planets

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

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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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Breakthrough! Do-it-Yourself Astronomy Pays Off: A Law of Nature (and a Bonus)

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In recent posts (here and here), I showed you methods that anyone can use for estimating the planets’ distances from the Sun; it just takes high-school trigonometry. And even more recently I showed how, using just algebra, you can easily obtain the planets’ orbital periods from their cycles as we see them from Earth, starting with one solar conjunction and ending at the next.

Much of this work was done by Nicolai Copernicus himself, the most famous of those philosophers who argued for a Sun-centered universe rather than an Earth-centered universe during the millennia before modern science. He had all the ingredients we have, minus knowledge of Uranus and Neptune, and minus the clues we obtain from telescopes, which would have confirmed he was correct.

Copernicus knew, therefore, that although the planetary distances from the Sun and their cycles in the sky (which astrologers [not astronomers] have focused on for centuries) don’t seem to be related, the distances and their orbital times around the Sun are much more closely related. That’s what we saw in the last post.

Let me put these distances and times, relative to the Earth-Sun distance and the Earth year, onto a two-dimensional plot. [Here the labels are for Mercury (Me), Venus (V), Mars (Ma), Jupiter (J), Saturn (S), Uranus (U) and Neptune (N).] The first figure shows the planets out to Saturn (the ones known to Copernicus).

Figure 1: The planets’ distances from the Sun (in units of the Earth-Sun distance) versus the time it takes them to complete an orbit (in Earth-years). Shown are the planets known since antiquity: Mercury (Me), Venus (V), Mars (M), Jupiter (J) and Saturn (S).

The second shows them out to Neptune, though it bunches up the inner planets to the point that you can’t really see them well.

Figure 2: Same as Figure 1, but now showing Uranus (U) and Neptune (N) as well.

You can see the planets all lie along a curve that steadily bends down and to the right.

Copernicus knew all of the numbers that go into Figure 1, with pretty moderate precision. But there’s something he didn’t recognize, which becomes obvious if we use the right trick. In the last post, we sometimes used a logarithmic axis to look at the distances and the times. Now let’s replot Figure 2 using a logarithmic axis for both the distances and the times.

Figure 3: Same as Figure 2, but now with both axes in logarithmic form.

Oh wow. (I’m sure that’s the equivalent of what Kepler said in 1618, when he first painstakingly calculated the equivalent of this plot.)

It looks like a straight line. Is it as straight as it looks?

Figure 4: Same as Figure 3, but with a blue line added to show how well a straight line describes the distance/time relationship of the planets, and with a grid added that passes through 1, 10, 100, 1000 Earth-years and distances equal to 1, 10, 100 Earth-Sun distances.

And now we see three truly remarkable things about this graph:

  • First, the planet’s distances to the Sun and orbital times lie on a very straight line on a logarithmic plot.
  • Second, the slope of the line is 2/3 (2 grid steps up for every 3 steps right) rather than, say, 7.248193 .
  • Third, the line goes right through the point (1,1), where the first horizontal and first vertical lines cross.

What do they mean?

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How You Can Determine Each Planet’s Year from its Distances and Cycles

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Whether you’re a die-hard who insists on measuring the distances between the planets and the Sun yourself (which I’ve shown you how to do here and here), or you are willing to accept what other people tell you about them, it’s interesting to look at the pattern among these distances. They are shown at right, starting with the smallest — Mercury (Me) — and proceeding to Venus (V), Mars (Ma), Jupiter (J), Saturn (S), Uranus (U) and Neptune (N), nearly 100 times further out than Mercury. The inner planets up to Mars are very close together, all bunched within 1.5 times the Earth-Sun distance, whereas the outer planets are much further apart.

Figure 1: (Left) the planet-Sun distances relative to the Earth-Sun distance; (Right) the time between solar conjunctions, when the planet disappears (roughly) behind the Sun from Earth’s perspective; “>” indicates the planet moves into the morning sky (east) after conjunction, whereas “<” means it moves into the evening sky (west).

Also shown in the figure are the lengths of the planet’s cycles. Remember, a cycle starts when a planet reappears from behind the Sun and ends when a planet again disappears behind the Sun… the moment of “solar conjunction,” or just “conjunction” for short in this post. Some planets have short cycles, others have long ones. Interestingly, now it is the outer planets that all bunch up together, with their cycles just a bit longer than an Earth year, whereas Mercury, Venus and Mars have wildly different cycles ranging from a third of an Earth-year to two Earth-years. In the figure I’m also keeping track of something that I didn’t mention before. As their cycles begin, Mercury and Venus initially move into the evening sky, in the west, setting just after sunset. I’ve indicated that with a “<” Meanwhile Mars, Jupiter and Saturn move into the eastern morning sky, rising just before sunrise, as indicated with a “>”. (Mars, Jupiter and Saturn just reappeared from behind the Sun this winter; that’s why they’re all in the morning sky right now.) This difference is going to prove important in a moment.

Before going on, let me make another version of the same figure, easier to read. This involves making a “logarithmic plot”. Instead of showing the step from 1 to 2 as the same as the step from 0 to 1, as we usually do, we replot the information so that the step along the axis from 1 to 10 is the same as the step from 0.1 to 1. It’s gives exactly the same information as the Figure 1, but now the planet-Sun distances don’t bunch up as much.

Orbits Vs. Cycles

Figure 2: exactly the same as Figure 1, but with the vertical axis in “log-plot” form.

Now, the cycles from one solar conjunction to the next, long beloved of astrologers, are not beloved of astronomers, because they involve a combination of two physically unrelated motions. A solar conjunction happens when a planet disappears behind the Sun from Earth’s perspective, so the time between one conjunction and the next combines:

  • the orbital motion of the planet around the Sun;
  • the yearly rotation of the line between the Sun and the Earth. (So far, we haven’t found evidence as to whether the Sun moves around the Earth or the Earth moves around the Sun — and we’ll remain agnostic about that today.)

So what astronomers want to know is the orbital period of each planet — it’s own year. That is, how long does it takes each planet to orbit the Sun, from the planet’s perspective, or from the Sun’s perspective. This is the time that an observer on the Sun would see for the planet to complete a circle relative to the fixed stars, and vice versa. (Remember we gathered evidence that the stars are fixed, or extremely slowly drifting from the perspective of the Earth, using a gyroscope, whereas either or both the Sun or the Earth are rotating relative to one another by about one degree per day. We also know the stars are much further than the Sun from our two measurements of the Moon’s radius.)

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How to Estimate the Distance to the Outer Planets Yourself

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Now, the last step in mapping out the other planets, before heading for more intriguing territory.

In a previous post I showed you how you can measure the distance between Venus and the Sun, RVS, relative to the distance between Earth and the Sun, RES. Under the assumption that Venus’s orbit around the Sun is circular (or nearly so), you can use the fact that when the angle between Venus and the Sun reaches its maximum (the moment of greatest elongation, and also approximately the moment when Venus appears half lit by the Sun), there’s a simple right-angle triangle in play. High school trigonometry then gives you the answer: RVS/RES ≈ 0.72 ≈ 1/√2. The same trick works for Mercury, which, like Venus, is a near Sun-orbiting planet, closer to the Sun than Earth.

But there’s no maximum angle for Mars, Jupiter, or the other far planets. These planets are further out than Earth and can even appear overhead at midnight, when they are 180 degrees away from the Sun. Fortunately there’s another right triangle we can use, again under the assumption of a (almost-)circular orbit, and that can give us a decent estimate.

The Triangle for the Far Sun-Orbiting Planets

Let’s focus on Mars first, although the same technique will work on the outer planets. Mars has a cycle in which it disappears behind the Sun, from Earth’s perspective, on average every 780 days. (That start of the cycle is called “solar conjunction,” or just “conjunction” when the context is clear.) About half a cycle later, after on average 390 days, it is at “opposition”: closest to Earth, largest in a telescope, appearing overhead at midnight, and at its brightest. But if we wait only a quarter cycle, on average 195 days after conjunction, then the Mars-Sun line is at a 90 degree angle to the Earth-Sun line. That means that Mars, Earth and the Sun form a right-angle triangle with the right angle at the location of the Sun.

So on the day of first quarter we should measure the angle on the sky between Mars and the Sun. That’s the angle A on the figure below. Then the Mars-Sun distance RMS and the Earth-Sun distance RES are the two sides of a right-angle triangle. That means they are related by the tangent function:

  • RMS/RES = tan A.
For a planet whose distance RPS from the Sun is greater than Earth’s, an estimate of its distance compared to the Earth-Sun distance RES can be made using the fact that (were its orbit perfectly circular) it would make a right-angle triangle after the first or third quarter of its cycle, with the right angle located at the Sun. Sizes of Earth, Sun and planet not shown to scale.

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Who Orbits Who, and Where? Check it Yourself

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So far the arguments given in recent posts give us a clear idea of how the Earth-Moon system works: Earth’s a spinning sphere of diameter about 8000 miles (13000 km), and the size of the Moon and its distance are known too (diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter). We also know that the Sun is much further than the Moon and larger than the Earth, though we don’t know more details yet.

What else can we learn just with simple observations? Since the stars’ daily motion is an illusion from the Earth’s spin, and since the stars do not visibly move relative to one another, our attention is drawn next to the motion of the objects that move dramatically relative to the stars: the Sun and the planets.  Exactly once each year, the Sun appears to go around the Earth, such that the stars that are overhead at midnight, and thus opposite the Sun, change slightly each day.  The question of whether the Earth goes round the Sun or vice versa is one we’ll return to.   

Let’s focus today on the planets (other than Earth) — the wanderers, as the classical Greeks called them.  Do some of them go round the Earth?  Others around the Sun?  Which ones have small orbits, and which ones have big orbits? In answering these questions, we’ll start to build up a clearer picture of the “Solar System” (in which we include the Sun, the planets and their moons, as well as asteroids and comets, but not the stars of the night sky.)

The Basic Patterns

If we make the assumption (whose validity we will check later) that the planets are moving in near-circles around whatever they orbit, then it’s not hard to figure out who orbits who. For each possible type of orbit, a planet will exhibit a different pattern of sizes and phases across its “cycle“ when seen through binoculars or a small telescope. Even with the naked eye, a planet’s locations in the sky and changes in brightness during its cycle give us strong clues. Simply by looking at these patterns, we can figure out who orbits who.

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Some Pre-Holiday International Congratulations

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I’m still kind of exhausted from the effort (see yesterday’s post) of completing our survey of some of the many unexpected ways that the newly discovered Higgs particle might decay. But I would be remiss if, before heading off into the holiday break, I didn’t issue some well-deserved congratulations. Congratulations, first, to China — to … Read more

Testing, Testing: 12/12/12 12:12:12

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This is a modified version of last year’s 11/11/11 article, in case you missed it.

Today is a special day — at least if you are fond of the number 12, and especially so if you’re willing to buy in to one of the oldest human pseudo-scientific pursuits: numerology. Oh, don’t get me wrong, I love numbers and I always have. When I was five years old I was mesmerized when my parents’ car reached 99,999.9 miles, and I think 12:34:56 on 7/8/90 is just a cool a time as anybody else does. But I do this with a sense of humor.

Unfortunately it happens that a few influential people attempt serious and consequential numerology involving the calendar — predicting disaster and convincing people to sell their homes and give away their belongings. Now that makes me mad. Outraged, in fact — because it’s often obvious from the way these predictions are generated that those who made them don’t understand much about the calendar, about time, about history and about astronomy or physics… and yet they speak with authority, an authority they haven’t earned and don’t deserve.

So as we celebrate this one-two-of-a-kind moment, let’s also remember, and enjoy, just how absurd it really is. Let us even count the ways.

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Physics and Curiosity on Mars

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The promised follow-up article on the workshop last week in Waterloo, Canada will have to wait til Monday; I had too many scientific activities and chores to take care of today, and I want to make sure the article, which is a bit complicated, is nevertheless clear.   But in the meantime, let’s celebrate Martian Curiosity!

First, a big congratulations to the NASA folks!  Very impressive, and fantastically cool.  I was a huge fan of the Spirit and Opportunity rovers, especially of their 3D photography.  Looking at those photos on a big screen, through red/green 3D glasses. brought me to sweeping Martian vistas and deep Martian craters — as vivid and as close as I’ll ever see them.  It was amazing stuff, and I look forward to more from the new rover.

Next: some perspective.

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