---
abstract: '@KB2004 reported a secular increase of the astronomical unit (AU) of $15$ meters per century. Recently, @Miura2009 proposed that a possible angular momentum transfer from the rotation of the Sun to the orbital motion of the solar system planets may explain the observed increase of the AU. They assumed that the tidal effect between the planets and the Sun is the cause of this transfer. Here we claim that tidal effect cannot be a cause of this type of the transfer to explain the increase of the AU.'
author:
- 'Yousuke <span style="font-variant:small-caps;">Itoh</span>'
title: Tidal mechanism as an impossible cause of the observed secular increase of the astronomical unit
---

It is well known that the mean separation between the Earth and the Moon has been increasing due to the tidal effect. The gravitational force of the Moon induces tidal deformation on the Earth and this deformation then exerts a tidal torque on the Moon. Then the orbital angular momentum of the Moon increases while the Earth loses the rotational one, thereby the orbital radius of the Earth-Moon system gets longer.

Recently, @Miura2009 proposed that this scenario may give an explanation on the secular increase of the astronomical unit (AU) of 15 meters per century reported by @KB2004. Namely, they assumed some tidal mechanism that transfers the angular momentum from the Sun rotation to the planets orbital motions and evaluated the necessary amount of the angular momentum transfer. They found that the decrease of the Sun’s rotational period is too tiny to be observed.

In this paper we explicitly evaluate the increase of the planet orbital radius $a_p$ due to the tidal effect. The theory of the secular increase of a satellite orbital radius around a planet is well known and given by, for example, Eq . (4.160) of @MD1999. Applying it to a planet-Sun sytem, the secular increase in $a_p$ becomes $$\begin{aligned}
\dot a_{p} &=& 
3\kappa_{2\odot}
\frac{m_p}{m_{\odot}}
\left(\frac{C_{\odot}}{a_p}\right)^5 n_p a_p
\sin2\epsilon_{\odot},
\label{eq:adot} \end{aligned}$$ where a circular orbit is assumed to obtain an order of magnitude estimate. The subscript “p” denotes a planet that induces the hypothetical tidal effect on the Sun and runs from the Mercury, the Venus, and so on. $m_p$ and $m_{\odot}$ are the mass of the planet and the Sun, respectively. $n_p$ is the orbital mean motion of the planet and the over-dot denotes a time derivative. $C_{\odot}$ is the mean radius of the Sun. $\epsilon_{\odot}$ is the tidal lag angle. $\kappa_{2\odot}$ is the tidal Love number of the Sun. The tidal Love number of the Sun itself may be difficult to measure, but can be estimated from the theory and the observation of apsidal motions of stars in eclipsing binaries. Both the theory and the observation shows $\kappa_2$ for a solar mass main sequence star having the solar metallicity is $\log \kappa_2 \sim - 1.1$ (See e.g., Sec. 18 of @Schwarzschild1965 and Sec. 7.2 and Fig. 12 (a) of @Torres2009. Note that the tidal Love number $\kappa_2$ is 3 times the apsidal motion constant denoted as $k$ in @Schwarzschild1965 or $k_2$ in @Torres2009).

The table \[tab:adot\] shows the values of $\dot a_{p}$ for the 5 inner-most planets. For all of those planets, $\dot a_{p}$ is well below the reported value of the increase of the AU. Note that the value of $\dot a_p$ for the Earth is about 100 times larger than that for the Mars. Note also that the most accurate observational data for the planetary motion is from the Earth-Mars distance measurement (e.g., @Standish2005 [@Pitjeva2005]). So let us consider the Earth-Mars distance. Then, whatever large $\kappa_{2\odot}\sin2\epsilon_{\odot}$ is, one should in principle see that the distance between the Mars and the Earth in opposition to the Sun secularly decreases and that in conjunction secularly increases. This is not the same as the increase of the AU, as it should cause homogeneous expansion of the planetary orbits. After all, it is clear from the orbital radius dependence in Eq. (\[eq:adot\]) that the tidal interaction cannot cause homogeneous expansion of the planetary orbits.

  --------- ------- -------- ---------- ---------
   Mercury   Venus   Earth      Mars     Jupiter
    $6.6$    $3.1$   $0.65$   $0.0068$   $0.024$
  --------- ------- -------- ---------- ---------

  :  The estimated increase rates in the orbital radii for the 5 inner-most solar planets due to the Sun’s tidal torque acting on those planets. Eq. (\[eq:adot\]) is used for those estimates. The values of $(d a_{p}/dt)/(\kappa_{2\odot}\sin2\epsilon_{\odot})$ in the unit of millimeters/century are shown.[]{data-label="tab:adot"}

As the AU is such a fundamental quantity in the solar system planetary physics (e.g., @Standish2005 [@Capitaine2008]), several works have attempted to explain the increase of the AU. However, (1) the real error of the value of the AU itself is 3 meters [@PS2009], (2) from the measurement of the planetary mean motion, the increase of the AU may be less than 5 meters per century [@KB2004] and (3) it seems that recent studies give smaller values for the increase of the AU ($\sim$ 1 meter/century. See @Noerdlinger2008). Hence the increase of the AU does not seem robust. On the other hand, radiative mass loss of the Sun would increase the planetary orbital radius of a meter level over a century [@Noerdlinger2008], and we expect to hear in the near future a result of such a fundamental test of physics, the equivalence of the radiative energy and the gravitational mass.

Acknowledgments {#acknowledgments .unnumbered}
===============

Prof. Umin Lee at Tohoku University let the author know the reference on the tidal Love number, for which the author is grateful to him. I thank the referee for carefully read this paper, corrected the errors I made and kindly gave me useful comments that have significantly improved this paper. The author is supported by the Japan Society of the Promotion of Science (JSPS) Global Center of Excellence (COE) Program (G01): Weaving Science Web beyond Particle-Matter Hierarchy at Tohoku University, Japan.

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