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CONTEXT

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.

QUESTION

(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?

UPDATE

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/

Interplanetary Magnetic Field

User /u/uhoh pointed out in a different question that the IMF has a major effect on the Sun's magnetic influence.

From https://en.wikipedia.org/wiki/Interplanetary_magnetic_field:-

The plasma in the interplanetary medium is also responsible for the strength of the Sun's magnetic field at the orbit of the Earth being over 100 times greater than originally anticipated. If space were a vacuum, then the Sun's magnetic dipole field — about 10−4 teslas at the surface of the Sun — would reduce with the inverse cube of the distance to about 10−11 teslas. But satellite observations show that it is about 100 times greater at around 10−9 teslas. Magnetohydrodynamic (MHD) theory predicts that the motion of a conducting fluid (e.g., the interplanetary medium) in a magnetic field induces electric currents, which in turn generates magnetic fields — and, in this respect, it behaves like an MHD dynamo.

The interplanetary magnetic field at the Earth's orbit varies with waves and other disturbances in the solar wind, known as "space weather." The field is a vector, with components in the radial and azimuthal directions as well as a component perpendicular to the ecliptic. The field varies in strength near the Earth from 1 to 37 nT, averaging about 6 nT.[5]

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  • $\begingroup$ 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. $\endgroup$ Jan 29, 2020 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, 2020 at 18:51
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    $\begingroup$ I calculated the magnetic force between the Earth and Sun to be 5 Newtons, about one pound weight. $\endgroup$ Jun 17, 2021 at 4:42
  • $\begingroup$ @Keith McClary Thanks. 5N is ~ 5/(3.6*10^22) = ~1.4*10^-22 of the gravitational force of the Sun on the Earth. Which is much less than the ~6*10^-11 force ratio required to account for the non-Newtonian Apsidal Rotation of the Earth's orbit predicted by GR. $\endgroup$
    – steveOw
    Jun 22, 2021 at 15:49

2 Answers 2

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We calculate the jet effect (“nongravitational force”) for active comets, and it’s readily observable. It was observable with 1800s instruments. I’ve never yet seen anyone calculate or search for the effect of the geocorona or magnetotail on a planet.

We calculate the Yarkovsky effect for small (~1 km or less) asteroids. If relevant, we try to measure it. Even then, it takes years to integrate into a measurable value. I’ve never personally seen any claim of a Yarkovsky detection on the large (hundreds of km) asteroids, comets (swamped by NGF, plus the issue of larger orbits), and not any significant (multi-km, and not irregular orbit) moon either. Planets? Hardly.

We calculate the radiation pressure and its net effect for the beta micrometeoroids. B -dust is a few micrometers ; radiation pressure falls exponentially with size and becomes irrelevant for most purposes before a body becomes (directly) visible in telescopes. I could repeat this exercise with Poynting-Robertson, but suffice it to say it’s fairly close. Is there some planet measured in micrometers, not kilometers, I don’t know about?

We don’t even bother with magnetic force. Not on asteroids which are, to a first approximation, magnetically static. Not on comet nuclei, which have an exosphere, ionosphere, and bow shock inflating their effective size. Planets? Please.

Thank you for stating your notion in the form of a question. If you had used the declarative (“The Interplanetary Magnetic Field Influences Planetary Orbits” period) I would have referred you to a different, more appropriate site.

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  • $\begingroup$ Thanks. It is interesting to know that magnetic force effects are so small relative to other effects that they are ignored in those various investigations. But I am interested in the absolute quantified magnetic forces (however small), as exampled by Keith McLary's calculation (for Sun to Earth), and as formulated by the Yung equation in my question (Update). $\endgroup$
    – steveOw
    Feb 12, 2023 at 1:48
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    $\begingroup$ BTW your tone comes over as slightly patronising/sarcastic/disrespectful, which reduces the motivation for further discussion. $\endgroup$
    – steveOw
    Feb 12, 2023 at 1:49
  • $\begingroup$ You asked for “quantitative effects”- observable, I presume. You got them (or their nonexistence) and asked again, with more scientism (“trawling through wikipedia”- we investigators rarely bother with that telephone game/roulette table). Free tip: productive people know when to drop an investigation that’s going nowhere, because… $\endgroup$ Feb 15, 2023 at 21:36
  • $\begingroup$ “If I have a thousand ideas and only one turns out to be good, I am satisfied.” - Alfred Nobel “The best way to have a good idea is to have a lot of ideas.” Linus Pauling ‘I have a thousand ideas, to have one good one.’ Ingvar Kamprad (paraphrased) Investigators and innovators- successful ones- don’t have ‘lightning strike’ ideas, they have the skill and experience to chug past 950 or so bad ideas, ~30 or so mediocres, and whew, a few of potential. Do you? It’s not really our job to help you from idea #40 to #400; clearly not to babysit you from #2 to #3, and not if you don’t know it’s #2. $\endgroup$ Feb 15, 2023 at 21:47
  • $\begingroup$ Hence, our (recognition of and) rejection of idée fixe . In this particular case, you admittedly did not mention “Electric Universe,” and I could not (and did not) fault you for that. But you’re awfully reminiscent of EU flabbajabba, and idée fixe insistences don’t give any better impressions and optics. Again: “I can’t tell you how many dozens or hundreds of my pet ideas have gone to the laboratory and died . And that’s valuable knowledge, you know, to realize which of your ideas are wrong or ideas are childish or ideas are naive.” -Prof. David Dunning… yes, THAT Dunning $\endgroup$ Feb 15, 2023 at 21:56
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UPDATE

So, you are (sheepishly) using the declarative sense:

“ 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 ”

At what point do you grasp that WE DO NOT BOTHER WITH THE IMF ON THE ORBITS OF THE PLANETS AND ASTEROIDS?

We aim our missions at the planets and asteroids, and could arrive with order 10,000 km accuracy using ‘60s technology and one course correction burn. This was when we barely understood the IMF, at least in any quantitative way hat held up. Since then, we perform a second correction (well after sep from launch vehicle, which had warranted the first burn), and reach the target with order 1,000 km accuracy… all without bothering with the IMF. In the case of orbiters and landers, we do additional burns for more accuracy… all WITHOUT BOTHERING WITH THE IMF.

You don’t need to look up any more data: you need to look up idée fixe .

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    $\begingroup$ What's the point of posting a separate answer? $\endgroup$ Feb 11, 2023 at 4:59
  • $\begingroup$ Some people can’t be told… or can’t be bothered to read the first answer. They UPDATE anyway. Henxe, la idée fixe .Science is not just a search for evidence; it’s a pursuit of well-posed questions that then permit proof AND disproof (…and occasionally one isn’t disproven- yay!) steveOw has finally admitted his question (“effects …on the orbits”) doesn’t hold up to industry practice- seventy years of best practice, by some of our top minds. $\endgroup$ Feb 15, 2023 at 21:24

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