Tag Archives: Q

This just in…

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With HAT-P-13c rapidly coming ’round the mountain, there was a very timely update on astro-ph last night. Josh Winn and his collaborators have obtained an additional slew of radial velocities which (1) demonstrate using the Rossiter-McLaughlin effect that the inner planet b’s orbit is likely well aligned with the stellar equator, (2) modify the orbital parameters, including the period of the outer massive planet, and (3) hint at a third body further out in the system.

How do these updates affect the unfolding story?

The Rossiter-McLaughlin measurement gives an estimate of the angle λ = -0.9°±8.5°, which is the angular difference between the sky-projected orbital angular momentum vector and sky-projected stellar spin vector. A non-intuitive mouthful. If we’re viewing the star edge-on, then λ = -0.9° amounts to a determination that the planet’s orbital plane is well-aligned with the star’s equator. (See this post for a discussion of what can happen if the star’s rotation axis is tipped toward the Earth). The good news from the measurement is that it’s a-priori more likely that planets b and c are coplanar — that happy state of affairs which will permit direct measurements of planet b’s interior structure and tidal quality factor. If, on the other hand, the planets b and c have a large mutual inclination, then b’s node will precess, and measurement of a small value for λ will occur only at special, relatively infrequent, times during the secular cycle. A close to co-planar configuration also increases the likelihood that the outer planet can be observed in transit.

With their beefed-up data set of out-of-transit Doppler velocities, Winn and his collaborators are able to get a better characterization of the planetary orbits. The best-fit orbital period and eccentricity of the outer planet are slightly modified when the new data are included. The best-guess center of the transit window for c has “slipped” to April 28, 2010, with a current 1-σ uncertainty of 2 days.

The later date, however, is not an excuse for procrastination! Measuring the TTV for this system is a giant opportunity for the whole ground-based photometric community, and a definitive result will require lots of good measurements of lots of transits starting now (or better yet, last month.) I’ll weigh in in detail on this point, along with the challenge posed by Mr. D very shortly…

Q: What’s Jupiter’s Q?

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With the flood of detail from extrasolar planets, one can forget that our knowledge of the worlds in our own solar system is literally centuries ahead of what we know about planets orbiting other stars. For example, careful naked-eye observations can be used to derive better orbital models for Venus et al. than we currently possess for any exoplanet (assuming, of course, that one owns a good watch and eyesight sufficient to resolve the disk of Venus when it transits the Sun). One of the best ways to learn about what’s out there is to learn as much as we can about what’s right here.

In this vein, an important paper came out in Nature last summer, in which Lainey et al describe a direct and unprecedentedly accurate measurement of the present value of Jupiter’s tidal quality factor, Q. The tidal quality factor encapsulates the ability of an object to dissipate disturbances raised by tidal gravity. The lower the Q, the more capable is the body at damping out the perturbations generated by tidal forcing. Q can depend quite sensitively on the frequency at which perturbations occur, and with a few notable exceptions (for example, the Earth and the Moon), it is notoriously tricky to determine. Previous estimates for Jupiter’s Q ranged from Q~60,000 to over a million. By extension, Q values for Jupiter-mass extrasolar planets are often assumed to lie in this range.

In order to directly measure the Jovian Q, Lainey et al. adopted a procedure that’s conceptually very similar to what goes on inside the systemic console. They first collected measurements of the positions of the galilean satellites that were obtained from 1891 all the way through 2007. They then constructed an orbital N-body model that includes the full gravitational forces acting on Jupiter and the galilean satellites, and which incorporates the non-axisymmetric gravitational pulls exerted by the tidal bulges of Jupiter and Io. The fitted parameters — that is, the initial conditions and undetermined constants — for their model are the osculating orbital elements of the moons, and the values of Q/k2 for Jupiter and Io. (The Love number, k2, is a measure of the degree of central concentration of a body, and has a value of k2~0.37 for Jupiter. For more, see these posts, one, two, from last summer).
Lainey et al. varied the parameters and repeatedly carried out new integrations until the the agreement between where the integrated orbital model said the moons should be located and where they were actually observed was optimized. For this type of direct integrations, goodness-of-fit is highly sensitive to the amount of tidal dissipation in Io and in Jupiter — the larger the dissipation, the larger the effect on the orbit. As a consequence, when a best-fit orbital model is attained, one has direct estimates for the Q‘s of both Jupiter and Io.

And the result? The integrations suggest that the current value of Jupiter’s Q is of order 30,000. This suggests that Jupiter is much more dissipative than has been assumed, and is indeed quite comparable to Neptune or Uranus in terms of its ability to damp out tidal disturbances. The measured Q is low enough, in fact, to suggest that Jupiter currently lies in a state where the tidal forcing by Io is leading to a historically large rate of dissipation. Over the past several billion years, as the orbital frequencies of Io, Europa and Ganymede evolved through a range of values, Jupiter’s Q was on average likely quite a bit higher than it is now.

Jupiter’s low Q hints that the transiting Neptune-mass planet Gliese 436b is even more mysterious than previously though. Gliese 436b has a significantly eccentric orbit whose non-circular figure can only be understood if (1) there’s a suitably influential perturber in the system, or (2) there was a relatively recent disaster, or (3) if the planetary Q has somehow stayed anomalously high through billions of years of orbital evolution. No matter which one of these possibilities turns out to be correct, it’ll be a very interesting story.

A look inside an extrasolar planet

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Image Source.

Cranking out a paper invariably takes longer than one expects. Last week, I was confident that Konstantin and Peter and I would have our HAT-P-13 paper out in “a day or so”, and then it ended up taking the whole week. As of ten minutes ago, however, it’s been shipped off to the Astrophysical Journal Letters. It’s also been submitted to astro-ph, hopefully in time to make tomorrow’s mailing.

In the meantime, here’s a link to (1) the .pdf of our text, and (2) the two figures (one, two) both in .gif format. The two figures are 800 pixels across, all the better for dropping in to presentations.

Put briefly, HAT-P-13 is an absolutely remarkable set-up. The presence of the outer perturbing body in its well-defined orbit allowed us to show that the system has undergone long-term evolution to a “tidal fixed point”. In this state of affairs, secular variations in the orbital elements of the two planets have been damped out by tidal dissipation, the apsidal lines of the orbits have been brought into alignment, and most importantly, the two orbits precess at the same rate. The paper shows how the eccentricity of the inner planet is a sensitive function of the planet’s interior structure, and in particular, the degree of central concentration (parameterized by the “Tidal Love Number”, k_2).

Here’s a schematic that shows what’s going on:

Right now, the eccentricity of the inner planet is determined to rather modest precision e=0.021 +/- 0.009. The system is transiting, however, and so when Warm Spitzer measures the secondary eclipse time, the error on the eccentricity measurement will drop dramatically. The situation will also benefit from an improved measurement of the planet’s radius. When improved measurements come in, it’ll be possible to literally read off the planet’s core mass and, in addition, the value of the much-discussed tidal quality factor Q.

Lucky 13

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In reviewing grant proposals and observing proposals that seek to study extrasolar planets, one notices that two cliches turn up with alarm-clock regularity. Number one is Rosetta Stone, as in this or that planetary system is a Rosetta Stone that will enable astronomers to obtain a better understanding of the formation and evolution of planetary systems. Number two is ideal laboratory, as in this or that system is an ideal laboratory for studying the processes that guide the formation and evolution of planetary systems.

A terse unsolicited e-mail from Gaspar Bakos always means that a big discovery is in the offing, and today was no exception:

Hello Greg,

You may like this.
http://xxx.lanl.gov/abs/0907.3525

Best wishes
Gaspar

Indeed! HAT-P-13b and c constitute a really exciting discovery. For a number of reasons, this system is a Rosetta Stone among extrasolar planets, and in large part, this is because the system is an ideal laboratory for studying processes such as tidal dissipation and orbital evolution.

HAT-P-13 harbors the first transiting planet that has a well-characterized companion planet. In this case, the outer companion has a P=428 day orbit, an Msin(i) of 15 Jupiter masses, and an eccentricity, e=0.7. In the following diagram, the orbits and the star are shown to scale; the small filled circles that delineate the outer orbit show the position of the outer planet at 4.28 day intervals.

Illustrator-editable PDF of the above

Of obvious interest is the question of whether planet c can be observed in transit. The a-priori probability is seemingly enhanced by the transit of the inner planet. (Give that one to the good Reverend Bayes). The next opporunity rolls around in April 2010, with the opportunity to observe secondary transit following a bit more than two months later.

It’ll be quite something if planet “c” does transit. A sense of the wide open spaces in the system can be obtained by plotting the star and the two planets to scale with their respective separations at the moment of inferior conjunction. Given the width restriction of the blog post format, one needs to present this plot vertically:

There’s a lot more to say about the HAT-P-13 system — so much in fact, that Peter Bodenheimer, Konstantin Batygin and I are furiously writing an ApJ letter. Should have it out the door in a day or so, with a roundup to follow here on oklo.org immediately thereafter…

HAT found a Neptune,

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and at 880K it’s close to ten times hotter (but likely the same color) as the original edition.

In the twenty months following Gillon et al.’s startling discovery that Gliese 436b is observable in transit, literally dozens of additional transiting planets have been found. New transiting hot Jupiters are now routine enough that they’re generally trotted out in batches. Reported cases of transit fever have also been on the decline, with symptoms often amounting to little more than a passing distraction.

That said, it’s been been a very long dry spell waiting for a second example of a transiting Neptune-mass planet, which makes HAT-P-11b both exciting and newsworthy. In a preprint that muscled its way to the top of today’s astro-ph mailing, Gaspar Bakos and collaborators have produced a admirably solid analysis of what’s definitely the toughest ground-based detection to date.

HAT-P-11b’s transit depth is 4.2 millimag, which is the smallest planet-produced dip yet detected by a photometric survey. (HD 149026b has a smaller transit depth, but it was discovered via the Doppler velocity method and then followed up photometrically for the transits during the time windows predicted by the orbital solution.) The HAT-P-11b analysis was further confounded by a photometrically variable parent star and ~5m/s stellar jitter on the radial velocity observations. The paper is definitely worth reading carefully.

HAT-P-11b is quite similar in mass and radius to Gliese 436b, and it’s actually somewhat larger than Neptune on both counts. When the mass and radius are compared to theoretical models, it’s clear that, like Gliese 436, it’s mostly made of heavy elements (that is, some combination of metal, rock and “ice”) with an envelope of roughly 3 Earth masses of hydrogen and helium). It’s completely dwarfed when placed next to an inflated hot Jupiter, HAT-P-9b, for instance:

Interestingly, HAT-P-11b seems to have a significant eccentricity, on the order of e=0.2. Drawn to scale with the parent star, the orbit looks like this:

The dots demarcating the orbit are not to scale. With 500 pixels of resolution, you can just barely see the planet. (I put one in front of the star, and tacked a copy onto the orbit for good measure.)

The e=0.15 eccentricity of Gliese 436b has caused a lot of consternation. For any reasonable value of the so-called tidal quality factor, Q, the circularization timescale for Gliese 436b’s orbit is considerably shorter than the age of the system. This has led to attempts (to date unfulfilled) to locate Gliese 436c. HAT-P-11b doesn’t have this problem. For a given Q, it’s circularization timescale is a full thirty times longer than that of 436b. The orbit will still be measurably eccentric even when the 0.8 solar mass primary starts to turn into a red giant.