Astrometers Are Wobble-Watchers

letter A Hi, Sy, what’s going on in Cathleen’s seminar?

You were right, Al.
It’s about exoplanets and how to find them.
Jeremy’s pitching astrometry.
That’s about measuring star locations in the sky.
I’ll fill you in later.

“So that’s my cultural colonialism rant, thanks for listening. On to the real presentation. Maria showed us how to look for exoplanets when they wobble along our line of sight. But what if they wobble perpendicular to that? Careful measurement should show that, right? The ancients thought that holy forces had permanently set the positions of all the stars except for the planets so they didn’t measure that close. Tycho Brahe took meticulous measurements with room‑sized instruments—”

<voice from the back> “Room‑sized? What difference does that make?”

“What if I told you that two stars are 3 millimeters apart in the sky?”

<another voice> “How far out’s your ruler? Sky stuff, you need to talk angles because that’s all you got.”

“Well there you go. That’s why Tycho went for maximum angle‑measuring accuracy. He built a sextant with a 5‑foot radius. He used an entire north‑south wall as a quadrant. His primary instrument was an armillary sphere three yards across.”

<first voice again> “Wait, a sphere, like a big bubble? Why north‑south? What’s a quadrant?”

  • I give him a nudge. “He’s just a kid, Mr Feder. Be nice. One question at a time.”
  • “But I got so many!”

“Think about Tycho’s goal. Like astrometers before him, he wanted to build an accurate map of the heavens. Native Americans a thousand years or more ago carved free‑hand star maps on cave ceilings and turtle shells. Tycho followed the Arabic and Chinese quantitative mapping traditions. There’s two ways to do that. One is to measure and map the visual angles between many pairs of stars. That strategy fails quickly because errors accumulate. Four or five steps along the way you’re plotting the same star in two different locations.”

<Feder’s voice again> “There’s a better way?”

“Yessir. Measure and map each star relative to a standard coordinate system. If your system’s a good one, errors tend to average out. The latitude‑longitude system works well for locating places on Earth. Two thousand years ago the Babylonians used something similar for places in the crystal sphere they thought supported the stars above us. Where the equinoctial Sun rose on the horizon was a special direction. Their buildings celebrated it. Starting from that direction the horizontal angle to a star was its longitude. The star’s latitude was its angle up from the horizon towards the zenith straight above. But those map coordinates don’t work for another part of the world. Astrometers needed something better.”

<Feder again> “So what did they do already?”

“They may or may not have believed the Earth itself is round, but they recognized the Pole Star’s steady position that the rest of the sky revolved around. They also noticed that as each month went by the constellations played ring‑a‑rosie in a plane perpendicular to the north‑south axis. Call that the Plane of The Ecliptic. Pick a star, measure its angle away from the Ecliptic and you’ve got an ecliptic latitude. Measure its angle around the Ecliptic away from a reference star and you’ve got a ecliptic longitude. Tycho’s instruments were designed to measure star coordinates. His quadrant was a 90° bronze arc he embedded in that north‑south wall, let him measure a star’s latitude as it crossed his meridian. His ‘Sphere’ was simply a pair of calibrated metal rings on a gimbal mounting so he could point to target and reference stars and measure the angle between them. If his calibration used degree markings they’d be about 25 millimeters apart. His work was the best of his time but the limit of his accuracy was a few dozen arcseconds.”

“Is that bad?”

“It is if you’re looking for exoplanets by watching for stellar wobble. Maria’s Jupiter example showed the Sun wobbling by 1½ million kilometers. I worked this example with a bigger wobble and a star that would be mid‑range for most of our constellations. Best case, we’d see its image jiggling by about 90 microarcseconds. Tycho’s instruments weren’t good enough for wobbles.”

~~ Rich Olcott

How Many Ways Can You Look at The Sky?

Cathleen and I were discussing her TRAPPIST-1 seminar in Al’s coffee shop when a familiar voice boomed over the room’s chatter.

“Hey, Cathleen, I got questions.”


“Yeah, Sy, he hangs out with the Astronomy crew sometimes.  You know him, too, huh?”

“From way back.  Long story.”

“What’re your questions, Vinnie?”

“I missed the start of your talk, Cathleen, but why so much hype about this TRAPPIST-1 system?  We’ve already found 3,500 stars with planets, right, and some of them have several.  What’s so special here?”

“You’re right, Vinnie, Kepler-90 has seven planets, just like TRAPPIST-1. (brandishes a paper napkin)  But that star’s more than 60 times further from us than TRAPPIST-1 is.  It’s just too far away for us to be able to learn much more about the planets than their masses and orbital characteristics.  This new system’s only 40 lightyears away, close enough that we’ve got a hope of seeing what’s in the planetary atmospheres.”

(another paper napkin)  “That ties in with the second thing that’s special.  The star’s surface temperature, 2550ºK, is so low that even though its planets orbit very close in, three of them are probably in the Goldilocks Zone.  They’re not too hot and not too cold for liquid water to exist on their surface.  IF there’s liquid water on one of them and IF there’s something living there, we should be able to detect traces of that biochemistry in the planet’s atmosphere.”

Star demographics
Observational data (dots) and four different models
of star count (vertical axis) versus temperature.
Hotter stars are to the left.

(napkin #3)  “The third special thing is that TRAPPIST-1 is the first-known planet-hosting star in its category — ultra-cool dwarf stars burning below 2700°K.  Finding those stars is hard — they’re small and dim.  No-one really knows how many there are compared to the other categories.  Some models say they should be rare, other models suggest they could be as common as G-type stars like our Sun.  IF there’s lots of ultra-cool dwarfs and IF they generally have planets like G-type stars do, then the category’s a new prime target for exoplanet hunters seeking life-signs.”

“Why’s that?”

“Because it’s easier to spot a small planet around a small star than around a big one.  Transits across TRAPPIST-1 dim its light by 1% or so.  A TRAPPIST-1 planet transiting our Sun would dim it by 1/100th of that.  The same problem hinders planet-finding methods fishing for stars that wobble because a planet’s orbiting around it.”

“Alright, I get that TRAPPIST-1 is special.  My other question is, I heard the part of your talk where you figured the odds on seeing its transits, but you lost me with the word steradian.  My dictionary says that’s an area on a sphere divided by the square of the sphere’s radius. What would that get me?  Where’d your numbers come from?”

“You need one additional piece of information.  If you take any sphere’s total surface area and divide that by r², you’ll always get 4π steradians.  You can use that to convert between absolute surface area and fraction of the sphere.  Mmm…  Sy, you own some land outside of town, yes?”

“A little.”

“And you have mineral rights?”

“Oh, yeah, that’s why I bought it.”

“And they go how far down?”

“All the way to the center of the Earth.”

“So your claim’s actually a pyramid 6370 kilometers deep.  When I moved here I learned it’s impolite to ask how much land someone has.  For round numbers I’ll assume 40 acres, which is about 1,000 square meters.  (tapping keys on her smartphone)  The Earth’s radius is 6.37×106 meters, so Sy’s claim is 1,000/(6.37×106)2 = 2.47×10-11 steradians.  Divide 4π by that and you get … 5.08×1011.  So Earth’s entire surface has room for 5.08×1011 patches matching Sy’s.  Visualize 5.08×1011 pyramids pointing in every direction from Earth’s center.  Now extend each pyramid outward to define a separate patch of sky.  Got that picture, Vinnie?”viewing cones

“Sort of.”

“TRAPPIST-1 is 3.74×1017 meters away.  TRAPPIST-1h’s orbit is a near-circle whose radius is 9.45×109 meters.  It covers π(9.45×109)2/(3.74×1017)2 = 2.00×10-15 steradians on a sphere centered on us. Divide 4π by 2.00×10-15 …  6.27×1015 sky-patches the size of TRAPPIST-1h’s orbit.  They had to pick the right patch to find TRAPPIST-1.”

“Long odds.”


~~ Rich Olcott

The Luck o’ The (insert nationality here)

“Afternoon, Al.  What’s the ruckus in the back room?”

“Afternoon, Sy.  That’s the Astronomy crew and their weekly post-seminar coffee-and-critique session.  This time, though, they brought their own beer.  You know I don’t have a beer license, just coffee, right?  Could you go over there and tell ’em to keep it covered so I don’t get busted?”

“Sure, Al.  … Afternoon, folks.  What’s all the happy?”

“Hey, Sy, welcome to the party.  Trappist beer, straight from Belgium!”

“Don’t mind if I do, Cathleen, but Al sure would like for you to put that carton under the table.  Makes him nervous.”

“Sure, no problem.”

“Thanks.  I gather your seminar was about the new seven-planet system.  How in the world do the Trappists connect to that story?”

“Patriotism.  The find was announced by a team from Belgium’s University of Liege.  They’ve built a pair of robotic telescopes tailored for seeking out rocks and comets local to our Solar System.  Exoplanets, too.  Astronomers love tying catchy acronyms to their projects.  This group’s proudly Belgian so they called their robots TRAnsiting Planets and Planetesimals Small Telescopes, ergo TRAPPIST, to honor the country’s 14 monasteries.  And their beer.  Mainly the beer, I’ll bet.”

“So the planets are a Belgian discovery?”

“Well, the lead investigator, Michaël Gillon, is at Liege, and so are half-a-dozen of his collaborators.  Their initial funding came from the Belgian government.  But by the time the second paper came out, the one that claimed a full seven planets spanning a new flavor of Goldilocks Zone, they’d pulled in support and telescope time from over a dozen other countries — USA, India, UK, France, Morocco, Saudi Arabia… the list goes on.  So it’s Belgian mostly but not only.”

“I love international science.  Next question — I see the planets are listed as TRAPPIST-1b, TRAPPIST-1c, and so on up to TRAPPIST-1h.  What happened to TRAPPIST-1a?”

“Rules of nomenclature, Sy.  TRAPPIST-1a is the star itself.  Actually, the star already had a formal name, which I just happen to have written down in my seminar notes somewhere … here it is, 2MASS J23062928 – 0502285.  You can see why TRAPPIST-1 is more popular.”

“I’m not even going to ask how that other name unwinds.  So what was the seminar topic this week?”

7 planets
TRAPPIST-1’s planets,
drawn to scale against their star. The
green ones are in the Goldilocks Zone.

“The low probability for us ever noticing those planets blocking the star’s light.”

“I’d think seeing a star winking on and off like it’s sending Morse code would attract attention.”

“That’s not close to what it was doing.  It’s all about the scale.  You know those cartoons that show planets together with their host sun?”

(showing her my smartphone) “Like this one?”

“Yeah.  It’s a lie.”

“How is it lying?”

“It pretends they’re all right next to the star.   7 planets perspectiveThis image is a little better.”  (showing me her phone)  “This artist at least tried to build in some perspective.  Even in this tiny solar system, about 1/500 the radius of ours, the star’s distance to each planet is hundreds to a thousand times the size of the planet.  You just can’t show planets AND their orbits together in a linear diagram.  Now, think about how small these planets are compared to their sun.”

“Aaaa-hah!   When there’s an eclipse, only a small fraction of the light is blocked.”

“That’s part of it.  Each eclipse (we call them transits) dims the measured brightness by only a percent or so.  But it’s worse than that.”

eclipses“How so?”

“All those orbits lie in a single plane.  We can’t see the transits unless our position lines up with that plane.  If we’re as little as 1½° out of the plane, we miss them.  But it’s worse than that.”

“How so?”

“During a transit, each planet casts a conical shadow that defines a patch in TRAPPIST-1’s sky.  You can tile TRAPPIST-1’s sky with about 150,000  patches that size.  There’s one chance in 150,000 of being in the right patch to see that 1% dimming.  In our sky there are over 6×1015 patches the size of TRAPPIST-1h’s orbit.  The team had to inspect the just right patch to find it.”

“With odds like that, no wonder TRAPPIST uses robots.”


~~ Rich Olcott

The New System’s in Tune

<We interrupt our running story line to bring you this important development…>

“Morning, Sy.  What can I get you?”

“My usual mugfull of black, Al.  What’s the Scone-of-The-Day?”

“I’m calling this The Trappist.  It’s got raspberry jam!”

“Why that name?”

“In honor of TRAPPIST-1, you know, that star they just found a bunch of planets around.”

“Your coffee shop being right next to the Astronomy building, I guess you’ve heard a lot about it.”

“Sy, you couldn’t believe.  The planetologists are going nuts of course, even though no-one’s actually seen the planets, and the astrometrics folks are lining up for telescope time ’cause they’ve got a whole new class of stars to monitor and of course the astrophysicists get to figure out how the system even works.”

“Astrometrics folks?  New class of stars?”

“Yeah, the high-precision star-measurers.  They didn’t used to pay attention to the small, dim stars because why bother.  But now … woo-hoo, whole new ballgame.”

“Nobody’s seen those planets?  How do they know they’re there?”

“Process of elimination, Sy.  The TRAPPIST telescopes picked up repetitive dark blips in the light coming from that star.  It’s a close, fast-moving star so there’s no sense supposing it’s like going behind or in front of a regular array of rocks or stars or something.  It’s not wobbling side-to-side like it would if it was a binary so it’s not traveling along with another star.  If the blips were sunspots going around as the star rotates there’d be only one rhythm, but these blips come in too complicated for that.  Besides, the star’s low-activity, too cool for lotsa sunspots.  Gotta be planets eclipsing it.”

NASA’s artistic (and cute) rendition
of the TRAPPIST-1 system
Note the close-in steam and the frost further out

“Sounds pretty good, but…”

“Hey Sy, there was something else, maybe you could explain it.  One astrophysics guy was real impressed that the planets had residences.  I didn’t understand that.”

“Residences?  That’s a new one on me.”

“Had something to do with the blip periods.  Yeah, here’s the paper napkin he wrote ’em all down on.”

Period, days
Actual /

“Oh, resonances! That I recognize, and yeah, those numbers are much more convincing.  Remember my post about gear logic?”

“Sorry, Sy, that must’ve been a long time ago and who has time to read?”

“I understand.  OK, that post explained how planets that survive the early chaos of a forming solar system tend to wind up in orbits whose relative year-lengths form ratios of small whole numbers.  In our system, for instance, the length of Pluto’s year is exactly 3/2 of Neptune’s, Neptune’s year is twice that of Uranus, and so on.  If a planet doesn’t synch up with its neighbors, it’ll collide with someone or be flung out of the system.  Put another way, a system’s not stable if its planetary orbit periods are just any old numbers.  Make sense?”

“I suppose, so…?”

“So look at this guy’s table.  The periods of each pair of adjacent objects follow that rule almost exactly.  Five times c‘s period is less than 0.25% away from eight times b‘s, and so on all the way out to h, which I take it has an uncertain period because the guy put in that question mark.  In fact, I think this system follows the rule more tightly than our Solar System does.  As far as I’m concerned that regularity in the periods makes the case for TRAPPIST-1 having planets.  You hear anything else?”

“Yeah, there was a lot of excitement about the middle three planets being in some kind of Goldilocks zone.  What’s that about?”

“Hah, I’d be excited, too.  If a planet’s too close to the star, like Mercury is to ours, it’ll be too hot for liquid water.  If the planet’s too far, any water it has would be frozen stiff.  Either way, not good for life to grow there.  In the Goldilocks zone, it’s…”

“Just right, huh, Sy?”

“On the nose, Al.  I’m going to have to read up on TRAPPIST-1.”

~~ Rich Olcott