Since the upcoming book is basically done, it’s time for me to launch the next phase of the project — the supplementary material, which will be placed here, on this website.
Any science book has to leave out many details of the subjects it covers, and omit many important topics. While my book has endnotes that help flesh out the main text, I know that some readers will want even more information. That’s what I’ll be building here over the coming months. I’ll continue to develop this material even after the book is published, as additional readers explore it. For a time, then, this will be a living, growing extension to the written text.
As I create this supplementary material, I’ll first post it on this blog, looking for your feedback in terms of its clarity and accuracy, and hoping to get a sense from you as to whether there are other questions that I ought to address. Let’s try this out today with a first example; I look forward to your comments.
In Chapter 2 of the book, I have written
Over two thousand years ago, Greek thinkers became experts in geometry and found clever tricks for estimating the Earth’s shape and size.
This sentence then refers to an endnote, in which I state
The shadow that the Earth casts on the Moon during a lunar eclipse is always disk-shaped, no matter the time of day, which can be true only for a spherical planet. Earth’s size is revealed by comparing the lengths of shadows of two identical objects, separated by a known north-south distance, measured at noon on the same day.*
Obviously this is very terse, and I’m sure some readers will want an explanation of the endnote. Here’s the explanation that I’ll post on this website:
A break from all these wormholes and strings; let’s take a moment to look at the sky. In the US, sadly, most of the country will be under cloud, but for those who aren’t, you have a spectacle tonight, at around 10-11pm Eastern Time in the US, roughly 5-6 am UT in Northern Europe. It’s … Read more
Sadly, the LunaH-MAP mini-satellite (or “CubeSat”) that I wrote about a couple of days ago, describing how it would use particle physics to map out the water-ice in lunar soil, has had a serious setback and may not be able to carry out its mission. A stuck valve is the most likely reason that its … Read more
The recent launch of NASA’s new moon mission, Artemis 1, is mostly intended to demonstrate that NASA’s incredibly expensive new rocket system will actually work and be safe for humans to travel in. But along the way, a little science will be done. The Orion spacecraft at the top of the giant rocket, which will actually make the trip to the Moon and back and will carry astronauts in future missions, has a few scientific instruments of its own. Not surprisingly, though, most are aimed at monitoring the environment that future astronauts will encounter. But meanwhile the mission is delivering ten shoe-box-sized satellites (“CubeSats“) which will carry out various other scientific and/or technological investigations. A number of these involve physics, and a few directly employ particle physics.
The use of particle physics detectors for the purpose of studying the not-so-empty space around the Moon and Earth is no surprise. Near any star like the Sun, what we think of as the vacuum of space (and biologically speaking, it is vacuum: no air and hardly any atoms, making it unsurvivable as well as silent) is actually swarming with subatomic particles. Well, perhaps “swarming” is an overstatement. But nevertheless, if you want to understand the challenges to humans and equipment in the areas beyond the Earth, you’ll inevitably be doing particle physics. That’s what a couple of the CubeSats will be studying, entirely or in part.
What’s more of a surprise is that one of the best ways to find water on the Moon without actually landing on it involves putting particle physics to use. Although the technique is not new, it’s not so obvious or widely known, so I thought I’d draw your attention to it.
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.
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.
Once you’ve convinced yourself the Earth’s a spinningsphere 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!
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.
