A Uniformly Derived Catalogue of Exoplanets from Radial Velocities

0.74 MB·English

by  Morgan D. J. Hollis

Mon.Not.R.Astron.Soc.000,1–9(2011) Printed4January2012 (MNLATEXstylefilev2.2)
A Uniformly Derived Catalogue of Exoplanets from Radial
Velocities
Morgan D. J. Hollis 1(cid:63), Sreekumar T. Balan2†, Greg Lever1‡, and Ofer Lahav1§
1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
2 2Astrophysics Group, Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, UK
1
0
2
4January2012

n
a
J ABSTRACT

3 A new catalogue of extrasolar planets is presented by re-analysing a selection of pub-
lished radial velocity data sets using exofit (Balan & Lahav 2009). All objects are

] treated on an equal footing within a Bayesian framework, to give orbital parameters
P
for 94 exoplanetary systems. Model selection (between one- and two-planet solutions)
E
is then performed, using both a visual flagging method and a standard chi-square
.
h analysis,withagreementbetweenthetwomethodsfor99%ofthesystemsconsidered.
p The catalogue is to be made available online, and this ‘proof of concept’ study may
- be maintained and extended in the future to incorporate all systems with publicly
o
available radial velocity data, as well as transit and microlensing data.
r
t
s Key words: planetary systems - stars: individual, methods: data analysis
a
[

1
v
1 INTRODUCTION significantnumberofknowncompanionstomakeinferences
8
about the correlation between orbital elements. Early dis-
6 Since the discovery of the first extrasolar planet in
6 cussions on this subject can be found in a series of articles
1995 (Mayor & Queloz 1995), the research on extrasolar
0 on the statistical properties of exoplanets by Udry et al.
planetshasundergoneexponentialexpansion.Awiderange
. (2003); Santos et al. (2003); Eggenberger et al. (2004) and
1 of search methods have been developed during this period,
Halbwachs et al. (2005). The statistical discussion in this
0 resulting in the discovery of more than 700 planets to date,
article is informed by the comparison to the published cat-
2 the majority of which have been from the radial velocity
1 alogues at http://www.exoplanet.eu (Schneider et al. 2011)
method. Traditional data reduction methods use a peri-
: and http://exoplanets.org (Wright et al. 2011).
v odogram (Lomb 1976; Scargle 1982) to fix the orbital pe-
The rest of this article is structured as follows: in Sec-
Xi riodandthentheLevenberg-Marquardtminimisation(Lev- tions 2 and 3 the Bayesian framework and exofit software
enberg 1944; Marquardt 1963) to fit the other orbital pa-
package are introduced. The data analysis pipeline is de-
r rameters. A catalogue of exoplanets has already been pub-
a scribedinSection4,modelselectionisdiscussedinSection5,
lished by Butler et al. (2006) using this method to extract
and the catalogue is presented in Section 6, and in the ta-
the orbital parameters of exoplanets. Recently, Bayesian
bles at the end of the article. The statistical properties of
MCMC methods have been introduced by Gregory (2005);
thedistributionsofvariousorbitalparametersarediscussed
Ford(2005);Ford&Gregory(2007)asareplacementforthe
in Section 7 and the results are summarised in Section 8.
traditionaldatareductionpipeline.exofit(Balan&Lahav
2009)isafreelyavailabletoolforestimatingorbitalparam-
eters of extrasolar planets from radial velocity data using a
Bayesian framework. Here are analysed 94 previously pub- 2 BAYESIAN FRAMEWORK
lisheddatasetsusingexofit,forminganew,uniformlyde-
TheBayesianframeworkprovidesatransparentwayofmak-
rived catalogue of exoplanets from a Bayesian perspective.
ing probabilistic inferences from data. It is based on Bayes’
Statistical properties of the distribution of orbital pa-
theorem, which states that for a given model H with a set
rameters are critical for explaining the planetary formation
of parameters Θ and data D, the posterior probability dis-
process. It has been argued that there is now a statistically
tribution of parameters Pr(Θ|D,H) is proportional to the
prior probability distribution Pr(Θ|H) times the likelihood
ofdata,Pr(D|Θ,H).Usingstandardmathematicalnotation
(cid:63) E-mail:[email protected] one can write,
† E-mail:[email protected]
‡ E-mail:[email protected] Pr(D|Θ,H)Pr(Θ|H)
Pr(Θ|D,H)= . (1)
§ E-mail:[email protected] Pr(D|H)
(cid:13)c 2011RAS
2 Morgan D. J. Hollis, Sreekumar T. Balan, Greg Lever, and Ofer Lahav
The denominator of the right hand side of the above use with Bayesian inference methods, as a radial velocity
equation is called the Bayesian Evidence. Since it is the es- data set will need to include at least half an orbital period
timationofparametersthatisofinteresthere,thistermcan of a potential planetary companion. Hence data sets with
beconsideredasanormalisingconstantandEquation1can notenoughmeasurementstogiveaccurateorbitalsolutions
be written as, werenotincluded.Also,theradialvelocitydataofanysys-
temswithmorethan2confirmedplanetswereignoredsince
Pr(Θ|D,H)∝Pr(D|Θ,H)Pr(Θ|H). (2) at present, exofit can only search for either one or two
ThekeystepintheBayesianapproachistoobtainthe planets.
posterior distribution of parameters accurately. The infer- Many more and different radial velocity data sets and
ences are then derived from the posterior distribution. The stellar mass estimates are now available (though some not
Markov Chain Monte Carlo (MCMC) method is a widely publicly), but for the sake of uniformity the original radial
employed technique for simulating the posterior distribu- velocitydatawereused(i.e.thosepubliclyavailable,frozen
tion (the left hand side of Equation 2). The basic steps in asof2009August21whentheoriginaldatawerecollected).
Bayesian parameter estimation can be summarised as fol- At a later stage the results can be improved by updating
lows: the original data sets to those which are now available, as
well as incorporating the data from the many hundreds of
(i) model the observed data, i.e. construct the likelihood
additionalplanetsthathavebeendetectedsincethestartof
function,
this study.
(ii) choose the prior probability distributions of parame-
To enable accurate calculation of the derivable orbital
ters,
parameters,themassesaswellastheradialvelocitiesofthe
(iii) obtain the posterior probability distribution,
associated stars were needed. These values were all taken
(iv) make inferences based on the posterior probability
from the published literature at http://www.exoplanet.eu,
distribution. frozen as of 2011 March 01. The input for exofit is in the
exofitisasoftwarepackagethatestimatestheorbital form of a simple text file with radial velocity, uncertainty,
parametersofextrasolarplanets,followingthestepsoutlined and the time of observation (in Julian Date format), where
above.Itshouldbenotedthatexofitdoesnotperformany theradialvelocityvaluesmustbeinms−1.TheJulianDate
Bayesianmodelselection-foradiscussionoftherelationof of the observation is offset to zero within exofit.
this aspect of the Bayesian framework to this study, the The publicly available statistical data analysis pack-
reader is directed to Section 5. age, R, from the R Project for Statistical Computing
(http://www.r-project.org), was used to analyse the output
of exofit. The output from R includes the mean, median
and standard deviation of the orbital parameters extracted
3 EXOFIT
fromtheposteriordistributionsamplesproducedbyexofit.
exofit (Balan & Lahav 2009) is a publicly available tool Themodalvaluesarealsoproduced,butwillonlyhavesig-
for extracting orbital parameters of exoplanets from radial nificanceintheeventoftheposteriorhavingmorethanone
velocitymeasurements.ItusestheMCMCmethodtosimu- peak.Posteriordistributionplotscanalsobeproducedwith
late the posterior probability distribution of the orbital pa- R,andthemarginaldistributionsofeachparametercanbe
rameters. The likelihood of data Pr(D|Θ,H) in Equation 2 found by plotting a histogram of the samples from the pos-
connectsthemathematicalmodeltotheobserveddata.The terior.Thefullposteriordistributionishelpfulinanalysing
radial velocity model and the corresponding Gaussian like- correlations between various parameters. Even though pa-
lihood function are given in Balan & Lahav (2009), where rameterdegeneracyispresentintheorbitalsolutions,highly
the choice of likelihood is based on Gregory (2005). The degenerate solutions are less common.
prior probabilities are as used by Ford & Gregory (2007). The calculation time required for exofit depends on
exofit then generates samples from the posterior distribu- computational resources available to the user. It scales lin-
tions of the orbital parameters in the mathematical model, early with the number of radial velocity entries input to
which can be analysed with the aid of any statistical soft- the code, and also depends significantly on the ease with
ware. Details of the algorithmic structure of the code, in- which exofit can converge the data. If exofit is pre-
cluding methods of controlling chain mixing and assessing sented with data from a non-converging posterior distribu-
convergence,canbefoundinBalan&Lahav(2009).Forfur- tion, it will take much longer than a larger data set with
therinformation,thereaderisdirectedtotheexofituser’s convergent orbital solutions. In technical terms, the mean
guide. average calculation time using exofit on 26 radial veloc-
ity data sets ranging from 10 to 50 data entries for a 1-
planet search was 44 seconds per data entry. For a 2-planet
search using 30 data sets with between 11 and 256 entries,
4 DATA ANALYSIS
the calculation time increased to 3 minutes and 40 sec-
As of 2009 August 21, when the data sets were ex- onds per data entry. These times were achieved on a 2.80
tracted from the literature (Butler et al. (2006), and ref- GHz dual core linux system. Multiple runs were performed
erences therein), there were 295 planetary systems de- in order to confirm the orbital solutions for each system
tected using the radial velocity method according to - these analyses were carried out using the UCL Legion
http://www.exoplanet.eu, with 346 individual planets in to- High Performance Computing Facility, details of which can
tal. Some radial velocity data sets from the literature have be found at http://www.ucl.ac.uk/isd/common/research-
fewer than ten entries and as such are not appropriate for computing/services/legion-upgrade.
(cid:13)c 2011RAS,MNRAS000,1–9
A Uniformly Derived Catalogue of Exoplanets from Radial Velocities 3
5 MODEL SELECTION 2-planetfitsforallsystemsanalysed.Theorbitalparameters
used to fit to the model were the systematic velocity offset
One of the most challenging aspects of the statistical in-
ofthedataV,theorbitalperiodoftheplanetT,theradial
ference procedure is the model selection problem. For the
velocitysemi-amplitudeK,theorbitaleccentricitye,thear-
analysisoftheradialvelocitydatathequestionofmodelse-
gument of periastron ω, and χ, parameterising the fraction
lectionreferstotheselectionofthecorrectnumberofplan-
of the orbit at the start time of the data along which the
etstofittheobserveddata.Ford&Gregory(2007);Gregory
planet has travelled from the point of periastron passage.
(2007)employedthermodynamicintegrationforcalculating
The final parameter, s, is a measure of all extra signal in
the Evidence and selecting the optimal number of planets
the data after the planetary fits have been accounted for,
that fit the data. On the other hand, Feroz et al. (2011b,a)
and hence a high value could indicate the presence of an
approached the situation as an object detection problem.
additional planet, or noisy data due to stellar activity, or
One of the most commonly employed model selection
the combined noise from all sources. The reader is referred
proceduresmakesuseofthechi-squarestatistic.Thisisone
toBalan&Lahav(2009)foramorecompletedescriptionof
ofthemostprominentmethodsforestimatingthegoodness
this parameter, which is not considered in any more detail
offitandithasbeenappliedtomanyastronomicalproblems
in this study. It should also be noted that the orbital pa-
includingtheanalysisofradialvelocitydata(seee.g.Butler
rameter χ is not related to the statistical measure χ2 used
et al. (2006)). Bayesian inference also offers a straightfor-
for determining the log likelihood ratio in Equation 3.
wardwayofperformingstatisticalmodelselection,basedon
The direct output values from exofit shown in the
Equation1andtheEvidence.Eventhoughthisapproachis
tables are the medians of the parameter posterior distri-
conceptually simple, its implementation is in general com-
butions, and the associated 68.3% confidence regions. The
putationallyexpensive,andexofitdoesnotcurrentlyhave
other displayed derivable parameters of the systems (mass
thefunctionalitytoperformsuchBayesianmodelselection.
and semi-major axis) were calculated by transforming the
Hence, in the present analysis, we make use of the tra-
orbital parameter posteriors using the standard relations,
ditional chi-square statistic as well as a visual flagging ap-
proach, discussing the relationship between the two in Sec- a = m∗a∗sini, (4)
tion8.Welimitourselvesto1or2planets,asperthecurrent p m sini
p
capabilitiesofthecode,butthismayofcoursebeextended
and,
in later studies. The rationale behind the visual flagging is
√
thatonecanidentifythepoorfitstothedatabycomparing K m2/3T1/3 1−e2
m sini ≈ ∗ ∗ , (5)
thepredictedradialvelocitycurvesforthe1-planetandthe p (2πG)1/3
2-planet solutions. The method involves assigning a ‘visual
assuming m (cid:28) m . The final values for these deriv-
quality flag’ by eye to each system, where ‘1’ signifies that p ∗
able quantities were again taken to be the medians with
the 1-planet fit is best, ‘2’ means that the 2-planet fit is
68.3% confidence regions, and are also displayed in the ta-
best,and‘3’meansthatboth1-and2-planetsolutionspro-
bles.
vide equally good (or equally bad) fits. The results of this
classification are shown in Table 6, next to the number of
planets currently confirmed to exist in that system, taken
by comparing the values on both http://www.exoplanet.eu 6.1 Choice of priors
and http://exoplanets.org (as these catalogues do not agree
The prior distributions and ranges used for the initial anal-
with each other in some cases).
ysiswereasshowninTables1and2.Thepriorforthesys-
Table6alsoshowstheloglikelihoodratioofthereduced
tematic velocity was dependent on the system - the mean
chi-square value of the 1-planet fit to that of the 2-planet
of the input RV data was calculated and used as the initial
fit, where we define the log likelihood ratio, value, and the allowed range was 10kms−1 symmetrically
1 about this. For some systems different sets of prior bound-
R ≡ − (χ2 − χ2 ) (3)
2 1p 2p aries were used in a second round of analysis - these stars
and the prior boundaries applied are listed in Table 3. Sys-
Hence a value of R > 0 indicates that the 1-planet fit
tems which did not return good fits using the normal prior
wasbest(hadasmallerreducedchi-squarevalue),andR <
boundaries were re-run with these ‘tight’ priors, where the
0 indicates that the 2-planet model provided the best fit to
period of the planet was constrained to be within a range
the data. For all bar one system (HD8574), every ‘1’ and
given by,
‘2’ quality flag assigned to the fits by eye was in agreement
with the calculated value of R, endorsing our method of
T ∈ [T −2σ ,T +2σ ], (6)
pub pub pub pub
assignation by visual inspection (see Figure 5). For only a
few systems there were not sufficient degrees of freedom to where Tpub is the published value of the period and
calculateavalueforR(duetoe.g.onlyhaving11datapoints σpub is the published error on the period, both taken from
for the 12 parameters), denoted by ‘-’ in the table. http://www.exoplanet.eu on 2011 August 01. Further con-
straintsmayalsobeapplied,forexample,systemswithnear
zeroeccentricitiesrequiretightpriorsontheorbitalparam-
eter χ in order to avoid multimodal distributions (see Sec-
6 CATALOGUE OF EXTRASOLAR PLANETS
tion8),whilstsystemswitheccentricitiesclosetounityneed
In this paper the catalogue of extrasolar planets generated tightpriorsontheorbitalperiodT inordertoachievecon-
using exofit is presented in Tables 4 and 5. These contain vergence of the MCMC chains. Examples of the output of
thebestestimatesoftheorbitalparametersforboth1-and exofit are shown in Figures 1 and 2.
(cid:13)c 2011RAS,MNRAS000,1–9
4 Morgan D. J. Hollis, Sreekumar T. Balan, Greg Lever, and Ofer Lahav
Table 1.Theassumedorbitalparameterpriordistributionsand Table3.Radialvelocitydatasetsanalysedwithdifferentperiod
their boundaries for a 1-planet model. The min and max values priorboundaries,forreasonsexplainedinSection6.2.Theinitial
forthesystematicvelocityparameterwerethemeanvalueofthe value is set to the published value of the period, the maximum
rawradialvelocitiesforthatdatafileminusandplus5000ms−1 value is the initial value plus twice the published error, and the
respectively. minimumistheinitialvalueminustwicethepublishederror.This
approach was generally necessary for those systems (e.g. WASP
and XO data sets) where the number of datapoints available at
thetimeofselectingthedatawaslow,thusrequiringtighterpriors
Parameter Prior Mathematicalform Min Max
toadequatelyconstrainthesolution.
V(ms−1) Uniform 1 - -
Vmax−Vmin
T1(days) Jeffreys (cid:18)1 (cid:19) 0.2 15000
T1ln TT11mmainx System Initialperiodvalue Minperiod Maxperiod
epsilonEri 2500 1800 3200
K1(ms−1) Mod.Jeffreys (cid:16)(K1+K10)−1 (cid:17) 0.0 2000 gammaCep 906 899.84 912.1
ln K10+KK110max GGJJ84896 11950.70649 11450.706412 21450.706568
e1 Uniform 1 0 1 HAT-P-9 3.92289 3.92281 3.92297
ω1 Uniform 21π 0 2π HD118203 6.1335 6.1323 6.1347
χ1 Uniform 1 0 1 HD12661 262.71 262.54 262.88
s(ms−1) Mod.Jeffreys (cid:16)(s+s0)−1 (cid:17) 0 2000 HD128311 924 913.4 934.6
ln s0+ss0max HD1H3D1616442 1395500 314826.88 325073.22
HD149143 4.072 4.058 4.086
HD162020 8.42820 8.428088 8.428312
HD168443 58.1121 58.111142 58.113058
HD169830 225.6 225.16 226.04
HD183263 627 624.8 629.2
Table 2.Theassumedorbitalparameterpriordistributionsand HD187123 3.096583 3.09656732 3.09659868
theirboundariesfora2-planetmodel.TheboundariesforV were HD189733 2.2185757 2.2185754 2.2185760
asdetailedpreviously. HD190360 2920 2862.2 2977.8
HD196885 1330 1300 1360
HD202206 255.87 255.75 255.99
HD20868 380.85 380.67 381.03
HD209458 3.5247486 3.52474784 3.52474936
Para. Prior Mathematicalform Min Max
HD217107 7.12682 7.1267318 7.1268882
V(ms−1) Uniform 1 - - HD219828 3.833 3.807 3.859
Vmax−Vmin HD28185 379 375 383
T1(days) Jeffreys (cid:18)1 (cid:19) 0.2 15000 HD330075 3.38773 3.38757 3.38789
HD33636 2128 2111.6 2144.4
T1ln TT11mmainx HD38529 2146 2134.98 2157.02
HD46375 3.02357 3.02344 3.0237
K1(ms−1) Mod.Jeffreys (cid:16)(K1+K10)−1 (cid:17) 0.0 2000 HHDD4570543969 4234060 02.3284.2 8265035.8
ln K10+KK110max HD5319 670 636 704
e1 Uniform 1 0 1 HD68988 6.2771 6.27668 6.27752
ω1 Uniform 21π 0 2π HHDD7734216576 12256200 12244960 12257540
T2(χda1ys) UJenffifroeryms (cid:18)11 (cid:19) 00.2 150100 HHDD7850268096 31.1510.9423767 31.1510.94137529 31.1510.94339785
T2ln TT22mmainx HD86081 2.1375 2.1371 2.1379
HD89307 2170 2094 2246
K2(ms−1) Mod.Jeffreys ln(cid:16)(KK22+0K+KK22020)m−a1x(cid:17) 0.0 2000 HItPaTu7rE5B4So5-o38 315..133111.21406 5301..3201.9224232 235..163121.2247888
e2 Uniform 1 0 1 WASP-2 2.152226 2.152218 2.152234
ωχ22 UUnniiffoorrmm 211π 00 21π WAXSOP--31 13..89446185334 31..98441648736 31..984416588348
s(ms−1) Mod.Jeffreys (cid:16)(s+s0)−1 (cid:17) 0 2000 XXOO--24 24..611255803883 24..611255802725 24..611255805941
ln s0+ss0max
though,andmanysystemswerethenre-runwithtightpriors
6.2 Ambiguous systems
on the period, given in Table 3. Some of these ambiguities
For some of the systems analysed there is a clear trend in were caused by data which were poor, or less accurate due
the radial velocities indicating the possibility of a second toage,ortoonoisyduetostellarjitter.Othersweresimply
planet,butthedataarenotinformativeenoughtoproperly due to the correlation between ω and χ, or data not good
constraintheorbitalparametersofsuchanobject.Plotting enough to constrain these two parameters. This resulted in
the resulting radial velocity curve and judging by eye can near-uniform posteriors for ω and χ, and hence fits that
help to assess and distinguish between the 1- and 2-planet match few of the measured datapoints as a result of being
fitsandevaluatethevalidityoftheorbitalsolution,though shifted in time. Estimates for masses and semi-major axes
such poorly-constrained orbits will lead to large error bars derived from these results are still valid however (provid-
on the estimates of the orbital parameters. ing reliable estimates for T, K and e are obtained, which
This did not always even lead to a clear classification wasgenerallythecase),asthesevalueshavenodependence
(cid:13)c 2011RAS,MNRAS000,1–9
A Uniformly Derived Catalogue of Exoplanets from Radial Velocities 5
Figure 1. The resulting marginal posterior distributions of the Figure 2. The resulting marginal posterior distributions of the
orbital parameters for a 1-planet fit to the BD-17 63 data, and orbital parameters for a 2-planet fit to the HD108874 data, and
thecorrespondingradialvelocitycurve. thecorrespondingradialvelocitycurve.
X2 s
s 0246 0.000.050.100.150.200.25 0.00.4 0 1 2 3 4
0.00.20.4 2 4 6 8 10 0510 e2 0.00.61.2 w2
w1 X1 0.000.050.100.150.200.25 0.0 0.5 1.0 1.5
Density 0.000.150510 5.0 5.1K1 5.2 5.3 0206004080120 0.42 e01.43 0.44 Density 0.01.02.00.00.20.4 2.5392 33.904wT321936.53984.0400 024680.00.2 03.04 360.13XK81204.02 420.344
165 170 175 180 0.52 0.53 0.54 0.55 0.56 0.57 K1 e1
0.2 V 0.4 T1 0.00.20.4 02468
0.03020 3025 3030 0.0 653 654 655 656 657 658 0.00.20.4 14 1146 1168V1280 2220 2242 0.0000.010 105.010 106.020T1017.300 108.400 109.500
(a) Posterior distributions of orbital parameters for BD-17 (a) Posterior distributions of orbital parameters for
63. HD108874.
31003150 llllll 60 l ll
l ll l l
l l l l
3050 l l l l 40 lll
Velocity (m/s) 29503000 Velocity (m/s) 20 l llll lllll lll
0 l lll
28502900 l ll llllll −20 lll l lll l ll ll lllllll
lll ll
53000 53500 54000 54500 11500 12000 12500 13000 13500
Time (days) Time (days)
(b) RadialvelocityplotforBD-1763. (b) RadialvelocityplotforHD108874.
on mean anomaly at epoch and the time evolution of the at very small values for T and K, and uniform for e, ω and
Keplerian orbit. χ (i.e. there is no single solution for a second planet from
The class 3 (both 1- and 2-planet fits equally good, or these data). From this an ‘Occam’s Razor’ approach could
equally bad) systems, as introduced in Section 5, are those be taken and the assumption made that the correct model
which were considered to be somewhat ambiguous even af- formostofthese‘3b’classsystemsisinfactthesingleplanet
ter being analysed with tighter priors. This category was one.Inafewcasesthoughtheremaytrulybeasecondplanet
subdivided further - in some cases these are distinct radial present, and the data used are simply not good enough to
velocity solutions which provide plausible fits for both 1- change the likelihoods of the parameters from the initial
and 2-planet models, and are classified as ‘3a’. However, ‘no-knowledge’ (uniform prior) situation. So for all class 3
therearealsosystemswherethe‘secondplanet’fitjustpro- systems,better(oratleastmoreup-to-date)dataandmore
duces small-amplitude variations on the 1-planet solution complete analyses (such as using the log likelihood ratio in
(see Figure 3 for an example), or where the 1- and 2-planet more detail to narrow down the classification) are required
fitsareidenticalbutthe‘secondplanet’posteriorsarepeaks to accurately determine the correct orbital solution.
(cid:13)c 2011RAS,MNRAS000,1–9
6 Morgan D. J. Hollis, Sreekumar T. Balan, Greg Lever, and Ofer Lahav
thefinalparametervaluesfromvaryinganalysistechniques.
Figure 3. The resulting radial velocity curve fits using the de-
rivedorbitalparametersforHD89307.Imposinga2-planetmodel This highlights the value of using a consistent technique to
on data with only one planet can have the result shown here, build up reliable databases of orbital parameters.
where the 2-planet fit is the same as the 1-planet fit with a su- Asmassandsemi-majoraxisvaluesarethemselvesde-
perimposedartificialsmall-amplitudeperiodicvariation. rivedfromperiodandeccentricity,anyinaccuraciesinalgo-
rithms used will propagate, and also present discrepancies
inthevaluesyieldedusing exofitandarelikelytoamplify
outliersintheseplots.Theseoutlierswillbeinvestigatedin
the future in order to assess the validity of the solutions.
There are some discrepancies in the global distribution
40 of parameter values between this catalogue and the pub-
l
lished literature, especially for the eccentricity parameter.
l
This may be partly due to poor or out-dated data, and is
20 l almostcertainlyaffectedbytheubiquitouseffectsofcertain
m/s) ll parameter correlations (as explained in Section 8). These
Velocity ( 0 l snhiqouuelds dbeevealnoapleydsetdoienxpmloorreetdheetapilarianmtehteerfustpuarcee, amnodreteecffih--
l l ciently and minimize or eradicate such dependencies.
−20
l
−40 l l l 8 DISCUSSION
Theprimaryobjectiveofthisarticleistoanalyseradialve-
11000 11500 12000 locity data sets uniformly, using a single platform for the
data analysis. Butler et al. (2006) produced a catalogue
Time (days)
of extrasolar planets using traditional methods (using pe-
(a) 1-planetradialvelocityplotofHD89307.
riodograms and Levenberg-Marquardt minimisation). We
have analysed a selection of radial velocity data sets us-
ingaBayesianparameterestimationprogramforextrasolar
planets. However, a model selection criteria is required for
completion of the statistical inference process, and for this
purpose,asdescribedinSection5,achi-squarestatisticwas
40
l employedaswellasavisualflaggingtechnique.Inconclusive
l resultsareobtainedforafewdatasets,butfromthoseanal-
20 ysedhereitcanbeseenthatbothmodelselectionmethods
l
m/s) ll pFeigrfuorrem5.well, agreeing in 99% of cases, as demonstrated in
Velocity ( 0 l ues dInevpeesntdigaotnintghefuprtohinetr,ewsteimfiantdesthoaftththeeocrhbii-tsaqlupaarreamvael--
l l
−20 ttehrespuossetdertioorcdonissttrriubcuttitohneipsruendiimctoeddarlasduicahlvaenloacpitpyrocaucrhvew.iIlfl
work flawlessly. However, posterior distributions of the or-
l
−40 l l l bitalparametersexhibitmulti-modalityonmanyoccasions.
For example the parameters ω and χ are extremely corre-
latedandtheirposteriordistributionsarebimodalformany
11000 11500 12000 datasets(anexampleofthisisshowninFigure6),especially
Time (days) for planets with e ≈ 0. This problem has also been noted
(b) 2-planetradialvelocityplotofHD89307. byGregory(2007),whoproposedre-parameterisationofthe
problem as a possible way of dealing with this situation.
Many data sets contain planetary signals whose period
isgreaterthanthespanoftheobservations,andsoobtaining
7 COMPARISON OF ORBITAL PARAMETERS
constraints on the orbital parameters of these objects is an
Figure4showsvaluesforspecifiedorbitalelementsfromthe extremelydifficulttask.Thereareseveraldatasetswhereit
literature against values yielded using exofit for each sys- was possible to obtain estimates for the orbital parameters
tem.Themass,semi-majoraxisandperiodvaluesallexhibit for one of the planets, but then the second signal could not
good correlations in general between the independently de- be constrained due to weak signal-to-noise. In most cases
rived values and those in the published literature - this is these signals appear to be a linear or quadratic trend in
unsurprising for the period as it has not been derived from the radial velocity data. Therefore, it becomes extremely
other quantities. The eccentricity however shows a greater difficult to classify these objects as planets, and this is one
spreadthanexpected-whereasouruniformlyderivedcata- of the reasons a visual flagging method was employed. One
logueisconsistentintakingthemedian,thepublishedvalues example of this is shown in Figure 6, the results of the 2-
use,ingeneral,manydifferentstatisticalmeasurestoextract planetfittothedataofthesystemHD190228.Thestrongest
(cid:13)c 2011RAS,MNRAS000,1–9
A Uniformly Derived Catalogue of Exoplanets from Radial Velocities 7
Figure 4.Orbitalparametervaluestakenfromhttp://www.exoplanet.eu,plottedagainstvaluesyieldedusing exofit.Plottedsystems
areonlythosewhereexofitgaveunambiguous(eitherclass1or2)results.
2000 3
1800
2.5
1600
1400
2
1200
ExoFit 1000 ExoFit 1.5
800
1
600
400
0.5
200
0 0
0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.5 1 1.5 2 2.5 3
Literature Literature
(a) Period(indays) (b) Semi-majoraxis(inAU)
8 1
7
0.8
6
5
0.6
ExoFit 4 ExoFit
0.4
3
2
0.2
1
0 0
0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1
Literature Literature
(c) Mass(inMJup) (d) Eccentricity
Figure 5.Theloglikelihoodratioforeachplanetassignedtovisualclass1or2isshownin(a).(b)showsthesamedataonasmaller
scale,aroundthethresholdatR = 0.Class2systems(opencircles)areallbelowthechi-squareambiguitythreshold,andclass1systems
(filledcircles)areallabove,withthesingleexceptionofHD8574(shownasaredtriangle),class1butlocatedjustbelowthethreshold
withavalueofR = −0.03.
1500 20
threshold threshold
class 1 class 1
class 2 class 2
1000 hd8574 15 hd8574
10
500
5
0
R R 0
-500
-5
-1000
-10
-1500 -15
-2000 -20
Planet Planet
(a) Log likelihood ratio with planet for all class 1 and 2 (b) Log likelihood ratio with planet for all class 1 and 2
systems. systems,intherange−20 ≤ R ≤ +20.
(cid:13)c 2011RAS,MNRAS000,1–9
8 Morgan D. J. Hollis, Sreekumar T. Balan, Greg Lever, and Ofer Lahav
Wehavealsofoundthatthealiasingeffects(seee.g. Dawson
Figure 6.Examplesofbimodalandambiguousposteriordensi-
ties,obtainedfromHD49674andHD190228. & Fabrycky (2010)) in observations can produce additional
peaksintheposteriordistributions,necessitatingtheuseof
the sharp prior on the period.
In summary, a brief overview of the Bayesian theory
hasbeengivenhere,alongwithadescriptionoftheMCMC
s approach used in order to estimate the orbital parameters
0.4 of extrasolar planets, more details of which can be found
0.2
0.0 in Balan & Lahav (2009). A new catalogue of extrasolar
2 4 6 8 planets is presented from the re-analysis of published ra-
w1 X1
Density 0.20.00.20.4 2 3 K41 5 6 7468100.01.02.03.0 0.2 0.4 0.6e1 0.8 1.0 dstsseoytiaamlsuntlpetdvimtoaenrilsodbscymfirtoeuayrddsui9denc4agtetosdabydoscsitthesehttim-sisna,qsggvuudiiaviessriurhneiavgblteeedcbctaohwottnenehgieqoan1ur-ueitsh,ananeigtfdiiosovorni2mlnu-mgptbileoagatnnoshiseosotd.fdooAraarngbneraideatcaetha--l
0.0 02 ment in 99% of the cases presented here. Improvements in
6 8 10 12 14 16 0.0 0.1 0.2 0.3 0.4 0.5 0.6 this ‘model selection’ area of the analysis may be made by
V T1
0.20.4 200400 t(a2k00in7g)ainntdoFaecrcoozunettaBl.ay(2e0si1a1nb,Eav);idmenorcee,riagsorsoeeunsainppGroreagcohreys
0.0 0 suchastheseareoutsidethescopeofthis‘proofofconcept’
−4 −2 0 2 4 4.942 4.944 4.946 4.948 4.950
study, but may be looked into in the future. Other further
workwillincludeupdatingthiscataloguetoincorporatethe
most up-to-date data, as well as extending exofit to be
(a)Posteriordensitiesfora1-planetfittotheHD49674data, abletousetransitandmicrolensingresults,tosearchforan
exhibitingsomebimodalityintheωandχorbitalparameter arbitrarynumberofplanets,andtolookintothepossibility
values. of accounting for interactions between planetary bodies.
X2 s
510 0.3
0 0.0 9 ACKNOWLEDGMENTS
0.35 0.40 0.45 0.50 0.55 0 2 4 6 8 10
e2 w2
02468 0246 TMhicehaaeultidheosr,s LwiosauldMelinkaehetom,thAanndkrePwauSltrGaonrgm,aRno,bRerotbeCrt.
0.4 0.5 0.6 0.7 0.8 0.9 4.0 4.5 5.0 5.5 Clouth,andMilroyTravassofortheirhelpinanalysingplan-
T2 K2
Density 0.0000.020 1050 110w011150 1200 0.000.06 100 150 X2010 250 300 edStetaunrdtysehnditapst,haSipTseaBtns.dacMaknHstouwidsleesndutgpsephsoipsrutfeprdopmobrytthafernoAmImsttrphoaepchIts/yaPsaieccrsrNGeenrwostutoupn-,
0.000.10 0.00.6 CetayveWnodlifsshonLaRbeosreaatrocrhy,ManerditOALwaacrkdn.oTwhleedgaeusthaorRsoaycaklnSoowcil--
0 2 4 6 0.0 0.2 0.4 0.6 0.8 1.0 edgetheuseoftheUCLLegionHighPerformanceComput-
K1 e1
0.08 1.0 ingFacility,andassociatedsupportservices,inthecomple-
0.00 0.0 tion of this work. This research has made use of the Exo-
0 50 100 150 0.0 0.2 0.4 0.6 0.8 1.0 planetOrbitDatabaseandtheExoplanetDataExplorerat
0.000.06 −50220 −V50180 −50140.00000.00150 0 5000T110000 15000 hatttph:tT/tp/h:e/ex/ocwapwtlaawnlo.eugtucs.le.oarwcg.i.lulkb/eexmopaldaenaetvsa/ielaxbolceatf.orpublicviewing
(b) Posterior densities for a 2-planet fit to the HD190228
data, showing ambiguity in the estimates for the ω, χ, and
evaluesforoneoftheplanets.
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(cid:13)c 2011RAS,MNRAS000,1–9
10 Morgan D. J. Hollis, Sreekumar T. Balan, Greg Lever, and Ofer Lahav
Table 4. Table of the orbital parameters for a 1-planet fit, both directly output from exofit and thence derived. The values of the
parametersT,K,eands(generatedfromexofit)arethemediansoftheparameterposteriordistributions,withtheassociated68.3%
confidenceregions.Theotherparameterswerecalculatedusingthesevaluesandstellarmassestakenfromthepublishedliterature.Note
thatsomeparametersareextremelywell-constrained,hencetheerrorsontheparameterestimatesaresosmallastoappeartobezero
tothetwodecimalplacesshowninthistable.Afulltableinmachine-readableformatwillbeprovidedonthewebsite,andthereaderis
directedthereifsuchdataarerequired.
planet m∗(Msol) T(days) K(ms−1) e s mp(MJup) a(AU)
BD-1763b 0.74 655.49+0.59 172.44+0.62 0.54+0.01 4.59+0.95 5.06+0.05 1.34+0.00
−0.62 −1.61 −0.01 −0.73 −0.05 −0.00
ChaHa8b 0.10 304.59+1.81 1221.89+186.78 0.15+0.15 32.67+130.79 8.60+1.12 0.41+0.00
−1.79 −128.02 −0.10 −30.48 −0.89 −0.00
epsilonErib 0.86 2503.68+57.36 17.83+1.93 0.16+0.16 9.44+0.91 1.05+0.10 3.43+0.05
−52.69 −1.81 −0.11 −0.82 −0.10 −0.05
epsilonTaub 2.70 597.53+12.02 96.16+3.93 0.13+0.04 8.85+2.48 7.66+0.30 1.93+0.03
−11.52 −3.85 −0.04 −1.95 −0.30 −0.02
gammaCepb 1.59 905.03+4.52 317.34+77.57 0.51+0.14 225.74+34.05 17.44+4.02 2.14+0.01
−3.68 −71.25 −0.16 −26.78 −4.15 −0.01
GJ3021b 0.90 133.70+0.20 167.02+3.87 0.51+0.02 15.86+2.34 3.36+0.08 0.49+0.00
−0.20 −3.95 −0.02 −2.05 −0.08 −0.00
GJ317b 0.24 672.33+8.26 90.96+46.14 0.45+0.20 15.63+4.38 1.36+0.40 0.93+0.01
−7.27 −12.10 −0.10 −3.16 −0.16 −0.01
GJ674b 0.35 4.69+0.00 9.50+0.99 0.11+0.10 3.55+0.58 0.04+0.00 0.04+0.00
−0.00 −1.02 −0.08 −0.46 −0.00 −0.00
GJ849b 0.49 2014.09+60.32 26.68+9.48 0.68+0.10 7.14+1.40 0.77+0.19 2.46+0.05
−61.27 −4.46 −0.09 −1.08 −0.11 −0.05
GJ86b 0.80 15.77+0.00 431.19+61.11 0.23+0.11 204.72+26.76 4.44+0.59 0.11+0.00
−0.00 −59.29 −0.12 −21.68 −0.59 −0.00
HAT-P-6b 1.29 3.85+0.00 115.69+3.99 0.04+0.04 8.73+3.25 1.06+0.04 0.05+0.00
−0.00 −4.17 −0.03 −2.46 −0.04 −0.00
HAT-P-8b 1.28 3.09+0.00 162.59+7.36 0.05+0.05 6.56+4.14 1.37+0.06 0.05+0.00
−0.00 −6.46 −0.03 −3.02 −0.05 −0.00
HAT-P-9b 1.30 3.92+0.00 84.50+10.56 0.12+0.14 4.09+9.61 0.77+0.09 0.05+0.00
−0.00 −9.37 −0.09 −3.40 −0.09 −0.00
HD101930b 0.74 70.58+0.40 17.99+0.89 0.08+0.05 1.92+0.65 0.30+0.01 0.30+0.00
−0.37 −0.91 −0.05 −0.46 −0.02 −0.00
HD108874b 0.95 395.16+5.60 34.93+3.82 0.05+0.08 13.00+1.65 1.21+0.13 1.04+0.01
−4.43 −3.57 −0.04 −1.38 −0.12 −0.01
HD11506b 1.19 1456.01+136.42 81.49+13.56 0.37+0.16 10.60+2.06 4.76+0.46 2.66+0.16
−85.10 −4.62 −0.10 −1.60 −0.23 −0.10
HD118203b 1.23 6.13+0.00 213.96+6.49 0.30+0.03 22.83+3.88 2.11+0.06 0.07+0.00
−0.00 −6.40 −0.03 −3.28 −0.06 −0.00
HD12661b 1.14 262.75+0.09 77.37+2.52 0.27+0.03 17.60+1.38 2.56+0.08 0.84+0.00
−0.13 −2.55 −0.03 −1.24 −0.09 −0.00
HD128311b 0.83 921.18+6.65 93.88+7.75 0.46+0.05 30.34+2.80 3.51+0.24 1.74+0.01
−5.13 −7.27 −0.05 −2.42 −0.24 −0.01
HD131664b 1.10 1976.18+32.94 356.10+24.90 0.64+0.02 5.11+0.79 18.03+0.85 3.18+0.04
−41.05 −18.59 −0.02 −0.66 −0.65 −0.04
HD132406b 1.09 1172.21+75.55 122.19+157.18 0.34+0.28 17.04+4.70 6.31+5.80 2.24+0.10
−49.55 −32.90 −0.19 −3.61 −1.47 −0.06
HD142b 1.23 344.05+2.12 32.00+7.12 0.19+0.16 20.29+2.53 1.24+0.24 1.03+0.00
−0.93 −6.14 −0.13 −2.15 −0.23 −0.00
HD142022b 0.90 1861.69+14.86 140.10+112.02 0.64+0.12 3.00+1.67 6.10+3.21 2.86+0.02
−13.47 −39.74 −0.09 −1.29 −1.36 −0.01
HD149143b 1.20 4.07+0.00 149.71+1.67 0.01+0.01 1.21+1.69 1.33+0.01 0.05+0.00
−0.00 −1.61 −0.01 −0.92 −0.01 −0.00
HD154345b 0.89 3332.50+84.05 14.10+0.84 0.05+0.05 2.84+0.37 0.96+0.06 4.20+0.07
−74.54 −0.85 −0.04 −0.32 −0.06 −0.06
HD155358b 0.87 194.26+0.88 31.86+1.98 0.21+0.06 9.69+1.03 0.81+0.05 0.63+0.00
−0.80 −1.97 −0.06 −0.89 −0.05 −0.00
HD162020b 0.80 8.43+0.00 1808.97+5.15 0.28+0.00 11.13+2.65 15.01+0.04 0.08+0.00
−0.00 −5.13 −0.00 −2.39 −0.04 −0.00
HD168443b 1.01 58.11+0.00 510.46+252.18 0.52+0.20 220.77+46.40 8.28+2.91 0.29+0.00
−0.00 −117.22 −0.18 −34.29 −1.94 −0.00
HD169830b 1.41 225.62+0.29 83.07+3.05 0.37+0.03 1.52+2.40 2.91+0.10 0.81+0.00
−0.31 −3.09 −0.03 −1.18 −0.10 −0.00
HD171028b 0.99 545.13+10.10 59.75+3.04 0.59+0.02 2.55+0.70 1.92+0.13 1.30+0.02
−12.19 −2.05 −0.02 −0.51 −0.10 −0.02
HD183263b 1.12 627.80+1.03 89.99+13.01 0.42+0.08 26.59+3.48 3.72+0.42 1.49+0.00
−1.64 −11.63 −0.09 −2.91 −0.41 −0.00
HD185269b 1.30 6.84+0.00 89.57+4.12 0.28+0.03 7.72+1.92 0.96+0.04 0.08+0.00
−0.00 −4.02 −0.04 −1.61 −0.04 −0.00
HD187123b 1.04 3.10+0.00 65.68+3.34 0.05+0.06 18.33+1.83 0.48+0.02 0.04+0.00
−0.00 −3.35 −0.04 −1.58 −0.03 −0.00
HD189733b 0.81 2.22+0.00 204.58+5.15 0.01+0.01 15.65+1.33 1.14+0.03 0.03+0.00
−0.00 −5.10 −0.01 −1.17 −0.03 −0.00
HD190228b 1.82 1141.21+15.40 92.26+4.58 0.52+0.04 1.23+1.87 6.07+0.17 2.61+0.02
−14.66 −3.48 −0.04 −0.94 −0.15 −0.02
HD190360b 0.98 2925.83+36.05 19.38+2.67 0.33+0.11 5.92+1.47 1.27+0.14 3.98+0.03
−41.53 −2.28 −0.10 −1.47 −0.13 −0.04
HD190647b 1.10 1038.09+5.27 36.78+1.19 0.17+0.02 0.97+0.70 1.92+0.06 2.07+0.01
−5.38 −1.17 −0.02 −0.65 −0.06 −0.01
HD195019b 1.02 18.20+0.00 270.12+1.54 0.01+0.01 10.42+1.19 3.54+0.02 0.14+0.00
−0.00 −1.55 −0.01 −1.14 −0.02 −0.00
HD202206b 1.07 255.90+0.06 585.94+6.24 0.42+0.01 30.18+2.61 17.41+0.16 0.81+0.00
−0.09 −6.13 −0.01 −2.31 −0.16 −0.00
HD20868b 0.78 380.79+0.13 97.02+7.97 0.61+0.04 32.81+4.04 2.31+0.18 0.95+0.00
−0.09 −7.95 −0.04 −3.35 −0.18 −0.00
HD209458b 1.13 3.52+0.00 84.33+0.87 0.01+0.01 3.34+0.69 0.69+0.01 0.05+0.00
−0.00 −0.87 −0.01 −0.67 −0.01 −0.00
HD212301b 1.05 2.27+0.00 56.31+5.83 0.08+0.08 17.70+3.65 0.37+0.04 0.03+0.00
−0.00 −5.94 −0.05 −2.73 −0.04 −0.00
HD217107b 1.11 7.13+0.00 140.71+2.35 0.15+0.02 22.70+1.23 1.41+0.02 0.08+0.00
−0.00 −2.41 −0.02 −1.12 −0.02 −0.00
HD219828b 1.24 3.84+0.01 3.11+7.83 0.58+0.32 15.98+2.97 0.02+0.05 0.05+0.00
−0.02 −2.55 −0.39 −2.27 −0.02 −0.00
HD221287b 1.30 458.77+8.13 69.77+8.66 0.11+0.10 10.16+1.97 3.14+0.38 1.27+0.01
−6.20 −6.77 −0.07 −1.50 −0.32 −0.01
HD224693b 1.30 26.73+0.03 39.92+1.52 0.04+0.04 1.92+1.07 0.70+0.03 0.19+0.00
−0.03 −1.53 −0.03 −1.10 −0.03 −0.00
HD23127b 1.13 1226.63+21.59 27.75+3.08 0.44+0.09 10.89+2.03 1.42+0.17 2.34+0.03
−21.71 −2.84 −0.10 −1.67 −0.16 −0.03
HD2638b 0.93 3.44+0.00 67.59+1.06 0.01+0.01 3.31+0.70 0.48+0.01 0.04+0.00
−0.00 −1.02 −0.01 −0.57 −0.01 −0.00
HD27442b 1.20 415.32+6.25 32.48+1.79 0.07+0.06 2.98+1.41 1.34+0.07 1.16+0.01
−5.74 −1.76 −0.04 −1.13 −0.07 −0.01
HD27894b 0.75 18.01+0.02 57.01+1.61 0.04+0.03 4.39+1.08 0.61+0.02 0.12+0.00
−0.01 −1.66 −0.02 −0.83 −0.02 −0.00
HD28185b 0.99 381.81+0.83 174.72+12.09 0.05+0.02 7.82+1.76 6.19+0.43 1.03+0.00
−1.32 −7.75 −0.02 −1.53 −0.28 −0.00
HD285968b 0.49 10.23+0.00 11.88+2.24 0.25+0.20 2.47+1.77 0.08+0.01 0.07+0.00
−0.00 −1.79 −0.17 −1.74 −0.01 −0.00
HD330075b 0.70 3.39+0.00 107.34+1.00 0.01+0.01 2.02+0.86 0.63+0.01 0.04+0.00
−0.00 −1.03 −0.00 −0.74 −0.01 −0.00
HD33636b 1.02 2127.74+11.40 389.98+156.81 0.90+0.03 0.57+0.69 11.02+2.19 3.26+0.01
−11.04 −152.41 −0.11 −0.42 −1.89 −0.01
HD3651b 0.88 60.36+0.04 9.60+1.91 0.54+0.15 7.73+0.68 0.14+0.02 0.29+0.00
−0.05 −1.60 −0.16 −0.61 −0.02 −0.00
HD38529b 1.48 2143.62+8.24 177.12+6.26 0.35+0.03 40.24+2.46 13.65+0.42 3.71+0.01
−6.24 −6.00 −0.03 −2.22 −0.42 −0.01
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