⎣
−(t−t)2
2θ22 − 2 sin2π(t−t) θ3
θ24
⎤⎥⎥⎥
⎥⎥⎦
(6.1) whereθ1is the amplitude of correlated noise;θ2is the lifetime of features which induce activity variations in RVs;θ3is a periodic signal; andθ4is the level of high-frequency structure in the GP model. Out of these hyperparameters for the quasi-periodic covariance kernel, there are two which can be reliably estimated prior to using the kernel: θ2 andθ3. The physical quantities they relate to are the spot lifetimes and stellar rotation period, which can both be found from an ACF of a light curve. Taking these and using them as priors for GP regression, it will give a more physically-motivated solution and spend less time investigating other possibilities which have values for the hyperparameters which are significantly different from the suggested prior.
6.1.1 Sampling of Radial Velocity Measurements
Whilst GPs are designed to infer the behaviour of stellar activity during gaps in RV observations, for them to work most effectively, there needs to be an observational strategy in place to ensure the sampling is adequate enough to cover the different aspects at play within RV observations.
There are three key signals which exist: (1) the quasi-periodic variation due to stellar rotation, (2) periodic variation – i.e. a planetary signal – and (3) the decay of spots. Given that all three need to be modelled, the effects of all three need to be well sampled. Often, a lot of focus is applied to sampling well the orbit of the planet in question, with a secondary goal of sampling well the stellar rotation. However, if the observing strategy was intensive sampling with a baseline only covering one or two stellar rotations (assuming Porb<Prot) then the GP will struggle to settle on the true spot lifetime. Therefore, when performing RV surveys or RV follow-up, it is vital to ensure the baseline of observations covers approximately an entire spot lifetime (which can be estimated from a light curve).
6.2 Starspot Lifetimes Used in Transiting Exoplanet RV Follow-Up
Starspot lifetimes have been used several times for constraining GP regression, either via the methods described in Giles et al. (2017) or by using the formula calculated (Equation 5.2).
Below are two cases where the ACF of the light curve was fitted to determine the stellar rotation period and spot lifetime.
6.2.1 Kepler-21b: A Rocky Planet Around a V = 8.25 Magnitude Star
A bright star observed byKepler, Kepler-21 was found to host a 1.6R⊕planet in a 2.78-day orbit byHowell et al.(2012). However they were only able to impose an upper limit on the mass, at 10M⊕. López-Morales et al.(2016) were able to constrain the mass by obtaining a further 82 RV observations with HARPS-N and, in conjunction with the pre-existing 14 RV observations from HIRES, use GP regression to obtain a mass of 5.1M⊕. For the GP regression, an ACF was generated from theKeplerlight curve to determine the spot lifetimes and stellar rotation periods. More details of the method and results can be found inLópez-Morales et al.(2016) whose first page is included below.
KEPLER-21b: A ROCKY PLANET AROUND AV=8.25 mag STAR*
Mercedes LÓpez-Morales1, Raphaëlle D. Haywood1, Jeffrey L. Coughlin2, Li Zeng3, Lars A. Buchhave4, Helen A. C. Giles5, Laura Affer6, Aldo S. Bonomo7, David Charbonneau1, Andrew Collier Cameron8, Rosario Consentino9, Courtney D. Dressing10,19, Xavier Dumusque5, Pedro Figueira11, Aldo F. M. Fiorenzano9,
Avet Harutyunyan9, John Asher Johnson1, David W. Latham1, Eric D. Lopez12, Christophe Lovis5, Luca Malavolta13,14, Michel Mayor5, Giusi Micela6, Emilio Molinari9,15, Annelies Mortier8, Fatemeh Motalebi5, Valerio Nascimbeni13, Francesco Pepe5, David F. Phillips1, Giampaolo Piotto13,14, Don Pollacco16, Didier Queloz5,17,
Ken Rice12, Dimitar Sasselov1, Damien Segransan5, Alessandro Sozzetti7, Stephane Udry5, Andrew Vanderburg1, and Chris Watson18
1Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 01238, USA;mlopez-morales@cfa.harvard.edu
2SETI Institute, 189 Bernardo Avenue Suite 200, Mountain View, CA 94043, USA
3Department of Earth and Planetary Sciences, Harvard University, 20 Oxford Street, Cambridge, MA 01238, USA
4Centre for Star and Planet Formation, Natural History Museum of Denmark & Niels Bohr Institute, University of Copenhagen, DK-1350 Copenhagen, Denmark
5Observatoire Astronomique de l’Université de Genéve, Chemin des Maillettes 51, Sauverny, CH-1290, Switzerland
6INAF—Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, I-90124 Palermo, Italy
7INAF—Osservatorio Astrofisico di Torino, via Osservatorio 20, I-10025 Pino Torinese, Italy
8SUPA, School of Physics & Astronomy, University of St.Andrews, North Haugh, St. Andrews Fife, KY16 9SS, UK
9INAF—Fundación Galileo Galilei, Rambla José Ana Fernandez Pérez 7, E-38712 Breña Alta, Spain
10Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
11Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
12SUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH93HJ, UK
13Dipartimento di Fisica e Astronomia“Galileo Galilei,”Universita’di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy
14INAF—Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
15INAF—IASF Milano, via Bassini 15, I-20133, Milano, Italy
16Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
17Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, UK
18Astrophysics Research Centre, School of Mathematics and Physics, Queens University, Belfast, Belfast BT7 1NN, UK Received 2016 August 15; revised 2016 September 14; accepted 2016 September 22; published 2016 December 2
ABSTRACT
HD 179070,akaKepler-21, is aV=8.25 F6IV star and the brightest exoplanet host discovered byKepler. An early detailed analysis by Howell et al. of thefirst 13 months(Q0–Q5)ofKeplerlight curves revealed transits of a planetary companion, Kepler-21b, with a radius of about 1.60±0.04Rand an orbital period of about 2.7857 days. However, they could not determine the mass of the planet from the initial radial velocity(RV)observations with Keck-HIRES, and were only able to impose a 2σupper limit of 10M. Here, we present results from the analysis of 82 new RV observations of this system obtained with HARPS-N, together with the existing 14 HIRES data points. We detect the Doppler signal of Kepler-21b with a RV semiamplitudeK=2.00±0.65m s1, which corresponds to a planetary mass of 5.1±1.7 M. We also measure an improved radius for the planet of 1.639+0.019/−0.015R, in agreement with the radius reported by Howell et al. We conclude that Kepler-21b, with a density of 6.4±2.1g cm3, belongs to the population of small,6Mplanets with iron and magnesium silicate interiors, which have lost the majority of their envelope volatiles via stellar winds or gravitational escape.
The RV analysis presented in this paper serves as an example of the type of analysis that will be necessary to confirm the masses of TESS small planet candidates.
Key words:planets and satellites: formation–planets and satellites: individual(Kepler-21b)–
stars: individual(HD 179070)–techniques: photometric–techniques: radial velocities–techniques: spectroscopic Supporting material:machine-readable table
1. INTRODUCTION
Results from NASA’s Kepler Satellite Mission have revealed an abundance of planets smaller than 2 R with orbital periods less than 100 days (Howard et al. 2012;
Dressing & Charbonneau 2013; Fressin et al.2013; Petigura et al.2013a,2013b; Foreman-Mackey et al.2014; Dressing &
Charbonneau2015; Silburt et al.2015). Although only a few of
those planets have measured masses, and therefore densities, those measurements have started to unveil an interesting picture. Below a radius of about 1.6 R most planets are consistent with bare rocky compositions without any significant volatile envelopes(Rogers2015). Moreover, when considering only planets with masses measured with precisions better than 20% via RVs, planets with masses smaller than about 6 M
appear to be rocky and have interiors composed mostly of iron and magnesium silicates in Earth-like abundances (26% Fe, 74% MgSiO3, on average, based on Zeng et al.2016), while planets more massive than about 7Mshow a wider range of densities (Dressing et al. 2015; Gettel et al. 2016). Such dichotomy suggests the possible existence of mechanisms by
*Based on observations made with the Italian Telescope Nazionale Galileo (TNG)operated on the island of La Palma by the Fundación Galileo Galilei of the INAF(Istituto Nazionale di Astrofisica)at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.
19NASA Sagan Fellow.
1
6.2.2 An Accurate Mass Determination for Kepler-1655b, a Moderately Irradiated World with a Significant Volatile Envelope
A 2.2R⊕ planet in a 11.87-day orbit was discovered around a Sun-like star from the Kepler mission. The follow-up consisted of 95 RV points with the HARPS-N spectrograph on the Telescopio Nazionale Galileo. The mass was measured using GP regression, and determined to be 5.0M⊕. For the GP, the light curve was split into three pieces and analysed with an ACF to determine the spot lifetimes and stellar rotation periods – with the average of the three used for the GP regression. More details of the method and results can be found inHaywood et al.
(2018) whose first page is included below.
An Accurate Mass Determination for Kepler-1655b, a Moderately Irradiated World with a Significant Volatile Envelope
Raphaëlle D. Haywood1,20 , Andrew Vanderburg1,2,20 , Annelies Mortier3 , Helen A. C. Giles4, Mercedes López-Morales1, Eric D. Lopez5, Luca Malavolta6,7 , David Charbonneau1 , Andrew Collier Cameron3, Jeffrey L. Coughlin8 , Courtney D. Dressing9,10,20 , Chantanelle Nava1, David W. Latham1 , Xavier Dumusque4 , Christophe Lovis4, Emilio Molinari11,12 , Francesco Pepe4, Alessandro Sozzetti13 , Stéphane Udry4, François Bouchy4, John A. Johnson1, Michel Mayor4, Giusi Micela14, David Phillips1, Giampaolo Piotto6,7 , Ken Rice15,16, Dimitar Sasselov1 , Damien Ségransan4,
Chris Watson17, Laura Affer14, Aldo S. Bonomo13, Lars A. Buchhave18 , David R. Ciardi19, Aldo F. Fiorenzano11, and and Avet Harutyunyan11
1Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 01238, USA;rhaywood@cfa.harvard.edu
2Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA
3Centre for Exoplanet Science, SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, UK
4Observatoire Astronomique de l’Université de Genève, Chemin des Maillettes 51, Sauverny, CH-1290, Switzerland
5NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA
6Dipartimento di Fisica e Astronomia“Galileo Galilei,”Universita’di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy
7INAF—Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
8SETI Institute, 189 Bernardo Avenue, Suite 200, Mountain View, CA 94043, USA
9Division of Geological Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
10Astronomy Department, University of California, Berkeley, CA 94720, USA
11INAF—Fundación Galileo Galilei, Rambla José Ana Fernandez Pérez 7, E-38712 Breña Baja, Tenerife, Spain
12INAF—Osservatorio Astronomico di Cagliari, via della Scienza 5, I-09047, Selargius, Italy
13INAF—Osservatorio Astrofisico di Torino, via Osservatorio 20, I-10025 Pino Torinese, Italy
14INAF—Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, I-90134 Palermo, Italy
15SUPA, Institute for Astronomy, Royal Observatory, University of Edinburgh, Blackford Hill, Edinburgh EH93HJ, UK
16Centre for Exoplanet Science, University of Edinburgh, Edinburgh, UK
17Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, Belfast, BT7 1NN, UK
18Centre for Star and Planet Formation, Natural History Museum of Denmark, University of Copenhagen, DK-1350 Copenhagen, Denmark
19NASA Exoplanet Science Institute, Caltech/IPAC-NExScI, 1200 East California Boulevard, Pasadena, CA 91125, USA Received 2017 July 10; revised 2018 March 16; accepted 2018 March 18; published 2018 April 20
Abstract
We present the confirmation of a small, moderately irradiated(F=155±7F⊕)Neptune with a substantial gas envelope in aP=11.8728787±0.0000085 day orbit about a quiet, Sun-like G0V star Kepler-1655. Based on our analysis of theKeplerlight curve, we determined Kepler-1655b’s radius to be 2.213±0.082R⊕. We acquired 95 high-resolution spectra with Telescopio Nazionale Galileo/HARPS-N, enabling us to characterize the host star and determine an accurate mass for Kepler-1655b of 5.0o2.83.1 M via Gaussian-process regression. Our mass determination excludes an Earth-like composition with 98% confidence. Kepler-1655bfalls on the upper edge of the evaporation valley, in the relatively sparsely occupied transition region between rocky and gas-rich planets. It is therefore part of a population of planets that we should actively seek to characterize further.
Key words:stars: individual(Kepler-1655, KOI-280, KIC 4141376, 2MASS J19064546+3912428)–planets and satellites: detection–planets and satellites: gaseous planets
1. Introduction
In our own solar system, we see a sharp transition between the inner planets, which are small(Rp1R⊕)and rocky, and the outer planets that are larger (Rp 3.88R⊕), much more massive, and have thick, gaseous envelopes. For exoplanets with radii intermediate to that of the Earth(1R⊕)and Neptune (3.88R⊕), several factors go into determining whether planets acquire or retain a thick gaseous envelope. Several studies have determined statistically from radius and mass determinations of exoplanets that most planets smaller than 1.6R⊕are rocky(i.e., they do not have large envelopes but only a thin, secondary atmosphere, if any at all; Lopez & Fortney 2014; Weiss &
Marcy 2014; Dressing & Charbonneau 2015; Rogers 2015;
Buchhave et al.2016; Gettel et al.2016; Lopez2017; Lopez &
Rice2016). Others have found that planets in less irradiated orbits tend to be more likely to have gaseous envelopes than
more highly irradiated planets (Hadden & Lithwick 2014;
Jontof-Hutter et al.2016). However, it is still unclear under which circumstances a planet will obtain and retain a thick gaseous envelope and how this is related to other parameters, such as stellar irradiation levels.
The characterization of the mass of a small planet in an orbit of a few days to a few months around a Sun-like star(i.e., in the incidentflux range ≈1–5000F⊕) is primarily limited by the stellar magnetic features acting over this timescale and producing RV variations that compromise our mass determina-tions. Magneticfields produce large, dark starspots and bright faculae on the stellar photosphere. These features induce RV variations modulated by the rotation of the star and varying in amplitude as the features emerge, grow, and decay. There are two physical processes at play: (i) dark starspots and bright faculae break the Doppler balance between the approaching blueshifted stellar hemisphere and the receding redshifted half of the star (Saar & Donahue 1997; Lagrange et al. 2010;
20NASA Sagan Fellow.
1