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K2-315b: Tight Orbits and the Joy of Numbers

The newly found planet K2-315b catches the eye because of its 3.14-day orbit, a catch from the K2 extension of the Kepler Space Telescope mission that reminds us of a mathematical constant. As I’m prowling through David Berlinski’s Infinite Ascent (Modern Library, 2011), a quirky and quite lively history of mathematics at the moment, the references to ‘pi in the sky’ that I’m seeing in coverage of the discovery are worth a chuckle. Maybe the Pythagoreans were right that everything is number. Pythagoras would have loved K2-315b and would have speculated on its nature.

After all, as Berlinski notes about Pythagoras (ca. 570 to ca. 490 BCE) and his followers, they were devoted to what he calls ‘a higher spookiness”:

The Pythagoreans never succeeded in explaining what they meant by claiming that number is the essence of all things. Early in the life of the sect, they conjectured that numbers might be the essence of all things because quite literally “the elements of numbers were the elements of all things.” In this way, Aristotle remarks, “they constructed the whole heaven out of numbers.” This view they could not sustain. Aristotle notes dryly that “it is impossible that [physical] bodies should consist of numbers,” if only because physical bodies are in motion and numbers are not. At some time, the intellectual allegiances of the sect changed and the Pythagoreans began to draw a most Platonic distinction between the world revealed by the senses and the world revealed by the intellect.

And we’re off into weird metaphysics, down a historical rabbit hole. But enough of the Pythagorean buzz with numbers remains that to this day we love the odd coincidence. Hey, K2-315b is the 315th planetary system discovered inside the K2 data, a near miss from 314. MIT’s Julien de Wit, a co-author of the paper on this discovery, points out that “everyone needs a bit of fun these days,” and it’s a reference to the paper’s playful title: “π Earth: A 3.14 day Earth-sized Planet from K2’s Kitchen Served Warm by the SPECULOOS Team.” MIT graduate student Prajwal Niraula is lead author of the paper, published in the Astronomical Journal.

Image: Scientists at MIT and elsewhere have discovered an Earth-sized planet that zips around its star every 3.14 days. Credit: NASA Ames/JPL-Caltech/T. Pyle, Christine Daniloff, MIT.

What we know about K2-315b is that its radius is about 0.95 that of Earth and, importantly, that it orbits a cool, low-mass star about a fifth of the Sun’s size. Its mass has yet to be determined, but as MIT press materials point out, its surface temperature is around 450 K, which is about where you want your oven to be if you’re baking an actual pie. There is little likelihood of any lifeforms on this planet capable of groaning at puns, though I do think the discovery is helpful because it’s yet another case of an ultracool dwarf star that may be a target for the James Webb Space Telescope. Large transit depths make for interesting studies of planetary atmospheres.

I try to keep up with SPECULOOS, another wonderful acronym: Search for habitable Planets EClipsing ULtra-cOOl Stars. Here we’re dealing with four 1-meter telescopes at Chile’s Paranal Observatory in the Atacama Desert, and a more recently included fifth instrument called Artemis in Tenerife, Spain. The observing effort is led by Michael Gillon (University of Liège, Belgium) and conducted in collaboration with various institutions including MIT and the University of Bern, along with the Canary Islands Institute of Astrophysics and the European Southern Observatory.

Image: The SPECULOOS project aims to detect terrestrial planets eclipsing some of the smallest and coolest stars of the solar neighborhood. This strategy is motivated by the unique possibility to study these planets in detail with future giant observatories like the European Extremely Large Telescope (E-ELT) or the James Webb Space Telescope (JWST). The exoplanets discovered by SPECULOOS should thus provide mankind with an opportunity to study the atmosphere of extrasolar worlds similar in size to our Earth, notably to search for traces of biological activity. Credit: SPECULOOS.

The K2-315b work spanned several months of K2 observation from 2017 in which 20 transit signatures turned up with a repetition of 3.14 days. At this point, closer examination relied upon tightening the transit time even further, as co-author Benjamin Rackham points out:

“Nailing down the best night to follow up from the ground is a little bit tricky. Even when you see this 3.14 day signal in the K2 data, there’s an uncertainty to that, which adds up with every orbit.”

Fortunately, Rackham had developed a forecasting algorithm to pin the transits down, and subsequent observations in February of 2020 with the SPECULOOS telescopes nailed three transits, one from Artemis in Spain and the other two from the Paranal instruments. The paper points out that differences in atmospheric “mean molecular mass, surface pressure, and/or
cloud/haze altitude will strongly affect the actual potential of a planet for characterization,” with ramifications for the study even of promising worlds like those circling TRAPPIST-1.

Nonetheless, K2-315b (referred to in the K2 data as EPIC 249631677) looks intriguing enough for JWST observations to be considered:

With an estimated radial velocity semi-amplitude of 1.3 m s−1 (assuming a mass comparable to that of Earth), the planet could be accessible for mass measurements using modern ultra-precise radial velocity instruments. Such possibilities and a ranking amongst the 10 best-suited Earth-sized planets for atmospheric study, EPIC 249631677 b will therefore play an important role in the upcoming era of comparative exoplanetology for terrestrial worlds. It will surely be a prime target for the generation of observatories to follow JWST and bring the field fully into this new era.

Note that reference to ‘comparative exoplanetology.’ Not all exoplanets singled out for atmospheric characterization are going to be ‘habitable’ in the sense of life as we know it. After all, we began using transmission spectroscopy to study atmospheres by working with ‘hot Jupiters’ like HD 209458b. We learn as we go, and firming up our methods by studying small planets around ultracool dwarf stars within 100 parsecs or so is part of the path toward finding a living world.

The paper is Niraula et al., “π Earth: a 3.14-day Earth-sized Planet from K2’s Kitchen Served Warm by the SPECULOOS Team,” Astronomical Journal Vol. 160, No. 4 (21 September 2020). Abstract / Preprint.

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{ 32 comments… add one }
  • Robert A Leigh September 23, 2020, 16:18

    I assume days means earth days here, in that we can’t detect the rotation of exoplanets? The coincidence would be a bit more interesting if it were the planet’s own days.

  • David Given September 23, 2020, 17:04

    Doing numerology by comparing dimensioned numbers with undimensional numbers is cheating — you can make anything fit by picking your units correctly. As generations of lunatic Egyptologists knew very well! Numerological coincidences that are dimensionally correct and work solely on relationships between numbers are much rarer.

    My favourite is c=gy, where c is the speed of light, g is the gravitational acceleration of Earth, and y is the length of the year of Earth. It’s independent of units. And I’ve seen people suggest that this must be true for any habitable planet…

    • Andrei September 23, 2020, 18:14

      I agree, and will not convert to numerology just because of this finding!
      When some planetary systems turned out to have the planets spaced to fit Titius – Bode’s “law” it got quoted all over. There’s however something to that idea, but better defined by the resonances between the planets. Our solar system could also be defined by the Fibonacci number series or several other means with amusing mathematical results.
      I did some short digging and found a post about it, where the author also managed to include atomic weights.
      https://www.reddit.com/r/space/comments/1uhka0/a_little_list_of_coincidence_in_the_solar_system/

  • Ron S. September 23, 2020, 19:27

    I was amused by the orbit length ring of digits, supposedly a vast number of them, only reaching down to around the 140th decimal. Sad to say I recognized it (but checked!) since when I was young I memorized pi to a few hundred digits. Of course that final set of digits is certain to reappear deeper down many times.

  • Ioannis Kokkinidis September 23, 2020, 19:43

    https://en.wikipedia.org/wiki/Speculaas Speculoos is a Belgian/Dutch cookie. I have eaten a few in my life. These are the same scientists as TRAPPIST and again they decided to put something iconically Belgian as their initials

    • Paul Gilster September 23, 2020, 20:29

      Ioannis, I had forgotten about that! Wish I had put it into the article. Thanks.

  • Henry Cordova September 23, 2020, 23:04

    When I was in high school, my trigonometry teacher drew a right triangle inside a unit circle and defined the sine and cosine functions as the ratios of the lengths of the opposite and adjacent sides of the triangle to the hypotenuse, vertical and horizontal values divided by 1.

    As the radius vector spun around, he plotted the sine and cosine values on a Cartesian coordinate system where the x axis represented the angle of the vector and the y was the value of the function. The sine and cosine curves, 90 degrees out of phase, marched off to the right, in an endless progression into infinity.

    In a terrifying flash of insight, I realized triangles had been playing these games with each other long before there were human minds to contemplate them, indeed, even before there was a universe at all, and that this simple mathematical trick could be used to model any oscillating system: springs, guitar strings, waves, alternating current, sound. This was the meaning of the display on an oscilloscope. All these “real” things, matter and energy, space and time, could be represented precisely in a human mind by a process of pure thought, and it was based on a concept that didn’t seem to need human minds at all–it was simply a property of triangles.

    I still haven’t come to grips with just what it all means.

    • Mike Serfas September 24, 2020, 16:57

      This is related to the solution of the Pythagoreans’ paradox. If numbers don’t move, how can moving things be made of numbers? Schroedinger brought “i” into the equation. With complex numbers, we represent waves that are endlessly moving even as they remain where they are.

  • ljk September 24, 2020, 10:14

    In Carl Sagan’s only science fiction novel, Contact, first published in 1985, the value of pi plays a huge role in the story: The mysterious ancient beings that made the cosmic wormhole tunnels that allow travel all over the Universe also left messages buried deep within the value of pi, implying that they made existence as well.

    This plot concept was absent from the 1997 film version, probably because the filmmakers were afraid of scaring off their audience with mathematics beyond teaching them prime numbers.

    Relevant links:

    http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf55

    https://arxiv.org/pdf/1209.2348.pdf

    http://adamslostdream.blogspot.com/2013/05/contact-carl-sagan-and-pi-code.html

    http://kasmana.people.cofc.edu/MATHFICT/mf55-spoiler.html

    http://pubs.sciepub.com/ijp/2/6/9/

    https://math.stackexchange.com/questions/1104660/patterns-in-pi-in-contact#:~:text=In%20Carl%20Sagan's%20novel%20Contact,messages%20embedded%20inside%20transcendental%20numbers.

    Humans want to think the Universe has a purpose or a reason for existing, even if it is by obscure aliens who behave like the God of the Deists: An all-powerful deity created the Cosmos, got it running, then left it alone for whatever reason.

    The idea that something as vast and complex as the Cosmos just happened, such as with the theory of everything suddenly popping into existence from a quantum fluctuation of nothing as one theory goes, is incomprehensible to most humans.

    However, I think most people have a conception of the Universe mired in ancient thinking about the world around us, which with few exceptions, is decidedly smaller than the reality we have known for the last few centuries – and in some aspects, just the last few decades.

    So could the Universe have been made by some kind of seriously advanced being? Well, there is the baby universe concept:

    http://live.iop-pp01.agh.sleek.net/2017/07/28/building-baby-universes/

    https://www.livescience.com/universe-machine-probes-dark-matter.html

    https://en.wikipedia.org/wiki/Black_Holes_and_Baby_Universes_and_Other_Essays

    • Henry Cordova September 24, 2020, 11:15

      The trouble with all these “The-universe-is-so-complex-it-must-have-been-created-by-some-advanced-being” theories is that they don’t really answer any questions, they just postpone them. Even if there is some supreme super-, para-, or metanautral being(s) that created everything, we are now faced with an even bigger question. WHO created He, Them or It? If the Creator is indeed real, then He must be even MORE complex, wonderful and impossible to arise spontaneously from nothingness than the multiverse itself. Like Carl Sagan himself said; “It’s turtles all the way down.”

      I prefer Mr Natural’s take on it.

      https://www.thirdmindbooks.com/pictures/medium/2226.jpg

    • Alex Tolley September 24, 2020, 11:58

      The idea that the universe is a simulation, possibly embedded in meta simulations, seems like a modern variant of “turtles all the way down” for supporting the [flat] Earth.

      Humans have a preference for “purpose” and hate randomness, even though science often uses randomness as the null hypothesis, especially as this is the basis for statistics. Ascribing random events to [God’s] purpose elevates the individual to some cog in “the plan” rather than a meaningless event.

      As Arroway says: “So if it is just us, it [the universe] it seems like an awful waste of space.” That reprises her father’s saying when she was a child, but she has met the ancient aliens, and knows better.

      The movie has spelled that out, and therefore doesn’t need to add the issue of “do they exist?” based on pi for the audience.

      • ljk September 25, 2020, 10:03

        Alex Tolley said on September 24, 2020, 11:58:

        As Arroway says: “So if it is just us, it [the universe] it seems like an awful waste of space.” That reprises her father’s saying when she was a child, but she has met the ancient aliens, and knows better.

        The movie has spelled that out, and therefore doesn’t need to add the issue of “do they exist?” based on pi for the audience.

        While it is obvious for us that Ellie met the ETI in the last parts of the film, the pi part in the novel was more about Sagan speculating on how we might go about finding a “message” from really advanced aliens, especially if they created the Universe. Embedding something in pi was one possibility.

        Both the novel and film versions of Contact spend a lot of time focusing on the whole science/reason versus faith debate. One chapter alone is devoted to a long meeting/debate Ellie has with two of the main religious characters in the novel, although we only get to meet Palmer Joss in the film substantially. Rankin was only given a few token lines and reactions by comparison.

        BTW, Joss’ character was far more endearing in the novel. You could definitely understand why Ellie was falling for him towards the end of the story. In the film, Joss seemed more like a mildly smarmy guy on the prowl who wasn’t quite as clever, smart, or sincere as he would like everyone to believe, including himself. Or maybe it is just the way the actor portrayed him.

        Is religion going to be a big part of the discovery of ETI? Of course, at least for humanity. Sagan tried to bring both sides into play, but of course religion was bound to be a straw man. He also tried hard to make the two sides find mutual ground using Ellie and Palmer, although I think he did better in the novel.

        This next article makes some interesting observations and good quotes from Contact that are quite relevant to this discussion:

        https://theconversation.com/contact-and-carl-sagans-faith-85150

        While I don’t think Sagan was actually trying to find God, at least not overtly so, I do believe he was trying very hard to get the two sides to reconcile or at least find some common ground, as he undoubtedly knew that religion wasn’t going to vanish in a puff of smoke if science confronted it with reason and evidence. I have always been amused at how Arthur C. Clarke took that very tack, especially in his novel The Fountains of Paradise, where he assumed Islam would vanish as humanity became more “reasonable.”

        At the least Sagan was far more understanding and tolerant regarding believers than many of the more militant atheists who have followed him since. People do not respond well to being told they are deluded fools who need to change their entire way of thinking and living at once.

    • kph September 25, 2020, 0:24

      Cicero’s “The Nature of the Gods” comes to mind when reading seemingly recent insights about the Universe’s origin, purpose.

      • ljk September 25, 2020, 9:24

        kph said on September 25, 2020, 0:24:

        “Cicero’s “The Nature of the Gods” comes to mind when reading seemingly recent insights about the Universe’s origin, purpose.”

        For those who would like to read this work…

        https://oll.libertyfund.org/titles/cicero-on-the-nature-of-the-gods

        I am also reminded of these lines in the 1960 film Spartacus:

        Julius Caesar: “I thought you had no reservations about the gods?”

        Sempronius Gracchus: “Privately I believe in none. Neither do you. Publicly I believe in them all.”

        https://en.wikipedia.org/wiki/Spartacus_(film)

        • Alex Tolley September 25, 2020, 12:55

          Sempronius Gracchus: “Privately I believe in none. Neither do you. Publicly I believe in them all.”

          A very similar situation in the US, where atheists are not considered suitable for public office (even excluded by state law in one state), with presidential candidates wearing their religion on their sleeves. The current POTUS even proclaiming being deeply religious to pander to the religious right, even as he has been overheard demeaning the sentiments of his religious advisors while in private. Perhaps another similarity to ancient Rome?

          The US is an anomaly in the world as a rich but also religious country, whilst most countries reduce religious beliefs as they get wealthier. For Example, Britain was an almost non-religious country by the 1960s, and today it is joked that most people attend church only 3 times in their lives – births (christening), marriage, and death (funeral).

          • ljk September 28, 2020, 12:24

            The United States has not undergone centuries of bloody religious wars as has Europe to become exceedingly tired of. Hopefully we won’t have to go through such trials in order to reach a state of cultural equilibrium.

            Speaking of ancient treaties speaking blasphemies about the gods, one cannot exclude this classic work, of which only ONE copy came down to us from antiquity:

            https://plato.stanford.edu/entries/lucretius/

      • ljk September 30, 2020, 9:35

        Interesting side note: I was just reading the Introduction to this work by Cicero and discovered that he dedicated it to one Marcus Junius Brutus.

        Yes, THAT Brutus. Et tu and all that.

  • Ron S. September 24, 2020, 13:07

    Might I humbly suggest that the code for rendering pages containing long URLs and other space-free text be improved. This comment page is now largely unreadable (using Firefox).

    • Paul Gilster September 24, 2020, 13:17

      I’m afraid I lack the expertise to do that, although perhaps others more conversant with the internals of WordPress would know how.

      • Ron S. September 24, 2020, 14:44

        Not to be a pest about it but there is a non-technical solution should there not be a good technical one. During moderation force a line break or even delete them. The latter is a crude instrument to train commenters to better deal with those giant URLs. It’s a trade off between allowing one commenter’s URL versus ensuring the comment thread is readable by everyone. I’ll shut up about it now.

        • Paul Gilster September 24, 2020, 19:38

          Yes, I’m willing to do that if it’s causing trouble, but I’m not seeing it on Firefox here, and I just saw that John Walker likewise has no trouble with Firefox. So I’m unsure what’s going on. Let’s get some more reports on this if possible.

          • Alex Tolley September 24, 2020, 22:49

            There is a problem. I just checked my FF c 81.0 on Windows 10 and the text overflows the white page area into the black border. This is even more obvious when opening a comment box – the grey box extends into the border. The blue “CANCEL REPLY” is readable, as the “EPLY” contrasts with the border color. The black “border” extends from the white page to the gray verticle separator between the text area and includes the right hand menu bar that renders links as blue text on a black backround and the black text in the sidebar as black on black. The result is that for the text in body of the post and some comments overflows into the border and becomes unreadable – black text on a black background.
            The Chrome browser renders exverything as white background from the left edge of the text display to the right edge of the seidebar.
            This may be a problem a problem with the rendering engine or some issue with the WP template. As Ron suggests, this appears to be a problem when URL links are long, as the rendering is fine on the previous post wion “Prenetrating radar mapping”.

            For the reader the best fix for now is use a different browser, which I know is annoying, especially if you use bookmarking a lot.
            The other fix is to train commenters to use html tags to format links, or to do the formatting during moderation, a major pain for the host.:

            HTML Links – Syntax

            • Paul Gilster September 25, 2020, 6:37

              Yes, and unfortunately not a task I have time to do on a daily basis. How often is this problem occurring? I’m not seeing it here so I have no idea.

    • John walker September 24, 2020, 18:40

      I don’t wish to be confrontational but, the page is rendered for me on Firefox with no technical problems and the URLs seem perfectly adequately sized.

    • Mike Serfas September 25, 2020, 10:15

      I see the same problem with the rendering, both in Firefox and in Tor (which is a Firefox derivative). Fortunately, Firefox has a great set of developer tools – I could track down that it was in fact ijk’s post that causes the phenomenon, but more importantly, the reader can fix it.

      As you’re reading this page, press control-shift-I to open the developer tools, and pick the tab Style Editor. At line 7 it says ‘word-wrap: break-word’. Change that text to ‘word-wrap: anywhere’ and then the rendering will be OK. (You don’t need to reload anything – I double-checked this works while I was typing this response) I’m not sure if the site operator(s) want to make that change on their end, since I don’t know what this breaks, but it only takes a moment for us to do if need be.

      • Alex Tolley September 25, 2020, 13:36

        That works nicely.

        Looking at my own Wordpress stuff, I see the word-wrap style is in 2 places – a static content file and a theme stylesheet file (style.css).

        It should be possible to make the appropriate fix at the host end as long as the problem can be replicated to test the changes.

        [This is how technological civilization ends, not with a bang, but with all cognition distracted fixing small problems in increasingly complex code. ;) ]

        • Mike Serfas September 25, 2020, 15:56

          From my end I’m editing a style sheet “css.css”, so I’m not sure where this came in, but it’s one of three rules: margin, padding, and word-wrap, which are applied to the entire document. You wouldn’t actually have to apply “word-wrap: anywhere;” to the entire document – you could put it under .comment only. But the global word-wrap overrides it, so you would at least have to delete that line there regardless. But in that case if the change breaks some other style on your page, like your article text, you could give *that* field a different word-wrap setting.

      • Ron S. September 25, 2020, 14:06

        I tried that and it works great but I won’t do it every time I navigate to one of those pages. As you say, Paul may be able to make it universal on his end if it’s doable and breaks nothing else (probably not).

        Presumably other browsers ignore or override the style, or it may be that many denizens of CD use the browser in full screen mode and so it takes a far longer URL to trigger the problem.

  • randomengineer September 24, 2020, 23:14

    How do we know every case of transit must be a planet? Is it possible that we’re sometimes seeing artificial objects constructed to orbit a star at a specific distance for other reasons? Overall I know that the transit frequency and duration is used to posit a planet size… i.e. if amount of light blocked is X at transit speed Y for duration Z then it’s said to be e.g. “1.3 earths hence must be a rocky planet” based on apparent size and distance. Is there also a detectable (barycentric?) wobble component confirming mass or is this always an assumed thing? I ask because of the continuing mystery of Tabby’s Star wherein the dips in light ought to coincide with barycentric wobble to confirm actual mass, but I’ve not seen this. Hence the question, thanks.

    • Ron S. September 25, 2020, 23:22

      “How do we know every case of transit must be a planet?”

      We know no such thing so I don’t understand why you claim this. Planets are, of course, the most likely object that is transiting and if the transit fits the model of a planet transiting it is likely a planet and not, say, an artificial construct purely on the grounds of statistical likelihood (Bayes prior). I don’t believe anyone claims that the transits of Tabby’s star are caused by a planet since the light curves don’t fit that explanation, even with the inclusion of some exotic addition such as a complex ring structure.

      “…detectable (barycentric?) wobble component…”

      That would be a radial velocity variation of the star caused by the orbit of a planet (star orbiting the barycenter), which is particularly useful for non-transiting planets (the majority). We are getting closer to resolutions that could reveal an Earth like planet, and we should achieve it in the coming years. I don’t follow this closely but others do and may want to jump in with more detail. It’s been covered extensively at CD so you can try a search.

  • AlexTru September 25, 2020, 4:04

    Too much speculations about ordinary event…
    Period 3.14 Earth days, why not? How it is different from 2.718 days or 1 day or 5 days period?
    Why should the Earth day length be important to Universe?
    This excitement by “Pi * earth_day“ number can be explained only by religious point of view to reality.

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