Tag Archives: Newton

Is it Meaningful to Say that Earth Goes Round the Sun, or Not? (And Why Is This So Hard…?)

Is the statement “The Sun Orbits the Earth” false? Not according to professor Richard Muller of the University of California, Berkeley, as I discussed yesterday. Muller argues that Einstein’s theory of general relativity implies that you can view the Sun as orbiting the Earth if you like, or that both the Sun and Earth orbit Venus, or a random point in space, or anything else for that matter. Meanwhile, every science textbook in our kids’ classrooms says that “The Earth Orbits the Sun“. But for all of our discussions yesterday on this subject, we did not yet collectively come to any conclusions as to whether Muller is right or wrong. And we can’t hope to find evidence that the Earth orbits the Sun if the reverse is equally true!

When we’re trying to figure out whether a confusing statement is really true or not, we have to speak precisely. Up to this stage, I haven’t been careful enough, and in this post, I’m going to try to improve upon that. There are a few small but significant points of clarification to make first. Then we’ll look in detail at what it means to “change coordinates” in such a way that would put the Sun in orbit around the Earth, instead of the other way round.

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From Kepler’s Law to Newton’s Gravity, Yourself — Part 2

Sometimes, when you’re doing physics, you have to make a wild guess, do a little calculating, and see how things turn out.

In a recent post, you were able to see how Kepler’s law for the planets’ motions (R3=T2 , where R the distance from a planet to the Sun in Earth-Sun distances, and T is the planet’s orbital time in Earth-years), leads to the conclusion that each planet is subject to an acceleration a toward the Sun, by an amount that follows an inverse square law

  • a = (2π)2 / R2

where acceleration is measured in Earth-Sun distances and in Earth-Years.

That is, a planet at the Earth’s distance from the Sun accelerates (2π)2 Earth-distances per Earth-year per Earth-year, which in more familiar units works out (as we saw earlier) to about 6 millimeters per second per second. That’s slow in human terms; a car with that acceleration would take more than an hour to go from stationary to highway speeds.

What about the Moon’s acceleration as it orbits the Earth?  Could it be given by exactly the same formula?  No, because Kepler’s law doesn’t work for the Moon and Earth.  We can see this with just a rough estimate. The time it takes the Moon to orbit the Earth is about a month, so T is roughly 1/12 Earth-years. If Kepler’s law were right, then R=T2/3 would be 1/5 of the Earth-Sun distance. But we convinced ourselves, using the relation between a first-quarter Moon and a half Moon, that the Moon-Earth distance is less than 1/10 othe Earth-Sun distance.  So Kepler’s formula doesn’t work for the Moon around the Earth.

A Guess

But perhaps objects that are orbiting the Earth satisfy a similar law,

  • R3=T2 for Earth-orbiting objects

except that now T should be measured not in years but in Moon-orbits (27.3 days, the period of the Moon’s orbit around the Earth) and R should be measured not in Earth-Sun distances but in Moon-Earth distances?  That was Newton’s guess, in fact.

Newton had a problem though: the only object he knew that orbits the Earth was the Moon.  How could he check if this law was true? We have an advantage, living in an age of artificial satellites, which we can use to check this Kepler-like law for Earth-orbiting objects, just the way Kepler checked it for the Sun-orbiting planets.  But, still there was something else Newton knew that Kepler didn’t. Galileo had determined that all objects for which air resistance is unimportant will accelerate downward at 32 feet (9.8 meters) per second per second (which is to say that, as each second ticks by, an object’s speed will increase by 32 feet [9.8 meters] per second.) So Newton suspected that if he converted the Kepler-like law for the Moon to an acceleration, as we did for the planets last time, he could relate the acceleration of the Moon as it orbits the Earth to the acceleration of ordinary falling objects in daily life.

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From Kepler’s Law to Newton’s Gravity, Yourself — Part 1

Now that you’ve discovered Kepler’s third law — that T, the orbital time of a planet in Earth years, and R, the radius of the planet’s orbit relative to the Earth-Sun distance, are related by

  • R3=T2

the question naturally arises: where does this wondrous regularity comes from?

We have been assuming that planets travel on near-circular orbits, and we’ll continue with that assumption to see what we can learn from it. So let’s look in more detail at what happens when any object, not just a planet, travels in a circle at a constant speed.

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How Einstein Trumped Newton

Sometimes I encounter people whose impression is that what Einstein’s 1905 theory of special relativity (the one that said no object’s speed can exceed the speed of light in vacuum, etc.) did in “overthrowing” the ideas of the past was somehow like what the Bolsheviks did to the Czars twelve years later– out with the old order, in with the new, and let nothing remain behind.  The widespread notion, inherited from philosopher and historian Thomas Kuhn, is that a new paradigm arose, and the old was swept away.  The truth is far different.  Important parts of the conceptual superstructure of 19th century physics had to be replaced, but the predictive mathematical core was not replaced, but rather was extended.   It was like tearing the roof and facade off a building while keeping the interior beams and columns, then extending the structure to make it much larger than before, and finally giving it a very novel external appearance.  That’s why the Einsteinian revolution was possible at all! Newton’s equations had already been used to design all sorts of real-world 18th and 19th century technology.  If Einstein’s equations hadn’t contained those of Newton and his descendants as a special case, they would have been in conflict with the real world… a no-no for a scientific theory.

Sine this is so important to understand, I’ve written an article illustrating how Einstein’s equations relating energy, momentum, mass and speed were an extension, not a replacement, of the equations that were previously in use.   It describes how Einstein unified two separate classes of equations, one set that could be used for massive objects moving slowly compared to the speed of light, and the other for light itself, into a single class of equations, one that not only included the two previous classes but made predictions for massive objects moving at speeds comparable to that of light.