The equation $F_D=\frac{GMm}{D^2}$ is a standard equation in Newtonian gravitation. It describes the centripetal force exerted, by a source mass$M$, on a target particle of mass $m$ located at distance $D$. Here $G$ is Newton's Universal Constant of Gravitation.

The Non-Newtonian Apsidal Rotation Anomalies ( aka "Perihelion Precession", first observed by LeVerrier for Mercury) observed in planetary orbits are explained by General Relativity. For example the formula $F_D=\frac{GMm}{D^2}*\left(1 + \frac{3GM.P}{C^2}\frac{1}{D^2} \right)$ where $P$ is the semi-latus rectum of the elliptical orbit of the target and $C$ is the speed of light.

However it is known (e.g. Wells 2011 ) that Apsides Rotation anomalies (in the context of perihelion precession, very similar to those from GR) can also be generated from an alternative, hypothetical, model by invoking a supra-Newtonian inverse-distance-cubed force, independent of $P$; as per the following equation:- $F_D=\frac{GMm}{D^2} *\left(1 + \frac{K}{D} \right)$ where $K=\frac{6.GM}{C^2}$. At the distance of Earth the additional force is ~ $6*10^{-11}$ times the Newtonian force.

According to wikipedia:-

The Sun's Dipole Magnetic Field of 50–400 μT (at the photosphere) reduces with the inverse-cube of the distance to about 0.1 nT at the distance of Earth. However, according to spacecraft observations the Interplanetary Magnetic Field at Earth's location is around 5 nT, about a hundred times greater (see Figs 11,12 in Svalgaard & Cliver 2010 ). The difference is due to magnetic fields generated by electrical currents in the plasma surrounding the Sun.

A paper by Laine & Lin 2011 considers long-term angular orbital momentum transfer through electromagnetic interactions between Stellar Magnetic Fields and close-in Super Earths.

The source of energy is the differential motion between the (close-in Super-Earth) planet and the magnetosphere of its host star. The Lorentz force on the planet and its host star leads to an evolution toward a state of synchronous rotation. Inside the corotation radius, planets tend to lose angular momentum and migrate inward and the opposite trend occurs outside the corotation radius. Consequently, planets inside corotation migrate inward and those outside corotation migrate outward.

Their analysis is too far beyond my current knowledge of physics for me to extrapolate to the case of the modern Solar System. But it does suggest to me that electromagnetic interactions between the Sun and its rocky satellites may lead to transfer of angular momentum from the former to the latter.


(a) How does the Interplanetary Magnetic Field (IMF) vary with distance from the Sun (e.g. does it decay in strength with distance in proportion to $1/D^3$ ?, what is the orientation of $B$?, ...)

(b) what quantitative effects does the present Solar System IMF have on the orbits of the planets and asteroids?


After a bit more trawling through wikipedia I guess that the problem can be modelled (initially) in terms of the force between two magnetic dipoles. The force exerted on $m_2$ is given by:- $$F = \frac{3\mu_0}{4\pi |r|^4}((\hat{r}*m_1)*m_2+(\hat{r}*m_2)*m_1-2\hat{r}(m_1. m_2)+5\hat{r}((\hat{r}*m_1).(\hat{r}*m_2))$$ where $r$ is the relative position vector, $m_1,m_2$ are the magnetic moment vectors and $\mu_0$ is the vacuum permeability or magnetic constant.

An alternative, equivalent formula given by Yung et al, equation 37, 1998 for the force exerted by $m_1$ on $m_2$ is:-

$$F = \frac{3\mu_0 |m_1| |m_2|}{4\pi |r|^4}\left( \hat{r}(\hat{m_1}.\hat{m_2}) +\hat{m_1}(\hat{r}.\hat{m_2}) +\hat{m_2}(\hat{r}.\hat{m_1}) -5\hat{r}(\hat{r}.\hat{m_1}).(\hat{r}.\hat{m_2}) \right)$$

Note that the magnitude of the dipole-to-dipole force varies in proportion to $1/r^4$.

The dipoles will also exert torques on each other. The torque exerted by dipole 1 on dipole 2 is given by:- $$\tau = m_2 * B_1$$.

Magnetic Dipole Moments for some Solar System Objects

0 3.5 * 10^29 N-m/T Sol

1 3.8 * 10^19 N-m/T Mercury

2 8.0 * 10^17 N-m/T Venus

3 7.98 * 10^22 N-m/T Earth

4 2.1 * 10^18 N-m/T Mars

5 N/A Ceres

6 1.55 * 10^27 N-m/T Jupiter

7 4.6 * 10^25 N-m/T Saturn

8 3.0 * 10^24 N-m/T Uranus

9 1.5 * 10^24 N-m/T Neptune

Source https://www.physicsforums.com/threads/dipole-moments-of-the-planets-and-the-sun.268157/

  • $\begingroup$ Dipole fields go as 1/r^3, so this will be very weak outside the innermost part of the solar system. Also, the solar field reverses every 11 years while planet fields do not: whatever the orbital interaction is, it will tend to average out unless there is a resonance. $\endgroup$ – Anders Sandberg Jan 29 '20 at 17:00
  • $\begingroup$ @Anders Sandberg A $K/r^3$ centripetal force is enough to account for observed Non-Newtonian planetary precession (out to Mars at least) so long as K has the right value. But is the larger-than-expected IMF field a simple dipole field? Good point about Solar Field reversal (and fluctuation in field intensity) - I suppose any mechanism of progressive, long-term, Sun to planet, angular momentum transfer is going to have to be rather more complex than a simple, steady, linear, moving-charged-particle-in-a-magnetic-field kind of effect. $\endgroup$ – steveOw Jan 29 '20 at 18:51
  • $\begingroup$ I calculated the magnetic force between the Earth and Sun to be 5 Newtons, about one pound weight. $\endgroup$ – Keith McClary Jun 17 at 4:42

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