Logarithms and units

One of the things that every intro calculus student learns is: $${d\ln x\over dx} = {1\over x}$$ This property of the logarithm leads to something else, which turns out to be useful to physicists and astronomers, but is never explicitly taught. If we rearrange this equation to read $${d\ln x} = {dx\over x}$$ we see that a given change in the logarithm (\(d\ln x\)) corresponds to a given fractional change in x. This equation also implies that the logarithm of anything is unitless, as follows:

• the right side of this equation, \({dx\over x}\), is unitless regardless of the units of x;
• therefore the left side, \(d\ln x\), must also be unitless; 
• \(d\ln x\) must have the same units as \(\ln x\);
• therefore \(\ln x\) must also be unitless, regardless of the units of x. 

Physics students keeping track of their units can be stumped: what units does the log of a current or a voltage have? This tiny bit of math helps us see that the answer is "none."

The fact that \(d \ln x\) specifies a fractional change in x has further repercussions in astronomy, because it is traditional to quote the measurement of a flux \(f\) in the magnitude system: $$m = -2.5 \log_{10} {f\over f_0}$$ where \(f_0\) is some reference flux. This means that a quoted uncertainty in the magnitude of a star or galaxy, \(dm\), specifies a fractional uncertainty in the flux. Let's work out the details: \(\log_{10} x\) is the same as \({\ln x \over \ln 10}\) so $$dm = -{2.5\over \ln 10}  d\ln{f\over f_0} $$ $$dm = -{2.5\over \ln 10} {df\over f} $$ Because \(\ln 10\approx 2.30\), we get \(dm \approx -1.086 {df\over f}\).  For quick estimation purposes, the magnitude uncertainty is about the same as the fractional uncertainty in flux.

This explains why a 0.1 mag uncertainty is about a 10% flux uncertainty, regardless of the magnitude. One should not say that a 0.1 mag uncertainty is a 1% uncertainty in an \(m=10\) star, nor a 0.5% uncertainty in an \(m=20\) galaxy.  For the quantity that matters---the flux of the object---a 0.1 mag uncertainty implies about a 10% uncertainty regardless of the flux.

SIRC Solar System

This is a summary of my Science in the River City workshop aimed
at fifth-grade solar system standards.

I started by projecting this opinion piece responding to the release
of a new survey showing that 26% of Americans answered the question
"Does the Earth go around the Sun, or does the Sun go around the
Earth?" incorrectly.  The opinion piece was a bit snarky and
unforgiving; educators know that everyday experience (eg, seeing the
Sun rise and set) is extremely powerful.  People who think the Sun
goes around the Earth are at least processing what they see and
building some kind of model to account for it. The vast majority of
people who "know" that the Earth goes around the Sun are probably just
memorizing something their teachers told them, and probably could not
cite evidence or build an argument to support this statement.  But the
latter is a much more valuable skill in today's society, and the Next
Generation Science Standards (NGSS) call for our kids to practice this skill.
So, I asked the teachers to role-play in pairs, one building an
argument that the Sun goes around the Earth and the other vice
versa. Try it...it's hard!

The #1 argument for Sun-around-Earth is, of course, rising and
setting, but there aren't a lot of obvious strong arguments for the
other way.  Kids will try to invoke authority, such as "NASA launches
rockets that can see what's going on" but I steer them away from that;
people concluded Earth-around-Sun long before rockets were built, so
they should be able to use simpler observations.  One pair of teachers
played out the two models using their bodies so they could see the
consequences of each, and that's exactly what I recommend whenever
possible.  They showed that the Sun's daily motion could be explained
by the Sun-around-Earth model or by a spinning-Earth model (which is
not the same as Earth-around-Sun).  Lesson: don't stop creating models
once you find one that fits, as other models might fit the data too!

If we had no other data, we might not be able to choose between these
models.  In such cases we should try to bring other observations to
bear, and/or deduce further consequences of each model. An example of
the latter: if the Earth is spinning, why don't we feel it?  We feel
dizzy when we spin on a merry-go-round....but if the merry-go-round
took 24 hours to complete one turn the effects might be too small to
feel.  So that was worth thinking about, but inconclusive. Can we
bring other data to bear? Well, just about everything in the night sky
rises and sets and rises again in about 24 hours, so the
Sun-around-Earth model has to become the
entire-universe-turns-around-Earth model.  But it's a lot easier to
believe that one thing (Earth) spins than that everything in the
universe contrives to revolve around Earth in 24 hours. Preferring the
simpler model is called Occam's razor, and we do that all the
time in real life. (Think of situations where someone is caught
red-handed doing something they shouldn't, and they say "This isn't
what it looks like" and tell a complicated story...do you believe the
simple story or the complicated story?)

The real physical proof that the Earth spins is quite subtle. One is the
Coriolis effect: we can measure the effects of being on a slowly spinning
"merry-go-round".  I showed the teachers a great kinesthetic activity
for this (described in a previous post).  Another is the Foucault
pendulum, which is not easy to demonstrate in a school but which kids
may have seen in science museums.  A third is even more modern:
astronomers can directly measure velocities of celestial objects using
the Doppler shift (the same principle is used by radar speed guns),
and we constantly have to correct for the velocity of the observatory.
Because of Earth's rotation, stars that are rising appear to be moving toward
us and stars that are setting appear to be moving away from us.

So the Earth spinning accounts for the daily (apparent) motions of Sun
and stars...so why do we think Earth goes around the Sun? (Do you see
how much reasoning we had to do to even get to this question?) Anyone
who looks at the stars periodically must have noticed that you can't
see the same stars all year round.  And because the presence of the
Sun in the sky defines when we can't see stars, that means that the
Sun moves relative to the stars.  Another way to say this is that,
unlike the Sun, the stars don't take exactly 24 hours to (appear to)
go around; they take 23 hours and 56 minutes.  So the Sun may be
opposite a certain star (ie the star is high up at midnight) at a
certain time of year, but over months will creep around the sky to
prevent us from seeing that star, and after 365 days the Sun will be
back to its original position relative to the stars.  Earth's rotation
period is either 24:00 or 23:56; it can't be both!  (It could be
neither, but considering that possibility may cause cognitive
overload.)

If we again use majority rule or Occam's razor, it's simpler to think
that the stars are fixed and that the Sun moves relative to the Earth
(note how little this conclusion has to do with the basic observation
that the Sun rises and sets each day, which is caused by Earth
spinning).  But there are at least two models which involve the Sun
moving relative to Earth, and again I had teachers play the roles of
Sun and Earth to demonstrate a model where Sun is stationary and Earth
moves, and vice versa.  Both of these models account for the
observations equally well (so far), and the moving-Earth model has the
disadvantage that we don't feel the Earth move.  This is one reason
many ancient Greek thinkers did not endorse the moving-Earth
hypothesis. But Galileo figured out that if you are in a laboratory
moving at constant velocity, you can't feel it move---think of a smooth plane
ride at 500 mph.  Earth doesn't move at constant velocity, but it changes its
velocity so slowly that the effects are really small.

The other thing the ancient Greeks figured out is that the
moving-Earth hypothesis means that we should see parallax. Think about
sitting in a moving car.  The roadside trees appear to rush by, but
the distant mountains appear to move very slowly.  If the Earth moves,
we ought to be able to see an effect like this by comparing nearby and
distant stars.  I taped some stars around the room and some teachers
played this out.  If Earth is still, we will not see parallax.  The
Greeks looked for parallax but did not see it, so they favored the
stationary-Earth hypothesis; they weren't stupid!  It just turned out
that even the nearest stars are so far away that the parallax effect is
tiny and could not be measured until modern times.

Using parallax we can determine that the nearest star is about 300,000
times more distant than the Earth-Sun distance (that comparison is natural
because it is Earth's motion over that distance that gives rise to the parallax
effect).  It's as if your car moved one mile but you were asked to discern the
difference in your view of mountains 300,000 miles away (eg, on the
Moon).  This can be illustrated dramatically by drawing a one-inch
Earth-Sun model on the board, and then drawing a long line
representing the distance to the next star and asking the kids to stop
me when they thought I had arrived.  Kids (even most adults) have no
idea how much 300,000 times is; they ask me to stop after 5 feet or
so, but I keep going.  When I run out of board, I get a roll of toilet
paper and start unrolling it, as a way to illustrate a very long line.
I keep going even when they tell me to stop. Then, when I run out of
toilet paper, I go to the back room and get a cart full of hundreds of
rolls of toilet paper!  It is really dramatic and fun. It's also fun
to write out the number of miles to the next nearest star on the
board: about 24,000,000,000,000 miles. (Kids and even most adults have
little idea what a "trillion" really means.)

Finally, because a light looks dimmer the greater its distance from
us, we can calculate how much light a typical star emits (as opposed
to the very small amount of its light that we receive).  Correcting
for this distance factor, it turns out that stars emit about as much
light as our Sun!  Some emit more, some less, but the bottom line is
that each star is a sun unto itself, or if you prefer, the Sun is just
another star.  In the 1600's, long before parallax was ever measured,
the astronomer Christiaan Huygens turned this argument on its head:
assuming the star Sirius is as luminous as our Sun, how far away would
it have to be to appear so dim?  His conclusion was 30,000 times as
far as the Sun; this is lower than the true value, but only because Sirius is
actually substantially more luminous than the Sun.  And his number was
certainly big enough to convey some of the vastness of the universe.

The Moon does go around the Earth, so we used that to address the
gravity standard.  I did the donutapult demo, which illustrates that
for anything to move in a circle there must be a force directed toward
its center; therefore the Moon is pulled on by a force directed toward
Earth's center.  We can relate that to gravity (ie, the force we feel
every day here on Earth's surface) by looking at a globe and noting
that, anywhere on Earth, "down" means toward the center of the Earth.
Therefore (Occam's razor again) we don't need to hypothesize about a
mysterious force keeping the Moon in its orbit; it could well be the
same force that makes apples fall. (Proving that it's the same force
goes beyond the fifth grade standards.)

Finally, we discussed scale models, as scale is a crosscutting concept
in the NGSS. There is no better way to make people appreciate how
empty space is than to build a scale model of the solar system! Rather
than write up our discussion, I refer you to my previous blog posts on
the scale model project I led at Peregrine School: intro, poster assignment,
and completion.

A few links you may find useful:

• my previous SIRC workshop on the solar system (8th grade)
• Powers of Ten video to help students appreciate scale
• interactive tool to do the same (Warning: this tool only shows relative sizes, not the huge distances between celestial objects)
• feel free to browse the labels at right

Mostly Harmless

In the Hitchhiker's Guide to the Galaxy, "mostly harmless" is the
Encyclopedia Galactica's assessment of Earth (which is not important
enough to merit a longer entry).  This made me think that looking at
the solar system through alien's eyes might help students learn about
it.  I conducted Science in the River City workshop for earth science
teachers based on this idea, and this is a list of resources for such
teachers.

First, I highlighted a graphing activity I had done with elementary
kids; that experienced is described in great detail here. (Feel free
to download and copy the graph.)  I extended the activity to graphing the
surface temperatures of the planets as a function of distance from the
Sun, which led to the greenhouse effect discussion below, but now it
occurs to me that a great way to extend this activity would be to jigsaw
it: assign one group of students to graph size vs distance from the Sun,
another to graph temperature vs distance from the Sun, another to graph
density vs distance from the Sun, etc, and then the groups come together
to think about what it all implies for the formation of the solar system.

Second, when discussing the formation of the solar system and

describing how small grains of dust started to stick together, I

wanted to show a video clip but had some technical difficulties.  Here

is the link; start at 3 minutes into the video and go for 2.5 minutes.

(If you have time, the whole episode is worth watching.  It's from the

How the Earth Was Made series, which has some really nice

visualizations and is constructed around evidence, which is a key

feature missing from many science documentaries.  It tells science

like the detective story it is.  That's generally a good thing, but in

this case the implication that this particular astronaut doing this

particular demonstration singlehandedly saved the theory is a bit of

an exaggeration.)

Pocket solar system: https://nightsky.jpl.nasa.gov/download-view.cfm?Doc_ID=392

Peppercorn Earth: http://www.noao.edu/education/peppercorn/pcmain.html

Extrasolar planets: http://exoplanets.org/ has the most up-to-date

info. Even better, they have built-in graphing tools so you and

your students can easily explore the data.

Earth's surface temperature: I got my plot from the most authoritative

source for modern temperatures, NASA's Goddard Institute for Space Studies.

This link only scratches the surface of climate change data because it deals

with modern temperature measurements (as opposed to long-ago temperatures

inferred from ice cores etc) but as the greenhouse effect was not the focus of

the workshop I won't try to compile a list of links here.  (For those

not attending the workshop: we graphed planets' surface temperatures

vs distance from the Sun, and we saw the general pattern that farther

from Sun equals colder, but we also saw that Venus is a real outlier

from this pattern.  That's because Venus has had a runaway greenhouse

effect.  Earth also has a natural greenhouse effect which keeps us

from being frozen, but which is now being augmented by a manmade

greenhouse effect.  I did tell the teachers that Earth has a "carbon

cycle" which will absorb the extra carbon dioxide through the oceans

into rocks, but I forgot to mention that it will take hundreds of

thousands of years; I didn't mean to imply that humans can carry on

regardless. Venus' greenhouse effect is "runaway" because its

carbon cycle shut down when its oceans boiled.)

Finally, a few links I didn't get time to show but which will help you

appreciate the size of the universe (and the sizes of things in it):

the classic Powers of Ten video and an interactive tool.

One Plus z

This marks the launch of a new series of posts, aimed at astronomy and physics majors. In the course of my teaching I've noticed a few topics---such as propagation of errors and reduced mass---which seem to fall through the cracks between classes.  Students hear a bit about reduced mass in more than one class, but never seem to get a satisfying explanation in any one class.  Their lab instructor taught them how to propagate errors but never made them think about why.  And so on.  This first post is much more specific---how to think about redshifts and velocity dispersions in cosmology---but fits the bill because it seems to fall through the cracks between textbooks.  Practitioners know that "you need to divide by 1+z" but documentation of this is hard to come by.  So here we go.

In cosmology, we often want to measure the rest-frame velocity dispersion of a galaxy cluster, but what we actually measure is the redshift dispersion. How are they related? Redshift z is defined in terms of emitted and observed wavelengths as

This means that 1+z is a stretching factor; it is the ratio of observed to emitted wavelengths.  So you will see the combination 1+z over and over, rather than z by itself.  Get used to thinking in terms of 1+z!

The Doppler shift formula tells us the wavelength stretching factor in terms of velocity:

You will often see this called the relativistic Doppler formula, as opposed to the simpler low-velocity approximation used in many situations. But I suggest thinking of this as the Doppler formula because  high velocities are common in astrophysics, and this correct version is simple enough to memorize. Habitually using the low-velocity approximation can get you in trouble.

The Doppler formula can be inverted to obtain

Now imagine two galaxies, one at rest¹ in the cluster frame (with velocity v₁ in our frame) and a second moving with some velocity v₂₁ relative to the cluster which implies some velocity v₂ in our frame.  According to the Einstein velocity addition law,

Substituting the inverted Doppler formula into this, we obtain a complicated-looking expression for v₂₁/c:

which we can simplify in a few steps:

Because of my poor equation formatting, I have to remind you here that this is an expression for v₂₁/c, where v₂₁ represents a velocity in the cluster frame rather than in our frame. This gets us close to our goal because we want to know the velocity dispersion in the cluster frame. But this is as far as we can go without an approximation. A useful approximation in this context is that so define and eliminate z₂ using :

Taylor expanding this about we obtain

This is true for any small redshift difference, so it must be true if delta represents the redshift dispersion of the cluster (thus making v₂₁ represent the velocity dispersion of the cluster). Therefore

However, there is a much more elegant way to derive the same result. Imagine a hypothetical observer on the first galaxy. Because of the definition of 1+z as the ratio of wavelengths, it must be true that 1+z₂ = (1+z₁)(1+z₂₁) where z₂₁ is the redshift of galaxy 2 as seen by galaxy 1 (z₁ and z₂ are, as before, redshifts seen by us). Therefore

Again we use an approximation: so that we can use the low-velocity approximation for the Doppler shift, . Therefore

which is the same result as before.  We don't actually need special-relativistic reasoning if we simply use the definition of redshift to isolate the one nonrelativistic velocity in the problem.

We can better expose the equivalence of these two approaches by taking the idea of daisy-chaining wavelength ratios and applying it directly to the Doppler law:

This just says that galaxy 2's wavelength ratio ("ratio'' here is relative to a laboratory standard) observed by us is its wavelength ratio observed by galaxy 1, times galaxy 1's wavelength ratio observed by us.   In a few lines of algebra, you can show that the above expression leads directly to the Einstein velocity addition law.  The addition law can be derived in more than one way, but to me this is the most intuitive way.  Thus, daisy-chaining Doppler factors and using the velocity addition law are not contrasting approaches; they are actually the same thing.

Exercise for the reader: show that the expression above does indeed lead to the Einstein velocity addition law.

Footnotes:
¹ I specify "at rest" here only so that later it will be easy to think of this galaxy’s redshift as the mean cluster redshift.

"Just" a Theory?

A recently published letter to the New York Times reminds us that relativity is "just a theory" and so is the Big Bang.  Scientists and science educators need to set the record straight on this "just a theory" meme any time we get a chance to discuss science with kids and grown-up nonscientists.  So here's my shot at it.

A good analogy is to think of facts as being like bricks: solid and dependable, but one or a few bricks are not very useful by themselves ("an electron passed through my detector at 11:58:32.01" or "the high temperature in Davis, CA on September 1, 2013 was 96 F").  Only when we assemble lots (lots) of bricks into a coherent structure do we get the benefits of having a building (the theory of relativity, or a climate model).  Not only is an isolated brick rather useless, but the building can easily survive the removal of a few bricks here and there.  A good theory integrates millions or billions of observations into a coherent whole.  Calling relativity "just a theory" is like calling the Great Wall of China "just a fence," the Panama Canal "just a ditch," or the Golden Gate Bridge "just a road."

There's a reason that calling the Great Wall of China "just a fence" sounds more outrageous than calling relativity "just a theory"---I used the word fence which connotes something less important than a wall.  There's a rich vocabulary to describe to describe barriers: from weak to strong we might use tape, rope, cordon, railing, fence, and wall. But most people don't use a similarly rich vocabulary to describe levels of sophistication of mental models.  From weak to strong I might suggest educated guess, working hypothesis, model, and theory, but most people in practice indiscriminately use the word theory for any of these.  So it's our duty as scientists to make clear that well-accepted scientific theories integrate an incredible range of observations into a structure which is so coherent that it is difficult to imagine all those pieces fitting into any other structure.  Maybe a better analogy to calling relativity "just a theory" is calling an assembled jigsaw puzzle "just one way to fit the pieces together."

Gotcha, the just-a-theory crowd says,  by making that analogy you are showing that you are rigid in your thinking and unwilling to accept alternative explanations.  Nonsense. Scientists are constantly trying to prove accepted theories wrong.  Anyone who succeeds in disproving relativity, the Big Bang, or evolution will win a Nobel Prize and eternal fame, so we'd be happy to do so.  But we know from experience that the most likely explanation for an isolated fact that seems to contradict relativity, the Big Bang, or evolution is that the fact itself was taken out of context or is not being properly interpreted, rather than that an extremely well-tested theory is wrong.

This doesn't mean that we will twist any fact to make it fit into our well-accepted theories. It does mean that surprising facts may end up extending the theory rather than replacing it.  For example, Newton's theory of gravity explains a ton of observations about the motions of the planets and stars, but in a few extreme circumstances (such as very close to the Sun) it doesn't predict exactly what is observed.  Einstein developed a theory of gravity (general relativity) which does correctly predict these situations. Einstein's theory is more complicated than Newton's, but in most situations the complicated parts of Einstein's theory have very little quantitative effect so we can simplify it a great deal and in those cases it turns out to be identical to....Newton's theory!  This almost had to be the case, because Newton's theory accounted so well for so many observations that it would be hard to imagine that it was wrong rather than incomplete. 

This example shows that a small number of facts can be critically important and that scientists do pay attention to facts which don't fit the theory.  But we don't modify or overturn theories willy-nilly.  When the planet Uranus didn't move exactly as Newton's theory predicted, modifications of the theory were considered but so was the possibility that some mass other than the Sun and the known planets was pulling on Uranus, and that led to the discovery of Neptune.  If we rejected well-established theories at the first hint of any discrepancy with new observations, we would be giving undue weight to the new observations and too little weight to the vast range of previous observations explained by the theory.  If you want to overthrow a theory because some new observation seems to contradict it, then give us a better theory which explains the new observation while still fitting the previous observations just as well as the old theory.  That latter part seems to be conveniently forgotten by people who want to reject well-established theories.

A closely parallel situation is that of criminal investigators and prosecutors who present their "theory of the crime" to a jury. ("Model of the crime" would better fit my vocabulary hierarchy, but this is the word actually used.)    A lot of facts may be introduced into evidence ("a car with the suspect's license plate was recorded crossing the Tappan Zee Bridge at 2:20am on August 31"), but by themselves they don't mean anything important.  A good theory of the crime provides a coherent explanation of so many different facts that the jury is forced to conclude that it is true beyond a reasonable doubt.  If you want to call it "just a theory" then offer us a different theory which fits the facts just as well.  The defense is given sufficient time and strong motivation to offer a good alternative theory, so failure to present one is damning. 

Great Balls of Fire

After learning about gravity and taking the midmorning break,  the Peregrine 3-4 graders and I worked on understanding nuclear fusion in the core of the Sun and where elements come from.

I started by setting the context.  The students had studied atoms and molecules the previous year so I started by drawing a molecule of water (two hydrogen atoms and one oxygen atom) and reminding them of the evidence for atoms and molecules.  Then we zoomed in to one hydrogen atom and discussed the Rutherford experiment showing that atoms are very fluffy; most of their volume is nearly empty while nearly all their mass is concentrated in a tiny volume in the center (nucleus).  Then we zoomed in further by a factor of 10,000 to the nucleus.  For a hydrogen atom, the nucleus is a single positively charged particle called a proton.  I held up a ping-pong ball as a proton and said that if protons really were that size, the atom would have to be the size of South Davis.

To reinforce the sense of scale, I showed the movie Powers of Ten.  This classic ten-minute movie should be seen by anyone wanting to understand the universe.  I also took the time to answer questions about it.

The basic rules of nuclear physics are actually understandable by anyone. Last year we investigated the effects of electrical charge, and concluded that like charges repel while opposite charges attract. Atoms beyond hydrogen in the periodic table have more protons.  But why do the protons stick together if they repel each other? There must be some form of glue.  I demonstrated two magnets  which repelled each other.  They were "donut" magnets threaded onto a rod so they didn't flop around and the repulsion was clear.  But when I turned the rod vertically and one magnet fell with enough speed onto the other one, they touched briefly.  That was enough for the velcro on their surfaces to attach and keep them together.  The velcro is a short-range force, like the strong nuclear force which keeps a nucleus together.

But protons alone can't generate sufficient strong nuclear force to keep nuclei together.  Another type of particle, with similar mass but no charge and called a neutron, provides the glue.  Nuclei need roughly equal amounts of protons and neutrons to be stable.  I modeled this with a bunch of ping-pong balls I had wrapped with velcro.  The "protons" had velcro hooks and the "neutrons" had velcro loops, so that you needed roughly equal numbers of each to build up a large nucleus.  (The different types were also different colors to make the idea plainly visible.)  Adding a neutron to a nucleus adds mass, but doesn't otherwise change the properties of the atom.  For example, a proton plus a neutron is still hydrogen, but we call it a different isotope of hydrogen.  Similarly, carbon-12 (usually written with a superscript 12 on the left) and carbon-14 are different isotopes of carbon which differ by two neutrons.

With that in mind, we can start building up more complicated elements from hydrogen.  Element number 2 (two protons) is helium, and we need two neutrons to provide the glue so the most common isotope of helium is helium-4.   The protons have to be smashed together at very high speed if they are to ever get close enough for the "velcro" of the strong nuclear force to make them stick, so we need very high temperatures to make this fusion process happen. (High temperature means the individual microscopic particles are wiggling or bounding around at high speed.)  We find it difficult to make these high temperatures on Earth, but the core of the Sun is 15 million degrees (Celsius; tens of millions of degrees if you think in Fahrenheit) and this happens quite routinely.  In fact, most stars turn hydrogen into helium in their cores.

Fusing helium into even heavier elements is harder, but most stars will do that as well by the ends of their lives.   It turns out that crashing two heliums together results in an unstable isotope of element 4 (beryllium), which quickly decays back into two helium-4 nuclei. But if you manage to crash a third helium into the two heliums before the two-helium complex has a chance to decay, you make carbon-12 (the most common isotope of element 6, carbon; again, equal amounts of protons and neutrons).  Then, if you crash another helium into that, you get element number 8: oxygen. Another helium into that produces element 10, argon.  These helium capture reactions are common in massive stars (substantially more massive than the Sun), and they create more of the even-numbered elements than the odd-numbered elements (nitrogen, fluorine, etc).  They go all the way up to iron (element 26).  I modeled all this with the velcro-covered ping-pong balls.

Have you noticed what we've done here?  We've explained the origin of the elements using basic, well-understood physical processes. That's pretty cool! Here's a graph of the observed abundances:

You can see that hydrogen is the most abundant, followed by helium, then the even-numbered elements carbon, oxygen....through iron.  But why are there elements beyond iron if stars only make up to iron? Well, stars make up to iron when they are in equilibrium.  But when they explode (a supernova), so much energy is released that even more complicated nuclei can be made.  I won't explain the details here, but the abundances of all the elements beyond iron are well understood as consequences of supernovae.  That we can understand all the features of the above plot is, to me, one of the most amazing things in all of science.

The supernova explosions are also what throw the newly-manufactured elements back into space, where they can mix into gas clouds that eventually collapse to form new stars. That means that the atoms in your body were once inside another star.  (Not from the Sun, because new atoms made in the Sun won't escape until the end of its life.)

Supernovae make some unstable elements, like uranium.  The most common type of decay for a heavy element is to violently eject a "bullet"  made of two protons and two neutrons, in other words a helium nucleus (again I modeled this with the ping-pong balls).  This is why there is helium on Earth; our gravity is too weak to hold on to helium gas, but helium produced by radioactive decays is trapped in rocks underground.  When we drill for natural gas, we can capture some of this helium and eventually use it to fill balloons.  When it escapes from the balloon, it eventually escapes into space.

Big Bang 

I left out one detail in the story above: most of the helium in the plot was actually made in the Big Bang.  Some of the kids had expressed interest in the Big Bang previously, so I used the remaining time to talk about that.  I used the usual balloon-with-stickers demo, and I also showed this interactive tool made by an undergraduate student of mine. The point of the tool is to show that although we see all galaxies moving away from us, observers in all other galaxies also see all galaxies moving away from them. So we are not at the center of anything. If we think back in time, all galaxies were closer to each other, so the universe was denser (and hotter).  Far enough back in time, the universe was so hot (everywhere) that a fair amount of hydrogen fused into helium.  This is called Big Bang nucleosynthesis (BBN). We can look at the abundance of various  BBN byproducts, like hydrogen-2 (aka deuterium) and confirm that this really happened.

Wrapping up

Most of this trimester we worked on understanding the immense size of space.  If this makes you feel insignificant, remember that you are made of atoms from another star.  You are a part of the universe which can actually understand itself. 

The Gravity of the Situation

Friday was my last day doing astronomy with the 3-4 graders at Peregrine School. The one standard I hadn't yet covered was gravity, so we did gravity before the break (this post) and after the break we discussed nuclear fusion in the Sun's core (next post).

I reviewed some ideas about motion we had discussed last year.  If you roll a marble, you expect it to go in a straight line unless something (another kid, perhaps, or a wall) interferes by pushing (exerting a force) on the marble. That's Newton's first law of motion. I then put a donut on a string and spun the donut in a circle over my head.   What will happen if the string is cut? Will the donut continue in a circle, fly off in a straight line, or fly off in a curve? We took a vote. I always clarify that the question is about what happens immediately, not about what happens eventually, like the donut falling due to the gravity in the room. This means that when we do the experiment, they have to really pay attention!

In reality I don't cut the string, but the string pulls through the soft donut, and it flies off in a straight line---Newton's first law again.  This is a pretty vivid demonstration that the Moon wouldn't keep going  around the Earth, nor the planets around the Sun, unless there was a force keeping them from flying off in a straight line.  Kids this age already know that we call that force gravity, but gravity is also the force that makes things fall when I drop them.  Why do we call these two forces by the same name?

I also have a tennis ball on a string so I can demonstrate circular motion as much as needed.  I do this and ask the kids what direction the force must be in.  It must be towards the center of the circle, where my fist is holding the string.  That's clear because the only direction a string can exert a force is pulling along the string! So whatever force is pulling on the Moon, it must be pointed toward the center of the Earth.  And that's exactly what we observe about gravity on Earth! (It helps to draw an Earth and how the arrow of gravity points in your location vs in, say, Australia.)  So it's quite plausible that these two forces are really the same force.

To bolster the argument that these are the same force, we should look not just at the direction, but also the strength.  I had the kids whirl the tennis ball on a string at various speeds, and feel whether the higher speed requires more force, less force, or the same force (the answer is more).  So let's look at the planets' speeds around the Sun and see if we can relate that to the force of gravity.  I asked the kids for suggestions as to what would affect the planet speed.  The two main suggestions were planet size, and planet distance from the Sun.  It would have been great to investigate both of these possibilities, but we were running short on time so we just did planet distance from the Sun.  I had the kids make graphs of planet speed vs planet distance from the Sun.  We took our time doing this right, figuring out how to draw the axes with reasonable scales, and adding planets one by one, starting with the most familiar ones.

A pattern did emerge: more distant planets are slower, as the graph below shows.

By our tennis ball experiment, slower circular motion implies a weaker pull (less acceleration). Therefore this graph implies that more distant planets feel a weaker pull, and planets closer to the Sun feel a stronger pull.  Does this make sense if the Sun's gravity is what keeps the planets from flying off in straight-line paths?  The kids agreed that it did.

[If we had also made the graph of speed vs planet size, we would not have seen such a clear pattern.  It happens that the outer planets tend to be bigger, so that there would be a tendency for bigger planets to be slower, but it would only be a tendency, not a law, because the biggest planet happens to be the nearest (fastest) of the outer four.  And the pattern would really be broken if we also included Pluto, which is a very distant (hence very slow), small object, providing a counterexample to the fast inner planets which happen to be small and which therefore might give someone the false impression that small means fast.]

I liked this 40-minute activity and I think it worked well.  I did simplify some details to avoid getting bogged down (eg the distinction between force and acceleration), but I think it was appropriate for 3-4 graders who wanted to focus on astronomy rather than physics. We also got in some more practice with graphs, which is important.  And we learned something which in Newton's time was revolutionary: the same laws of physics which we can deduce here on Earth also apply to objects in the sky.   This was one of the most wonderful discoveries in the history of science, and it's what allows us to understand the universe.

Light and Telescopes

In the second half of this morning's activities with the 3-4 graders, we discovered some things about light and telescopes.  I handed out diffraction gratings and we looked at the spectrum of the Sun and of the fluorescent lights in the room, discovering that white light is actually composed of many colors. We also looked at discharge tubes filled with different elements, with mercury and helium being the stars.    We found that each element emits a unique "fingerprint" of spectral lines.  To see a great 2-minute video of everything the kids saw, check this out. This is how we know what stars and other planets are made of.

We then discussed how the colors always appear in a certain order in a rainbow or a diffraction grating: red, orange, yellow, green, blue, violet.  Could there be any light which appears before red?  Yes, it's called infrared, and we can build cameras to see it even though our eyes can't.  I showed this nice video demonstrating the properties of infrared light.  Could there be any light which appears after violet?  Yes, ultraviolet, and after that would be X-rays and finally gamma rays.  We talked about X-rays for a while because some kids were worried about it being dangerous.  (Like many other things, they are safe if used properly, but dangerous if not.  A yearly dental X-ray is ok, but how do we protect the parts of our bodies which don't need to be X-rayed?  And how do we protect the workers who administer dozens of X-rays each day?) I extended that discussion to the ultraviolet and sunlight.

All this was a springboard for discussing telescopes, which is one of the last astronomy standards I hadn't covered yet.  Specialized telescopes are built to look at all kinds of light, from gamma rays to the infrared and radio. I showed pictures of some of the big telescopes I have used in my research, and that led to all kinds of interesting questions. We ran out of time, so I may start next Friday by answering more telescope questions.

Scale Model Solar System Complete!

This morning I guided the 3-4 graders through assembling our  scale model solar system.  I wanted them to really think about how to make a scale model, so I returned to each student the graph they had made last time  and I asked them to use the graph to figure out where they would put their planet, given that I had put Teacher Moné's beautiful Earth poster 2.5 meters from the Sun poster.  Of course, I found that I needed to break this task into smaller chunks for them to process.  We began by revisiting some of the steps we had done last week. Each child identified his/her planet on the graph, read its distance off the graph, and then we thought about what that distance means.  For example, Jupiter is at a distance of 5 on the graph.  Five what? The graph doesn't say.  But the graph itself is a scale model of the solar system.  We don't really care what  the actual distance is because we are simply stretching this scale model to become a larger scale model which will fill the school.  All we need to do is choose a reference point and stretch everything else accordingly.  The graph made this easy because it shows Earth as being at a distance of 1.  So if Jupiter is at 5, we simply need to put Jupiter 5 times farther from the Sun than Earth is from the Sun; in other words 5x2.5 meters or 12.5 meters.

To help the kids visualize this, I took a rubber band and marked three dots on it, representing Sun, Earth, and Jupiter.  This is a scale model much like the graph (if we ignore the vertical dimension of the graph).  If I stretch the rubber band, will Jupiter still be 5 times more distant from the Sun than the Earth is from the Sun?  Some kids said no and some said yes, so we took a vote.  Having to commit to a vote made the kids think harder and they voted overwhelmingly yes.  After the vote I did stretch the rubber band and I did get a bigger scale model.  In principle, if we got a really long rubber band, I could mark all the planets' distances at the scale of the graph and then stretch it out to get a giant scale model as big as the school, and that would tell us where to put each planet poster.  But since that's impractical, we do the math instead.

This seems to have been more or less the right level of conceptual challenge and the right level of math for the kids.  They found it a bit of a challenge, but a doable one that became satisfying rather than frustrating. After looking over each child's computation, we practiced some metacognition.  Alex was concerned that his number didn't make sense given what he knew about the relative positions of the Sun, Earth and Venus.  It turned out that he was misinterpreting his number as the Earth-Venus distance, but the point was a really important one: always check that your numerical results make sense! I have had so many students make a mistake punching numbers into a calculator, and get a number that obviously doesn't make sense given a moment's thought, but blithely write down the number as if any number displayed by a calculator must be correct.  In this case we wrote out the multiplication rather than use a calculator, but the same principle applies: check that the results actually make sense! This goes not only for numbers that you compute, but also for numbers that other people compute for you.

An especially effective way to double-check your number is to perform some completely different procedure; if you just perform the original procedure again, you may easily make the same mistake again.  So I thought of a way we could all check our numbers without recomputing anything.  I made a list of the students' results, starting with the closest planet and proceeding outward.  If the distance numbers didn't increase steadily, that would be a smoking gun indicating a mistake.  And we did find a mistake this way, so it was instructive.

Once we had our final numbers, we split into groups to measure off the distances and attach the posters to the walls.  We couldn't quite fit Neptune into the school grounds, and Orcus wasn't even close, but we put them up at the far end with a note saying where they should really be.  Even after choosing a scale so large that the orbit of Neptune was just outside the fence, the sizes of the planets are really small, smaller than a grain of sand for most planets.  Even Jupiter is only 2.4mm across.  Space is really big!

Looking at the finished product, I am really happy we did it and spent enough time on it to do it right. We certainly appreciate the solar system much better now, but we also learned new ways of thinking.

Planet Posters

Two weeks ago each student chose a planet (or other solar system object) to research and make a poster about. Today they brought in their posters, and each student told the class what they learned in their research.  The kids were very engaged and asked so many good questions that we spent all morning doing this.  So next week we will put up the posters at the appropriate distances from the Sun poster (which I made and put up near the school entrance today) to make a scale model of the solar system.   The discussions today were so full, frank, and wide-ranging that I can't hope to capture them in a blog post.  I will simply leave you with a short video with amazing images of Jupiter's moon Europa.

I think the posters were quite successful as a learning experience. The kids learned by researching and making them, but they also learned by listening to other kids talk about their posters, and they all learned when I answered numerous questions in more depth as they arose.  I think a key to real learning is that the posters should not be just a laundry list of facts, but should really be based on the students' questions.  When I issued the assignment, I offered some questions they might be interested in answering:

• What would it be like to visit?  What is the temperature?  Is there a solid surface? Would the Sun look bright from that distance?  If the temperature is extreme, think about ways to convey how extreme it is.
• Does the planet have moons or rings? If you chose a moon to begin with, briefly describe the host planet.
• What are seasons like on that planet? This depends on how tilted the planet is with respect to its orbit.
• How long is the year on that planet?  How long is a day?
• Are there volcanoes? Rocks? Rivers/lakes/oceans?  (If so, are they made of water or some other substance? Moons of Jupiter and Saturn are especially interesting in this respect.) Clouds? Earthquakes? Storms? Lightning?
• Could you possibly find life there? 

I think these questions helped prevent a "laundry list" result.  One thing I would like to do better next time is have some kind of first draft with feedback and then a final draft. I don't know how to do this with posters, but I would like to give kids feedback before it's too late to change the final result.

Our Solar System, Graphs, and Classification Schemes

Following the previous week's intro to the solar system, on Friday May 17 I visited the 3-4 grade room and used the solar system as a context for practice with graphs.  We used the graphs in turn as a tool for helping us think about how to classify solar system objects.  By establishing several clearly different classes of solar system objects, we raised questions about how the solar system might have formed these different classes, and we even began to answer those questions.  I think this worked quite well as a coherent activity while asking the students to practice a variety of skills.

The centerpiece was a graph (technically a scatterplot*) of size vs distance from Sun for various solar system objects.  My first idea was to help the kids make their own graphs from a table of data, but I discarded that idea as requiring too much time before we got to any science.  So I made this graph and handed out a copy to each student:

I still wanted students to graph some data, so I planned to make them analyze and understand this graph as a gateway to getting them to add more points and do more analysis.  I think this plan went well.  I started with the question: can you identify any of the points?  This required them to think about the meaning of the axes, and once they understood, they started saying things like "the top one must be Jupiter, because it's the biggest planet" and "the one most to the right must be Neptune because it's most distant from the Sun."  Once they grasped that, they were able to label more and more points until we eventually got them all. (The word "eventually" hides a lot of time spent one-on-one with kids, helping them with the reasoning.  Eg, Earth and Venus are almost exactly the same size, but Earth is a bit bigger, so which point is Earth?  Double-check your conclusion by looking at distance from the Sun.  Does it make sense? Etc.)

This was an excellent activity to make them think about the meaning of the graph rather than getting caught up in big numbers which wouldn't mean much to them anyway.  (Jupiter is 90,000 miles across?  How big is that?)  But now let's think about the numbers.  The graph says Earth's distance from the Sun is 1.  What is that? One foot?  One billion miles?   The only unit that makes sense is units of "Earth-Sun distance."  In other words, the graph makes it easy to read off the relative distances of the planets.  It's a scale model. Again, this makes it easy to think about what the solar system is without getting caught up in a bunch of meaningless numbers.  We repeated that exercise with the vertical axis.

Then we looked at whether the planets form any distinct groups.  The graph makes it clear that there are two groups: small and close to the Sun, vs large and far from the Sun. What other differences might these groups have? It turns out that the large ones are made of different stuff (mostly gas vs rock), so maybe we should really think of two types of planets (gas giants and rocky planets) rather than thinking that all things called "planet" are similar things.

Next, I took them back to the year 1801 when a new planet was discovered: Ceres.  I gave them the Ceres-Sun distance in units of the Earth-Sun distance (2.77) and Ceres' size in Earth-size units (0.07) and asked them to put Ceres on the graph.  For the faster students, I gave them three more planets which were discovered soon after Ceres (Pallas**, Juno, and Vesta, which have similar distances and sizes) while the teachers helped the slower students with the graphing.  After graphing these, it's clear that they form a distinct group: a group of very small things between Mars and Jupiter.  Today we call these things main-belt asteroids, but when they were discovered they were simply called new planets.  It was only after discovering many of them that people began to think that maybe we shouldn't call all new discoveries planets, and especially not these new discoveries which clearly form a separate group.  The way we think about things is highly dependent on how much information we have.

This took until the break.  After the break, we added Pluto to the graph.  When Pluto was discovered, it was immediately called a planet because it was much larger than any asteroid, and there was no other category it could have been assigned to.  But it does seem a bit out of place on the graph, being substantially smaller than any of the eight planets we started with, and also breaking the pattern of the larger planets being farther from the Sun. Well, it took 60 years, but eventually astronomers started discovering lots of other things roughly as far from the Sun and roughly the same size. I gave the kids data for these new objects: Eris, Sedna, Quaoar, and Orcus to start with.

Just as with the asteroids, it became clear that things like Pluto form a new category: the Kuiper Belt.  This is even more clear when we realize that all these things are made of ices***, which is not like the inner planets or the outer planets. Once this new category was recognized, it became silly to continue calling Pluto a planet, just as in the 1800's it became silly to continue calling Ceres, Pallas, Juno, and Vesta planets.  Perhaps Pluto should have been in a category of its own from the start, but there was no available category other than "planet," and why create a new category just for one object?  Another illustration that the way we think about things depends on how much information we have.

[A side note: astronomers created the additional category "dwarf planet" to describe a body which, regardless of its location, is large enough that its gravity pulls it into a round shape (but smaller than the eight planets).  Thus Pluto is both a Kuiper Belt object and a dwarf planet just as I am both a teacher and a father---they are not exclusive categories.  But  "Kuiper Belt object" is a much more descriptive term because it implies being made of ice, being a certain distance from the Sun,  etc, whereas  "dwarf planet" implies only that the size is neither very large nor very small.]

Next, we talked about how the solar system might have formed in order to form these different classes of objects. I showed clips from the Birth of the Earth episode of the series How the Earth Was Made.  It has some really nice visualizations, and it is constructed around evidence, which is a key feature missing from most science documentaries.  It tells science like the detective story it is.  We spent probably half an hour on this, but I won't write much here because it's already a long blog post.

To cap off this intense morning, I brought some liquid nitrogen to demonstrate how cold the outer planets are. I froze a racquetball and shattered it just by trying to bounce it off the floor; I froze a banana and showed how it can be used as a hammer (until it shattered), and I made a balloon shrink and then expand again as I warmed it up.  LN2 is always a great hit with the kids.  On Pluto summers can be just warm enough to vaporize some nitrogen, but right about now Pluto is in early fall, and it will get so cold that nitrogen will not only liquify, it will freeze.

Notes

*Notice that this graph is not a histogram, which seems to be the only type of graph elementary teachers ever work with.  I see that kids start working with graphs around second grade if not earlier, so by the time they get to college, they should be highly proficient.  But in my college classes that students are typically far from proficient.  My guess is that much of the time spent on graphs in school is wasted because students are never introduced to the idea of graphing the relationship between two different abstract quantities, which is absolutely key to data analysis and science.

**I got the idea for some of this activity when I saw that the element palladium was so named because for a long time it was fashionable to name newly discovered elements after recently discovered planets. I was long aware of uranium, neptunium, and plutonium being named this way, but I had never made the connection to cerium and palladium.  People really thought that asteroids were planets until enough asteroids were discovered.

***Ices includes ice made of materials other than water, such as methane, ammonia, etc.

Solar system

Today we blasted off from our Earth-Moon base and explored the other planets.  I started with this image of the terrestrial planets, which accurately depicts their relative sizes but not their distances. I brought in a big yoga ball to represent the Sun and we went in order from the Sun (ie from the left in that image).  For each planet I elicited what they already knew or thought they knew about each planet, and then enriched it as best I could.  For example, they knew Mercury is hot because it's close to the Sun...but what about the side away from the Sun (ie the night side, which is not always the same side)? It is actually very cold; why would that be?  To put it another way, why is the day/night temperature variation on Earth not very extreme? That led to a discussion of atmospheres, which further led to a discussion of cratering, which further led to comparisons between Mercury and our Moon (similar size, both airless and cratered, extreme day/night temperature variation).  I won't try to document each planet's discussion here, but 45 minutes flew by. (Here are links to a similar image comparing some asteroids in the asteroid belt, one comparing the gas giant planets (aka Jovian planets) and an image comparing the dwarf planets outside Neptune's orbit.)  As we went, I filled in a table of planet sizes (diameters) and distances from the Sun, for later reference.  I rounded the numbers quite a bit so kids would more easily see the comparisons.  For example, rounding the Sun's diameter to 800,000 miles and Earth's to 8,000 we easily see that the Sun is about 100 times bigger across.  This is way easier to understand than listing the exact numbers and doing the exact computation to find that it is 109 times bigger across.

Just before the break, I addressed why Pluto is no longer considered a planet. Short answer: it became clear that Pluto was just one of many smallish iceballs which are very unlike terrestrial planets and also very unlike Jovian planets, so they deserve their own class.  When Pluto was the only known example, it didn't occur to anyone to put it in its own class.  A nice example of how the way we classify things can change as we get more data.

After the break, we worked on understanding the distances and sizes by building scale models. First, we did the pocket solar system to understand the relative distances. It's quite amazing to see how relatively jam-packed the inner solar system is compared to the outer solar system, yet even in the inner solar system there are many tens of millions of miles between planets.

Next, the sizes. With the 65-cm-diameter yoga ball as the Sun, I pulled balls of various sizes out of my box: softball, baseball, tennis ball, ping-pong ball, etc.  Because I had two ping-pong balls, students suggested they could be Earth and Venus, which are nearly the same size.  Does this accurately depict how much smaller than the Sun these two planets are? Well, Earth is 100 times smaller than the Sun, so on this scale it should be 0.65 cm across, or 6.5mm (1/4 inch).  That's way smaller than a ping-pong ball, so I had to rummage around in my kit, where I found some allspice.  Allspice varies in size, but we did find some which were 6mm across.  That's right, if the Sun is a yoga ball, Earth is the size of an allspice!

Whenever we do a calculation, we have to double-check it.  I held up the yoga ball and the allspice and asked the kids if they thought 100 allspice would fit across the yoga ball.  Yes, it looks about right.  Out of curiosity, how many would fit in the yoga ball? Some of them guessed 100x100, because the yoga ball is 100x bigger in each of the two dimensions which are easily seen.  But the yoga ball is also 100x bigger in the third dimension, so its volume is 100x100x100 or 1,000,000 (a million) times bigger. One million Earths could fit into the volume of the Sun.  (The Sun's density is a bit less than Earth's, so the Sun's mass is "only" 318,000 times bigger than Earth's.  For older kids, adding density and mass to this whole discussion might make sense.)

OK, so now we have Earth and Venus.  What about Jupiter? Using the same reasoning, we found a ball about Jupiter's size (a small whiffleball, not much bigger than a ping-pong ball), and Saturn is just a bit smaller. Uranus and Neptune could be represented by small marbles.  Mars could be a small allspice or an average peppercorn, and Mercury could be a mustard seed.  Amazing! (If you're a teacher who would like to do this kind of activity, check out the peppercorn Earth website for some supporting materials.)

Finally, if these are the sizes of the planets in our scale model, what are the distances between planets? The Earth-Sun distance is about 100 Sun diameters, so we need 65 meters or about 200 feet.  That's about the distance from our classroom to the far side of the playground.  Jupiter is 5 times farther, so maybe we could put it at the KFC a block or so away.  Pluto is 40 times further than Earth from the Sun, so that would be 8,000 feet or 1.6 miles, the distance from school to home for some of the kids.  Imagine...all that space in between would be empty.  Even Mercury, closest to the Sun, would be about 80 feet away and the size of a mustard seed!

At the end, I asked the students to choose a favorite planet or moon, learn more about it, and make a poster over the next two weeks.  We'll put the posters up all over school at the appropriate distances to make a scale model.  At the center of each poster will be a small object size to match the scale model.  To fit the scale model into the school, some of them will have to be very small objects, like a grain of sand.  Teacher Brittany will work with the students on the math for that, and I'll report back on the scale model in a few weeks.

Earth, Moon, Sun

Last Friday I had my second astronomy session with the 3-4 graders.  In the first one, we spent a lot of unplanned time on why we don't feel the Earth moving, and also it had been 3 weeks since my last visit, so I spend the first block of time recapping how we know the Earth rotates and how we know the cause of the seasons,  and building on that to analyze the Earth-Sun motion.

Do we go around the Sun or does it go around us?  We know that one of these two things is happening, because a given star rises at intervals of 23 hours and 56 minutes, whereas the Sun does at 24-hour intervals.  (Maven alert: 24 hours is an average which varies with the seasons, but that's too much detail here.) So each day the Sun gets 4 minutes "behind" the stars and over the course of a year it appears to make a complete circuit around the sky relative to the stars.  Ancient people knew this without having accurate clocks; they simply observed that the stars they could see at night (ie when the Sun was below the horizon) shifted slowly throughout the year.  We also know (as a boy mentioned last time) that the Sun's apparent size varies slightly throughout the year, thus indicating that our distance from the Sun varies slightly throughout the year.  [We happen to be closest to the Sun in January; if this shocks you, read about the cause of the seasons.]

I drew two models on the board: one with the Sun going around us in an ellipse (thus varying the distance) and the other with Earth going around the Sun in an ellipse.  What would be the observable differences between these two scenarios? This is a tough question!

Think about sitting in a moving car.  The roadside trees appear to rush by, but the distant mountains appear to move very slowly.  If the Earth moves, we ought to be able to see an effect like this by comparing nearby and distant stars.  I had taped some stars around the room and we had a small circular carpet to orbit around, so we practiced that.  You could also do this activity in the schoolyard. This effect is called parallax; if Earth is still, we will not see it.  The ancient Greeks thought of this, they looked for parallax and didn't see it, so they leaned toward Earth being still.  It turns out that even the nearest stars are so far away that the parallax effect is tiny, and was not measured until modern times.  So the ancient Greeks were not at all ignorant; they just didn't have precise enough tools to measure this really small effect. 

It turns out that the nearest star is about 250,000 times more distant than the Earth-Sun distance, ie the distance which Earth moves. It's as if your car moved one mile but you were asked to discern the difference in your view of mountains 250,000 miles away (eg, on the Moon).  I illustrated this dramatically by asking the kids to drawing a one-inch Earth-Sun model on the board, and then drawing a long line representing the distance to the next star and asking the kids to stop me when they thought I had arrived.  Kids (even most adults) have no idea how much 250,000 times is; they ask me to stop after 5 feet or so, but I keep going.  When I run out of board, I get a roll of toilet paper and start unrolling it, as a way to illustrate a very long line.  I keep going even when they tell me to stop. Then, when I run out of toilet paper, I go to the back room and get a cart full of hundreds of rolls of toilet paper!  It is really dramatic and fun. I also wrote out the number of miles to the next nearest star on the board: about 24,000,000,000,000 miles. (Kids and even most adults have little idea what a "trillion" really means.)

Knowing these distances, we can answer a few questions about the nature of the Sun and stars.  Lights which are further away appear to be fainter, so if we compensate for the enormous distance of the stars, we find that their intrinsic brightness (aka luminosity) is about the same as the Sun.  The Sun is just another star!  And we can compute that luminosity in terms of watt, just like a light bulb.  The Sun's luminosity turns out to be 400,000,000,000,000,000,000,000,000 watts.  It is the ultimate source of nearly all the energy we use on Earth.  If it burned fuel like coal or oil to produce its energy, it would rapidly run out of energy.  Astronomers were stumped for years on what the source of energy could be, until they discovered nuclear fusion (which we may address in more detail next time).

After the break, we looked at kids' observations of the Moon over the past three weeks, and we figured out what model could explain these observations, using as many different ways as possible: kinesthetic activity, going into a dark room with a blacklight and ping-pong balls so each student could move the "Moon" around his/her own head; and a mechanical model I had borrowed. I really think all these ways (and more) are necessary for most people to really get it.  I don't have time to write up this activity now, but if you are interested there are plenty of internet resources to help you understand it.  I just want to say that the kids' own observations over the previous three weeks were key to demolishing the misconceptions that the Moon is only visible at night, and/or that the Moon is always visible at night.  Finally, I want to leave you with some insanely cool pictures of eclipses.

This is a picture of the Moon's shadow falling on Earth during a solar eclipse.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This one shows the Moon at three different times throughout a lunar eclipse (when Earth's shadow falls on the Moon).  The ancient Greeks were able to determine from this that the Moon is about 1/4 the diameter of Earth, or about 2000 miles across. Furthermore, they know that for a 2,000-mile-diameter object to look as small as the Moon looks to us, it would have to be about 240,000 miles away.

There is a season

At the end of last Friday's session with the 3-4 graders we discussed
the seasons. It's natural to think that winter is when we are farther
from the Sun, but is that true?  What evidence can we reason with
here?

Well, for one, we know that when it's winter in the northern
hemisphere, it's summer in the southern hemisphere.  That's a pretty
good clue that the seasons are not caused by the Earth getting closer
to and farther from the Sun.  What else?  One boy, to my amazement,
specifically mentioned that if/when we are closer to the Sun, it
should appear to be bigger, and it doesn't appear to be bigger in
(northern) winter.  I was pretty surprised, but then I recalled that
he had recently seen the movie Agora.  This is a trenchant
observation: the Sun actually looks smallest (as seen from anywhere on
Earth) in January, indicating that January is when Earth is furthest
from the Sun. (You can't see this just by looking, because the Sun is
so blindingly bright and it's a fairly small change, but you can use a
pinhole camera to do it.)


Most people (these kids included) "know" that seasons are caused by
"tilt," but what does that really mean?  It's very useful to take a
globe (on a standard stand tilted by the right amount) and make it
orbit a light source to see how this plays out.  The tilt actually
points to the same place in the sky (judging by the stars) all the
time, but since Earth goes around Sun (or vice versa, it doesn't
matter for this point), for part of the orbit the tilt points the
north pole toward the Sun.  This is northern summer, when there is
midnight Sun near the north pole, and longer daylight hours in
northern latitudes generally.  (You can see this by spinning the globe
in the aforementioned model. Make sure to keep the north pole always
pointing toward the same place in the room, like the clock on the wall,
even as it goes around the "Sun.") Six months later, the Sun is on the
opposite side of Earth (relative to the stars) so it's the south's
turn to have summer and the north pole has constant darkness.  In
between those two extremes, neither hemisphere is favored.

So we are NOT tilted closer to the Sun.  The tilt merely allows the

Sun to shine a greater or lesser fraction of the time on a certain

hemisphere, depending on where the Sun is relative to this constant

tilt.  (This fits the evidence that whatever hemisphere has summer, it

does not see the Sun as closer and bigger.)  Translated into what we

see on the ground, it means that summer is when the Sun rises earlier

and sets later, and goes higher in the sky at noon.  You can see all

these things with the globe and light setup.  I had had the kids keep

a sketch of the position and time of sunset or sunrise each week for

the past six weeks, and these data match the model I just described.

I'll analyze that in a bit more detail, and explore the Moon's orbit

around Earth, in my next visit to the school.

Turn! Turn! Turn!

This spring I am assigned to work with the Peregrine School 3-4
graders on astronomy, and last Friday was my first day, so I started
with basics like how we know the Earth spins.  We tend to feel
superior to people in the past who believed that the Sun went around
the Earth but, really, how can you use basic observations to show that
it doesn't?  I suspect that most people on the street would be stumped
by this if I didn't allow "satellites" or "NASA" as an answer.

If we only had the observation that the Sun rises and sets every 24
hours, we wouldn't be able to conclude anything.  Each star also rises
and sets in roughly (later we'll see why I say roughly) 24 hours, so
based on pure majority rule, it might be easy to attribute the
apparent motion of the Sun and stars to Earth spinning.  This model
invokes only one thing (Earth) moving, vs the other model invokes a
grand conspiracy of everything else in the universe circling us at an
agreed-upon rate of once every 24 hours.  Sounds like a no-brainer,
but why don't we feel Earth moving?

The kids had lots of ideas in response to this question. It moves so
slowly we can't feel it? No, its circumference is about 24,000 miles
so if it spins in 24 hours its equator must move 1,000 mph.  It moves
so quickly we can't feel it? Gravity?  Centrifugal force? It's so big
we don't feel it move?  There were so many ideas about this that I
decided to explore Galilean relativity: if you are in a laboratory
moving at constant velocity, there is no experiment you can do to
prove you are not actually stationary.  Think about a smooth flight in
an airplane.  If you drop something does it fly backward, indicating
that you are actually traveling at 500 mph?  No, it falls straight
down.  Unless you look out the window, you can't tell that you're
moving---there is NO experiment which will tell you this.  If you do
look out the window, all you can conclude is that you are moving
relative to Earth...there is no experiment you can do that says Earth
is stationary and you are the one who is moving.

This was a pretty new and shocking idea for the kids, so we spent a
long time discussing it.   I remembered a great video I had seen
demonstrating one aspect of this, so I sketched it out and asked them
to predict what would happen.  Say you have a pitching machine which
shoots baseballs at 100 mph, but you mount this machine in the back of
a pickup truck which goes 100 mph the other way.  What will the ball's
speed be, relative to people on the ground? 200 mph? 100 mph?  Zero?
We analyzed this until break time, then after break I showed them the
video.  Mythbusters also did a similar thing, which you can see much more clearly.

The bottom line is that velocities are relative.  So if everything in
your vicinity is moving together at the same speed, it can all be
considered stationary.  This applies to your vicinity on Earth:
although Earth's rotation causes different parts of Earth to move in
different directions and speeds, the part that you experience at any
one time is so small that to high precision it's all moving at the
same speed in the same direction. (When winds move air over hundreds
of miles, that air does eventually feel the effect of Earth's
rotation, the Coriolis effect.)

So not feeling Earth's spin is not a valid argument that it must be
still.  But how do we prove it spins? The Coriolis effect is one way,
but I considered that too advanced for this audience.  Instead, I
explained the Foucault pendulum, which many of them had seen but
probably didn't realize the significance at the time.

Next, we tackled Earth's motion through space.  Spinning is not enough
to explain all our observations, because the stars rise and set every
day slightly faster than the Sun, which means that over time the Sun
loses more and more ground to the stars, and over the course of a year
we see the Sun make one complete circle around the sky relative to the
stars.  What could explain this?  Well, maybe the Sun does go around
Earth, in addition to Earth spinning. But maybe the Earth goes around
the Sun. How could we tell the difference?  I'll get to the bottom of
that next time I visit the school, but for the time being I wanted to
focus on other changes throughout the year and tie them all together
into a coherent model.  I'll post that part of our discussion soon.