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The Mass-Period Diagram

December 21st, 2006

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When J. Edgar Hoover was getting on in years, his aides would often tell scheduled visitors to his office that he was unable to meet with them because he was “in conference”. In reality, this meant that Hoover was napping at his desk.

It might seem that the refrain of, “we’re busy working on the systemic back-end” is an equally convenient euphemism for long lapses between posts on the front end. Nevertheless, we have been busy getting the new oklo xserve quad xeon up and running. The whole site has now been replicated and tested, the server is live and on air, and very shortly, we’ll be flipping the switch. Can’t wait, man!

With the vast increase in processing power afforded by the xserve, we’ll be able to provide a much more extensive suite of research tools to oklo visitors. In particular, it’ll be possible to dynamically generate the kinds of correlation diagrams that are currently only available from our estimable continental competition: exoplanet.eu.

It’s always interesting to look through the latest versions of the correlation diagrams to see whether the various trends and hints of trends are holding up. The a-e plot is worth examining, as is the plot that charts the number of planetary discoveries per year over the past decade. As of today, exoplanet.eu lists 192 planets that have been detected with the radial velocity method. Plotting the masses of these planets against their periods on a log-log plot (and running the resulting screenshot through Illustrator) yields the following:

latest mass-period diagram

For Keplerian orbits, the relationship between the radial velocity half-amplitude of the parent star and the orbital period of the planet is given by:

equation for radial velocity half-amplitude

If we assume that the mass of the planet is negligible in comparison to the mass of the star and if we further assume edge-on, circular orbits around solar mass stars, then we get the dashed lines in the figure that show detection thresholds for K=3 m/s and K=1 m/s. The three planets orbiting HD 69830 stand out in this diagram as the most striking discoveries of 2006.

To the eye, there are two curious clusters of planets in the diagram. At short periods (P~3d) we have the hot Jupiters. Most of these have masses (times the sine of the unknown inclination) somewhat less than Jupiter. At longer periods (P>100d) we have a second prominent clump of planets. These are the Eccentric Giants, and their masses average out at a significantly higher value (between 2 and 3 times the mass of Jupiter). Part of the difference in mass is due to selection bias, but nevertheless there is a real effect. Like the planet-metallicity connection, this effect is telling us something about either planet formation or planet migration (probably the latter).

Anyone got an idea regarding what’s going on? Let’s get a discussion going in the comment section. Over the past week, I’ve been flooded by depressingly clumsy attempts at comment spam from single-minded robots with mechanical enthusiasms for satellite TV service and online poker, e.g. “Great blog, keep it comming.” It’d be nice to see some signal in the noise…

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  1. TheoA
    December 21st, 2006 at 18:14 | #1

    OK. Let me take a first feeble minded stab at this.

    I think its a big give away that the Jupiter+ planets are Eccentric. It hints at the eccentricity being induced very early on in the planet formation process. This eccentricity dissipates the gas that encourges circularization and further falling in. The process is size dependent and the bigger planets accomplish this faster while the smaller ones are not as efficient and hence spiral in. The effect must be very marginal as one can argue that the bigger planets need more time and a denser gas cloud to form, both of which should counter this effect and induce circularization.

    The other thought I’ve always had, is that there might be a window before the star ignites and begins to exert outward pressure when a gas giant could potentially begin to form close in. This is cut off fairly quickly and hence the size limitation. Science has always frowned on this but as I said this is a feeble attempt.

    There is also a possiblity that larger planets evaporate down to this size. I can’t see much proof for this as larger planets are actually smaller in size and hence more efficient in retaining mass.

    As long as we are taking long shots, I’ll put out the idea that SEDNA type extraneous objects could enter the planetary disc along the plane of the spinning gas, slow down enough to be captured and grow a gas cloud and yet keep their eccentricity. The similarity in paths with comets chould not be ignored.

  2. darin
    December 22nd, 2006 at 05:41 | #2

    First, I would like to point out that the lower mass end of the large-period “eccentric Jupiters” might be missing due to observational difficulties (long baselines and high accuracy are both required) and not becuase of a true lack of intermediate mass planets with periods measured in years.

    Next, may I note that plotting the planet-to-star mass ratio (a more physically relevant quantity than just the mass of the planet) versus period preserves the two clusters, but shows a more smooth transition region. Of course, we only have minimum masses, so some of the data points would jump up if we knew their true masses.

    Perhaps this plot actually illustrates: 1) a lack of high-mass short-period planets and 2) a general lack of planets with 10-100 day periods. For the first, I agree with TheoA that evaporation of the largest planets is an unlikely explanation. I also find it interesting that there are no Hot Brown Dwarfs and that nearby companions appear to be limited to Jupiter-size planets and stars (there are many known stellar binaries with such short periods).

    The increase in planets starting at about 1-year periods may simply be due to the presence of the snow line: at these distances from the stars, the disk is cool enough for ices to condense, highly enhancing the solid fraction of the disk and allowing for core accretion to work much more effectively. Planets with shorter periods would need to migrate inwards. If the disk dissipates during the migration, the planet stops before it reaches the star. If the planet makes it all the way to a ~4-day orbit, it’s migration halts (since the disk which torques it inwards is truncated by the hot star) creating the pile-up. (This is following the original argument by Doug Lin and others for the explanation of Hot Jupiters and their periods.) Ignoring this pile-up, it actually looks like there are equal numbers of planets in logarithmic period bins all the way up to the snow line.

    Just some general comments to think about.

    Also, may I mention that capturing interstellar comets or other material in the primordial gas disk is pretty ineffective. As a rule of thumb, to be captured, you must intersect an amount of gas/dust that has of order the same amount of mass as you do. I’ve done this calculation for Sedna, and it is much too large to be captured in this way. Also, eccentric orbits are caused by dynamical mechanisms that can be insensitive to mass, so that planets and comets can share similar orbits, but not because one is formed from the other.

  3. December 22nd, 2006 at 17:50 | #3

    This may well be irrelevant to the discussion (I’m kind of thinking out loud here), but does the clustering still occur when other properties of the host star are taken into consideration? (For example, a plot of exoplanet mass versus the radiation flux density at its semi-major axis.)

    I too am wondering if a transition zone (such as the frost line) in the composition of the proto-planetary disc was a factor in the dynamical as well as the physical evolution of these planets.

  4. andy
    December 22nd, 2006 at 18:08 | #4

    Might this be something to do with the timescale of migration? If migration typically dumps a planet in a ~3 day orbit, the only planets that end up in the 10-100 day region are those that migrated late and happened to end up in this reason when the dust disc disappeared – maybe they started off quite a long way out, or formed relatively late.

    As for short-period massive planets, could this be a tidal effect? A planet in such an orbit would be orbiting faster than the star rotates, and thus the tidal bulges it raises on the star would tend to make it fall downwards (like Phobos is predicted to do around Mars). Massive planets would raise bigger tides on the star and hence fall faster.

    Not sure either of those is reasonable…

  5. TheoA
    December 22nd, 2006 at 22:32 | #5


    Greg has admitted that there might be a selection effect, nevertheless he is convinced there is a real effect here.

    On further thought there might be a mechanism through which evaporation might be brought into play. If large mass planets find a way to end up closer to the star gradual evaporation should allow the planet to slowly recede till an optimum ‘parking’ location is found presumably ~Mjup atthe 3-4 day location. The mass of the star should not affect this process too much. If you look at the chart there is an extremely weak trend towards larger hot jupiters being closer in. The tidal effect may actually enchance this mechanism.

    Thanx for doing the calc on SEDNA. Is it possible that multiple passes might help. or maybe lower mass might improve the gas drag. It just seems too much of a coincidence that these vast numbers of planetisimals have existed since planet formation and yet did not take part in the process! Esp. since we know they intruded in the solar system enough to be captured multiple times by our gas giants.

    Isn’t the consensus right now that hot jupiters all form beyond the snow line and spiral inwards. Why would this process not encourage even smaller hot jupiters. Why cluster at that particular size!

    Andy, delayed migration is challenged by many systems where the outer planet from a hot jupiter is much larger. Maybe the migration is limited to closer in planets or is initiated as the gas density drops?

    It is going to be very difficult to rely on star mass as the vast majority of stars selected so far were similar to the sun and hence similar in mass. The selection effect could overwhelm any signal.

  6. December 23rd, 2006 at 23:08 | #6

    I think that hot Jupiters (also called roaster, pegasid or Pegasi planet) at shorts periods (P~3d) can be explained by Type II migration. The reason why I say this is because gas-giants could not possibly have formed so close to their parent star, simply because there would not be enough material there to form such massive objects. The stellar winds from the parent star would have blown the the lighter material such as hydrogen and helium several AU away before a gas-giant would have a chance to form that close to it’s star. I believe that this is the explantion for the first cluster of the 2006 Mass-Period Diagram.
    As for the second cluster with longer orbital periods (P>100d), I believe that the higher masses of those group of planets are due more to planet formation rather than planet migration. The reason why I say this is because Tristan Guillot of the Observatoire de la Cote d’Azur in France discovered a correlation between the metallicity of a planet’s parent star and the metaliticity of Jupiter-like extrasolar planet. The correlation is quite simple: the lower the metallicity of the parent star the larger the volume of the planet as whole, but the smaller is its core. Conversely, the higher the metallicity of the parent star, the smaller the volume of the planet as a whole, but the larger is it’s metal rich core, thus making it more massive, even though it’s smaller.
    I believe the reason why the planets in the second cluster have such highly eccentric orbits is because 1) either there is a companion which is causing a Kozai effect or 2) the planetary system at some time early in the formation of the system, underwent some kind of catastrophic perturbative event. I believe that these may have started out as metal-rich planets like Mercury and when there orbits were perturbed, they over time accumulated enough hydrogen and helium from the outer reaches of the newly formed planetary system to form atmospheres by means of simple gravitational attraction, thus in time becoming Jovian-like gas-giants over many orbital periods. It’s just conjecture though.

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