Shadow Play

“Uncle Sy! Uncle Sy! You’re back! Didja see the red moon?”

“Hi, Teena. Good to be home. No, I didn’t get to see the red moon. Where I was it didn’t even get red.”

“I saw it! I saw it! Mommie put me to bed early so I could wake up to see it earrrly in the morning. I saw the red part but the Moon looked smaller than it does coming up from behind the houses and they said it was going to be sooo big but it wasn’t. Anyway, I didn’t stay awake. Why was it red?”

“Was it really red red like your favorite crayon?”

“Mm-no, more like orange-y red.”

“Sunset color, right?”

“Uh-huh. Was it sunset on the Moon?”

“Sort of. The sunsets we see on Earth are red mostly because our air absorbs the Sun’s blue light when we’re looking across the atmosphere. Only the red light gets through to our eyes. Remember the solar eclipse we saw, when the Moon came exactly between us and the Sun? Moon eclipses are inside out from that. We come between the Moon and the Sun. The only light getting past us has gone across our atmosphere just like sunset light does so it’s orange‑y red like a sunset.”

“Oooh … does the Sun ever get between us and the Moon?”

“Don’t worry, Sweetie. We’re far, far from the Sun. Mr Newton’s Laws of Motion say that we and the Moon will be waltzing out here for a long, long time.”

“Whee, we’re dancing around the Sun! MOMmie, Uncle Sy’s here!”

“Hi, Sis. You saw the eclipse, then.”

“Mm-hm. I realized while I was watching it that lunar phase shadows work differently from eclipses.”

“Oh? How so?”

“The shadow shapes are different, for one. The edge of the lunar phase shadow always passes through both poles. In a solar eclipse the shadow only reaches the poles at totality, and in a lunar eclipse there’s this almost straight shadow arc that marches across the whole face.”

“Interesting. You said ‘for one,’ so what else?”

“Eclipse shadows move in the wrong direction. Starting from a full moon, the shadow comes in from the right until you get to new moon, then it falls away to the left until you get back to full moon. Agreed?”

“I always get confused. I’ll take your word for it.”

“I looked it up. In two places. Anyhow, in both kinds of eclipse the shadow creeps from left to right. Just backwards from the lunar phases. I wonder if that has anything to do with ancient societies thinking that an eclipse is somehow evil.”

“Mommie, you know you’re not supposed to use words I don’t know unless you’re keeping secrets. What’s lunar faces?”

“Sorry, Teena, not secret. Lunar means Moon. Sy, can you show her phases on Old Reliable?”

“Sure. Here’s a quick sketch, Teena. Pretend that the little ball is the Moon going around the Earth. The Sun is off to the right. You know the Moon goes around the Earth and it always keeps the same side towards us, right?”

“That’s the Man In The Moon except it’s really mountains and stuff pointing at us.”

“That’s what the little triangle shows, like it’s his nose. See how sometimes it’s in the light and sometimes it’s in shadow? The big ball is what we see when the Moon is in each position. When the Man is facing straight towards the Sun we call that the Full Moon phase. When he’s completely in shadow that’s the New Moon phase. There’s names for other special positions, and all of the special positions are phases, OK?”

“I suppose you have a logical explanation for the shadows?”

“Sure, Sis. It’s all about where the shadow’s being cast and how the shadow caster is moving at the time. This diagram tells the story. Nearly everything in the Solar System runs counterclockwise—”

“Widdershins.”

“… Right. Every orbit runs left‑to‑right half the time, right‑to‑left the other half. The two kinds of eclipse happen in opposite halves. The geometry works out that we see both eclipse shadows move left‑to‑right. See?”

“Cool.”

~~ Rich Olcott

  • Thanks to Alex for the question, and to Lori for the shadow observation, which I hadn’t seen discussed before.

A Perspective on Gravity

“I got another question, Moire.”

“Of course you do, Mr Feder.”

“When someone’s far away they look smaller, right, and when someone’s standing near a black hole they look smaller, too.  How’s the black hole any different?”

“The short answer is, perspective depends on the distance between the object and you, but space compression depends on the distance between the object and the space-distorting mass.  The long answer’s more interesting.”

“And you’re gonna tell it to me, right?”

“Of course.  I never let a teachable moment pass by.  Remember the August eclipse?”

“Do I?  I was stuck in that traffic for hours.”

“How’s it work then?”

“The eclipse?  The Moon gets in front of the Sun and puts us in its shadow. ‘S weird how they’re both the same size so we can see the Sun’s corundum and protuberances.”

“Corona and prominences.  Is the Moon really the same size as the Sun?”

“Naw, I know better than that.  Like they said on TV, the Moon’s about ¼ the Earth’s width and the Sun’s about 100 times bigger than us.  It’s just they look the same size when they meet up.”

“So the diameter ratio is about 400-to-1.  Off the top of your head, do you know their distances from us?”

“Millions of miles, right?”

“Not so much, at least for the Moon.  It’s a bit less than ¼ of a million miles away.  The Sun’s a bit less than 100 million miles away.”

“I see where you’re going here — the distances are the same 400-to-1 ratio.”

“Bingo.  The Moon’s actual size is 400 times smaller than the Sun’s, but perspective reduces the Sun’s visual size by the same ratio and we can enjoy eclipses.  Let’s try another one.  To keep the arithmetic simple I’m going to call that almost-100-million-mile distance an Astronomical Unit.  OK?”

“No problemo.”

“Jupiter’s diameter is about 10% of the Sun’s, and Jupiter is about 5 AUs away from the Sun.  How far behind Jupiter would we have to stand to get a nice eclipse?”

“Oh, you’re making me work, too, huh?  OK, I gotta shrink the Sun by a factor of 10 to match the size of Jupiter so we gotta pull back from Jupiter by the same factor of 10 times its distance from the Sun … fifty of those AUs.”

“You got it.  And by the way, that 55 AU total is just outside the farthest point of Pluto’s orbit.  It took the New Horizons spacecraft nine years to get there.  Anyhow, perspective’s all about simple ratios and proportions, straight lines all the way.  So … on to space compression, which isn’t.”

“We’re not going to do calculus, are we?”

“Nope, just some algebra.  And I’m going to simplify things just a little by saying that our black hole doesn’t spin and has no charge, and the object we’re watching, say a survey robot, is small relative to the black hole’s diameter.  Of course, it’s also completely outside the event horizon or else we couldn’t see it.  With me?”

“I suppose.”

“OK, given all that, suppose the robot’s as-built height is h and it’s a distance r away from the geometric center of an event horizon’s sphere.  The radius of the sphere is rs.  Looking down from our spaceship we’d see the robot’s height h’ as something smaller than h by a factor that depends on r.  There’s a couple of different ways to write the factor.  The formula I like best is h’=h√[(r-rs)/r].”

“Hey, (r-rs) inside the brackets is the robot’s distance to the event horizon.”

“Well-spotted, Mr Feder.  We’re dividing that length by the distance from the event horizon’s geometric center.  If the robot’s far away so that r>>rs, then (r-rs)/r is essentially 1.0 and h’=h.  We and the robot would agree on its height.  But as the robot closes in, that ratio really gets small.  In our frame the robot’s shrinking even though in its frame its height doesn’t change.”

“We’d see it getting smaller because of perspective, too, right?”

“Sure, but toward the end relativity shrinks the robot even faster than perspective does.”

“Poor robot.”

~~ Rich Olcott

  • Thanks to Carol, who inspired this post by asking Mr Feder’s question but in more precise form.

Teena Meets The Eclipses

“Don’t look up until it suddenly gets really dark, Teena.  I’ll tell you when it’s time.”

“OK, Uncle Sy.  Oooo, look at the house where our tree makes a shadow!  It’s all over crescents!”

“Yep, wherever leaves overlap to make a pinhole, it’s like the one we made in our cardboard.  See, those crescents are just like the one our pinhole beams onto the sidewalk.”

“Yeah.  ‘Cause it’s the same Sun, right?”

“Sure is.”

“Are other little kids seeing the eclipse all over the world?  They’ve got the same Sun, too.”

“No, just the ones who happen to be on the shadow stripe that the Moon paints on the Earth.”

“How many kids is that?”

“Hard to tell.  Some families live where the shadow passes through, some families travel to be there, lots of other families just stay where they are.  No-one knows how many of each.  But we can make some not-very-good guesses.”

“The crescent’s going so slow.  Let’s make guesses while we’re waiting.”

“OK.  Let’s start by imagining that all the world’s people are spread evenly over the land and sea.”

“Even on the ocean?  Like everyone has a little boat?”

“Yep, and sleds or whatever on polar ice, people everywhere.  In our city there are eight blocks to a mile, so if we spread out the people there’d be one person every other block.”

“Every other block.  Like just on the black squares on our checker board.”

“Uh-huh.  The Moon’s shadow today will be a circle about 80 miles across and it’ll travel about 2500 miles across the whole country.  The stripe it paints would cover about 6½ million spread-out people.  Maybe 10 million if you count the people in little boats, ’cause the eclipse starts and ends over the ocean.”Local eclipses

“Lots of people.”

“Yes, but only about one person out of every thousand people in the world.”

“We’re pretty lucky then, huh?”

“Oh, yeah.”

“Are there eclipses on other planets?”

“Of a sort, but only for planets that have a moon.  Poor Mercury and Venus don’t have moons so they never see an eclipse.”

“Aww. … Wait — you said ‘of a sort.’  Are there different kinds of eclipses?”

“You’re very alert this morning.  And yes, there are.  Two that get the publicity and two that we never see on Earth.  It has to do with perspective.”

“Per … perspec…?”

“Perspective.  The word originally meant very careful looking but it’s come to be about how things look from a particular point of view.  See that tree across the street?”

“Yeah.”

“Think your hand is bigger than the tree?”

“Of course not.  I climb that tree.”

“OK, put your hand between your eyes and the tree.”

“Oh!  My hand covers the whole tree!”

“Yup.  Nearer things look big and farther things look small.  That’s perspective.  Eclipses are all about perspective.”How big is the Sun

“How come?”

“The perspective principle works in the Solar System, too.  If you were to travel from Earth to Mars to Jupiter and so on, the Sun would look smaller at each planet.”

“Like the far-away trees look smaller than the close trees.  But what does that have to do with eclipses?”

“A planet gets an eclipse when one of its moons comes between it and the Sun.  That’s what’s happening right now here.  Our Moon is moving between us and the Sun and blocking its light.”

“But I don’t see the Moon, just the carved-out piece.”

“That’s because we’re looking at the unlit side of the Moon.  It’s so dim compared to the rest of the sky.  Anyway, the Moon’s width we see is just about the same as the Sun’s width.  The moons on the other planets don’t match up that well.  On Mars, for instance, its moon Phobos appears less than half the width of the Sun even though the Sun appears only 2/3 as wide as we see it.  Phobos can never cover the Sun entirely, so no true eclipse, just a transit.”

“Can the planet’s moon be bigger?”

“Sure.  On Jupiter, Europa’s width completely blocks out the Sun.  That’s called an occultation.  You can look up now.  Jupiter people can never see that corona.”

“Oooooo, so pretty.  We’re lucky, aren’t we?”

“In more ways than you know, sweetie.”

~~ Rich Olcott