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.”

“Vinnie?”

“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.”

(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?”

“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.”

“Yep.”

~~ Rich Olcott