Staying Relevant

Mildly out-of-date computer.

It has been nearly 20 years since the discovery of the planet orbiting 51 Pegasi. What followed over the rest of the late 90s were the landmark discoveries of the first eccentric giant planets at 16 Cygni B, and 70 Vir, and the first two-planet system at 47 Ursae Majoris. As new discoveries are made that push the boundary of what is known, prior ones fade into distant memory.

The public interest in these objects also varies with time. It seems odd to think it today, but in the early 1800s, 61 Cygni was wildly more popular than Alpha Centauri. This was merely because at the time, only the former’s distance had been measured, but there does seem to be a correlation between the public interest in an object and its scientific importance. Consider for example three landmark discoveries, the first planet orbiting a sun-like star, the first confirmed brown dwarf, and the first known transiting planet (with stellar hosts 51 Pegasi, Gliese 229 and HD 209458, respectively).

Trends of interest in three landmark discoveries

51 Pegasi becomes wildly famous, and rightfully so being the first of its kind known. Even today most people with a casual interest in astronomy know why 51 Pegasi is important. Gliese 229 has never really reached the prestige of 51 Pegasi — brown dwarfs just aren’t as exciting, and as time went on, interest faded. What started out as just another hot Jupiter became the most important when it was found to transit, and interest in it has continuously increased over the timeframe allowable to me by Google Ngrams.

As time went on, new planets stopped grabbing people’s attention unless they were set apart by some level of spectacularity. From memory alone, what do you know about the planet HD 290327 b? If you’re like me, absolutely nothing. Still, over time new planets and planetary systems were announced that were genuinely interesting. At the turn of the century, the first super-Earths at Gliese 876 and 55 Cancri held our attention for a while, followed by our first transiting Neptune-mass planet at Gliese 436. HD 69830 and HD 40307 gave us our first multi-planet systems made up of sub-Jovians in the mid-to-late 2000s. CoRoT broke ground with the first transiting super-Earth at the end of the decade and a multi-planet system was imaged at HR 8799.

Throughout this evolution of the kinds of things that have kept our attention, it is truly remarkable to pause and realise how numb we seem to have become to some discoveries. The discovery of Earth-sized planets now occurrs so often that it does not even raise an eyebrow anymore. The time between when a type of discovery goes from immensely exciting to just-another-day-at-arXiv seems to be only on the order of a couple years or so. It almost appears that there seems to be a sort of Moore’s Law at hand for extrasolar planet discoveries as there is with computers.

Earlier this month, the Kepler team made public about 700 new planets. Keep in mind we only just recently achieved a total of a thousand known planets. Now we’re knocking on the door of two thousand known planets. These planets are all in multi-planet systems, which is the foundation of the statistical argument used to validate their existence — a single transiting planet candidate can be any number of false positives, but having multiple candidates in a system is much harder to emulate by a non-planetary phenomenon. Many of the planets are Earth-sized and super-Earth sized, with considerable gains in transiting Neptune-sized planets.

New Kepler Planets

To further drive home the point, among the new Kepler planets are four new habitable planet candidates (at Kepler-174, Kepler-296, Kepler-298 and Kepler-309). At least that’s what they’re being called — it is my assertion that their radii are much more consistent with being low-mass, low-density “mini-Neptunes” or “micro-Jovians.” The combined interest in these four new habitable zone planets is less than half the public interest in Kepler-22 b, for example.

Much closer to home, RV studies on M dwarf stars have yielded eight new planets in the solar neighbourhood, and constrained the frequency of planets around M dwarf stars.

According to our results, M dwarfs are hosts to an abundance of low-mass planets and the occurrence rate of planets less massive than 10 M⊕ is of the order of one planet per star, possibly even greater. …

They, too, report new habitable planet candidates, but their minimum masses are, again, consistent more with being more closely reminiscent of Neptune than Earth. Regardless, it is my opinion that this is actually more interesting than the 700 new planets from Kepler. By now, we know that planets are common. The Galaxy is drowning in planets and while new planets are great for population statistics, individual planet discoveries don’t count for anywhere near what they used to. We are moving from an era of having the attention and focus on planet detection and discovery to an era of planet characterisation. We’re hungry for planets that are actually accessible to HST, Spitzer, Keck and soon(-ish) JWST for transmission spectroscopy and eclipse photometry. New planet discoveries in the solar neighbourhood count for far more than Kepler planets because the nearby planets are the ones that we have a shot at studying in-detail from direct imaging in the near future.

They also report the existence of a Neptune-mass planet in a fairly circular, 400-day orbit around Gliese 229, bringing perhaps a little more relevance and attention to a star that saw its moment of fame twenty years ago.

A Thousand Planets

Depending on where you get your information from and how much weight you lend it, we have reached a thousand known planets.

Some of the semi-official sites like exoplanet.eu and more official sites like NASA’s Exoplanet Archive show less than this number. In the case of the latter because it appears they only accept planets that have made it past peer review, which is a reasonable, if not high, standard. In the case of exoplanet.eu, while it has been a valuable asset since 1995, it has missed a few planets here and there as time has gone on (especially during a recent overhaul of the site). There’s a number of other anomalies there, but it’s a site run by a guy in his spare time so there’s a limit to how much you can expect of it. That being said, it’s still a very valuable resource.

There exists a fairly small group of people, myself shamelessly included, who keep tabs on extrasolar planet news and developments nearly religiously. The count varies from person to person, but I am not alone in asserting that there are now 1,000 known planets. By my count, we’ve passed that a couple months ago, but I’ve decided to give it more time to help cover some margin for error in the planet count.

Where does this margin of error arise? There’s a number of planets whose disposition is not very clear. They have been proposed and later disputed, but not fully disproven. There are planets that are unconfirmed, but confident enough that they can be talked about as real planets. And lastly there are Kepler candidates that have been determined to be planets, but in some cases have not even been included in a preprint on arXiv yet. As such, it is not possible for me or anyone to point to a specific planet and say “this is the thousandth known planet.”

In the big picture, humanity’s first thousand planets is only the top layer of H2O molecules of the iceburg of the planet population in the Galaxy. It is severely plagued by biases in favour of short-period and/or high-mass planets due to the nature of our detection methods and completeness of our detection surveys. We have found many hot Jupiters, but we know full well that this is a minority (less than 1% of stars have a hot Jupiter). It’s clear that small planets are more prevalent, it’s just a matter of detecting them.

Recently, it was announced that the nearby M dwarf GJ 667C hosts three super-Earths in its habitable zone. Taken together with the two habitable planet candidates at Kepler-64 and single habitable planet candidates in other systems, we have about a dozen targets for a search for life. Some of these planets are better candidates than others, and I won’t encourage any undue optimism by refraining from being outright by saying that some of them appear pretty unlikely candidates – a few of them look like we’re scraping the bottom of the barrel in desperate hope (I’m looking at you, HD 40307 g, GJ 163 c, Kepler-22 b, GJ 581 d).

Still, the fact that our first thousand planets contains at least a few planets where it’s not impossible for life to exist there is encouraging, especially when considering how biased our detection methods are against them. Combined with Kepler data that tells us that habitable planets are ubiquitous in the Galaxy, I am actually quite optimistic about the odds for there being a second biosphere in the solar neighbourhood.

We have learned so much in the first thousand planets, detected at a slow rate at first, but growing to over a hundred per year. It has taken us 20 years to detect the first thousand exoplanets. I would not be surprised if the next thousand come in only five years and feature many more habitable planet candidates.

Lastly, I have been dealing with some events in my “personal life” that have kept me busy, and so I have had less time to focous on extrasolar planet science and writing about it here. This is partly why this post doesn’t have a lot of meat to it. I look forward to writing more enlightening posts in the near future.

Probability of Transit

Transiting Planets. Credit: NASA

Transiting planets are valuable items to explore the properties of planetary atmospheres. Planet searches like Kepler that focous on fields of sky tend to reap rewards amongst dimmer stars simply because there are many more dim stars in a given patch of the sky than bright ones. Transiting planets around bright stars are of particular value, though, as the increased brightness makes the system easier to study.

Radial velocity surveys tend to monitor brighter stars since spectroscopy is even more severely limited by stellar brightness than photometry, but it is not limited to observing patches of sky – telescopes performing Doppler spectroscopy tend to observe a single object at a time due to technical and physical limitations. Radial velocity surveys are also much less sensitive to the inclination angle of a planet orbit with respect to the plane of the sky. The planet doesn’t have to transit to be spectroscopically detectable. As such, radial velocity surveys tend to generate discoveries of planet candidates with unknown inclinations and true masses, but around much brighter stars than those planets discovered by the transit method.

As such, planet candidates discovered by radial velocity, especially planet candidates in short orbital periods are excellent targets for follow-up observations to attempt to detect transits. Transiting planets that have been discovered first through radial velocity have been of great scientific interest due to their host stellar brightness and thus ease of study. If more such systems are found, it would be of great benefit to understanding extrasolar planet atmosphere. While only a hand-full of transiting planets have been discovered first through radial velocity, they all orbit bright stars and are some of the best-characterised planets outside our solar system.

The probability that a planet will transit is, as has been discussed previously, given by
$\displaystyle P_{tr} = \frac{R_*}{a}$
where a is the semi-major axis of the planet orbit. This is the distance between the centre of the star and the centre of the planet. However, due to the inclination degeneracy – the reoccurring evil villain constantly plaguing radial velocity science – the star-planet separation is unknown. Remember that the period of the RV curve gives only the orbital period of the planet. If the orbital period is held constant, increasing the mass of the planet increases the star-planet separation. An increase in the total system mass requires greater separation between the two bodies to preserve the same orbital period.

For example, if radial velocity observations of a star reveal the presence of a mp sin i = 1 ME planet candidate, but the inclination is actually extremely low such that the true mass of the companion is in the stellar regime, then because the mutual gravitational attraction between the two stars will be much greater than the mutual gravitational attraction between the star and an Earth-mass planet at the same period, the two stars must have a wider separation, otherwise their orbital period would be smaller.

Mathematically, the true semi-major axis is given by
$\displaystyle a = \left(\frac{G[M_*+M_{\text{pl}}(i)]}{4\pi^2}\right)^{1/3}T^{2/3}$
Where G is the gravitational constant, and Mpl(i) is the mass of the planet at a given inclination i, and T is the period of the system. It is worth noting that the true semi-major axis is not significantly different from the minimum semi-major axis as long as the mass of the star is much greater than the mass of the planet – which is typically the case.

The fact that the true semi-major axis is a function of the unknown inclination makes for an interesting clarification: The probability that a planet of unknown inclination will transit is not simply given by Rstar/a, but is only approximated by it. If we assume that the distribution of planet masses is uniform (and extending through into the brown dwarf mass regime), then you would expect a planet with a minimum mass equal to Earth to have a much greater chance of being a bona-fide planet than a planet with a minimum-mass of 10 MJ, simply because there is a greater range of inclinations the former planet can be while still remaining in the planetary mass regime. Taking this a step further, even if both the Earth-mass planet candidate and the 10 Jupiter-mass planet candidate have the same orbital period, the probability that the latter planet transits ends up being less than the Earth-mass planet simply because of its high mass. Since its inclination is unknown, the probability that its mass is so high that the true semi-major axis is noticeably larger than the minimum semi-major axis is much higher, resulting in a likely lower transit probability.

Except it turns out that the mass distribution of planets and brown dwarfs isn’t constant. Earth-sized planets are significantly more common than Jupiter-sized planets, and super-Jupiters appear rare. It isn’t clear yet what the mass distribution planets actually is, with significant uncertainty in the sub-Neptune regime, but it is clear that for a highly accurate estimate of the transit probability, the inclination distribution cannot be thought of as completely random as it is fundamentally tied to the planet mass distribution.

Planet Mass Distribution given by Ida & Lin (Left) and Mordasini (Right)

Consider the case of a super-Jovian planet candidate, perhaps with a minimum mass of 7 or 8 Jupiter-masses. Because a significant fraction of physically allowable inclinations would place the true mass planet into a mass regime that is in reality sparsely populated, it is less likely that the planet candidate’s orbit is in those inclinations. It is thus more likely that the planet candidate’s orbit is edge-on than would be expected from the probability function of randomly oriented orbits. As such, the transit probability of a super-Jovian planet is actually boosted by ~20 – 50% over what you would expect from Ptr = Rstar/a. If this is the case, then we would expect to find an excess in the fraction of transiting planets in this mass regime then would be expected purely from the standard transit probability function. Indeed this is what we see.

Candidate planets with masses in the terrestrial planet regime are similarly affected, with broadened transit probabilies owing to the fact that terrestrial planets are more common than higher mass planets, arguing in favour of a higher inclination than the random inclination distribution function.

On the other hand, planet or brown dwarf candidates of minimum masses in the most sparsely populated region of the mass distribution are unlikely to truly have that mass. They are quite likely in orbits with low inclinations and with much higher true masses. The transit probability for companion candidates with minimum masses in this mass regime are actually reduced from the standard transit probability function.

Geometric and a posteriori transit probabilities

In the table above, taken from this preprint, we see that the geometric transit probability, Ptr,0, can be much less than the a posteriori transit probability, Ptr. The transit probability for 55 Cnc e, for example, jumps up from 28% to 36%. With these higher a posteriori transit probabilities, these short-period low-mass planets should be followed-up for transits. If transits are found, it would be of significant benefit to the extrasolar planet field.

In summary, there are various additional effects that can cause the a posteriori transit probability to be significantly different from the geometric transit probability. Planets with only minimum masses known can be more accurately assigned a transit probability when taking into account the uneven planetary mass distribution. Low-mass planets and super-Jupiters are more likely to transit than their geometric transit probability because a significant range of the inclination space is consumed by planets of masses that are simply rare. These planet candidates are more promising targets for transit follow-up than, for example, Jupiter-mass planets or intermediate-mass brown dwarfs.

2012 Review

An Earth-mass planet orbiting Alpha Centauri B. Credit:ESO

2012 brought us yet another remarkable year of extrasolar planet science. While the planet catch for 2012 was a little less than last year’s, the quality and importance of planets revealed this year was amazing. By far the most major results have been the discovery of an ~Earth-mass planetary companion orbiting the secondary component of the nearest star system to our own, Alpha Centauri (see here), and evidence for a system of planets around the nearby star Tau Ceti (see here). I hesitate to draw conclusions from a small amount of data, but the discovery of a terrestrial planet at none other than our nearest neighbour seems to really emphasize the point that terrestrial planets are likely as common as dirt.

A nice system of planets was reported at Gliese 676A consisting of super-Earths and Jovian planets, HATnet and SuperWASP produced more hot Jupiters, and interestingly, a couple sub-Earths may have been found around the nearby star Gliese 436. Spitzer provided us with the first detection of thermal radiation from a super-Earth (see here). A pair of M giants also became the first known to have planets, with planets reported around HD 208527 and HD 220074.

Circumbinary planets were announced around RR Cae, NSVS 14256825, Kepler-34 and Kepler-35 and Kepler-38, which is notable as the first Neptune-sized circumbinary planet.

Kepler results picked up en masse this year. At first it started out nice and slow, with small groups of planets being announced in batches (See here, here, here and here), followed by dozens and dozens of planets.

Interesting Kepler results included Kepler-64, the first quadruple-star system with a planet. The planet is a circumbinary planet, no less. But easily the most important circumbinary planet find was Kepler-47, the first transiting multi-planet circumbinary system. Multi-planet circumbinary systems have been found before but this is the first to have multiple planets transiting. This allows not only for their existence to be much more certain (non-transiting circumbinary planets still suffer from the mass-inclination degeneracy), but allows us to test for coplanarity. The Kepler-47 system demonstrates conclusively that short-period binary stars can host full systems of planets. Another pair of planets with very close orbits to each other, yet very dissimilar densities were reported at Kepler-36. The orbits of the planets in the Kepler-30 system were shown to be well-aligned with their host star’s equator, showing us that systems of planets are, like ours, often neatly arranged and not chaotically scattered.

Good news and bad news about the Kepler spacecraft. The good news is that the mission is extended for another three years. The bad news is that unfortunately, a reaction wheel on the Kepler spacecraft failed, and the mission’s continued usefulness now rests on all of the other reaction wheels remaining operational.

Kepler also unveiled a system of three sub-Earth planets huddled around a dim red dwarf, Kepler-42, which is very similar to Barnard’s Star, as well as a possible small terrestrial planet being evaporated away due to the heat from its star (see here). One of these three planets is Mars-sized(!).

We gained more evidence that the Galaxy is just drowning in planets both from continued Kepler results, HARPS results, and from gravitational microlensing data. Kepler showed us that hot Jupiter systems are frequently lacking in additional planets.

Last but not least, habitable planet candidates were reported around Gliese 163 and HD 40307, with unconfirmed habitable planet candidates reported at Tau Ceti and Gliese 667 C – with two more planets possibly occupying the star’s habitable zone. If GJ 667 Ce is confirmed, then it would be the most promising habitable zone candidate to date, based on its low mass.

At the end of 2011, I gave some wild guesses as to how the extrasolar planet landscape would look like at the end of 2012. Here we are and how have those predictions held up?

The Extrasolar Planets Encyclopaedia lists 854 planets as of the time of this writing, however it is missing quite a few. My own count has us at 899 planets.

• The discovery of a ring system around a transiting planet

There are hints of ring systems (or perhaps rather circumplanetary disk systems) around Fomalhaut b, β Pictoris b, and 1SWASP J140747.93-394542.6 b (see here) but none of these are confirmed. So I’m calling it a missed prediction.

• More low-mass planets in the habitable zone from both radial velocity and transit

Two new habitable planet candidates from radial velocity, none from transit.

• Confirmation of obvious extrasolar planet atmospheric variability (cloud rotations, etc).

I was counting on continued monitoring of the HR 8799 planets to search for atmospheric variability, but it simply didn’t happen (or rather, if it did happen, the results are still pending). So I’m calling this a miss.

2013 could be a very interesting year, especially for Kepler. It seems we are on the verge of finding a true Earth analogue. The detection rate of candidate habitable planets is picking up and we’re really starting to get a list of targets to follow-up in the next decade. Here’s some more brave guesses for the end of 2013:

• 1200 Confirmed planets and planet candidates
• A satellite of an extrasolar planet (an “exomoon”)
• A confirmed ring system around an extrasolar planet
• Phase curve mapping of a sub-Jovian planet

Making Waves

Photo by Mila Zinkova, 2007

We looked at how the Doppler effect can tell you about the radial velocity of a star (here), and how you can use this information to detect an orbiting companion (here). Let’s now look a bit more at some of the later parts of this process. You’ve already finished your observations, you’re done with the telescope time and you have a collection of data points. What do you do with them?

The first thing you will want to do is to create a model that fits the data. This is done using the equations in the post referred to in the above paragraph. The model will simply manifest itself as a plot of the radial velocity behaviour you would expect for a planet with a given mass and orbit. The closer your model fits the data, the more likely your model is a good representation of reality. For example, let’s look at radial velocity data of the giant star Iota Draconis (Credit: ESO).

Iota Draconis (HR 810) Radial Velocity Data

The individual points are the actual data, whereas the sine wave is the model of a 2.26 MJ planet with a 320-day period. We see that the proposed model fits the data well (being very close to the data), and is thus likely representative of reality. There are a few stray points and the model is not a perfect match to the data, but the data points can be prone to intrinsic noise, such as flares or granulation on the star’s surface, or instrumental noise with the spectrometer.

We see that the radial velocity data set covers several orbits of the planet around the star. As the planet continues to orbit the star (as it will do so for aeons) and the radial velocity data continues to come in (as we would hope would occur, more data is more opportunities to discover something new), the graph could become extremely long and complicated. For example, consider a set of radial velocity data for HD 192263 (source).

Unfolded HD 192263 RV data

What we could do would be to fold the entire data set by a certain period of time, specifically the orbital period (taking the time of the measurement modulo the orbital period). Since the model is periodic, that is, repeating each orbital period, the data points (provided they do have the same periodicity that you’re folding by) will fold up into a nice, neat, singular sine curve.

HD 192263 Folded RV data

Here it is quite convincing that the model well fits the data. But how do we quantify how good of a fit to the data our model is? A value exists called $\chi^2$(read “chi squared”).

$\displaystyle \chi^2 = \sum \frac{(O-C)^2}{\sigma^2}$

That is, calculate the difference between each observed data point and the model (or “observed minus calculated”), divided by $\sigma^2$, which represents the variance of the data. Then total up each of these. The higher the $\chi^2$, the poorer the model fits the data. The variance, $\sigma^2$,can be determined with

$\displaystyle \sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(v_i - \bar{v})^2$

Which is simply an average of the differences of each point, $v$, from the average values of the points $\bar{v}$. To put this in other terms (and this is more of a discussion about general statistics than it is specific to radial velocity data), all of the radial velocity measurements can be averaged together. The average of the differences of each point from that average radial velocity value is the variance of the data, $\sigma^2$, and is a representation of how scattered the data is. Our $\chi^2$value is simply the total of each point’s difference from our model, while dividing it by the variance to compensate for highly scattered data. Models that are not rigidly affixed to the data points (high total OC) are given more forgiveness of the data itself has a lot of variance ($\sigma^2$) anyway. If there is low variance, then only models that are closer to the datapoints (low OC) are treated well, or given a low $\chi^2$ value.

We can obtain another value called the “reduced chi-squared” which takes into account the number of freely adjustable variables (or, the “degrees of freedom”). This is obtained simply by dividing $\chi^2$by the degrees of freedom, $d$.

$\displaystyle \chi^2_{red} = \frac{1}{Nd} \sum \frac{(O-C)^2}{\sigma^2} = \frac{\chi^2}{d}$

The need to do this stems from the fact that a quantitative analysis of data must be undertaken to seriously distinguish between two models whose goodness of fit to the data may not be as easily discriminated between in our “chi-by-eye” technique of just looking at them.

Large reduced chi-squared values indicate a poor model fit, and a value of 1 is typically sought but can be extremely difficult or impossible to achieve due to instrumental or intrinsic noise that are out of one’s ability to control. Values of the reduced chi squared that are less than 1 are typically a sign that you are “over-fitting” the model to the data. This can result from trying too hard to fit the model to noise, or overestimating the variance.

HD192263 RV Residuals

Above are the residuals to the HD 192263 dataset. This is keeping the x-axis (the time of the measurement) and plotting the O-C on the y-axis, instead of the radial velocity value for that data point. In this example, there is a trend in the data that has been modelled out which could potentially be due to a second orbiting companion that is far enough away that only a very small section of its orbit has been observed.

Other than the trend, there is no apparent structure in the residuals, and so there is no reason to suspect the existence of additional orbiting bodies.

One can calculate the standard deviation, or variance, of the residuals to get a measure of how scattered the residuals are. This is typically represented with $\sigma$ or written as ‘rms’ in literature, and is an algebraically obvious transformation of one of the above equations,

$\displaystyle \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N}(v_i - \bar{v})^2}$

The radial velocity jitter of a star scales with both its mass and chromospheric activity. A quiet M dwarf may have a jitter as low as 2 to 3 metres per second, whereas a sun-like star may have a jitter on the order of 3 – 5 metres per second. F-type stars have jitters close to 10 metres per second, and the jitter sky-rockets from there, where A-type stars are nearly unusable for using radial velocity to detect extrasolar planets. As of the time of this writing, no planets have ever been discovered around an A-type star through the use of the radial velocity method. So there is an amount of variance in the residuals we should expect to have given the star we’re studying. If the variance is much higher than expected for the star, then it could be evidence of either unusually high stellar chromospheric activity, or as-yet undetected planets with small amplitude signals.

The systems we have looked at thus far throughout this blog in the context of radial velocity have been systems whose radial velocity data graphs are dominated by a single periodicity. Indeed, our own solar system is like this, with Jupiter dominating the sun’s radial velocity profile. But not all planetary systems come so conveniently organised.

I leave you with an image of how daunting this can be and why we are all thankful that computers are here to help us: An image of the 2008 three-planet fit for HD 40307 (since discovered to have three more planets).

HD 40307 Three-Planet Fit

Update (15 Apr 2013): I have corrected a math error in the equation for variance and standard deviation. In both cases the $(v_i-\bar{v})$ term should have been squared.