Tag Archives: moon

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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Who Orbits Who, and Where? Check it Yourself

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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Which is Bigger, the Sun or the Earth?  Check it Yourself!

Once you’ve convinced yourself the Earth’s a spinning sphere of diameter about 8000 miles (13000 km), and you’ve estimated the Moon’s size and distance (diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter), it’s easy to convince yourself the Sun’s bigger than the Earth, and much further than the Moon.  It just takes a couple of triangles, and a bit of Moon-gazing.

Since that’s all there is to it, you can guess that the ancient Greek astronomers, masters of geometry, already knew the Sun’s the larger of the two.  That said, they never did quite figure out how big and far the Sun actually is; we need modern methods for that.

It’s Just a Phase

The Moon goes through a monthly cycle of phases, lasting about 291/2 Earth days, in which the part that glows brightly with reflected sunlight grows and shrinks, from crescent to full and back again.  The phases arise because there are two simple ways of dividing the Moon in half:

  • At any moment, the half of the Moon that faces Earth — let’s call it the near half of the Moon — is the only half that we can potentially see. (We’d only be able to see the far half, facing away from Earth, if the Moon were transparent, or a big mirror was sitting beyond the Moon.)
  • At any moment, the half of the Moon that faces the Sun is brightly lit — let’s call it the lit half.  The other half is dark, and its presence can only be detected by the fact that it can block stars that it moves in front of, and through a very dim glow in which it reflects sunlight that first reflected from the Earth (called “Earthshine.”)  

The phases arise because the lit half and the near half aren’t the same, and the relationship between them changes from night to night.   See the diagram below. When the Moon is more or less between the Sun and the Earth (it rarely passes exactly between, because its orbit is tilted by a few degrees out of the plane of the drawing below) then the Moon’s lit half is its far half, and the near half is unlit. We call this dark view of the Moon the “New Moon” because it is traditionally viewed as the start of the Moon’s monthly cycle. 

Figure 1: The Moon’s phases, assuming the Sun’s much further than the Moon. When the Moon is roughly between the Earth and Sun, its near half coincides with the unlit half, making it invisible (New Moon). As the cycle proceeds, more of the near half intersects with the lit half; after 1/4 or the cycle, the Moon’s near half is half lit and half unlit, giving us a “half Moon.” At the cycle’s midpoint, the near side coincides with the lit half and the Moon appears full. The cycle then reverses, with the other half Moon occurring after 3/4 of the cycle.

When the Moon is on the opposite side of the Earth from the Sun (but again, rarely eclipsed by Earth’s shadow because of its tilted orbit), then its near side is its lit side, and that creates the “Full Moon”, a complete white disk in the sky. 

At any other time, the near side of the Moon is partly lit and partly unlit. When the line between the Moon and Earth is perpendicular to the Earth-Sun line, then the lit side and unlit side slice the near side in half, and the Moon appears as a half-disk cut down the middle.

When I was a child, I wondered why half this half-lit phase of the Moon, midway between New Moon (invisible) and Full Moon (the bright full disk), was called “First Quarter”, when in fact the Moon at that time is half lit.  Why not “First Half?”  Two weeks later, the other half of the near-side of the Moon is lit, and why is that called “Third Quarter” and not, say, “Other Half”?

This turns out to have been an excellent question. The fact that a Half Moon is also a First Quarter Moon tells us that the Sun is large and far away!

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How to Figure Out the Distance to the Moon Yourself

Last time I described an easy way for you to determine the size of the Moon — easier than the famous techniques used by the classical Greeks. (We don’t need to know the Earth’s circumference, as they did, if we’re ok with a moderately precise estimate.) Once you’ve done that, there’s an simple method, well known since classical times, for figuring out how far away the Earth’s companion is. That’s what I’ll describe in this short post.

(What’s not so easy is to determine the distance and size of the Sun. The classical Greeks failed in their efforts. We’ll need a more modern approach… but that’s for next week.)

Size Versus Distance

Even the early classical Greeks knew something about the Sun, just from the fact that the Moon and Sun appear roughly the same size to our eyes — that is, they occupy about the same amount of sky. If the Sun is twice as far away as the Moon, its diameter must be twice as big, in order that it appear the same size. That’s illustrated in the figure below. If it is ten times as far away, its diameter must be ten times as big. If it’s four hundred times as far away, its diameter is four hundred times as big. (Spoiler: that last one’s the truth; but we’ll get to it later.)

If the Moon is a distance L away from you, and another object twice as far away appears to be the same size in the sky, then that object’s diameter must be twice the Moon’s diameter D. This logic applies more generally to objects further and nearer than the Moon.

You can run this logic in the other direction; if something perfectly blocks the Moon, then if it’s ten times closer than the Moon its diameter must be ten times smaller. If it’s a billion times closer than the Moon, it must be a billion times smaller.

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How to Figure Out the Size of the Moon Yourself


Having confirmed we live on a spherical, spinning Earth whose circumference, diameter and radius are roughly 25000, 8000, and 4000 miles (40000, 13000, and 6500 km) respectively, it’s time to ask about the properties of the objects that are most obvious in the sky: the Sun and Moon. How big are they, and how far away?

If the Moon were close to Earth, then at any one time it would only be visible over a small part of the Earth, as indicated in light blue. But in fact (except at new moon) about half the Earth can see it at a time.

Historically, many peoples thought they were quite close. With our global society, it’s clear that neither can be, because they can be seen everywhere around the world. Even the highest clouds, up to 10 miles high, can only be seen by those within a couple of hundred miles or so. If the Moon were close, only a small fraction of us could see it at any one time, as shown in the figure at right. But in fact, almost everyone in the nighttime half of the Earth can see the full Moon at the same time, so it must be much further away than a couple of Earth diameters. And since the Moon eclipses the Sun periodically by blocking its light, the Sun must be further than the Moon.

The classical Greeks were expert geometers, and used eclipses, both lunar and solar, to figure out how big the Moon is and how far away. (To do this they needed to know the size of the Earth too, which Eratosthenes figured out to within a few percent.) They achieved this and much more by working carefully with the geometry of right-angle triangles and circles, and using trigonometry (or its precursors.)

The method we’ll use here is similar, but much easier, requiring no trigonometry and barely any geometry. We’ll use eclipses in which the Moon goes in front of a distant star or planet, which are also called “occultations”. I’m not aware of evidence that the Greeks used this method, though I don’t know why they wouldn’t have done so. Perhaps a reader has some insight? It may be that the empires they were a part of weren’t quite extensive enough for a good measurement.

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An Experience of a Lifetime: My 1999 Eclipse Adventure

Back in 1999 I saw a total solar eclipse in Europe, and it was a life-altering experience.  I wrote about it back then, but was never entirely happy with the article.  This week I’ve revised it.  It could still benefit from some editing and revision (comments welcome), but I think it’s now a good read.  It’s full of intellectual observations, but there are powerful emotions too.

If you’re interested, you can read it as a pdf, or just scroll down.

 

 

A Luminescent Darkness: My 1999 Eclipse Adventure

© Matt Strassler 1999

After two years of dreaming, two months of planning, and two hours of packing, I drove to John F. Kennedy airport, took the shuttle to the Air France terminal, and checked in.  I was brimming with excitement. In three days time, with a bit of luck, I would witness one the great spectacles that a human being can experience: a complete, utter and total eclipse of the Sun. Continue reading

Some Pre-Holiday International Congratulations

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.

The Jade Rabbit rover on the surface of the Moon, 15 December. Credit:Xinhua

Congratulations, first, to China — to the scientists and engineers who’ve managed to put a lander and a rover on the Moon. If you think that’s easy… think again! And they succeeded on their first attempt, a real coup. Now let’s see what science they can do with it, exploring a region of the Moon that apparently may offer answers to important questions about the Moon’s history. Specifically, by accident or by design, the rover is going to be able to explore an area of considerable geological importance, involving one of the Moon’s giant lava flows, a relatively young one (1-2.5 billion years rather than 3 billion or more).

Soyuz VS06, with Gaia, lifted off from Europe’s Spaceport, French Guiana, on 19 December 2013. Copyright: ESA – S. Corvaja, 2013

Congratulations, next, to the scientists and engineers of the European Union, who’ve put a fantastic telescope into space, destined to orbit the sun. The Gaia mission is aimed at doing the extraordinary: mapping, with ultra-high precision, the locations and motions of no less than 1 billion stars within our galaxy — nearly 1% of the total number. The distance to each of these stars will be determined by parallax — looking at how the positions of stars wobble, from the perspective of the spacecraft as it orbits the sun — and the real motions of the stars will be determined by how they drift across the sky, and by the Doppler effect for light.  This wealth of information will help scientists figure out the shape and history of the galaxy to a degree never previously possible.  Meanwhile, Gaia will also be able to do a lot of other science, picking up distant supernovas outside our galaxy, nearby asteroids orbiting our sun, and signs of planets around other stars, as well as brown dwarfs (small failed stars) that may be floating around between the stars. Gaia can even check some aspects of Einstein’s theory of gravity! Read here about all the wonderful things this mission can do.

Congratulations also to the scientists and engineers in Iran, who’ve apparently moved their rocketry program, and its potential application to human space flight, among other things, another step forward. A second monkey has made the trip to the edge of space, a suborbital trip. (Did the first survive? it’s not clear, and admittedly Iran is known for photo-shopping reality into supporting the story it wants to tell. Not that it matters; it took the US several tries, back over 60 years ago, before a monkey survived the trip, and the survival rate continued to be poor for a while. )  Anyway, it puts Iran well on its way toward its goal of a human in space by 2018.

And finally, congratulations to my own country, the United States, for having passed a budget deal. Not out of the woods yet, but at least it was bipartisan, and we’re not yet talking about another damaging government shutdown, or worse, default. Politics isn’t rocket science. We’ll have to hope our politicians can learn something from China: that it’s good to find some common and worthy goals to work toward together, rather than to fight about absolutely everything and bring the nation’s operations to a halt.

No Comet, But Two Crescents

I’m sure you’ve all read in books that Venus is a planet that orbits the Sun and is closer to the Sun than is the Earth. But why learn from books what you can check for yourself?!?

[Note: If you missed Wednesday evening’s discussion of particle physics involving me, Sean Carroll and Alan Boyle, you can listen to it here.]

As many feared, Comet ISON didn’t survive its close visit to the Sun, so there’s no reason to get up at 6 in the morning to go looking for it. [You might want to look for dim but pretty Comet Lovejoy, however, barely visible to the naked eye from dark skies.] At 6 in the evening, however, there’s good reason to be looking in the western skies — the Moon (for the next few days) and Venus (for the next few weeks) are shining brightly there.  Right now Venus is about as bright as it ever gets during its cycle.

The very best way to look at them is with binoculars, or a small telescope.  Easily with the telescope, and less easily with binoculars (you’ll need steady hands and sharp eyes, so be patient) you should be able to see that it’s not just the Moon that forms a crescent right now: Venus does too!

If you watch Venus in your binoculars or telescope over the next few weeks, you’ll see proof, with your own eyes, that Venus, like the Earth, orbits the Sun, and it does so at a distance smaller than the distance from the Sun to Earth.

The proof is simple enough, and Galileo himself provided it, by pointing his rudimentary telescope at the Sun 400 years ago, and watching Venus carefully, week by week.  What he saw was this: that when Venus was in the evening sky (every few months it moves from the evening sky to the morning sky, and then back again; it’s never in both),

  • it was first rather dim, low in the sky at sunset, and nearly a disk, though a rather small one;
  • then it would grow bright, larger, high in the sky at sunset, and develop a half-moon and then a crescent shape;
  • and finally it would drop lower in the sky again at sunset, still rather bright, but now a thin crescent that was even larger from tip to tip than before.

The reason for this is illustrated in the figure below, taken from this post [which, although specific in some ways to the sky in February 2012, still has a number of general observations about the skies that apply at any time.]

A planet (such as Mercury or Venus) with an orbit that is smaller than Earth's has phases like the moon but grows and shrinks during its orbit round the sun due to its changing distance from earth.  It is always largest when a crescent and smallest when full, and is brightest somewhere in between.

A planet (such as Mercury or Venus) with an orbit that is smaller than Earth’s has phases like the Moon.  The portion of Venus that is lit is a half-sphere (shown in light color); the portion of Venus we can see is a different half-sphere (dashed red lines); the overlap is shaped like the wedge of an orange and looks like a crescent in the sky.  But unlike the Moon, which is at a nearly fixed distance from Earth, such a planet appears to grow and shrink during its orbit round the Sun, due to its changing distance from Earth. It is always largest when a crescent and smallest when full, and is brightest somewhere in between.

So go dig out those binoculars and telescopes, or use Venus as an excuse to buy new ones! Watch Venus, week by week, as it grows larger in the sky and becomes a thinner crescent, moving ever closer to the sunset horizon.  And a month from now the Moon, having made its orbit round the Earth, will return as a new crescent for you to admire.

Of course there’s another proof that Venus is closer to the Sun than Earth is: on very rare occasions Venus passes directly between the Earth and the Sun.  No more of those “transits” for a long time I’m afraid, but you can see pictures of last June’s transit here, and read about the great scientific value of such transits here