}+r^{ 2}-2r r{'}\cos \theta {'}_2 } +\cos \theta {'}_2 \left( {\cos ^{2}\theta {'}_2 -1} \right) r^{3}\\&\qquad \times \,\ln \left[ {2\left( {\sqrt{r{'}^{ 2}+r^{ 2}-2r r{'}\cos \theta {'}_2 }+r{'}-r\cos \theta {'}_2 } \right) } \right] \end{aligned}$$

and, after summing,

$$\begin{aligned} I_{r{'}}= & {} +\frac{1}{3}\left[ {r{'}^{ 2}+r^{ 2}\left( {3\cos ^{2}\theta {'}_2 -2} \right) +r r{'}\cos \theta {'}_2 } \right] \\&\times \sqrt{r{'}^{ 2}+r^{ 2}-2r r{'}\cos \theta {'}_2 } -\cos \theta {'}_2 \sin ^{2}\theta {'}_2 r^{3}\\&\times \, \ln \left[ {2\left( {\sqrt{r{'}^{ 2}+r^{ 2}-2r r{'}\cos \theta {'}_2 }+r{'}-r\cos \theta {'}_2 } \right) } \right] \\&-\,\frac{1}{3}\left[ {r{'}^{ 2}+r^{ 2}\left( {3\cos ^{2}\theta {'}_1 -2} \right) +r r{'}\cos \theta {'}_1 } \right] \\&\times \,\sqrt{r{'}^{ 2}+r^{ 2}-2r r{'}\cos \theta {'}_1 } +\cos \theta {'}_1 \sin ^{2}\theta {'}_1 r^{3}\\&\times \,\ln \left[ {2\left( {\sqrt{r{'}^{ 2}+r^{ 2}-2r r{'}\cos \theta {'}_1 }+r{'}-r\cos \theta {'}_1 } \right) } \right] \end{aligned}$$

Once $I_{r{'}}$ is evaluated at the integration limits, the following result is found

$$\begin{aligned} \Delta g_{P} =\frac{G\rho }{r^{2}}\Delta \lambda {'}I \end{aligned}$$

with

$$\begin{aligned} I= & {} +\frac{1}{3}\left[ {\left( {R+h} \right) ^{ 2}+r^{ 2}\left( {3\cos ^{2}\theta {'}_2 -2} \right) +r \left( {R+h} \right) \cos \theta {'}_2 } \right] \\&\times \,\sqrt{\left( {R+h} \right) ^{ 2}+r^{ 2}-2r \left( {R+h} \right) \cos \theta {'}_2 }\\&-\,\frac{1}{3}\left[ {R^{ 2}+r^{ 2}\left( {3\cos ^{2}\theta {'}_2 -2} \right) +r\, R\cos \theta {'}_2 } \right] \sqrt{R^{ 2}+r^{ 2}-2 r R\cos \theta {'}_2 } \\&-\,\frac{1}{3}\left[ {\left( {R+h} \right) ^{ 2}+r^{ 2}\left( {3\cos ^{2}\theta {'}_1 -2} \right) +r\left( {R+h} \right) \cos \theta {'}_1 } \right] \\&\times \,\sqrt{\left( {R+h} \right) ^{ 2}+r^{ 2}-2r \left( {R+h} \right) \cos \theta {'}_1 }+\frac{1}{3}\left[ R^{ 2}+r^{ 2}\left( {3\cos ^{2}\theta {'}_1 -2} \right) \right. \\&\left. +\,r\, R\cos \theta {'}_1 \right] \sqrt{R^{ 2}+r^{ 2}-2 r R\cos \theta {'}_1 } -\cos \theta {'}_2 \sin ^{2}\theta {'}_2 r^{3}\\&\times \, \ln \left[ {\frac{2\left( {\sqrt{\left( {R+h} \right) ^{ 2}+r^{ 2}-2r \left( {R+h} \right) \cos \theta {'}_2 }+\left( {R+h} \right) -r\cos \theta {'}_2 } \right) }{2\left( {\sqrt{R^{ 2}+r^{ 2}-2 r R\cos \theta {'}_2 }+R-r\cos \theta {'}_2 } \right) }} \right] \\&+\,\cos \theta {'}_1 \sin ^{2}\theta {'}_1 r^{3}\\&\times \, \ln \left[ {\frac{2\left( {\sqrt{\left( {R+h} \right) ^{ 2}+r^{ 2}-2r \left( {R+h} \right) \cos \theta {'}_1 }+\left( {R+h} \right) -r\cos \theta {'}_1 } \right) }{2\left( {\sqrt{R^{ 2}+r^{ 2}-2 r R\cos \theta {'}_1 }+R-r\cos \theta {'}_1 } \right) }} \right] \end{aligned}$$