Bayesian Methods for Exoplanet Science

Exoplanet research is carried out at the limits of the capabilities of current telescopes and instruments. The studied signals are weak, and often embedded in complex systematics from instrumental, telluric, and astrophysical sources. Combining repeated observations of periodic events, simultaneous observations with multiple telescopes, different observation techniques, and existing information from theory and prior research can help to disentangle the systematics from the planetary signals, and offers synergistic advantages over analysing observations separately. Bayesian inference provides a self-consistent statistical framework that addresses both the necessity for complex systematics models, and the need to combine prior information and heterogeneous observations. This chapter offers a brief introduction to Bayesian inference in the context of exoplanet research, with focus on time series analysis, and finishes with an overview of a set of freely available programming libraries.Comment: Invited revie

Constellations of co-orbital planets: horseshoe dynamics, long-term stability, transit timing variations, and potential as SETI beacons

Co-orbital systems contain two or more bodies sharing the same orbit around a planet or star. The best-known flavors of co-orbital systems are tadpoles (in which two bodies' angular separations oscillate about the L4/L5 Lagrange points $60^\circ$ apart) and horseshoes (with two bodies periodically exchanging orbital energy to trace out a horseshoe shape in a co-rotating frame). Here, we use N-body simulations to explore the parameter space of many-planet horseshoe systems. We show that up to 24 equal-mass, Earth-mass planets can share the same orbit at 1 au, following a complex pattern in which neighboring planets undergo horseshoe oscillations. We explore the dynamics of horseshoe constellations, and show that they can remain stable for billions of years and even persist through their stars' post-main sequence evolution. With sufficient observations, they can be identified through their large-amplitude, correlated transit timing variations. Given their longevity and exotic orbital architectures, horseshoe constellations may represent potential SETI beacons.Comment: 10 pages, 10 figures. Published in MNRAS. YouTube playlist with animations of horseshoe constellation systems here: https://www.youtube.com/playlist?list=PLelMZVM3ka3F335LGLxkxrD1ieiLJYQ5N . Blog post here: https://planetplanet.net/2023/04/20/constellations-of-co-orbital-planets

Survival and dynamics of rings of co-orbital planets under perturbations

In co-orbital planetary systems, two or more planets share the same orbit around their star. Here we test the dynamical stability of co-orbital rings of planets perturbed by outside forces. We test two setups: i) 'stationary' rings of planets that, when unperturbed, remain equally-spaced along their orbit; and ii) horseshoe constellation systems, in which planets are continually undergoing horseshoe librations with their immediate neighbors. We show that a single rogue planet crossing the planets' orbit more massive than a few lunar masses (0.01-0.04 Earth masses) systematically disrupts a co-orbital ring of 6, 9, 18, or 42 Earth-mass planets located at 1 au. Stationary rings are more resistant to perturbations than horseshoe constellations, yet when perturbed they can transform into stable horseshoe constellation systems. Given sufficient time, any co-orbital ring system will be perturbed into either becoming a horseshoe constellation or complete destabilization.Comment: 5 pages, 4 figures. Re-submitted to MNRAS. Blog post about co-orbital constellations here: https://planetplanet.net/2023/04/20/constellations-of-co-orbital-planets

Mathematical encoding within multi-resonant planetary systems as SETI beacons

How might an advanced alien civilization manipulate the orbits within a planetary system to create a durable signpost that communicates its existence? While it is still debated whether such a purposeful advertisement would be prudent and wise, we propose that mean-motion resonances between neighboring planets -- with orbital periods that form integer ratios -- could in principle be used to encode simple sequences that one would not expect to form in nature. In this Letter we build four multi-resonant planetary systems and test their long-term orbital stability. The four systems each contain 6 or 7 planets and consist of: (i) consecutive integers from 1 to 6; (ii) prime numbers from 2 to 11; (iii) the Fibonacci sequence from 1 to 13; and (iv) the Lazy Caterer sequence from 1 to 16. We built each system using N-body simulations with artificial migration forces. We evaluated the stability of each system over the full 10 Gyr integration of the Sun's main sequence phase. We then tested the stability of these systems for an additional 10 Gyr, during and after post-main sequence evolution of the central stars (assumed to be Sun-like) to their final, white dwarf phase. The only system that was destabilized was the consecutive integer sequence (system i). The other three sequences therefore represent potential SETI beacons

SOPHIE velocimetry of Kepler transit candidates. V. The three hot Jupiters KOI-135b, KOI-204b and KOI-203b (alias Kepler-17b)

We report the discovery of two new transiting hot Jupiters, KOI-135b and KOI-204b, that were previously identified as planetary candidates by Borucki et al. 2011, and, independently of the Kepler team, confirm the planetary nature of Kepler-17b, recently announced by Desert et al. 2011. Radial-velocity measurements, taken with the SOPHIE spectrograph at the OHP, and Kepler photometry (Q1 and Q2 data) were used to derive the orbital, stellar and planetary parameters. KOI-135b and KOI-204b orbit their parent stars in 3.02 and 3.25 days, respectively. They have approximately the same radius, Rp=1.20+/-0.06 R_jup and 1.24+/-0.07 R_jup, but different masses Mp=3.23+/-0.19 M_jup and 1.02+/-0.07 M_jup. As a consequence, their bulk densities differ by a factor of four, rho_p=2.33+/-0.36 g.cm^-3 (KOI-135b) and 0.65+/-0.12 g.cm-3 (KOI-204b). Our SOPHIE spectra of Kepler-17b, used both to measure the radial-velocity variations and determine the atmospheric parameters of the host star, allow us to refine the characterisation of the planetary system. In particular we found the radial-velocity semi-amplitude and the stellar mass to be respectively slightly smaller and larger than Desert et al. These two quantities, however, compensate and lead to a planetary mass fully consistent with Desert et al.: our analysis gives Mp=2.47+/-0.10 M_jup and Rp=1.33+/-0.04 R_jup. We found evidence for a younger age of this planetary system, t<1.8 Gyr, which is supported by both evolutionary tracks and gyrochronology. Finally, we confirm the detection of the optical secondary eclipse and found also the brightness phase variation with the Q1 and Q2 Kepler data. The latter indicates a low redistribution of stellar heat to the night side (<16% at 1-sigma), if the optical planetary occultation comes entirely from thermal flux. The geometric albedo is A_g<0.12 (1-sigma).Comment: submitted to Astronomy and Astrophysic

Methane in the atmosphere of the transiting hot Neptune GJ436b?

We present an analysis of seven primary transit observations of the hot Neptune GJ436b at 3.6, 4.5 and $8~\mu$m obtained with the Infrared Array Camera (IRAC) on the Spitzer Space Telescope. After correcting for systematic effects, we fitted the light curves using the Markov Chain Monte Carlo technique. Combining these new data with the EPOXI, HST and ground-based $V, I, H$ and $K_s$ published observations, the range $0.5-10~\mu$m can be covered. Due to the low level of activity of GJ436, the effect of starspots on the combination of transits at different epochs is negligible at the accuracy of the dataset. Representative climate models were calculated by using a three-dimensional, pseudo-spectral general circulation model with idealised thermal forcing. Simulated transit spectra of GJ436b were generated using line-by-line radiative transfer models including the opacities of the molecular species expected to be present in such a planetary atmosphere. A new, ab-initio calculated, linelist for hot ammonia has been used for the first time. The photometric data observed at multiple wavelengths can be interpreted with methane being the dominant absorption after molecular hydrogen, possibly with minor contributions from ammonia, water and other molecules. No clear evidence of carbon monoxide and dioxide is found from transit photometry. We discuss this result in the light of a recent paper where photochemical disequilibrium is hypothesised to interpret secondary transit photometric data. We show that the emission photometric data are not incompatible with the presence of abundant methane, but further spectroscopic data are desirable to confirm this scenario.Comment: 19 pages, 10 figures, 1 table, Astrophysical Journal in pres

Water in HD 209458b's atmosphere from 3.6 - 8 microns IRAC photometric observations in primary transit

The hot Jupiter HD 209458b was observed during primary transit at 3.6, 4.5, 5.8 and 8.0 microns using the Infrared Array Camera (IRAC) on the Spitzer Space Telescope. We detail here the procedures we adopted to correct for the systematic trends present in the IRAC data. The light curves were fitted including limb darkening effects and fitted using Markov Chain Monte Carlo and prayer-bead Monte Carlo techniques, finding almost identical results. The final depth measurements obtained by a combined Markov Chain Monte Carlo fit are at 3.6 microns, 1.469 +- 0.013 % and 1.448 +- 0.013 %; at 4.5 microns, 1.478 +- 0.017 % ; at 5.8 microns, 1.549 +- 0.015 % and at 8.0 microns 1.535 +- 0.011 %. Our results clearly indicate the presence of water in the planetary atmosphere. Our broad band photometric measurements with IRAC prevent us from determining the additional presence of other other molecules such as CO, CO2 and methane for which spectroscopy is needed. While water vapour with a mixing ratio of 10^-4-10^-3 combined with thermal profiles retrieved from the day-side may provide a very good fit to our observations, this data set alone is unable to resolve completely the degeneracy between water abundance and atmospheric thermal profile.Comment: 14 pages, 6 tables, 10 figures, Accepted for publication in MNRA

Exoplanets transmission spectroscopy: accounting for eccentricity and longitude of periastron. Superwinds in the upper atmosphere of HD209458b?

Context: Several studies have so far placed useful constraints on planetary atmospheric properties using transmission spectrsocopy, and in the case of HD209458b even the radial velocity of the planet during the transit event has been reconstructed opening a new range of possibilities. AIMS. In this contribution we highlight the importance to account for the orbital eccentricity and longitude of periastron of the planetary orbit to accurately interpret the measured planetary radial velocity during the transit. Methods: We calculate the radial velocity of a transiting planet in an eccentric orbit. Given the larger orbital speed of planets with respect to their stellar companions even small eccentricities can result in detectable blue or redshift radial velocity offsets during the transit with respect to the systemic velocity, the exact value depending also on the longitude of the periastron of the planetary orbit. For an hot-jupiter planet, an eccentricity of only e=0.01 can produce a radial velocity offset of the order of the km/s. Conclusions: We propose an alternative interpretation of the recently claimed radial velocity blueshift (~2 km/s) of the planetary spectral lines of HD209458b which implies that the orbit of this system is not exactly circular. In this case, the longitude of the periastron of the stellar orbit is most likely confined in the first quadrant (and that one of the planet in the third quadrant). We highlight that transmission spectroscopy allows not only to study the compositional properties of planetary atmospheres, but also to refine their orbital parameters and that any conclusion regarding the presence of windflows on planetary surfaces coming from transmission spectroscopy measurements requires precise known orbital parameters from RV.Comment: Accepted for publication in A&A Letter

HAT-P-26b: A Low-Density Neptune-Mass Planet Transiting a K Star

We report the discovery of HAT-P-26b, a transiting extrasolar planet orbiting the moderately bright V=11.744 K1 dwarf star GSC 0320-01027, with a period P = 4.234516 +- 0.000015 d, transit epoch Tc = 2455304.65122 +- 0.00035 (BJD), and transit duration 0.1023 +- 0.0010 d. The host star has a mass of 0.82 +- 0.03 Msun, radius of 0.79 + 0.10 - 0.04 Rsun, effective temperature 5079 +- 88 K, and metallicity [Fe/H] = -0.04 +- 0.08. The planetary companion has a mass of 0.059 +- 0.007 MJ, and radius of 0.565 + 0.072 - 0.032 RJ yielding a mean density of 0.40 +- 0.10 g cm-3. HAT-P-26b is the fourth Neptune-mass transiting planet discovered to date. It has a mass that is comparable to those of Neptune and Uranus, and slightly smaller than those of the other transiting Super-Neptunes, but a radius that is ~65% larger than those of Neptune and Uranus, and also larger than those of the other transiting Super-Neptunes. HAT-P-26b is consistent with theoretical models of an irradiated Neptune-mass planet with a 10 Mearth heavy element core that comprises >~ 50% of its mass with the remainder contained in a significant hydrogen-helium envelope, though the exact composition is uncertain as there are significant differences between various theoretical models at the Neptune-mass regime. The equatorial declination of the star makes it easily accessible to both Northern and Southern ground-based facilities for follow-up observations.Comment: 16 pages, 9 figures, 5 tables, submitted to Ap