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# Pole Tide

Fundamentals
Title Pole Tide
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Intermediate
Year of Publication 2011

The instantaneous earth rotation axis shifts inside a square of about $\displaystyle{ 20 }$ meters in relation to a point with fixed coordinates on the earth (i.e., Chandler wobble with a period of $\displaystyle{ 14 }$ months). This entails a varying elastic response of the earth's crust. This has an effect smaller than $\displaystyle{ 2.5 }$ centimetres in vertical and $\displaystyle{ 0.7 }$ centimetres in horizontal, but must be taken into account if the observations are carried out over periods longer than two months.

From the IERS Conventions [Denis et al., 2004] [1], pages 83-84, the following expression [footnotes 1] can be derived for the displacement at a point of geocentric latitude $\displaystyle{ \displaystyle \varphi }$ and longitude $\displaystyle{ \displaystyle \lambda }$:

$\displaystyle{ \begin{array}{ll} \delta_{\hat{\mathbf r}}=&-\frac{\omega^2_E \, R_e}{2\,g}\,h \, \sin 2\varphi \left ( m_1 \cos \lambda + m_2 \sin \lambda \right ) \,\hat {\mathbf r}\qquad \\[0.3cm] \delta_{\hat{\mathbf \lambda}}= & -\frac{\omega^2_E \, R_e}{g}\, l \, \sin \varphi \left (-m_1 \sin \lambda + m_2 \cos \lambda \right ) \, \hat{\mathbf \lambda} \qquad \\[0.3cm] \delta_{\hat{\mathbf \varphi}}= & -\frac{\omega^2_E \, R_e}{g}\, l\, \cos 2\varphi \left (\, m_1 \cos \lambda + m_2 \sin \lambda \right )\, \hat{\mathbf \varphi}\qquad \end{array} \qquad\mbox{(1)} }$

where ($\displaystyle{ \displaystyle m_1 }$,$\displaystyle{ \displaystyle m_2 }$) are the displacements (in meters) from the 1903.0 CIO, pole position, and $\displaystyle{ \displaystyle h= 0.6027 }$, $\displaystyle{ \displaystyle l=0.0836 }$ are the Love numbers.

Taking the earth's angular rotation $\displaystyle{ \omega_E= 7.29\cdot 10^{-5} rad/s }$, the earth's equatorial radius $\displaystyle{ R_e=6378\cdot 10^3\, m }$ and the gravitational acceleration $\displaystyle{ g=9.8\, m/sec^2 }$, it follows:

$\displaystyle{ \begin{array}{ll} \delta_{\hat{\mathbf r}}=&-32 \, \sin 2\varphi \left ( x_1 \cos \lambda + x_2 \sin \lambda \right ) \, \hat {\mathbf r}\qquad \mbox{(mm)} \\[0.3cm] \delta_{\hat{\mathbf \lambda}}= & -9 \, \sin \varphi \left (-x_1 \sin \lambda + x_2 \cos \lambda \right ) \, \hat{\mathbf \lambda} \qquad \mbox{(mm)} \\[0.3cm] \delta_{\hat{\mathbf \varphi}}= & -9 \, \cos 2\varphi \left (\, x_1 \cos \lambda + x_2 \sin \lambda \right )\, \hat{\mathbf \varphi}\qquad \mbox{(mm)} \end{array} \qquad\mbox{(2)} }$

where ($\displaystyle{ \displaystyle x_1 }$,$\displaystyle{ \displaystyle x_2 }$) are the displacements (given in seconds of arc). Pole displacements can be found at ftp://hpiers.obspm.fr/iers/eop/eop.others.

The displacement $\displaystyle{ \displaystyle \delta }$ is given in the radial, longitude and latitude $\displaystyle{ (\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi}) }$ vectors (positive upwards, eastwards and northwards, respectively). Thus, the displacement vector in the (x, y, z) ECEF Cartesian coordinates is given by

$\displaystyle{ \left[ \begin{array}{c} \Delta x \\ \Delta y \\ \Delta z \end{array} \right]= \underbrace{\left( \begin{array}{ccc} \cos \lambda \cos \varphi & -\sin \lambda & -\cos \lambda \sin \varphi \\ \sin \lambda \cos \varphi & \cos \lambda & -\sin \lambda \sin \varphi \\ \sin \varphi & 0 & \cos \Phi \end{array} \right)}_{{\mathbf R}_3(-\lambda )\cdot {\mathbf R}_2(\varphi )} \left[ \begin{array}{l} \delta_{\hat{\mathbf r}}\\ \delta_{\hat{\mathbf \lambda}}\\ \delta_{\hat{\mathbf \varphi}} \end{array} \right] \qquad\mbox{(2)} }$

Figure 1: Transformation from UEN $\displaystyle{ (\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi}) }$ to TRS $\displaystyle{ (x,y,x) }$ coordinates.

where $\displaystyle{ {{\mathbf R}_3(-\lambda )\cdot {\mathbf R}_2(\varphi )} }$ are the rotations in latitude (1) and longitude (2) indicated in figure 1 (see Reference Frames in GNSS).

## Notes

1. ^ Notice the use of latitude $\displaystyle{ \displaystyle \varphi }$ in equations (1 and 2), instead of the co-latitude $\displaystyle{ \displaystyle \theta }$ used in the IERS equations.

## References

1. ^ [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32.. IERS Convention Center., Frankfurt am Main.