APP Template V1.01 Article id: apj513330 Typesetter: MPS Date received by MPS: 19/05/2015 PE: CE : LE: UNCORRECTED PROOF The Astrophysical Journal, 00:000000 (28pp), 2015 Month Day © 2015. The American Astronomical Society. All rights reserved. TRUE MASSES OF RADIAL-VELOCITY EXOPLANETS Robert A. Brown Space Telescope Science Institute, USA; [email protected] Received 2015 January 12; accepted 2015 April 14; published 2015 MM DD ABSTRACT We study the task of estimating the true masses of known radial-velocity (RV) exoplanets by means of direct astrometry on coronagraphic images to measure the apparent separation between exoplanet and host star. Initially, we assume perfect knowledge of the RV orbital parameters and that all errors are due to photon statistics. We construct design reference missions for four missions currently under study at NASA: EXO-S and WFIRST-S, with external star shades for starlight suppression, EXO-C and WFIRST-C, with internal coronagraphs. These DRMs reveal extreme scheduling constraints due to the combination of solar and anti-solar pointing restrictions, photometric and obscurational completeness, image blurring due to orbital motion, and the “nodal effect,” which is the independence of apparent separation and inclination when the planet crosses the plane of the sky through the host star. Next, we address the issue of nonzero uncertainties in RV orbital parameters by investigating their impact on the observations of 21 single-planet systems. Except for two—GJ 676 A b and 16 Cyg B b, which are observable only by the star-shade missions—we find that current uncertainties in orbital parameters generally prevent accurate, unbiased estimation of true planetary mass. For the coronagraphs, WFIRST-C and EXO-C, the most likely number of good estimators of true mass is currently zero. For the star shades, EXO-S and WFIRST-S, the most likely numbers of good estimators are three and four, respectively, including GJ 676 A b and 16 Cyg B b. We expect that uncertain orbital elements currently undermine all potential programs of direct imaging and spectroscopy of RV exoplanets. Key words: methods: statistical – planetary systems – planets and satellites: detection – techniques: radial velocities 1. INTRODUCTION relevant because we plan follow-on spectroscopy as an adjunct to the mass-determination program. To date, hundreds of exoplanets have been discovered by the In the current work, which is a point calculation, we define radial-velocity (RV) technique. Not only have these objects the science requirement on mass determination to be changed the course of the history of astronomy and opened rich fi new elds of study, the known RV planets are special objects σmp of further research, mainly because they are relatively nearby Fmp ≡=0.10 (1) m and because they revolve around bright, well-known stars. p Without doubt, scientists will one day know each of these RV planets intimately as individual objects, each with unique where Fmp is the fractional uncertainty of the estimate of mp, origins, attributes, and environments, just like the solar-system and σmp is the standard deviation of mp. The WFIRST-C and = planets today. We can now look ahead to the next steps toward EXO-C study reports use Fmp 0.10, and EXO-S and WFIRST- = that goal, which are to determine the true masses ()m p of S use Fmp 0.20. known RV planets and to obtain simple diagnostic spectra of In the case of spectroscopy, we are guided by the treatment their atmospheres. To support those ends, this paper presents an of Maire et al. (2012), which discusses the spectroscopic analytic framework for mass determination by direct astro- opportunities for SPICES, a notional 1.5 m telescope with the metry. We use that framework to estimate the performance of purpose of atmospheric characterization of cold exoplanets. For four space optical systems currently under study. These giant planets, they identify methane bands at 620, 730, and missions—which proponents say could be implemented in 790 nm, in order of increasing strength. Shallower water bands the 2020s—are WFIRST-C, a 2.4 m telescope with an internal are located at 720 and 820 nm. Longer wavelengths are coronagraph (Spergel et al. 2015), EXO-S, a 1.1 m telescope disadvantageous due to reduced spatial resolution and, in the with a star shade (Seager et al. 2015), EXOC, a 1.4 m telescope case of CCD detectors, falling sensitivity due to silicon with an internal coronagraph (Stapelfeldt et al. 2015), transparency. We therefore adopt I band as the nominal range WFIRST-S, a 2.4 m telescope with an external star shade for RV planet imaging and spectroscopy, and 760 nm as the (Seager et al. 2015; Chapter 8). nominal wavelength for calculations. Maire et al. found that a Two types of research motivate the science requirement for “spectral resolution of at least 50 is required to identify the determining mp. The first type investigates the formation and main bands of the spectra of giant planets as well as super- evolution of planetary systems. Just as explaining the Earths.” We have chosen spc = 70 and a nominal target distributions of planetary mass and orbital distance has long signal-to-noise ratio (S/N) of 10 in the continuum for been a focus of solar-system studies, today such research has spectroscopy been taken to the next level, addressing those distributions Our observing plan envisions starting the spectroscopic broadly—ultimately, everywhere in the universe. The second exposure immediately after directly imaging a possible RV type of research investigates planetary atmospheres, where the planet. The reason for acting quickly is that the source may planetary mass enters into the scale height. Knowing the scale shortly disappear behind the central field stop or otherwise height is important for interpreting planetary spectra, which is become unobservable. (From Tables 3–6, we expect most 1 The Astrophysical Journal, 00:000000 (28pp), 2015 Month Day Brown detections to occur near the inner working angle (IWA), which nearby RV planets, estimates i from the apparent separation s, is the nominal radius of the field stop.) between the planet and the host star, where s is measured by In the case of true-mass determination, our framework is direct astrometry on a coronagraphic image. In this case, we somewhat complicated, but also new and interesting. Even with will show that a single measurement of s is, in principle, perfect knowledge of the RV orbits—which we assume until sufficient to estimate i. Section 7—successfully planning the observations is a We are not aware of any other way to measure mp and take a challenging task. When we introduce errors into the RV orbital follow-on spectrum. Nevertheless, various statistical parameters, we find that estimating mp with any acceptable approaches have been offered for estimating the true distribu- accuracy becomes impossible except for a few exoplanets tion of a quantity “projected” by a factor sin i, including at most. rotation rates of unresolved stars (Chandrasekhar & To the best of our knowledge, no science requirements on Münch 1950), microlensing (Giannini & Lunine 2013; ) ( the accuracy of estimating mp have yet been proposed in print, Brown 2014 , and RV planets Jorisson et al. 2001; Zucker nor have any measurement requirements been derived from any & Mazeh 2001; Marcy et al. 2005; Udry & Santos 2007; such requirements, nor has the performance of any candidate Brown 2011; Ho & Turner 2011; Lopez & Jenkins 2012). mission so far been tested taking into account all key factors. Nevertheless, Brown (2011) shows that a usefully constrained This paper provides the needed framework, which includes distribution of the unprojected quantity may demand a sample measurement requirements. We find those requirements depend of thousands or tens of thousands of projected values; certainly strongly on the orbital phase of the planet, especially near node more than the mere hundreds we have in the case of currently crossings as well as solar-avoidance restrictions, variable known RV planets. obscuration near the central field stop, the systematic limit to To outline the sections ahead, in Section 2, we derive our the depth of search exposures, variation of local zodi with method of estimating the value of S/N required to estimate mp ecliptic latitude and longitude, observational overheads, and the with fractional accuracy Fmp. In Section 3, we assemble a mise blurring of images due to the orbital motion of the planet. en place of DRM basics: in Section 3.1, we document the Traub et al. (2014) uses a simplified model to calculate how baseline parameters and assumptions for the four missions many RV planets could be detected in how many days of under study; in Section 3.2, we describe our input catalog of potentially observable RV planets; and, in Section 3.3,we exposure time by various coronagraph designs for WFIRST-C. fi The Traub et al. results are not relevant to this report because describe our exposure-time calculator. In Section 4,wede ne a cube of intermediate information they do not treat true-mass estimation. , which realizes a mission and After developing our framework for estimating m by direct generates diagnostic reports. In Section 5, we introduce the p concept of the “analytic” DRM, including scheduling algo- astrometry on coronagraphic images, we calculate analytic rithm, merit function, union completeness (UC), and median design reference missions (DRMs) for the four missions under exposure times. In Section 6, we compare the baseline study. These DRMs assume perfect knowledge of the RV performance of four missions sharing the goal of determining orbital parameters for each exoplanet. In Section 7, we relax m for known RV planets. Section 7 investigates the negative this assumption and investigate the effect of current uncertain- p impact on estimating mp due to the current levels of uncertainty ties in orbital parameters.