## Super-Earths and Mini-Neptunes

Low-Mass Habitable Zone Planets (artist images)

Our Solar System did not prepare us for what we would discover orbiting other stars. Instead, it told us that planets fall into neat categories: Gas giants made mostly of hydrogen and helium (of which Jupiter and Saturn are the archetypes), ice giants made mostly of water (for which Uranus and Neptune are representatives), and solid terrestrial planets with comparatively thin atmospheres — that would be the planets of the inner solar system and the one right under your feet). Since the discovery of thousands of planets orbiting other stars, and the measurement of their masses and densities, it has become clear that not all planets fit into this paradigm. Significantly, unless rocky worlds have an optimistically high abundance, what may be the most abundant type of planet in the Galaxy is a sort of mix between low-density, volatile-rich Neptune-like planets and rocky terrestrial planets. The Solar System features no such planet — after Earth, the next most massive planet is Uranus at ~14.5 times as massive. A casual look at the entirety of discovered transiting planet candidates discovered by Kepler reveals the magnitude of this problem.

While Kepler is no longer observing its original field, the massive amount of data can still be combed through to reveal new planet candidates. Here, previously discovered planet candidates are blue dots, and newly announced planet candidates are yellow. A few things are noteworthy. Firstly, the overwhelming majority of the newly discovered planet candidates have reasonably long orbital periods. This can be expected as shorter period planets have been detectable in the existing data for longer, and have had time to be spotted already. Secondly, and not really the point of this post… they’re still finding warm Jupiters in the data? Wow! What’s up with that? I would have thought those would have been found long ago.

With the obvious caveat that lower regions of that diagram feature harder to detect planets leading to that part being less populated than would be the case if all planets were detected, it would appear that there is a continuous abundance of planets from Earth-sized to Neptune-sized. While radius and mass may only be loosely related, it may also be that there is a continuous abundance of planets from Earth-mass to Neptune-mass, as well. Not having an example of such an intermediate planet in the Solar System, we really don’t know what to expect for what these planets are composed of. As such we began to call them (sometimes interchangeably) super-Earths or Mini-Neptunes. Are they enormous balls of rock with Earth-like composition extending up toward maybe 10 Me? Are they dominated by mass by a rocky core with a thick but comparatively low-mass hydrogen envelope? Do they have some fraction of rock, water and gas? Are they mostly entirely water with a minimal gas envelope? Answering this question would require some constraints on the masses of these planets, as it would allow one to know their density.

The first data point was CoRoT-7 b, the first transiting super-Earth — discovered before Kepler. The host star is very active, leading to a lot of disagreement in the literature about its mass, but further work seems to have settled on a rocky composition for the planet with ~5 Me. Great! Next data point was the transiting super-Earth orbiting GJ 1214, a ~6.5 Me planet with a much lower density, which is too low to be explained by even a pure water composition. This is decidedly not Earth-like. Additional measurements by highly precise spectrometers (namely HARPS and SOPHIE) of Kepler discovered planets have allowed for more data to be filled in, and an interesting trend can be seen.

Mass-Radius Diagram of Extrasolar Planets with RV-Measured Masses

Interestingly, planets less than ~1.6 Earth-radii seem to have not only solid, but Earth-like compositions. It’s worth noting that only planets where the mass measurement is acquired through Doppler spectroscopy are shown here. Planets like the Kepler-11 family where the masses have been derived by transit timing variations are not shown. If these planets are added, the adherence to the Earth-like composition is much less strict. This may imply that planets which have masses measurable by detectable transit timing variations have had a different formation history and therefore a much lower density. Further data will be very useful in addressing this issue.

On a somewhat unrelated topic, several new habitable planet candidates have been validated by ruling out astrophysical false positives. Among them is Kepler-442 b, which appears to me to be a more promising habitable planet candidate than even Kepler-186 f. Some newly discovered but not yet validated habitable planet candidates have been found as well, including one that appears to be a near Earth-twin.

New Kepler habitable planet candidates

## 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.

## 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.

## The Real Ones

On the last post, we looked at recovering a periodic signal from a radial velocity plot and interpreting it as a planet. Now let’s look at a few of the complications involved in this.

A powerful statistical tool used to get an idea of what kind of periodic signals are in your radial velocity data set is a Lomb–Scargle periodogram (the mathematical details for the interested reader may be found here, but it is sufficiently complex to warrant skipping over in the interests of maintaining reader attention and reasonable post length). In the interests of brevity, further references to the Lomb-Scargle periodogram will be shortened to simply “periodogram.”

The purpose of this periodogram is to give an indication of how likely an arbitrary periodicity is in a data set whose data points need not be equally spaced (as is frequently the case in astronomy for a variety of reasons). Periodicities that are strongly represented in the data are assigned a higher “power,” where periodicities that are not present or only weakly present are given a lower power.

Let’s look at an example using a radial velocity data set for BD-08 2823 (source) If we calculate a periodogram for the data set, we come up with this

BD-08 2823 RV Data Periodogram

The dashed line represents a 0.1% false alarm probability (FAP). A clear, obvious peak is seen at 1 day, 230 days and ~700 days, implying that periodicities of 1 day, 230 days, and ~700 days are present in the data. Creating a one-planet model with a Saturn-mass planet at 238 days produces a nice fit. After subtracting this signal from the data, we’re left with the residuals. Now we may run up a periodogram of the residuals and see what’s left in the data.

BD-08 2823 Periodogram of Residuals

We see three noteworthy things. First and foremost is the emergence of a new peak in the periodogram that was not strongly present before at 5.6 days. We also see that the peak at 1 day remains. Lastly we see that the peak toward 700 days has weakened and moved further out. It would seem to suggest the 700-day signal is perhaps not real, or was an artifact of the 238-day signal.

Why was the 5.6-day signal not present in the first periodogram? The answer may lie in it’s mass: the planet has a mere 14 Earth-masses. It’s RV signal is completely dominated by the Saturn-mass planet. The giant planet forces the shape of the RV diagram and the signal of the second planet is just dragged along, superimposed on the larger signal.

On the radial velocity data plot, the two-planet fit we have come to looks like this:

BD-08 2823 Two-Planet Fit

It is important to realise that the obvious sine curve is not necessarily a bold line, but there is a second periodicity in there going up and down frantically, once every 5.6 days, compared to the Saturn-mass planet, at 237 days.

The fit has a reduced chi-squared of χ2 = 3.2, and a scatter of σO-C = 4.3 m s-1. There’s no obvious structure to the residuals and the scatter is not terribly bad, so any new signals will likely indicate planets of low mass. Let’s check in on the periodogram of the residuals to the two-planet fit and see what may be left in the data.

Periodogram of Residuals to 2-Planet Fit

That signal out toward a thousand days is stubbornly refusing to go away, despite a low χ2. It may either not be real, or it may be indicative of a low-amplitude signal with a rather long period.

Also noteworthy is that the periodicity at one day continues to exist, rather strongly. This periodicity is what’s known as an alias. Because the telescope observes only at night, the observations are roughly evenly spaced – there are (on average) twelve hour gaps between each data point. Therefore a sine curve with a period of 24 hours can be made to fit the data. To illustrate this, consider this (completely made up) data set:

Fake Signal

There’s no doubt that the data is well-fitted by the sine curve, but there is no real evidence that the periodicity proposed by it arises from a real, physical origin. What’s more, a sine curve with half this period could also equally well fit the data. So could a sine curve with a third of this period, and so on. There are mathematically an infinite number of aliases at ever-shortening periods that can be fit to this data.

Generally, if you observe a system with a frequency of $f_o$, and there exists a true signal with a frequency of $f_t$, then aliases will exist at frequencies $f_{t+i} * f_o$, where i is an integer.

Therefore we see that these aliases are caused by the sampling rate. If we could get data between the data points already available, if we could double our observation frequency, we could break this degeneracy. But the problem for telescopes on Earth is that the star is not actually up in the sky more than half the day, and a given portion of the time it is up could be during daylight hours. Therefore the radial velocity data sets of most stars can be plagued with short-period aliases since there is typically a small window of a few hours to observe any given star. It must be noted that as the seasons change and the stars are in different places in the sky at night, that window of availability will shift around a bit, allowing one some leverage in breaking these degeneracies. Ultimately, telescopes in multiple locations around the world (or one in space) would sufficiently break these degeneracies.

A real example of aliases exist in this example from an Alpha Arietis data set. In this case, the alias is not nearly so straightforward. Two signals of periods 0.445 days and 0.571 days can be modelled to fit the data.

Alpha Arietis RV Alias

So which of these two signals correspond to an actual planet? It turns out neither of them do: these radial velocity variations are caused by pulsations on the star – contracting and expansion of the star produces Keplerian-like signals in radial velocity data, too. That’s yet another thing to watch out for. This can be detected with simultaneous photometry of the star. If there is a photometric periodicity that is equivalent to your radial velocity periodicity, avoid claiming a planet at this period as if your academic credibility depends on it.

Additional observations could easily break this degeneracy, provided they are planned at times where the two signals do not overlap.

We see therefore that it is important to keep in mind that a low FAP speaks only to whether or not the signal is real, and not where or what it actually came from. The one-day periodicity is surely present in the data, but it is not of physical origin. It can also be extremely hard to tell whether or not a signal at a given period is actually an alias of another, more real period. There are times when the peak of an alias in the periodogram can be higher than the actual, real period. For reasons that include these, radial velocity fits must be considered fairly preliminary. New data may provide drastic revisions to the orbital periods of proposed planets if signals are exposed to be aliases.

Confusion over aliases have occurred before in literature. HD 156668 b and 55 Cnc e have both had their orbital periods considerably revised after it was realised that their published periods were, in fact, aliases. In the case of 55 Cnc e, the new, de-aliased orbital period ended up being vindicated after transits were detected). The GJ 581 data set, for example, is severely limited by sampling aliases that have spawned controversies over the possible existence of additional planets in that system.

In summary, periodograms are a useful tool to provide the user with a starting point when fitting Keplerian signals to radial velocity data, but they cannot distinguish real signals from aliases. Many observations with a diverse sampling rate are necessary to disentangle aliases from true planetary signals. Ultimately, a cautious approach to fitting signals to radial velocity data works best.

## 2011 Review

Arguably the most important discovery of 2011: Earth-sized exoplanets

2011 was a banner year for extrasolar planet science with the Kepler results really beginning to come in. Among the more interesting:

Kepler results:

• Kepler-10: Kepler’s first rocky super-Earth, with a transiting Neptune further out.
• Kepler-11: A system of six transiting super-Earths with anomalously low density.
• Kepler-14: A massive hot Jupiter in a binary system.
• Kepler-16: The first transiting circumbinary planet around an eclipsing binary star.
• Kepler-18: A system of three planets: A super-Earth and two inflated Neptunes in a 2:1 resonance. Very similar to Kepler-9 but scaled down in masses.
• Kepler-19: A transiting sub-Jovian planet and the first case of the discovery of a second planet through transit timing variations in the transiting planet.
• Kepler-20: A system of five planets, two of which are ~Earth-sized.
• Kepler-21: A transiting super-Earth around a bright (V = 8.27) star.
• Kepler-22: A transiting “mini-Neptune” in the habitable zone, and the first transiting planet in the habitable zone of any star.
• KOI-423: First transiting planet around a subgiant star.
• KOI-730: A remarkable system of four planets in a 1:2:4:8 resonance.
• KOI-55: What appears to be two remnant cores of gas giants engulfed by their parent star during its red giant phase.
HARPS results:

• 82 Eri: Three low mass planets only a few times the mass of Earth.
• HD 136352: Three super-Earth/sub-Jovian planets.
• HD 39194: Three super-Earths.
• HD 134606: Three Neptunes.
• HD 215152: Two lower-mass super-Earths.
• Gliese 667 C: A second planet of a few Earth-masses in the habitable zone.
• HD 85512: A super-Earth on the inner edge of the habitable zone.

On the orbital dynamics front, in January, it was found that the HD 37124 system, which hosts three intermediate period gas giant planets of roughly equal mass, may have a 2:1 resonance for the orbit of two of its planets. Furthermore, the planet candidate orbiting Rho Coronae Borealis, one of the first planet candidates, was proven to have a true mass far outside the planetary regime. The recovery of the planets of HR 8799 in old HST data has permitted the architecture of the system to be much more constrained.

A planet around a naked-eye giant star Alpha Arietis was reported in April. Later, in August, it was revealed that a new planetary mass object has been found orbiting a pulsar. On the subject of post-main sequence stars, a candidate planet was imaged orbiting a white dwarf. On the other side of the main sequence, a planet had been found in the late stages of formation at LkCa 15.

Gravitational microlensing provided us some constraints on the abundance of rogue giant planets as well as another cold super-Earth. It turns out that rogue giant planets may be twice as frequent as main sequence stars.

Several planets around eclipsing binaries were found this year, including planets at UZ For, HU Aqr, Kepler-16, and NY Vir.

Without a doubt, one of the most exciting stories of 2011 is the discovery that one of the first known super-Earths, the innermost planet at 55 Cancri, transits its star. This is the brightest star known to have a transiting planet, and it will prove very useful for the study of these kinds of planets. While it was initially thought that the planet was rocky and iron-rich, later observations suggest that the planet must have a significant envelope of volatiles.

In summary, 2011 has been an astounding year. The focus has shifted away from gas giant planets to sub-Jovian planets — Neptunes, mini-Neptunes and super-Earths. Below is a graph that shows this year’s total planet catch compared to previous years. One thing is clear: Not only is the galaxy full of planets, but we can look forward to seeing a huge number more in the near future.

Now for some fun, some predictions for what we might have by the end of 2012:

• 1,000 Planets on the Extrasolar Planets Encyclopaedia
• The discovery of a ring system around a transiting planet
• More low-mass planets in the habitable zone from both radial velocity and transit
• Confirmation of obvious extrasolar planet atmospheric variability (cloud rotations, etc).