Central Magnetic Force between Earth and Moon


by jay.yoon314
Tags: central, earth, force, magnetic, moon
jay.yoon314
jay.yoon314 is offline
#1
May8-12, 01:40 PM
P: 22
Hey guys I have a question about central magnetic forces between the Earth-Moon system and how they would work at a very large scale similar to that of the central gravitational force. I don't know about electromagnetism as well as I do basic celestial mechanics, so I'm at a loss as to how I would approach this problem. I'll phrase it in the form of a "thought experiment," although the part in the small-sized font can be ignored.

Imagine that the Earth were to have a magnetic field that was a strong as its gravitational field at the distance of the Moon's orbit with the field lines having the opposite directionality, but with the same torus-like shape as the actual field does (as it would if there was a magnetic reversal), and that the Moon would also have a very large magnetic field; essentially, both the Earth and the Moon would be extremely powerful "permanent magnets" whose mutual attractive magnetic force would be on the same order of magnitude of/identical to their mutual attractive gravitational force.

Aside from the extremely large fields and forces, this scenario is physically possible, and it can be shown by the Lorentz force law that the magnetic force would be attractive and central, provided that the Earth's magnetic field underwent a reversal in polarity. The next few paragraphs show that the force is indeed central and attractive in that case (it would be central and repulsive with the present magnetic polarity), but they can be skipped without losing my main argument.

Let the Earth's magnetic field lines emerge from near the North geographic pole and terminate near the South geographic pole, as they would if the Earth's magnetic field underwent a reversal in polarity. In the imaginary plane that cuts through the equator of the Earth, the field lines will thus intersect this plane orthogonally, with the magnetic vector field downward and antiparallel to the axial line going from the South pole to the North pole (geographic poles). This imaginary plane is called the equatorial plane, and is to be distinguished from the ecliptic plane, which is the plane that is coplanar with the Earth's orbit around the Sun.

Now the Moon's orbit is actually much more aligned with the ecliptic than it is with the Earth's equatorial plane, which is in contrast to the alignment of most other satellites with their respective planets. But for the sake of this thought experiment, let the Moon's orbit be coplanar with the equatorial plane instead. In that case, the Moon will be moving, in its orbit, through that same plane in which the Earth's magnetic field, as it was shown in the previous paragraph, will be orthogonal to that plane. Of course, the velocity vector of the Moon's orbit will always be both contained in that plane (the equatorial plane of the Earth, namely), and perpendicular to the Moon's position vector.


As was shown by Newton, and is quite obvious today, the gravitational force between the Earth and the Moon is a central force, as well as an "action-at-a-distance" force. This mutual attractive force doesn't lead to the Moon approaching the Earth because the Moon's tendency to move in a straight line at constant speed along the tangent to its orbit is counterbalanced (almost) perfectly by its tendency for the gravitational force to make the Moon deviate from its inertial path so that the resulting orbit is elliptical with a low eccentricity.

If this large attractive central magnetic force, equal to that of the (also central, of course) gravitational force, also existed between the Earth and the Moon, would the resulting "stable" orbit of the Earth-Moon system be preserved? Or would the introduction of such a magnetic force, even though it is a central one, lead to an unstable system in which long-term orbital regularity would be impossible?

Would the eventual result be the Moon spiraling inward to the Earth? If so, what is it about the gravitational force that allows stable orbits between two bodies and the combination of a magnetic and gravitational force, both central, that prevents stability?
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