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Correction to: Determining parameters of Moon’s orbital and rotational motion from LLR observations using GRAIL and IERS-recommended models

  • Dmitry A. PavlovEmail author
  • James G. Williams
  • Vladimir V. Suvorkin
Correction
  • 102 Downloads

1 Correction to: Celest Mech Dyn Astr (2016) 126:61–88  https://doi.org/10.1007/s10569-016-9712-1

Correction 1. Equation (15) must be
$$\begin{aligned} \frac{I}{m}= & {} \frac{2R_M^2 {\tilde{J}}_2}{2\beta -\gamma +\beta \gamma }\left[ \begin{array}{ccc}1-\beta \gamma &{} 0 &{} 0 \\ 0 &{} 1+\gamma &{} 0 \\ 0 &{} 0 &{} 1+\beta \end{array}\right] - \frac{I_c}{m} \\&-\, k_2\frac{\mu _E}{\mu _M}\left( \frac{R_M}{r}\right) ^5 \left[ \begin{array}{ccc} x^2-\frac{1}{3}r^2 &{} xy &{} xz \\ xy &{} y^2-\frac{1}{3}r^2 &{} yz \\ xz &{} yz &{} z^2-\frac{1}{3}r^2 \end{array}\right] \\&+\, k_2\frac{R_M^5}{3\mu _M}\left[ \begin{array}{ccc} \omega _x^2-\frac{1}{3}(\omega ^2-n^2) &{} \omega _x\omega _y &{} \omega _x\omega _z \\ \omega _x\omega _y &{} \omega _y^2-\frac{1}{3}(\omega ^2-n^2) &{} \omega _y\omega _z \\ \omega _x\omega _z &{} \omega _y\omega _z &{} \omega _z^2 - \frac{1}{3}(\omega ^2+2n^2) \\ \end{array}\right] \end{aligned}$$
In the original, the bottom-right diagonal element of the last term had \((\omega ^2 - n^2)\).
Correction 2. Equation (24) must be
$$\begin{aligned} {\varvec{\Delta }}_{\mathrm {solidmoon}}= & {} \frac{\mu _A R_M^4}{\mu _M r_{\mathrm {MA}}^3} \left[ \right. \frac{h_2}{2}\left( 3\left( {\varvec{{\hat{r}}}}_{\mathrm {MA}}\cdot {\varvec{\hat{l}}}\right) ^2 - 1\right) {\varvec{\hat{l}}} \\&+ \left. 3 l_2 \left( {\varvec{\hat{r}}}_{\mathrm {MA}}\cdot {\varvec{\hat{l}}}\right) \left( {\varvec{\hat{r}}}_{\mathrm {MA}} - \left( {\varvec{\hat{r}}}_{\mathrm {MA}}\cdot \varvec{{\hat{l}}}\right) \varvec{{\hat{l}}}\right) \right] \end{aligned}$$
In the original, \({\varvec{\hat{r}}}_{\mathrm {MA}}\) was given instead of \(\varvec{{\hat{l}}}\) at the end of the first line before the plus.

Correction 3. In Table 8, lines 7–9, the first columns should be \({\varvec{\omega }}_c\).x, \({\varvec{\omega }}_c\).y, \({\varvec{\omega }}_c\).z. In the original, there were dots above the omegas.

Notes

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Dmitry A. Pavlov
    • 1
    Email author
  • James G. Williams
    • 2
  • Vladimir V. Suvorkin
    • 1
  1. 1.Institute of Applied Astronomy RASSt. PetersburgRussia
  2. 2.Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadenaUSA

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