Ships, Planes And, Granularity

My concept of granularity has a lot of significance for ships and aircraft. This is relatively simple but I cannot see it referred to anywhere. 

Granularity is when something is finite but we tend to perceive it as either infinite or infinitesimal. One example is a drawer full of an equal number of right and left gloves. Suppose that there are ten gloves altogether that are well-mixed and, without looking, you reach in and pull out two gloves. What are the odds that you will have a pair of a left and right gloves?

Your first reaction may be to answer fifty percent, since there are an equal number of right and left gloves. But the answer is actually 5/9. Since there is a finite number of gloves once you pick the first one the number of right and left gloves is no longer equal. Once you pick the first glove there will be 9 left, of which 5 will make a pair. So the odds are 5/9. The odds would be fifty percent only if there was an infinite number of gloves.

Another example of granularity is games of chance. Suppose that you are playing a game in which the odds of winning is 1/20, and you have played 19 times in a row. You are now "due for a win", right? Unfortunately, wrong. The game is not obligated to give a win matching the odds for any finite number of plays. It only has to conform to the odds of the game is played an infinite number of times. Casinos must really like people who confuse the finite and the infinite and believe that they are "due for a win".

The example of granularity that applies to ships and planes is center of gravity. The manifestation of gravity that we see is weight. Mass and weight are not exactly the same thing. Mass is the amount of matter and weight is the effect of gravity on that matter. Mass is constant but weight is relative, according to the presence of a gravitational field. If we took an object from the earth to the moon it's mass would remain constant but it's weight would be only 1/6 as much.

Just as mass and weight are not exactly the same thing so the center of mass and the center of gravity are not exactly the same thing. The center of mass is constant but the center of gravity, like weight, is relative.

At first glance it might seem that the earth's center of mass and it's center of gravity are one and the same, and that we can use the two terms interchangeably. But this is another example of granularity. The earth is not an infinitesimal point, it has a certain volume. Gravity operates by the Inverse Square Law but since the near half of the earth is closer to us than the further half, the closer half must have a greater gravitational effect on us than the further half. 

This can only mean that the center of gravity of the earth must be closer to us than it's center of mass. It is only if the earth were an infinitesimal point or if we were an infinite distance from it that the center of mass and the center of gravity would be one and the same.

So we appear to be pulled by gravity toward the earth's center of mass but the nearer parts of the earth have a much greater gravitational effect on us because gravity operates by the Inverse Square Law, falling off with distance in proportion to the square of the distance. I find that this has an important implication for ships and planes.

The earth is composed of three basic layers, the core, the mantle and the crust. The density gets heavier the deeper we go. The earth's crust, the layer just below the surface averages about 2.6 times the density of water.

What this means is that a ship must weight less when it is in deep water than when it is in shallow water, with other factors being equal, and a plane weighs less when it is over water, especially deep water, than when it is over land. We know that weight gets less with altitude, because of the Inverse Square Law, but that must also apply to water because it is less dense than the rock of the crust, 2.6 times less dense. Remember that everything on the earth's surface seems to be pulled toward it's center of mass, but the earth has it's own finite volume and the parts of the earth that are closer, including the lighter water, have a greater effect than the parts that are further away.

For the massive oil tankers and cargo ships nowadays this weight loss, meaning less fuel required to drive the ship, in deep water can be significant. The same applies to the large aircraft propelled by expensive fuel. I am really surprised not to see this referred to anywhere.

Another thing that is explained by this principle of granularity is the gaps between Saturn's rings. Here is another look at it.

THE GAPS IN SATURN'S RINGS

Saturn is the planet that is known for it's spectacular ring system. All of the outer planets actually have rings around them, but those of Jupiter, Uranus and, Neptune are faint. Saturn's rings are not visible from earth with the unaided eye, but are easily visible in a small telescope. Saturn has several rings, with gaps between them.

The current Wikipedia article on "Saturn's Rings" give the reason for the gaps in Saturn's rings, other than "gravitational resonance" with Saturn's moon's, as "unexplained".

https://en.wikipedia.org/wiki/Rings_of_Saturn#/media/File:Saturn_and_its_3_moons.jpg

The gaps in Saturn's rings are actually simple to explain if we use the concept of granularity.

Without thinking further we might presume that the "center of mass" and the "center of gravity" of a planet are the same. But they aren't.

The "center of mass" of a planet is constant. It is the point from where the planet's concentration of mass is equal in all directions. We should expect that the center of mass of the planet will be just about exactly the same as it's geometric center.

But the "center of gravity" of the planet is relative, and not constant. According to the Inverse Square Law, the force of gravity is inversely proportional to the square of the distance. In other words, an object at three times the distance will exert 1 / 9 of the gravitational force.

The reason that the center of gravity is relative is that, if we are at a finite distance from a planet, the close half of the planet will have a greater gravitational effect on us than the far half of the planet. This means that the center of gravity would be closer to us than the center of mass. The closer were to the planet the greater the difference between the center of mass and the center of gravity. It is only if the planet were infinitesimal in scale, or if we were an infinite distance from the planet, that the center of gravity would be the same as the center of mass.

If we were in a spacecraft, orbiting a planet at a finite distance from the planet, the planet's center of mass would remain constant but the center of gravity would be continuously changing. The center of gravity of the planet would follow our orbit, within the planet, and always closer to us than the center of mass.

It is very unlikely that a planet will be of uniform density throughout. Almost certainly the innermost parts of the planet will be the most dense. The earth, for example, consists of a heavy iron core, above which is the mantle which consists of dense rock but not as dense as the core. Above the mantle is the less-dense crust.

The closer the spacecraft, or observation point, is to the planet, the closer the effective center of gravity is to the surface of the planet, meaning the furthest from the center of the planet. This is simply because the closer we are to the planet the greater the gravitational effect of the close half of the planet, relative to the far half.

Now, back to Saturn's rings. The rings are made mostly of particles of ice. The particles of ice closest to the planet have their center of gravity in the least-dense outermost part of the planet. But those particles of ice in orbit at a somewhat higher altitude have their center of gravity in the denser layer beneath that.

The effect of this is as if the particle at the higher altitude is in orbit around a more dense planet. An orbit around a more dense planet would mean that an object in orbit, at the same altitude around the heavier planet, would have a higher orbital energy. 

In orbit around the same planet, a higher altitude means a higher orbital energy. The orbital energy is governed by the same Inverse Square Law that governs gravity. If there is an object in orbit, and we give it 3x the orbital energy, it would then orbit at 9x the altitude, but would move at only 1 / 3 the speed.

So if one of the particles of ice composing Saturn's rings is in orbit, with it's effective center of gravity in a less-dense outer part of the planet, and a particle in a little bit higher orbit has it's effective center of gravity lower than that, in a more-dense inner part of the planet, the higher particle will have to have more orbital energy, than that which would be proportional to altitude, if the planet were of uniform density.

Since orbital energy is proportional to altitude, with a higher orbit having a higher orbital energy, this means that the particle in higher orbit, with it's real center of gravity in the lower and denser part of the planet, will have to gain more altitude to reflect it's higher orbital energy because it is, in effect, in orbit around a more-dense planet.

This is why there are gaps in Saturn's rings. The gaps are a reflection of the layers of material composing the planet, with the denser layers deeper inside the planet making necessary a higher orbital energy for the particles of ice with their effective centers of gravity within those denser layers. Since a higher orbital energy means a higher orbital altitude, this creates the gaps in the rings that reflect the layers of different density within the planet.