2.2 Radial Velocity Follow-Up
2.2.3 Joint Fitting
Once RV follow-up has been obtained, covering the full phase of the exoplanet candidate’s orbit (either by intense sampling during one planetary orbit or less-intense sampling over several orbits), the planetary properties need to be determined. This can be done individually for each set of data – modelling the light curve to obtain a set of parameters which can be fed as priors into the modelling of the RVs, whose results can then be fed back into the light curve model.
Essentially iterating between the two until they converge to one set.
On the other hand, it is possible to tackle both the light curve and RVs simultaneously by joint-fitting. In essence, this takes both data sets, some known or estimated stellar properties and attempts to fit everything at once (for example with a Monte Carlo Markov Chain). By taking
vr(t) =K [cos(2πφ(t)+ω)+ecosω]+vr,∗ (2.1) where
K = 2πG
Porb
1/3 Mpsini
M∗+Mp2/3 1
1−e21/2 (2.2)
which describes the planetary orbit through the changes in RVs and combines them with complex, analytic light curve models (such as those in Mandel & Agol 2002;Giménez 2006;
Abubekerov & Gostev 2013;Kreidberg 2015). The light curve models take not only the transit shape (based on planetary radius, orbital distance and period etc.), but the models also consider the stellar properties such as limb darkening in different forms (these typically depend on the spectral type of the star).
There are various tools and packages for performing joint-fitting analysis, but two popular choices are the EXONAILER andEXOFAST (two versions available) packages. EXONAILERfrom Espinoza et al.(2016) is a straightforward python package which can fit either a light curve or RVs or both. EXONAILERis also able to fit using different telescope observations for both photometry and RVs, by allowing each to have their own set of instrument-based parameters (e.g.
‘jitter’ term, absolute radial velocity value μetc.). It uses the light curve modelling provided by the packageBATMAN(Kreidberg 2015) and RV fitting with RadVel (Fulton et al. 2018), all performed through an MCMC package, emcee (Foreman-Mackey et al. 2013). EXONAILERdoes not require stellar parameters or return any stellar parameters (though to assist in defining the transit shape you can provide the stellar density if known).
EXOFAST(Eastman et al. 2013) is different by not only performing joint-fitting between the light curve and radial velocity (though, like EXONAILERit is possible to fit each individually),
it also investigates and determines likely stellar parameters. To determine stellar parameters EXOFASTneeds estimates of the stellar effective temperature and the metallicity, which can be measured from the stellar spectra. For the light curve modelling, EXOFAST uses the analytic calculation fromMandel & Agol(2002), with limb darkening defined by the work fromClaret
& Bloemen (2011). The stellar parameters are determined using the results ofTorres et al.
(2010). Although coded in IDL, a version of EXOFASThas been made available as an online resource at the NASA Exoplanet Archivea. More recently, a new version ofEXOFASThas been made available (Eastman 2017) which has the added ability of handling multi-planet systems and analysing single-transit- event light curves.
ahttps://exoplanetarchive.ipac.caltech.edu/cgi-bin/ExoFAST/nph-exofast
Chapter 3
Application to K2
‘Science is a quest for understanding.’
Prof. Dame Jocelyn Bell Burnell
3.1 Transit Detection for the K2 Mission
The methods described in Chapter2were applied to theK2mission, the follow-up mission of Keplerafter the second of four reaction wheels failed. K2observed inCampaignsof different regions of the sky along the ecliptic for 80 days at a time (as opposed to the 4 years which Keplerobserved for continuously).
For the analysis ofK2data, all of the steps detailed in Chapter2were followed. All targets labelled ‘Stars’ withlong cadence light curves (i.e. 29.4 minute sampling) were downloaded when each Campaign had its public data release. The light curves were flattened (see Sec-tion2.1.2) and had significant outliers removed (using the function which fitted the flattening polynomial) – the majority of these outliers were due to instrumental causes. The flattened light curves were then analysed by BLS (Section 2.1.1). For any which showed to have a significant SNR, they were reanalysed by the BLS at a higher resolution around the detected orbital period, and a diagnostic document was created (and stored) containing details such as:
right ascension and declination; stellar magnitudes and other stellar properties such as radius and effective temperature (if known); the BLS output such as orbital period, transit duration
51
Table 3.1:A table detailing important details of each Campaign.
Campaign RA Dec
Length of Campaign [days]
Number of Targets†
Number of BLS Results
Number of Final Targets 11a/b* 17h21m33.1s -23◦58’33.45" 23.3/47.7 27080 1704/1695 6
12 23h26m38.1s -5◦6’8.4" 78.9 26392 4992 11
13 4h51m11.3s 20◦47’13.47" 80.6 21339 3457 18
14 10h42m43.7s. 6◦51’3.35" 79.7 18205 6466 16
15 15h34m28.2s. -20◦4’45.26" 88.0 22579 5136 31
16 8h54m50.3s. 18◦31’31.42" 79.6 22880 5582 20
17 13h30m11.9s. -7◦43’15.87" 67.1 28551 5278 10
18 8h40m38.6s 16◦49’40.31" 50.9 18482 4963 37
* C11 was split into two due to an error in the initial roll-angle used to minimise the solar effect.
†Number of ‘Stars’ observed with Long Cadence per Campaign.
and depth; as well as plots of the different stages of light curve analysis (raw, flattened, outliers removed), of the BLS power spectrum and the folded light curve showing the transit (including two zoom-ins of the odd and even transits). An additional step was included after the analysis was complete, which was to rank the results by their SNR – higher SNR (and therefore stronger candidates) were given higher ranks than those with low SNR. Also, to aid in the follow-up, stars which had Kepler magnitudes of 14 or less were deposited into an additional directory.
For swift follow-up with the Swiss-Euler telescope, targets with magnitudes of 14 or less were the only instrumentally achievable options.