Are the “Difffering distance” and the “Centrifugal Force” explanations for twice daily tides mathematically equivalent?
My intuition tells me that they ought to be if they are both valid (and I know that Cecil gave the thumbs down to the centrifugal notion). Although both explanations seem very different on their faces, they also have some important similarities. Both yield stronger tides with larger orbiting masses and with closer (shorter) orbits.
I suppose the skater analogy falls through for Earth’s solar tides, as the center of rotation in that case lies far from the Earth, and “centrifugal force” might be expected to cause only a nightside tidal bulge. But maybe, nearside/middle/farside differences in centrifugal force could cause a similar dual tide effect as invoked by the “differing distance” gravitational explanation.
Anyway, I was hoping someone more knowledgable about orbital mechanics could weigh in on this.
The short answer is that all the various verbal explanations work out to the same math, and that all tides are qualitatively the same (assuming that you work only two bodies at a time). Here’s another way to think of it; if both sides didn’t bulge equally, then the orbit would necessarily collapse, because the near side, bulging alone, would move the Earth’s center of mass closer and closer to the Sun. Orbital mechanics wouldn’t work at all (except with infinitely rigid bodies, I suppose).
To simplify, picture the orbital arrangement in 2 dimensions, a circle (the earth) orbiting a point (the center of mass of the system; basically, the center of the sun). There are 2 equal forces affecting the earth-circle, gravity from the point, and opposing-direction centrifugal force from orbital motion. The 2 forces tend to pull the earth-circle into an ellipsis.
This would be the situation with a fluid blob of a planet, an ellipsis doing its revolutions around the point. However, the earth is a solid (less fluid, really) object encased in a fluid ocean layer. Thus, the ocean layer deforms to form the ellipsis while the solid earth keeps the circle shape, more or less. A circle inside an ellipsis will have 2 bulges on opposite sides. These are tides.