Dynamical Masses of Young M Dwarfs: Masses and Orbital Parameters of GJ 3305 AB, the Wide Binary Companion to the Imaged Exoplanet Host 51 Eri
Benjamin T. Montet, Brendan P. Bowler, Evgenya L. Shkolnik, Katherine M. Deck, Ji Wang, Elliott P. Horch, Michael C. Liu, Lynne A. Hillenbrand, Adam L. Kraus, David Charbonneau
DDraft version Monday 12 th March, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
DYNAMICAL MASSES OF YOUNG M DWARFS: MASSES AND ORBITAL PARAMETERS OF GJ 3305 AB,THE WIDE BINARY COMPANION TO THE IMAGED EXOPLANET HOST 51 ERI ∗ Benjamin T. Montet , Brendan P. Bowler , Evgenya L. Shkolnik
4, 5 , Katherine M. Deck , Ji Wang
1, 6 ,Elliott P. Horch , Michael C. Liu , Lynne A. Hillenbrand , Adam L. Kraus , David Charbonneau (Dated: Monday 12 th March, 2018, 05:54)
Draft version Monday 12 th March, 2018
ABSTRACTWe combine new high resolution imaging and spectroscopy from Keck/NIRC2, Discovery Chan-nel Telescope/DSSI, and Keck/HIRES with published astrometry and radial velocities to measureindividual masses and orbital elements of the GJ 3305 AB system, a young ( ∼
20 Myr) M+M binary(unresolved spectral type M0) member of the β Pictoris moving group comoving with the imagedexoplanet host 51 Eri. We measure a total system mass of 1 . ± .
04 M (cid:12) , a period of 29 . ± . . ± .
14 AU, and an eccentricity of 0 . ± .
02. The primary componenthas a dynamical mass of 0 . ± .
05 M (cid:12) and the secondary has a mass of 0 . ± .
05 M (cid:12) . Therecently updated BHAC15 models are consistent with the masses of both stars to within 1 . σ . Giventhe observed masses the models predict an age of the GJ 3305 AB system of 37 ± Subject headings: astrometry — binaries: close — stars: fundamental parameters — stars: individual(GJ 3305 AB) INTRODUCTION
Loose associations of young, nearby ( <
70 pc) starswith common ages, kinematics, and origins have beena subject of increasing interest (Zuckerman et al. 2004;Shkolnik et al. 2012; Malo et al. 2013). Because of theirproximity to Earth, these young moving groups (YMGs)are excellent targets to study pre-main sequence (PMS)stellar and substellar evolution, protoplanetary and de-bris disk structure, and giant planet formation at agesbetween distant star-forming regions and old field stars(e.g. Close et al. 2005; Nielsen & Close 2010). About 10YMGs containing hundreds of objects between 8 and 120million years old are known (e.g. Torres et al. 2008).As these moving groups are amenable to numerous agedating methods, including kinematic techniques, theyprovide the opportunity to measure dynamical masses [email protected] Cahill Center for Astronomy and Astrophysics, CaliforniaInstitute of Technology, Pasadena, CA, 91125, USA Harvard-Smithsonian Center for Astrophysics, Cambridge,MA 02138, USA Caltech Joint Center for Planetary Astronomy Fellow School of Earth and Space Exploration, Arizona State Uni-versity, Tempe, AZ 85287, USA Lowell Observatory, 1400 West Mars Hill Road, Flagstaff,AZ, 86001, USA Department of Astronomy, Yale University, New Haven, CT06511, USA Department of Physics, Southern Connecticut State Univer-sity, 501 Crescent Street, New Haven, CT 06515, USA Institute for Astronomy, University of Hawaii, 2680 Wood-lawn Drive, Honolulu, HI 96822, USA Department of Astronomy, The University of Texas atAustin, Austin, TX 78712, USA * Some of the data presented herein were obtained at the W.M.Keck Observatory, which is operated as a scientific partnershipamong the California Institute of Technology, the University ofCalifornia and the National Aeronautics and Space Administra-tion. The Observatory was made possible by the generous finan-cial support of the W.M. Keck Foundation. of PMS low-mass binary objects and test stellar evolu-tion models (Stassun et al. 2014). Generally, PMS stel-lar masses are inferred by comparing a star’s temper-ature, luminosity and metallicity to model predictions(e.g. Schaefer et al. 2014). These models are poorlyconstrained by observations and may induce systematicoffsets (Dupuy et al. 2009, 2014). Worse yet, differentmodels predict disparate masses, primarily due to un-certainties in the treatment of convection in low-gravityatmospheres (Baraffe et al. 2002), stellar accretion his-tory (Baraffe & Chabrier 2010), and molecular line lists(Baraffe et al. 2015). In some cases, model-predictedmasses can differ by a factor of two or more (Hillenbrand& White 2004; Schlieder et al. 2014). Dynamical massmeasurements of binary stars with known ages are essen-tial to test models.Recently, Macintosh et al. (2015) presented 51 Eri b,the first exoplanet discovery from the Gemini Planet Im-ager. The planet has a mass of ≈ Jup (assuming a hotstart model), a projected separation of 13 AU, a temper-ature of 600-750 K, and a T4.5-T6 spectral type. GJ 3305is known to be a binary with combined spectral type M0(Kasper et al. 2007). Feigelson et al. (2006) identifiedGJ 3305 and 51 Eri as an F0-M0 common proper motionpair, separated by 66 (cid:48)(cid:48) or ∼ β Pictoris moving group, an approximate age ofthe system is known (24 ± Hipparcos mea-sured to a precision of 1%. Combining this parallax with15 years of imaging and RV data enables us to determinethe system orbital parameters, elucidating the architec-ture of this 4—or more—body system.In this paper, we combine RV and astrometric observa- a r X i v : . [ a s t r o - ph . S R ] O c t tions of GJ 3305 AB to measure orbital parameters andmasses for each star. We compare these masses to modelpredictions and discuss the possible implications of thisbinary pair on the long-term evolution of the orbit of51 Eri b. DATA COLLECTION AND REDUCTION
GJ 3305 AB has been imaged and resolved many times(Kasper et al. 2007; Bergfors et al. 2010; Delorme et al.2012; Janson et al. 2012, 2014). The system was also im-aged with NIRC2 (Wizinowich et al. 2000) in one unpub-lished epoch in 2001 available in the Keck ObservatoryArchives (KOA, PI Zuckerman). In this work, we com-bine these data with five observations from 2002 to 2015,three using Keck/NIRC2 and one with the DifferentialSpeckle Survey Instrument (DSSI, Horch et al. 2009) atthe Discovery Channel Telescope at Lowell Observatory.All NIRC2 data were obtained with the narrow cam-era mode, which has a field of view of 10 . (cid:48)(cid:48) × . (cid:48)(cid:48) − (Yelda et al. 2010).All images were flat fielded and cleaned of bad pixelsand cosmic rays. Astrometry and relative photometry ofGJ 3305 was derived by simultaneously fitting three bi-variate Gaussians to each component following Liu et al.(2010).DSSI allows for simultaneous observations in two fil-ters. We use the DSSI R and I filters, with central wave-lengths 692 and 880 nm and FWHMs of 40 and 50 nm.We obtained 1000 40-ms exposures in each channel simul-taneously. The data were then reduced following Horchet al. (2015). Specifically, the autocorrelation of eachframe was calculated and summed over all exposures, andthe near-axis subplanes of the image bispectrum werecalculated. To create a reconstructed image, the Fouriertransform of the autocorrelation of the binary was di-vided by that of a nearby point source (HR 1415). Thesquare root of this value is taken, and the result com-bined with a phase function derived from the bispectralsubplanes. The pixel scale (19 mas pixel − in R and20 mas pixel − in I ) and orientation of the detector werefound by observing several widely separated binaries withknown astrometry. Our astrometry is listed in Table 1.The GJ 3305 binary system has also been monitoredspectroscopically. One Keck/HIRES spectrum from 2003exists in the KOA (PI Zuckerman); we measure the RVfollowing Kraus et al. (2015). We combine this spectrumwith nine additional spectra from Bailey et al. (2012),Shkolnik et al. (2012), and Elliott et al. (2014). In allcases, the RVs were calculated treating the system as anSB1. We take the reported RV and uncertainty for eachobservation, but assume the flux from the secondary isnon-negligible, as explained in Section 3. ANALYSIS
We infer the orbital parameters of GJ 3305 AB by com-paring the astrometric and RV data to a Keplerian orbitmodel at each of the observation times. A parallax, as-trometric orbit, and SB1 RV data can be combined tomeasure individual masses of each star (e.g. Bean et al.2007). There is no measured parallax for GJ 3305, sowe adopt the Hipparcos distance to 51 Eri/HIP 21547:29 . ± .
30 pc (van Leeuwen 2007). These two comov-ing systems have a projected separation of 1940 ± √ e cos ω and √ e sin ω , the time of periapse t P , the period P , the totalmass M + M , the inclination i , and the longitude ofthe ascending node Ω. We parameterize the eccentricityvector in this manner following Eastman et al. (2013).The RV data can provide additional information aboutseveral of these (not M + M or Ω directly), also allow-ing us to fit the systemic RV γ and the secondary mass M .We include ten additional terms to account for possi-ble systematics in the datas. This star has been imaged,resolved, and published by four different groups. Weaccount for the possibility each group may have under-estimated their uncertainties on the orbital separationand position angle by a multiplicative factor by includ-ing a systematic error term on the measured positionsfrom each group, allowing outlier points to be down-weighted without manually choosing specific points todownweight. We do the same with our reductions ofboth archival and new data, allowing for separate sys-tematic error terms on our data from Keck/NIRC2 andDCT/DSSI, providing a total of six systematic errorterms. We allow the uncertainties on each dataset tobe inflated up to a factor of five.Similarly, we allow for the possibility that the uncer-tainties in the RVs may be underestimated, possibly dueto stellar variability (Moulds et al. 2013), errors in sys-temic RVs of standard stars, or drifts in the stability ofthe spectrographs. As our RV data originate from foursources, we allow each to have its own systematic errorterm, analogous to the jitter term commonly applied inRV orbit fits of exoplanets (e.g. Johnson et al. 2011):log L ∝ − (cid:88) i (cid:20) log (cid:113) σ o,i + σ s + 0 . (cid:18) ( f i ( t ) − v i ( t )) σ o,i + σ s (cid:19)(cid:21) . (1)Here, L is the likelihood of the data given some under-lying physical model, σ o,i is the observed uncertainty onthe i th data point, σ s the systematic error associatedwith each particular set of observations, f i ( t ) the RVmodel evaluated at time t , and v i ( t ) the observed RV ateach t . Maximum likelihood jitter values range from 0 . − for the 2003 HIRES data to 0 .
57 km s − for theUVES data, suggesting stellar jitter is significant in theRV data, as expected for young stars.In all cases, one set of lines are observed because theRV separation is smaller than the line width. We ex-pect each RV measurement to be the flux-weighted sumof the two individual RVs. At each step, we calculatethe RVs for each star, weighting them according to theirexpected flux contribution in each bandpass, using theobserved flux ratios in the visible and near-IR as priorsand assuming an additional 0.1 mag of variability in theoptical and 0.05 mag in the near-IR.We neglect the possibility that 51 Eri could contributesignificantly to the observed RV signal. Following Equa-tion 1 of Montet et al. (2015), the maximum RV acceler- TABLE 1Data for GJ 3305 AB
Epoch Bandpass RV Contrast Separation Position Angle Source(Year) (km s − ) (∆ mag) (mas) (deg)2001.910 H ( ν =1–0) 1 . ± .
02 286 ± . ± . H . ± .
02 275 . ± . . ± . K . ± .
05 225 ± . ± . Kasper et al. (2007)2003.195 H . ± .
01 217 ± . ± . L (cid:48) ± ± L (cid:48) . ± .
28 93 ± . ± . z (cid:48) . ± .
16 218 ± . ± . i (cid:48) . ± .
05 218 ± . ± . i (cid:48) + z (cid:48) ± . ± . L (cid:48) ± . ± . L (cid:48) ± . ± . z (cid:48) . ± .
01 284 ± . ± . i (cid:48) . ± .
01 Janson et al. (2012)2010.81 SDSS z (cid:48) ± . ± . L (cid:48) ± . ± . z (cid:48) ± . ± . z (cid:48) ± . ± . γ . ± .
01 244 ± . ± . R . ± .
04 239 ± . ± . I . ± .
03 240 ± . ± . K . ± .
01 199 ± . ± . H . ± .
01 198 ± . ± . J . ± .
01 199 ± . ± . Y . ± .
03 200 ± . ± . V . ± .
38 This work2004.884 NIRSPEC K . ± .
05 Bailey et al. (2012)2005.862 NIRSPEC K . ± .
06 Bailey et al. (2012)2005.971 HIRES V . ± .
30 Shkolnik et al. (2012)2006.014 NIRSPEC K . ± .
05 Bailey et al. (2012)2006.016 NIRSPEC K . ± .
05 Bailey et al. (2012)2006.019 NIRSPEC K . ± .
05 Bailey et al. (2012)2011.778 UVES Blue 24 . ± .
04 Elliott et al. (2014)2001.994 UVES Blue 23 . ± .
02 Elliott et al. (2014)2012.022 UVES Blue 23 . ± .
02 Elliott et al. (2014)
Note . — In some previous analyses, contrast ratios were not listed for specific epochs. Observations withoutlisted separations correspond to simultaneous multiband photometry. Observations published without uncertainty estimates; we choose conservative values. ation expected from 51 Eri is 3 cm s − yr − , well belowour sensitivity.We calculate posterior distributions for all parametersusing emcee (Foreman-Mackey et al. 2013), an imple-mentation of the affine-invariant Markov Chain MonteCarlo ensemble sampler of Goodman & Weare (2010).After performing a local optimization to determine amaximum-likelihood fit, we move 3000 walkers each 4000steps. We discard the first 2000 steps of each walkeras burn-in, and use the test of Geweke (1992) and vi-sual inspection to verify the system has converged. Thedata and allowed orbits are shown in Figure 1. Summarystatistics for the orbital parameters are given in Table 2.We note the fitted systemic RV of 20 . ± .
18 km s − is consistent with the measured RV for 51 Eri, 21 . ± . − (Bobylev 2006) and the UVW velocities are con-sistent with Mamajek & Bell (2014). Our samples areavailable online. We estimate bolometric luminosities for both starsby integrating the CFHIST2011 2015 model spectra ofBaraffe et al. (2015). We use the 3700 and 3500 K mod-els with log g = 4 . ∼ btm/research/gj3305.html TABLE 2Parameters for GJ 3305 AB
Parameter Median Uncertainty(1 σ ) √ e cos ω ± √ e sin ω -0.406 ± ± ω [deg] -69 ± ± ± (cid:12) ] 0.67 ± (cid:12) ] 0.44 ± (cid:12) ] 1.11 ± M B /M A ± i [deg] 92.1 ± a [AU] 9.78 ± ± γ [km s − ] 20.76 ± K A [km s − ] 4.01 ± − ] -13.76 ± − ] -16.40 ± − ] -9.71 ± (cid:12) ] 0.112 ± (cid:12) ] 0.043 ± magnitudes and 0.05 mag in the near-IR to account forstellar variability. -300 -200 -100 0 100 200 300 dRA (mas) d D e c ( m a s ) NE R a d i a l V e l o c i t y ( k m s − ) Year -2-1010100200300 O r b i t a l S e p a r a t i o n ( m a s ) Year -20-10010 -180-90090180 P o s i t i o n A n g l e ( d e g ) Year -20-10010
Fig. 1.— (Top Left) Astrometry for GJ 3305 AB. Data points correspond to the observations listed in Table 1. Blue lines correspondto random draws from the posterior distributions of orbital elements. The red, dashed line corresponds to the maximum likelihood orbit.(Top Right) RV data for GJ 3305 A from the literature. The published uncertainties are in black; in gray are the best-fitting uncertainties,incorporating an RV jitter model. The red, dashed line corresponds to the maximum likelihood orbit. The blue shaded regions correspondto the 1 − , 2 − , and 3 σ uncertainties in the RV of GJ 3305 A. (Bottom Left) Measured separations for GJ 3305 AB and residuals from themaximum likelihood model. Each feature on the plot retains its meaning from the previous subplot. (Bottom Right) Measured positionangles for GJ 3305 AB and residuals from the maximum likelihood model. COMPARISON WITH BHAC15 EVOLUTIONARYMODELS
Given the known distance to the system from
Hippar-cos we can test if theoretical stellar evolution modelsaccurately predict the inferred stellar masses and age ofthe β Pic moving group. Combined-light photometryspanning from B (0.4 µ m) to Ks (2.3 µ m) was mea-sured by the APASS, 2MASS, and WISE surveys (Ta-ble 3). We add an uncertainty of 0.03 mag in quadra-ture to the listed APASS uncertainties due to the pres-ence of systematics in APASS DR9 at that level (Hendenet al. 2012). We also have obtained one epoch of differ-ential photometry in two visible-light bandpasses withDSSI and two near-IR bandpasses ( H and Br γ ) withKeck/NIRC2.We compare the observed brightness of GJ 3305 AB tothat predicted by the BHAC15 models of Baraffe et al.(2015) for two stars of masses consistent with those in-ferred during our analysis as a function of age. Wefind models of 25 Myr old stars accurately predict thecombined-light near-IR flux for these stars, although the TABLE 3Photometry for GJ 3305 AB
Bandpass Source Magnitude UncertaintyCombined B APASS DR9 11.94 0.03 V APASS DR9 10.56 0.05 g (cid:48) APASS DR9 11.27 0.03 r (cid:48) APASS DR9 10.03 0.07 J H K W W W W Keck/NIRC2 1.00 0.02∆Br γ Keck/NIRC2 0.92 0.01∆ H Keck/NIRC2 1.00 0.02 models predict brighter V magnitudes than those ob-served (Figure 2). However, star B is brighter than thesesame models predict: a 25 Myr old GJ 3305 B would besignificantly brighter than what is observed. Assumingthe stars are coeval, the models then predict a mass forGJ 3305 B that is 20% lower than the observed mass.We create a simulated spectral energy distribution foreach star, given the measured masses and the averageage of β Pic as measured from higher-mass stars. We in-terpolate absolute magnitudes predicted by the updatedBHAC15 models of Baraffe et al. (2015) along isochronesand isomass contours to predict apparent magnitudes forthese stars in each bandpass. We find that the total re-ceived flux is lower than predicted by the BHAC15 mod-els in each bandpass. While the flux for GJ 3305 A is con-sistent with the model predictions, GJ 3305 B is fainterthan predicted.Given the observed masses, we then vary the age ofthe system, assuming both stars are coeval, to determinewhich system age would be predicted by these modelsgiven the observed combined and differential magnitudes.We apply a flat prior on the age of the system, finding theBHAC15 models predict an age of 37 ± ± DISCUSSION
We have measured the masses and orbits ofGJ 3305 AB, finding both to be consistent with theBHAC15 models at the 1 . σ level. In the futureGJ 3305 AB and the gravitationally bound 51 Eri Ab willbe able to act as an isochronal test as a coeval, co-metallicity quadruple system spanning stellar to plan-etary mass regimes.The derived period of GJ 3305 (29 . ± .
50 year) islonger than the 21 year found by Delorme et al. (2012).The authors of that paper did not have sufficient data tofit all orbital parameters, so they fixed the total systemmass to 1 . (cid:12) . Given our lower mass measurement,it is not surprising that our measured orbital period islonger. Current Limitations
It is possible that an unseen very low-mass star orbrown dwarf orbiting GJ 3305 B could cause us to overes-timate its mass, causing the observed 20% discrepancy.For the system to be stable over 20 Myr, such a com-panion would have to be in a close (
P <
50 day) orbit.The companion would then have to be in a nearly face-on( i < ◦ ) orbit to evade RV detection. Such companionscould be found through continued astrometric monitor-ing of GJ 3305. Such a companion would not affect ourastrometry due to its small separation from GJ 3305 Band would likely not affect our photometry due to itslow luminosity relative to the other stars in the system.Most PMS M dwarfs have distance measurements to aprecision no better than 5%, meaning the total mass can-not be measured to better than 15% (e.g. Shkolnik et al.2012). The uncertainty in the mass of GJ 3305 AB is only4%: the dominant source of uncertainty in this value isthe 1% Hipparcos parallax to 51 Eri, making this sys-tem an ideal low-mass benchmark. With a Gaia parallaxforthcoming in the next few years, parallaxes for low-mass PMS stars will be improved substantially. Long- term astrometric and RV monitoring of wide M dwarfsis essential as parallaxes are obtained over the next fewyears.The uncertainty in the individual mass of each star isdominated by the uncertainty in the Doppler semiampli-tude. While additional astrometric observations will notsignificantly improve the measured physical propertiesof GJ 3305, additional RV observations will be impor-tant. RV observations behind AO would be especiallybeneficial, as the RV from each star could be measuredseparately, instead of a flux-weighted RV centroid.
Dynamical Effects on 51 Eri b
GJ 3305 AB and 51 Eri Ab exist in a dynamical config-uration that may be susceptible to Kozai-Lidov oscilla-tions (Kozai 1962; Lidov 1962), as suggested by Mac-intosh et al. (2015). In this scenario, the planet-starbinary (51 Eri Ab) interacts secularly with GJ 3305 AB,leading to oscillations in inclination and eccentricity ofthe planet-star sub-system. The timescale for such aninteraction is τ ≈ P planet M (cid:63) M pert (cid:18) a pert a planet (cid:19) (1 − e ) / (2)where P planet is the orbital period of a planet with asemimajor axis of a planet about a host of mass M (cid:63) , M pert is the mass of a distant perturber, and a pert and e pert are the semimajor axis and eccentricity of theperturber/planet-star “binary” orbit (see e.g. Holmanet al. 1997).Although we have limited information about this sys-tem, we can estimate the timescale for Kozai-Lidov cyclesshould the mutual inclination of the 51 Eri Ab system and(51 Eri Ab)-(GJ 3305 AB) system satisfy 140 ◦ (cid:46) i m (cid:38) ◦ . Taking the instantaneous sky-projected separationsas a proxy for the semimajor axes and inferred massesof M (cid:63) = 1 .
75 M (cid:12) (Simon & Schaefer 2011) and M pert =1 . (cid:12) yields a timescale of τ ∼ × yr (1 − e ) / .Therefore, unless the eccentricity of GJ 3305 about the51 Eri subsystem satisfies e pert (cid:38) .
9, the timescale forKozai-Lidov oscillations is longer than the age of the sys-tem, so we do not expect the Kozai-Lidov mechanism tohave had time to induce a large eccentricity or spin-orbitmisalignment within the 51 Eri sub-system. If future ob-servations indicate non-zero spin-orbit misalignment ora high eccentricity for the orbit of 51 Eri b, a primordialorigin unrelated to the distant perturbers would be sug-gested.B.T.M. is supported by the National Science Foun-dation Graduate Research Fellowship under Grant No.DGE-1144469.This research has made use of the Keck ObservatoryArchive (KOA), which is operated by the W. M. KeckObservatory and the NASA Exoplanet Science Institute(NExScI), under contract with the National Aeronauticsand Space Administration.These results made use of Lowell Observatory’s Dis-covery Channel Telescope. Lowell operates the DCT inpartnership with Boston University, Northern ArizonaUniversity, the University of Maryland, and the Univer-sity of Toledo. Partial support of the DCT was providedby Discovery Communications.
System Age (Myr) R e l a t i v e L i k e li h oo d Wavelength (microns) A pp a r e n t M a g n i t u d e Wavelength (microns) D i ff e r e n t i a l M a g n i t u d e Wavelength (microns) F l u x ( − e r g c m − s − − ) Combined LightResolved LightModel
GJ 3305 A Mass ( M fl ) G J B M a ss ( M fl ) DataModel H − K M H fl fl fl fl fl Fig. 2.— (Top) (Left) Combined-light, unresolved and (Right) differential, resolved photometry for GJ 3305 AB (black) compared topredictions (red) of the BHAC15 models as a function of age given the observed masses and parallax. The data are consistent with anage larger than 25 Myr. Plotted bars along the abscissa correspond to the width of each filter and are meant to guide the eye: they donot represent an uncertainty. (Middle left) SED for the system, assuming a 24 ± ± σ confidence regions. The BHAC15 models predicta mass for GJ 3305 B consistent with the mass inferred from the data, but underpredicts the mass of GJ 3305 A by 20%. (Bottom left)CMD showing the absolute H magnitudes and H − K colors of GJ 3305 AB compared to theoretical models. The models provide a moreaccurate fit for GJ 3305 A than GJ 3305 B. (Bottom right) Posterior probability distribution on the age of the GJ 3305 system, calculatedby marginalizing the joint mass-age posterior over all allowed masses, assuming both stars are the same age. The BHAC15 models predictan age of 37 ± β Pictoris system.
This publication was made possible through the sup-port of a grant from the John Templeton Foundation.The opinions expressed in this publication are those ofthe authors and do not necessarily reflect the views ofthe John Templeton Foundation.The authors wish to recognize and acknowledge the very significant cultural role and reverence that the sum-mit of Maunakea has always had within the indigenousHawaiian community. We are most fortunate to have theopportunity to conduct observations from this mountain.
Facilities: