Planets Across Space and Time (PAST). I. Characterizing the Memberships of Galactic Components and Stellar Ages: Revisiting the Kinematic Methods and Applying to Planet Host Stars
Di-Chang Chen, Ji-Wei Xie, Ji-Lin Zhou, Su-Bo Dong, Chao Liu, Hai-Feng Wang, Mao-Sheng Xiang, Yang Huang, Ali Luo, Zheng Zheng
DDraft version February 23, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Planets Across Space and Time (PAST). I. Characterizing the Memberships of Galactic Componentsand Stellar Ages: Revisiting the Kinematic Methods and Applying to Planet Host Stars
Di-Chang Chen,
1, 2
Ji-Wei Xie,
1, 2
Ji-Lin Zhou,
1, 2
Su-Bo Dong, Chao Liu,
4, 5
Hai-Feng Wang, Mao-Sheng Xiang,
7, 8
Yang Huang, Ali Luo, and Zheng Zheng School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China Key Laboratory of Modern Astronomy and Astrophysics, Ministry of Education, Nanjing 210023, China Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China Key Lab of Space Astronomy and Technology, National Astronomical Observatories, CAS, 100101, China University of Chinese Academy of Sciences, Beijing, 100049, China. South-Western Institute for Astronomy Research, Yunnan University, Kunming, 650500, China; LAMOST Fellow National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China Max-Planck Institute for Astronomy, K¨onigstuhl 17, D-69117 Heidelberg, Germany South-Western Institute for Astronomy Research, Yunnan University, Kunming, 650500, China Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112
ABSTRACTOver 4,000 exoplanets have been identified and thousands of candidates are to be confirmed. Therelations between the characteristics of these planetary systems and the kinematics, Galactic compo-nents, and ages of their host stars have yet to be well explored. Aiming to addressing these questions,we conduct a research project, dubbed as PAST (Planets Across Space and Time). To do this, oneof the key steps is to accurately characterize the planet host stars. In this paper, the Paper I of thePAST series, we revisit the kinematic method for classification of Galactic components and extend theapplicable range of velocity ellipsoid from ∼
100 pc to ∼ ,
500 pc from the sun in order to cover mostknown planet hosts. Furthermore, we revisit the Age-Velocity dispersion Relation (AVR), which allowsus to derive kinematic age with a typical uncertainty of 10-20% for an ensemble of stars. Applying theabove revised methods, we present a catalog of kinematic properties (i.e. Galactic positions, velocities,the relative membership probabilities among the thin disk, thick disk, Hercules stream, and the halo)as well as other basic stellar parameters for 2,174 host stars of 2,872 planets by combining data fromGaia, LAMOST, APOGEE, RAVE, and the NASA exoplanet archive. The revised kinematic methodand AVR as well as the stellar catalog of kinematic properties and ages lay foundation for future studieson exoplanets from two dimensions of space and time in the Galactic context.
Keywords: (stars:) planetary systems — Galactic position and spatial motion — Galactic components— kinematic age — catalogs INTRODUCTIONIt has been a quarter century since the discovery ofthe first exoplanet. To date, over 4,000 exoplanets havebeen discovered and thousands of candidates are yetto be confirmed (NASA Exoplanet Archive, EA here-after; Akeson et al. 2013). There is a clear trend (shownin Figure 1, data from http://exoplanet.eu) that ourknowledge of exoplanets is expanding in the Galaxy. Be-
Corresponding author: Ji-Wei [email protected] fore 2005, most known exoplanets were confined in thesolar neighborhood with distance less than ∼ ∼ a r X i v : . [ a s t r o - ph . E P ] F e b Galactic velocity (e.g., McTier & Kipping 2019; Bashi& Zucker 2019).One of fundamental questions in studying exoplan-ets in the Galactic context is: what are the differencesin the properties of planetary systems at different po-sitions in the Galaxy with different ages? The answerof this question will provide insights on the formationand evolution of the ubiquitous and diverse exoplanetsin different Galactic environments. Aiming to address-ing the question, in a series of papers from here on, weconduct statistical studies of planets at different posi-tions in the Galaxy with different ages, a project thatwe dub as PAST (Planets Across Space and Time). discovery year d i s t a n ce ( p c ) Radial velocityTransitOther methods
Figure 1.
The distance of planet host stars to the Sunvs. the year of discovery. Planets discovered by differentmethods are plotted in different colours.
To this end, the first step is to figure out where theexoplanets are in the Galaxy. Specifically, for a givenexoplanet host star, we would like to know which Galac-tic component (i.e., the thin disk, the thick disk or thehalo) it belongs to. One of well-established methods todistinguish these Galactic components is the kinematicapproach as different components generally have differ-ent kinematic characteristics. For example, the thin diskhas a smaller vertical scale-height (Bovy et al. 2012a; Wuet al. 2018), but the thick disk is generally kinematicallyhotter with larger velocity dispersions (Gilmore et al.1989; Reddy et al. 2003; Bensby et al. 2003, 2014; Buderet al. 2018). By comparing the kinematic properties ofa given star to the typical kinematic characteristics ofa Galacic component, one may calculate the likelihoodthat the star belongs to this component (e.g. Bensbyet al. 2003). However, the kinematic characteristics ofthis method were obtained with data in the Solar neigh-borhood within ∼
100 pc (Bensby et al. 2003, 2014),and thus limiting this kinematic method to a relativelysmall range of area. Thanks to the recent large scale star surveys both from space (e.g., Gaia, Gaia Collabo-ration et al. 2016, 2018a,b) and ground (e.g., LAMOST,Wang et al. 1996; Su & Cui 2004; Cui et al. 2012; Zhaoet al. 2012; Luo et al. 2012), we are now allowed to ex-tend the kinematic method to beyond 1,000 pc in orderto characterize the majority of exoplanet host stars.The second step is to obtain the ages of exoplanethost stars since most of exoplanet host stars have no(accurate) age estimates. Stellar ages can hardly bemeasured but only be inferred or estimated indirectlythrough a number of techniques, which have their ownstrength and weakness (Soderblom 2010). For exam-ple, the widely-used isochrone placement method is ap-plicable for estimating ages of a large range of stars,but it usually suffers from relatively large uncertainty( ∼
50% typically) for main sequence stars, which arethe bulk of exoplanet hosts (e.g. Berger et al. 2020b).Asteroseismology is significantly better than any otherage-dating method, which can deliver age estimates forindividual stars with uncertainties of ∼ −
20% (e.g.Gai et al. 2011; Chaplin et al. 2014). However, thismethod requires observation with sufficiently accurate,high-cadence photometric measurements and it can onlybe applicable for stars with a limited range of spec-tral types that exhibit prominent oscillations. In ad-dition, the carbon and nitrogen abundances have beensuggested to be age indicators, but it is usually applica-ble for giant stars, and the reported age has achieved aprecision of ∼ REVISITING THE KINEMATIC METHOD TOCLASSIFY THE GALACTIC COMPONENTSIn this section, we revisit the kinematic method toclassify stars into different Galactic components (e.g.,thin/thick disks). The key is revising the characteristickinematic parameters (section 2.3), an extension fromthe solar neighborhood ( ∼
100 pc) to ∼ ,
500 pc inorder to cover most planet hosts as shown in Figure 1.2.1.
Space Velocities and Galactic Orbits
We calculated the 3D Galactocentric cylindrical co-ordinates (
R, θ, Z ) by adopting a location of the Sunof R (cid:12) = 8.34 kpc (Reid et al. 2014) and Z (cid:12) = 27 pc(Chen et al. 2001). The Galactic rectangular velocitiesrelative to the Sun ( U, V, W ) and their errors were cal-culated by the right-handed coordinate system based onthe formulae and matrix equations presented in Johnson& Soderblom (1987). Here, U is positive when point-ing to the direction of the Galactic center, V is positivealong the direction of the Sun orbiting around the Galac-tic center, and W is positive when pointing towards theNorth Galactic Pole. Cylindrical velocities V R , V θ , and V Z are defined as positive with increasing R , θ , and Z ,with the latter towards the North Galactic Pole. To ob-tain the Galactic rectangular velocities relative to thelocal standard of rest (LSR) ( U LSR , V
LSR , W
LSR ), weadopted the solar peculiar motion [ U (cid:12) , V (cid:12) , W (cid:12) ] = [9.58,10.52, 7.01] km s − (Tian et al. 2015).2.2. Classification of Galactic Components
We adopted the widely-used kinematic approach as inBensby et al. (2003, 2014) to classify the stars in oursample into different Galactic components, e.g., thinand thick disk stars. This method assumes that theGalactic velocities ( U LSR , V
LSR , W
LSR ) in different com-ponents (the thin disk, the thick disk, the halo, and the Hercules stream) follow a multi-dimensional Gaussiandistribution as f ( U, V, W ) = k × exp (cid:18) − ( U LSR − U asym ) σ U − ( V LSR − V asym ) σ V − W σ W (cid:19) , (1)where the normalization coefficient k = 1(2 π ) / σ U σ V σ W . (2)Here, σ U , σ V , and σ W are the characteristic velocitydispersions, and V asym and U asym are the asymmetricdrifts.For V asym , following Binney & Tremaine (2008) weadopted V asym = ¯ V θ − V c , (3)where V c is the circular speed of LSR as 238 km s − (Sch¨onrich 2012) and ¯ V θ is the mean value of azimuthalvelocities for a given component. For U asym , we adopted U asym = 0 for the disk and halo components and U asym = − − for Hercules stream component(Binney & Tremaine 2008; Bensby et al. 2014).The relative probabilities between two differentcomponents, i.e., the thick-disk-to-thin-disk ( T D/D ),thick-disk to halo (
T D/H ), the Hercules-to-thin-disk(
Herc/D ), and the Hercules-to-thick-disk (
Herc/T D )can be calculated as
T DD = X TD X D · f TD f D , T DH = X TD X H · f TD f H , (4) HercD = X Herc X D · f Herc f D , HercT D = X Herc X TD · f Herc f TD , (5)where X is the fraction of stars for a given component.Then for stars in our planet host sample, we calcu-lated their above probabilities, and classified them intodifferent Galactic components by adopting the same cri-teria as in Bensby et al. (2014), which are (1) thindisk: T D/D < . Herc/D < .
5, (2) thick disk:
T D/D >
T D/H >
Herc/T D < .
5, (3) halo:
T D/D >
T D/H <
Herc/T D < .
5, and (4)Hercules:
Herc/D >
Herc/T D >
Revision of Characteristic Kinematic Parameters
One of the important steps during the above classi-fication procedure is to obtain the characteristic kine-matic parameters for each Galactic component, i.e., σ U , σ V , σ W , U asym , V asym , and X . In the solar neighbour-hood within ∼
100 pc from the Sun, these parametersare available from Bensby et al. (2014), which are listin Table 1 here. However, as shown in Figure 1, thestars in our sample are located in a much wider zone
Table 1.
Characteristics for stellar components in the Solarneighbourhood from Bensby et al. (2014). † σ U σ V σ W U asym V asym X ———– [km s − ] ———–Thin disk 35 20 16 0 −
15 0.85Thick disk 67 38 35 0 −
46 0.09Halo 160 90 90 0 −
220 0.0015Hercules 26 9 17 − −
50 0.06 † σ U , σ V , and σ W are the velocity dispersions for the differentcomponents; U LSR and V LSR are the asymmetric drifts in U and V relative to the LSR; and X is the normalisation fractions for eachcomponent in the Solar neighbourhood (in the Galactic plane).Values are taken from for the thin disk, thick disk, the stellar halo,and the Hercules stream (Bensby et al. 2007, 2014). (up to several kpc from the Sun). It has been foundthat the velocity ellipsoids change with the Galactic po-sition (Williams et al. 2013). Therefore, the values ofthese characteristic kinematic parameters for each com-ponent should be revised and extended (section 2.3) sothat they are applicable for stars in a larger range of Z (e.g., | Z | < . . < R < . calibration sample To revise the values of characteristic kinematic pa-rameters, we rely on a calibration sample based on theLAMOST and Gaia data. The LAMOST main-sequenceturn-off and subgiant (MSTO-SG) star sample of (Xi-ang et al. 2017a) provides the estimates of stellar age,mass, and RV for 0.93 million Galactic-disk stars fromthe LAMOST Galactic spectroscopic surveys. The typ-ical uncertainty in age is 34%.To construct the calibration sample, we first cross-matched the above LAMOST MSTO-SG catalog withthe Gaia DR2 catalog. This was done by using the X-match service of CDS. We set a critical distance of 1.25arcseconds for position match. We carried out a magni-tude cut, which was set as the G magnitude differenceless than 2.3, to ensure our cross-matches were of similarbrightness. The G magnitudes for the LAMOST MSTO-SG stars were calculated by using the XSTPS − GAC g,r, and i color − color polynomial fits in Table 7 of Jordiet al. (2010). For stars with multiple matches, we keptthose with the smallest angular separations. After theabove cross-match, we have 863,663 stars left.We then applied the following filters to further cleanthe calibration sample.(1) Binary filter. We removed binary star sys-tems because their kinematics contain additional mo-tions (Dehnen & Binney 1998). This was done by choos- Figure 2.
Top panel: Galactocentric radius ( R ) vs. height( Z ) for the calibration sample. (8.34 kpc, 0.027 kpc) marksthe position of Sun. Bottom panel: Galactocentric radius( R ) vs. angle ( θ ) for the calibration sample. (8.34 kpc, 0)marks the position of Sun. ing stars flagged as ’Normal star’ (i.e., single and withspectral type of AFGKM) in the LAMOST MSTO-SGcatalog (Xiang et al. 2017a).(2) Parallax precision filter. Following Dehnen & Bin-ney (1998), we removed stars with relative parallax er-rors larger than 10 percent as reported in the Gaia DR2.(3) Age precision filter. We removed stars with ageolder than 14 Gyr or the error of age larger than 25%or blue straggler stars ( | Z | > . ∼ / (0 . / .
1) = 1 .
54 kpc. We therefore removedstars with distance larger than this limit. Such a cut at1.54 kpc also makes the distance distribution of the cal-ibration sample closer to that of the planet host sample(Figure 13).After applying the above filters, we are left with130,403 stars. Figure 2 shows the location of stars inthis calibration sample. As can be seen, these stars aremainly located at 7 . < R < . | z | < . binning and examining the calibration sample In order to calculate the characteristic kinematic pa-rameters for each Galactic component as a function of( R , Z ) in the Galaxy, we binned the calibration sampleas follows. For | Z | , we set 8 bins with boundaries at | Z | = 0, 0.1, 0.2, 0.3, 0.4, 0.55, 0.75, 1.0, and 1.5 kpc,resulting similar sizes ( ∼ ∼ R , we set 5 bins with boundaries at R = 7.5, 8.0, 8.5,9.0, 9.5, and 10 kpc. In total, there are 5 × R − Z space. Two of the grids ( R : 9 . − . | Z | : 0 . − . . − . < S , follows S = ( 23 σ tot2 ) / = [ 23 ( σ R + σ θ + σ Z )] / . (6)The result of this examination is shown in Figure 3. Ascan be seen, the calibration sample, either as a whole oras being divided into various grids, generally obeys theabove relation, indicating the sample is kinematicallyunbiased.2.3.3. classifying the calibration sample Next, we classify stars in the calibration sample intodifferent Galactic components. Following Bonaca et al.(2017), we identify halo stars if V tot = ( U + V + W ) / >
220 km s − . Stars with V LSR ≈ − ± − and ( U + W ) (1 / ≈ −
70 km s − are selected as the Hercules stream (Famaey et al. 2005;Bensby et al. 2007, 2014). We adopt the age-definedthin and thick disc components with a boundary at
40 50 60 70 80 σ tot (km s − ) S ( k m s − ) Whole calibration sample Subsamples
Figure 3.
The dispersion in proper motions, S , of the cali-bration sample as a function of the total dispersion in veloc-ity, σ tot . The black dashed line represents where S and σ tot obey Equation 6. Figure 4.
The Toomre diagram of the calibration sam-ple for different Galactic components. The diagram iscolour-coded to represent different components. Dashedlines show constant values of the total Galactic velocity V tot = 100 , , and 300 km s − . V tot (cid:46) − ) are in thin disk, while those withmoderate velocities ( V tot ∼ −
180 km s − ) are mainlyin thick disk (e.g. Feltzing et al. (2003); Adibekyan et al.(2013); Bensby et al. (2014). ).2.3.4. revising the velocity ellipsoid We then revise the velocity ellipsoid of each Galacticcomponent, namely calculate σ U , σ V , σ W , and V asym ofeach grid in the R − Z plane. Here, we revise those val-ues only for the thin and thick disk components. For thehalo and Hercules components, their numbers in each σ U ( k m s − ) σ V ( k m s − ) | Z | (kpc) σ W ( k m s − ) | Z | (kpc) | Z | (kpc) | Z | (kpc) | Z | (kpc) Thick diskThin disk
Figure 5.
The velocity dispersions as functions of position (R, Z) in the Galaxy for the calibration sample. The black line ineach panel denotes the result of the best fit of Equation 7 using the coefficients in Table 2. grid are too low for the revision, thus we adopt the ve-locity ellipsoid values as in Bensby et al. (2007, 2014) in-stead. The calculated values of σ U , σ V , σ W , and V asym for the thin and thick disk components are tabulated inTable 9 and visualized in Figure 5 and Figure 6.For velocity dispersion, according to Williams et al.(2013), it generally follows a simple formula: σ = b + b × R kpc + b × ( Z kpc ) km s − . (7)We therefore fit σ U , σ V , σ W in this formula. To ob-tain the uncertainty of each fitting parameter, we as-sumed that the Galactic velocity following the Gaussiandistribution N ( V, err V ), where err represents the cor-responding uncertainty. Then, we resampled Galacticvelocities based on these Gaussian distributions. Afterthat, we refit the resampled data in the formula of Equa-tion 7. We repeated the above resampling process 1,000times and obtained 1,000 sets of best fits. The uncertain-ties (one-sigma interval) of the fitting parameters are setas the range of 50 ± . σ U , σ V , σ W ) as a function of Z in each R bin. The best fitsare over-plotted as the black solid curves. The values of Table 2.
Fitting parameters of the velocity dispersion asfunctions of (
R, Z ), i.e. Equation 7. b b b σ D U . +1 . − . − . +0 . − . . +0 . − . σ D V . +0 . − . − . +0 . − . . +0 . − . σ D W . +1 . − . − . +0 . − . . +0 . − . σ TD U . +6 . − . . +0 . − . . +0 . − . σ TD V . +5 . − . − . +0 . − . . +0 . − . σ TD W . +1 . − . − . +0 . − . . +0 . − . fitting parameters and their one-sigma uncertainties aresummarized in Table 2. As expected, velocity disper-sions generally increase with Z for both thin and thickdisks in all the R bins.For asymmetric velocity, according to Robinet al. (2003); Binney & Tremaine (2008),it generally follows the relation V asym = σ U [km s − ] -30-25-20-15-10-50 V a s y m ( k m s − ) Thin disk σ U [km s − ] -70-65-60-55-50-45-40-35-30 V a s y m ( k m s − ) Thick disk
Figure 6.
The asymmetric velocity, V asym as a functionof σ U for the thin disk (left panel) and thick disk (rightpannel). The black lines denote the results of the best fitusing Equation 8. − σ U V LSR (cid:104) ∂ ln ρ∂ ln R + ∂ ln σ U ∂ ln R + (cid:16) − σ V σ U (cid:17) + (cid:16) − σ W σ U (cid:17)(cid:105) .Therefore, we use the following formula to calculate V asym , i.e., V asym = σ U /C . (8)Figure 6 shows V asym as a function of σ U . The bestfits are over-plotted as the black solid lines. The valuesof C are − . +1 . − . km s − and − . +2 . − . km s − forthe thin and thick disks respectively, which are generallyconsistent with the theoretical estimate ( − ± − )(Binney & Tremaine 2008).2.3.5. revising the X factor As defined in Equation 2.2, X is the fraction of starsfor a given component. For halo and Hercules stream,their number density distributions and structures arenot quite clear yet. Thus, we set their X as the observedfractions, i.e., X H = N H N tot , X Herc = N Herc N tot , (9)where N H , N Herc and N tot are the numbers of Herculesstream stars, halo stars, and total stars in each ( R − Z )grid.For thin and thick disks, the number density is mod-eled as the following formula (Chen et al. 2001; Binney& Tremaine 2008): n ( R, Z ) = n × exp( − R − R (cid:12) h R )exp( − | Z | h Z ) , (10)where h Z and h R are the scale height and scale length ofthe disk, respectively. Here we take ( h R , h z ) as (3.4, 0.3)kpc for the thin disk and (1.8, 1.0) kpc for the thick disk(Binney & Tremaine 2008; Cheng et al. 2012; Bovy et al.2016). Then, the ratio of thick/thin disk star numbersin each R − Z grid can be calculated as X TD / D = X TD /X D = (cid:82) R (cid:82) Z n TD ( R, Z )2 πR d R d Z (cid:82) R (cid:82) Z n D ( R, Z )2 πR d R d Z . (11)
Table 3.
The kinematic characteristics for stellar compo-nents in the Solar neighbourhood from Bensby et al. (2014)and this work. X D X TD X H X Herc
Bensby et al. (2014) 0.85 0.09 0.0015 0.06This work 0.84 0.10 0.0013 0.06 σ U D σ V D σ W D V asymD Bensby et al. (2014) 35 20 16 -15This work 34 21 16 -14 σ U TD σ V TD σ W TD V asymTD Bensby et al. (2014) 67 38 35 -46This work 65 39 35 -44
In practice, we first calculated nominal ratio, i.e., n /n =0.098 (first term of the right-hand side ofEquation 10) by solving Equation 11 with X TD /X D =0 . / .
85 for the solar neighbourhood ((( R − R (cid:12) ) + Z ) / = 100 pc, Bensby et al. (2014)). Then, we ap-plied this nominal ratio to Equation 11 to calculate theratio of thick/thin disk star numbers in each R − Z grid.Finally, the X of thin and thick disks were calculatedas: X D = (1 − X H − X Herc ) ×
11 + X TD / D ,X TD = (1 − X H − X Herc ) × X T D/D X TD / D . (12)The results of these revised X values in all the R − Z grids are tabulated in Table 9. Figure 7 shows the Xvalues of various Galactic components as functions ofGalactic radius R and absolute value of height, | Z | . Asexpected, X D ( X TD ) generally decrease (increase) with | Z | in all the R bins.2.3.6. comparison to Bensby et al. (2014) So far, we have calculated the characteristic parame-ters (i.e., σ U , σ V , σ W , U asym , and V asym ) as functionsof R and Z (Table 9). In Table 3, we then compareour results in the solar neighbourhood (here bin with R : 8 . − . | Z | < . REVISITING THE AGE-VELOCITYDISPERSION RELATION TO DERIVEKINEMATIC AGESWhen the stars in the solar neighbourhood are binnedby age, the velocity dispersion of each bin increases X D [ % ] X T D [ % ] | Z | (kpc) X H e r c [ % ] R: 7.5-8.0 kpcR: 8.0-8.5 kpcR: 8.5-9.0 kpcR: 9.0-9.5 kpcR: 9.5-10.0kpc | Z | (kpc) X H [ % ] Figure 7.
The normalisation fraction X of stars for each component as a function of Galactic radius R and absolute value ofheight | Z | . The different colours denote subsamples of stars with different Galactic radii. with its age. This age velocity relation (AVR) has beenknown and studied for decades (Str¨omberg 1946; Pare-nago 1950; Wielen 1977; Holmberg et al. 2009). Similarrelationship has also been inferred for external Galacticdisk (Aumer et al. 2016; Robin et al. 2017). Here, werevisit the AVR with the calibration sample constructedin section 2.3.1. 3.1. Fitting AVR
In our study, we divided the foregoing calibration sam-ple (section 2.3.1) into 30 bins with approximately equalsizes ( ∼ σ tot = ( σ U + σ V + σ W ) . (13)Figure 8 shows the velocity dispersion as a function ofthe median age of each bin. As can be seen, all thecomponents of velocity dispersion ( U LSR , V
LSR , W
LSR )and the total velocity dispersion ( V tot ) increase with age.Following Holmberg et al. (2009); Aumer et al. (2016),we fit the AVRs shown in Figure 8 by using a simple power law formula, i.e., σ = k × (cid:18) t Gyr (cid:19) β km s − , (14)where t is stellar age, σ is the velocity dispersion, k and β are two fitting parameters.We used the Levenberg-Marquardt algorithm (LMA)to find the best fit. To obtain the uncertainty of eachfitting parameter, we assumed that the Galactic velocityand age follows the Gaussian distribution N ( V, err V )and N ( t, err t ), where err represents the correspondinguncertainty. Then, we resampled Galactic velocities andstellar ages based on these Gaussian distributions. Afterthat, we refit the AVR by using the resampled data. Werepeated the above resampling process 1,000 times andobtained 1,000 sets of best fits. The uncertainties (one-sigma interval) of the fitting parameters are set as therange of 50 ± . β =0 . +0 . − . , 0 . +0 . − . , 0 . +0 . − . , and 0 . +0 . − . , for U LSR , V
LSR , W
LSR , and V tot , respectively, which are con-sistent with the values derived from previous studies(Holmberg et al. 2009; Aumer & Binney 2009, see sec-tion 3.5 for detail comparisons).3.2. AVRs of Different Galactic Components
Our calibration sample consists of 98,486 (75.51%)thin disk stars 23,572 (18.08%) thick disk stars, 8,152(6.25%) Hercules stars, and 193 (0.15%) halo stars. Toexplore the difference of AVRs between different Galac-tic components, for each component, we divided starsinto bins with approximately equal size according totheir ages. Due to sample size, the bin numbers are setas 20, 10, 10, and 5 for thin disk, thick disk, Herculesstream, and halo respectively. For each component, weperformed the same method as in section 3.1 to obtainthe AVR.Figure 9 displays the AVRs of different components.As can be seen, the AVRs obtained from the thin andthick disk components fit well to the power-law AVRderived from the whole calibration sample. For the Her-cules stream, the dispersion of velocity component σ W generally follows the power-law AVR, but other compo-nents, i.e, σ U and σ V seem to be irrelevant with age.For the halo, the velocity dispersions are much largerthan those predicted by the power-law AVR, and thereis no clear trend between velocity dispersions and ages.Therefore, we conclude that the power-law AVR canonly apply to the thin/thick disk components.3.3. Radial Variation of AVR
To explore how the AVR varies with Galactic ra-dius, we divided the foregoing calibration sample (sec-tion 2.3.1) into to five subsamples according to theirGalactic radii, i.e., R = 7 . − . R = 8 . − . R = 8 . − . R = 9 . − . R = 9 . −
10 kpc. For each subsample, we performedthe same method as in section 3.1 to fit the AVR. Figure10 shows the fitted AVRs for the five subsamples. Thefitting parameters of the five AVRs are summarized inTable 4. As can be seen, the AVRs are generally lowerwith the increase of R , which is caused by the decreaseof velocity dispersion with R (Equation 7). Therefore,the typical uncertainties in k , β of AVRs obtained fromthe whole sample are generally larger than the five sub-samples due to the radial variation. However, the valuesof k and β differ mildly with R (mean value: ∼
5% for k and 4% for β ). As can be seen in Figure 10, the AVRs forthe five subsamples are generally within 1 − σ range ofthat for the whole sample (grey regions). This is consis-tent with the result derived from simulations in Aumeret al. (2016), which showed that the shape of the AVRis almost independent of R when R (cid:38) Vertical Selection Effect of AVR
Selecting stars from a limited vertical volume intro-duces phase correlations between stars which influencethe values of velocity dispersions at the time of selectionand before (Aumer et al. 2016). For example, when starsare selected close to Z = 0, they are all close to theirmaxima in | W LSR | . Tracking them back in time, theirvertical velocity dispersions thus have to be lower thanat the time of selection (Aumer & Sch¨onrich 2015). Toexplore the vertical selection effect in our sample, fol-lowing the method in Aumer et al. (2016), we comparedthe AVRs for stars with | Z | <
100 pc and for all starsirrespective of the Z position. The result is displayedin Figure 11. As can be seen, there is no significantdifference between the AVR for | Z | <
100 pc (red) andthat for all stars (grey region). We conducted KS testbetween the velocity dispersions of stars with | Z | < U LSR , V
LSR , W
LSR , and V tot , respec-tively. Such high p-value values demonstrate the verticalselection effect has little influence in AVR.3.5. Comparison to Previous Works
Here, we compare our fitted AVR (section 3.1) to thoseAVRs fitted with different samples in previous studies.The result is plotted in Figure 12. As can be seen, ourresult is in good agreement with those of Holmberg et al.(2009); Yu & Liu (2018); Mackereth et al. (2019). Never-theless, we note that our result differs significantly fromthose of Binney et al. (1997); Bovy et al. (2012b), andSharma et al. (2014) on the younger and older ends,respectively.All these studies fit their AVRs with different stel-lar samples. Gomez et al. (1997) used 2,812 stars fromthe Hipparcos data. Holmberg et al. (2009) used 2,640FG main-sequence stars from the Geneva-CopenhagenSurvey (GCS) data. Bovy et al. (2012b) used 3,365stars from the APOGEE data. Sharma et al. (2014)used 5,201 stars from the GCS and RAVE data. Yu& Liu (2018) used 3,564 sub-giant/red giant stars fromthe LAMOST-Gaia data. Mackereth et al. (2019) used65,719 stars from the APOGEE and Gaia data.Our study used 130,403 stars which were selected fromthe LAMOST DR4 value added catalog with well con-strained kinematics and ages (section 2.3.1). Due tothe upgrade of sample size as well as the quality of starcharacterizations, the uncertainties of the AVR param-eters have been largely reduced in this work. To betterdemonstrate this point, we performed a more detail com-parison to the AVR of Holmberg et al. (2009), which arewidely used and very close to our best fit of the AVR(Figure 12). We adopted the same procedures (section0 t (Gyr) V e l o c i t y d i s p e r s i o n ( k m s − ) t (Gyr) σ V σ U σ W σ tot Figure 8.
The Velocity dispersions for U LSR , V
LSR , W
LSR and V tot vs. age for the selected calibration star sample. The 30 binshave approximately equal numbers of stars ( ∼ Table 4.
Fitting parameters of the Age-Velocity dispersion Relation (AVR, Equation 14). ——— U ——— ——— V ——— ——— W ——— ——— V tot ——— k β k β k β k β section 3.1: AVR of the whole calibration sampleWhole sample 23 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . section 3.3: AVR of different radii R : 7 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . R : 8 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . R : 8 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . R : 9 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . R : 9 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . section 3.4: AVR of | Z | < . | Z | < . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . AVR of Holmberg et al. (2009) Holmberg et al. (2009) 22 . +2 . − . . +0 . − . . +1 . − . . +0 . − . . +1 . − . . +0 . − . . +2 . − . . +0 . − . Note 1: Here the parameters and their uncertainties were obtained by fitting the data of Holmberg et al. (2009) with our methods shown insection 3.1. σ U ( k m s − ) thin diskthick diskHaloHerculesAVR of calibration sample σ V ( k m s − ) t (Gyr) σ W ( k m s − ) t (Gyr) σ t o t ( k m s − ) Figure 9.
The AVRs of differnt Galactic components. The different colours denote subsamples of stars with different compo-nents. The black lines represents the best fit of AVR obtained from the whole sample.
Figure 10.
The radial variation of AVRs. The differentcolours denote subsamples of stars with different Galacticradii. The grey region represents the 1 − σ range of AVRobtained from the whole sample. marized at the bottom part of Table 5. As can be seen,the typical errors in k and β decrease from ∼ − − and 0 . − .
08 to ∼ . − and 0 .
02, respectively.The reduction of uncertainties in AVR parameters leads
Figure 11.
Effect of vertical selection on velocity disper-sions. The velocity dispersions are plotted as the solid redline for | z | <
100 pc. The grey region represents the 1 − σ range of AVR obtained from the whole sample. to significant improvement in the precision of kinematicage (section 3.6).3.6. Kinematic Age and Uncertainty Figure 12.
The vertical AVRs from different studies. Thedashed lines present results from previous studies. The greyregion denotes the 1 − σ range of the best fit obtained fromour calibration sample. For a group of stars, the typical kinematic age canbe derived by using the AVR (solving the Equation 14),which gives t = (cid:16) σk km s − (cid:17) β Gyr , (15)By means of error propagation, the relative uncertaintyof kinematic age can be estimated as :∆ tt = (cid:115) ( ∂ ln t∂β ∆ β ) + ( ∂ ln t∂k ∆ k ) + ( ∂ ln t∂σ ∆ σ ) = (cid:115)(cid:18) ln t Gyr (cid:19) (cid:18) ∆ ββ (cid:19) + 1 β (cid:18) ∆ kk (cid:19) + 1 β (cid:18) ∆ σσ (cid:19) , (16)where ∆ represents the absolute uncertainty. For thesake of simplicity, here we assume k , β , and σ are in-dependent of each other. In other word, we neglect thecovariances between them. It can be seen from the for-mula: the first term (due to the uncertainty of β ) haspositive correlation with the age ( t ), while the latter twoterms are independent with age. Therefore, the relativeuncertainty in age generally increases with age itself. Forthis reason, we set t=1 Gyr and t=10 Gyr to estimatethe range of the relative uncertainty of age.Based on Equation 16, we analyzed the budget of kine-matic age uncertainty derived from AVR. The resultsare listed in the left part of Table 5. The uncertaintiesin AVR fitting parameters, i.e., ∆ k and ∆ β , were calcu-lated as the half width of the 1-sigma interval of k and β as listed in Table 4. The relative uncertainties in veloc-ity dispersions (∆ σ/σ ) were adopted as the median rel-ative uncertainties of velocity dispersions in our planethost stellar sample, which are 2.0%, 3.5%, 4.1%, and Table 5.
The typical relative uncertainties of parametersand kinematic ages derived by the Age-Velocity Relationsusing data from Holmberg et al. (2009) and this work.
This work Holmberg et al. 2009 U ∆ k/k β/β σ/σ t/t . − .
1% 31 . − . V ∆ k/k β/β σ/σ t/t . − .
1% 23 . − . W ∆ k/k β/β σ/σ t/t . − .
9% 27 . − . V tot ∆ k/k β/β σ/σ t/t . −
16% 27 . − . σ U , σ V , σ W and σ tot , respectively. Puttingall the above uncertainties into Equation 16, we ob-tained the relative age uncertainties ∆ t/t ∼ . − . t/t ∼ . − . t/t ∼ . − . t/t ∼ . − .
0% for σ U , σ V , σ W , and σ tot , respectively. Thelower and upper values are calculated by assuming t = 1Gyr and t = 10 Gyr respectively in Equation 16.For comparison, we repeated the above budget cal-culation but using AVR of Holmberg et al. (2009, seethe bottom part of Table 4). The results are listed inthe right part of Table 5. As can be seen, for AVRfrom Holmberg et al. (2009), the uncertainties of k , β are much larger and thus dominant, leading to a muchlarger (by a factor of ∼
3) uncertainty in the derivedkinematic age.As discussed in section 3.2, AVRs are not significantfor halo and Hercules stream, therefore we suggest thatthis method to obtain kinematic age is only suitable forstars belonging to the Galactic disk components. APPLICATION TO PLANET HOST STARSIn this section, we apply the above revised kinematicmethod (section 2) and AVR (section 3) to a sampleof planet host stars (section 4.1), providing a catalogof their kinematic properties (section 4.2), with focuson the Galactic components (section 4.3) and kinematicages (section 4.4). 4.1.
Data samples
This subsection describes how we constructed theplanet host sample for further kinematic characterizing.34.1.1. initializing planet host sample from EA
We initialized our planet host sample with the con-firmed planets table and the Kepler DR 25 catalog fromEA. The Kepler catalog contains 8,054 Kepler Ob-jects of Interest (KOIs) in DR25. Here, we excludedKOIs flagged by ‘False Positive’ (FAP, Thompson et al.2018), leaving 4,034 planets (candidates) around 3,069stars. Besides Kepler, there are 1,728 non-Kepler plan-ets flagged by ‘Confirmed’ around 1,387 stars. Wealso removed potential binaries because additional mo-tions caused by binary orbits could affect the resultsof kinematic characterization. Specifically, for Keplerplanet host stars, we eliminated stars with Gaia DR2re-normalized unit-weight error (RUWE) > . (cid:54) = 0 (flag in-dicating whether the planet orbiting a binary flag, 0 forno) in EA. In total, we are left with 4,126 stars hosting5,331 planet candidates in our initial sample.4.1.2. obtaining five astrometric parameters from Gaia Next, we cross-matched our initial planet host samplewith Gaia to obtain astrometric parameters. The sec-ond Gaia data release (DR2, Gaia Collaboration et al.(2018a)) includes five astrometric parameters: posi-tions on the sky ( α, δ ), parallaxes, and proper motions( µ α , µ δ ) for more than 1.3 billion stars, with a limitingmagnitude of G = 21 and a bright limit of G ≈
3. Thecross-matching was done by using the X match serviceof the Centre de Donnees astronomiques de Strasbourg(CDS, http://cdsxmatch.u-strasbg.fr). The separationlimit of the cross-matching was chosen as where the dis-tribution of separations displayed a minimum, ∼ . obtaining RV from various sources We obtained radial velocities from the following fivecatalogs: the APOGEE data release (DR) 16 catalog,the LAMOST DR4 value-added catalog, the RAVE DR5catalog, Gaia, and EA. • APOGEE
The APOGEE DR16 provides a cat-alog of 437,485 unique stars, which contains theinformation of radial velocity (RV), effective tem-perature ( T eff ), surface gravity (log g ), and chem- ical abundance (e.g,. [Fe / H] and [ α/ Fe]) (Ahu-mada et al. 2020). We cross-matched it with theplanet host sample obtained in section 4.1.2. Here,we applied the following quality control cuts: (1)STARFLAG = 0 to only select star with no warn-ings on the observation; (2) ASPCAPFLAG = 0 toensure parameters have converged and no warning;and (3) SNR >
80 to ensure high SNR (Holtzmanet al. 2018), leaving 692 stars hosting 956 planets. • RAVE
The fifth data release (DR5) of RAVEprovides radial velocities with a precision of ∼ . − and physics properties ( T eff , log g ,[Fe / H], etc.) from a magnitude-limited (9 < I <
12) survey for 457,588 randomly-selected stars inthe southern hemisphere (Kunder et al. 2017). Tocross-match it with the planet host sample ob-tained in section 4.1.2, we applied the followingquality cuts: (1) Algo Conv K = 0 to ensure thatthe stellar parameter pipeline has converged; (2)SNR >
40; (3) spectroscopic morphological flags(c1, c2, c3) are n; (4) Alpha C > − .
99 (Kunderet al. 2017). The RAVE DR5 contains stars only ina range of declination from −
88 to +28 deg, thusthe Kepler field which ranges from 36 to 53 deg indeclination is not covered. Therefore, the abovecross-matching with RAVE returned only 30 starshosting 37 non-Kepler planets. • Gaia
Gaia DR2 also includes radial velocities formore than 7.2 million stars with a magnituderange of G ∼ −
13 and a T eff range of about3550 to 6900 K. The quality control cut is set asthat the ratio of radial velocity and its uncertaintyshould be larger than 3. After cross-matching withthe planet host sample obtained in section 4.1.2,we obtained 1,143 stars hosting 1,479 planets (268of them are stars with 371 planets in the Keplerfield). • LAMOST
The LAMOST survey has several com-ponents focusing on different Galactic aspects,e.g., the Galactic halo (Deng et al. 2012), stellarclusters (Hou et al. 2013), the Galactic anticenter(LSS-GAC; Liu et al. (2014)), the Kepler fields(De Cat et al. 2015), and etc. The LAMOSTDR4 value-added catalog contains parametersderived from a total of 6.5 million stellar spectrafor 4.4 million unique stars (Xiang et al. 2017b).RVs, T eff , log g , and [Fe / H] have been deduced us-ing both the official LAMOST Stellar parameter http://dr4.lamost.org/v2/doc/vac Table 6.
Compositions of our planet host sample. ————– Space-based ————– ————– Ground-based ————–Kepler K2 CoRot RV Transit Otherssection 4.1.1: Initializing Planet Host Sample from EAWithout FAP& binary: 4126 (5331) 2737 283 29 590 366 121(3620) (389) (31) (775) (384) (132)section 4.1.2: Obtaining Five Astrometric Parameters from GaiaWith Gaia Astronomy: 3912 (5069) 2571 282 29 568 364 98(3409) (388) (31) (752) (382) (107)section 4.1.3: Obtaining RV from Various SourcesAPOGEE: 692 (956) 628 30 2 21 9 2(874) (42) (3) (24) (11) (2)RAVE: 30 (37) 0 4 0 6 20 0(0) (6) (0) (10) (21) (0)Gaia: 1143 (1479) 268 145 3 497 223 7(371) (204) (4) (659) (234) (7)LAMOST: 1059 (1421) 951 38 0 24 45 1(1292) (54) (0) (26) (47) (2)EA: 1303 (1737) 215 185 20 529 338 16(371) (266) (23) (709) (351) (17)section 4.1.4: Finalizing the Planet Host SampleCombined: 2174 (2872) 1134 179 20 516 306 19(1562) (249) (22) (699) (319) (21)
The numbers without and with brackets represent that of the stars and planets during the process of sample selection in section 4.1.
Pipeline (LASP; Wu et al. (2011)) and the LAM-OST Stellar Parameter Pipeline at Peking Univer-sity (LSP3; Xiang et al. (2015)). The typical un-certainties for RVs, T eff , log g , and [Fe / H] are 5.0km s − , 150 K, 0.25 dex, and 0.15 dex respectively.After applying a quality cut of SNR >
10 andcross-matching with planet host sample obtainedin section 4.1.2, we obtained 1,059 stars hosting1,421 planets. The majority (951) are stars withKepler planets (1,292). • EA The NASA Exoplanet Archive (EA) also re-ports RVs for a portion of stars. We thus cross-matched these stars with planet host sample ob-tained in section 4.1.2. The quality control cut isalso set as that the ratio of radial velocity and itsuncertainty should be larger than 3, which yields1,303 stars hosting 1,737 planets. The majority(1,088) are stars with non-Kepler planets (1,366)from various ground based RV and transit surveys. Note, most RV data in EA are collected from var-ious literatures and thus are inhomogeneous.4.1.4. finalizing the planet host sample
We finalize the planet host sample by combining var-ious matched samples in section 4.1.3. For stars withmultiple RV measurements from different sources, wetake the order of precedence as APOGEE, RAVE, Gaia,LAMOST, then EA. This generally follows the order ofspectral resolution and thus the RV uncertainty. Herewe set the EA as the lowest priority because the mostRVs from EA are collected from various sources, whichare inhomogeneous. For the sake of reliability, we ex-clude the stars if the differences in their RVs from dif-ferent sources are larger than three times of the uncer-tainties. We also apply the same cut as in the calibra-tion sample, i.e. distance < .
54 kpc, corresponding to(7 < R <
10 kpc, θ <
10 deg, and | Z | < . Figure 13.
Galactocentric radius ( R ) vs. height ( Z , toppanel) and angle ( θ , bottom panel)for the combined planethost sample. The diagram is colour-coded to represent dif-ferent discovery methods and facilities. (8.34 kpc, 0, 0.027kpc) marks the position of Sun. the sample after each step mentioned above. Figure 13shows the location of stars in our planet host sample.4.2. A Catalog of Planet Hosts with KinematicCharacterizations
Applying the methods described in section 2 and sec-tion 3 to the planet host sample (section 4.1), we ob-tained a catalog (Table 10) of 2,174 planet hosts withkinematic characterizations, e.g., Galactocentric veloci-ties to the LSR ( U LSR , V LSR , and W LSR ) and the rela-tive membership probabilities between different Galacticcomponents (
T D/D , T D/H , Her/D , and
Her/T D ).For the sake of completeness, we also put in the cat-alog the stellar parameters that used during the pro-cess of our kinematic characterization (e.g., parallax,proper motion, and RV) and other basic stellar parame-ters (e.g., T eff , log g , [Fe / H], and [ α/ Fe]). As mentionedbefore, for stars with multiple sources of RV, the or-der of precedence follows the order of spectral resolu-tion and thus the uncertainty, i.e. as APOGEE, RAVE,Gaia, LAMOST then EA. While for other stellar pa-rameters, because the estimates of stellar parametersfrom APOGEE are only reliable for relatively cool stars4000 < T eff < Galactic Components of Planet Hosts
With the derived relative membership probabilitiesbetween different Galactic components (
T D/D , T D/H , Her/D , and
Her/T D in Table 10), we then classify the2,174 planet host stars into four Galactic components,i.e., thin disk, thick disk, Hercules stream, and halo fol-lowing the method as mentioned in section 2.2. For starsnot belonging to the above four components, followingBensby et al. (2014), we classify them into a categorydubbed as ‘in between’.The results of the classification are summarized in Ta-ble 7, which lists the numbers of stars in different cat-egories. As can be seen, about 87.1% (1,894/2,174) ofstars in our sample are in thin disk and about 5.2%(114/2,174) stars are in thick disk. 45 stars in theplanet host sample are affiliated to Hercules stream,which has been speculated to have a dynamical originin the inner parts of the Galaxy and then kinematicallyheated by the central bar (Famaey et al. 2005; Bensbyet al. 2007). The fraction of halo stars is ∼ ∼ V tot (cid:46)
50 km s − ) arein the thin disk, while those with moderate velocities( V tot ∼ −
180 km s − ) are mainly in thick disk. Thevelocity of the only halo star is larger than 220 km s − .We summarized the median values of velocities andchemical abundances for different component in Table8. As expected, the halo star has the highest Galacticvelocity and poorest [Fe / H], and the thick disk stars arekinematically hotter, metal-poorer ( ∼ . α -richer ( ∼ . α/ Fe] measurementand thus the median and 1 − σ interval of [ α/ Fe] are not6
Table 7.
The numbers of stars (planets) of our planet host sample in different Galactic components
Total Thin disk Thick disk Hercules Halo In betweenRadial Velocity 516 (699) 440 (602) 25 (33) 20 (30) 1 (1) 30 (33)Transit Kepler 1134 (1562) 982 (1363) 68 (89) 15 (20) 0 (0) 69 (90)K2 179 (249) 156 (221) 8 (9) 5 (9) 0 (0) 10 (10)CoRoT 20 (22) 19 (21) 1 (1) 0 (0) 0 (0) 0 (0)Ground-based 306 (319) 278 (290) 12 (12) 5 (5) 0 11 (12)Other methods 19 (21) 19 (21) 0 (0) 0 (0) 0 (0) 0 (0)All 2174 (2872) 1894 (2518) 114 (144) 45 (64) 1 (1) 120 (145)
The numbers without and with the brackets represent that of stars and planets.
Table 8.
The summary of Galactic velocities and chemical abundances for the combined sample of planet host stars ——— V tot (kms − ) ——— ——— [Fe / H] (dex) ——— ——— [ α/ Fe] dex ———value 1 − σ interval value 1 − σ interval value 1 − σ intervalThin disk 34.8 (19.1, 56.0) 0 .
00 ( − . , .
21) 0 .
01 ( − . , . − .
20 ( − . , .
11) 0 .
15 (0 . , . − .
89 NA NA NAHercules 73.5 (62.7, 91.1) − .
05 ( − . , .
13) 0 .
07 ( − . , . Figure 14.
The Toomre diagram of combined planet hostsample for different Galactic components. The top panelshows the full range of velocities while the bottom panelzooms in the region where the majority of the data points arelocated. The diagrams are colour-coded to represent differentcomponents. Dashed lines show constant values of the totalGalactic velocity V tot = ( U + V + W ) / , in stepsof 100 and 50 km s − respectively in the two panels. provided. In Figure 15, we plot the total velocity V tot ,[Fe / H], and [ α/ Fe] as a function of
T D/D . As expected, V tot , [ α/ Fe] increase with
T D/D , while [Fe / H] decreaseswith
T D/D . Kapteyn’s star (or GJ 191, HD 33793) is the only halostar in our sample. This star has been also identified asthe closest halo star to the Sun at a distance of only3.93 pc with Hipparcos data (van Leeuwen 2007). Itis a 11.5 Gyr old M0 type star with a temperature of3,550 K and estimeated mass of 0.28 solar mass (Wylie-de Boer et al. 2010; Anglada-Escude et al. 2014). In-terestingly, it is orbited by a confirmed super-Earth(Kapteyn c) with a period of 121.5 days and a can-didate super-Earth (Kapteyn b) with a period of 48.6days. Furthermore, Kapteyn b lies within the habitablezone (Anglada-Escude et al. 2014) and could probablysupport life at the present stellar activity level (Guinanet al. 2016). If confirmed, Kapteyn b will become theoldest habitable planet known so far. The existence ofsuch a multiple planetary system around a halo star mayprovide peculiar insights into the planetary formationand evolution at the early time of the Milky way.4.4.
Kinematic Ages of Planet Hosts
In this section, by using our catalogs of kinematicproperties for planet host star, we divided planetary hoststars into various groups to study their kinematic ages.For each group, we calculated the velocity dispersion toderive the corresponding kinematic age from Equation15. To access the uncertainties of the kinematic ages, wetook a Monte Carlo method by resampling the AVR pa-rameters ( k and β ) and velocity dispersion ( σ ) based ontheir uncertainties. For k and β , their values and uncer-tainties were adopted from Table 4. For σ , its value anduncertainty were calculated by resampling each star’sGalactic velocities from a normal distribution given itsvalue and uncertainty. Finally, the age uncertainty was7 Figure 15.
Top panel: The relative probabilities for the thick-disk-to-thin-disk
T D/D vs. total velocity V tot ; Middle pannel: T D/D vs. [Fe / H]; Bottom panel:
T D/D vs. [ α/ Fe]. Medians and 1 − σ dispersions are marked in the red color. Histogramsof V tot , [Fe / H] and [ α/ Fe] are shown in the right panels. Histogram of
T D/D is displayed in the toppest. The vertical dashedlines represent where
T D/D = 1. set as the 50 ± kinematic ages derived withtotal velocity and velocity components As shown in Figure 8 of section 3, the dispersions ofvelocity components, i.e., σ U , σ V , σ W , and total veloc-ity σ tot all increase with age and fit well with power- law functions. Nevertheless, the fitting parameters( k and β in Table 4) are different for different veloc-ity components, which in turn could give different kine-matic ages. Here, we compare different kinematic agescalculated from the total velocity and different velocitycomponents. To do this, we first sorted the plant hostsample by the relative probabilities for the thick-disk-to-thin-disk, i.e., T D/D . Next, according to
T D/D , we8 t σ U (Gyr) t σ t o t ( G y r ) t σ V (Gyr) t σ W (Gyr) (a) (b) (c) Figure 16.
Comparisons of the kinematic ages calculated from dispersions of V tot with ages from U LSR (a), V LSR (b) and W LSR (c) for the planet host sample. The black line in each panel indicates where the horizontal and vertical coordinates are equal toeach other. t (Gyr) F r e q u e n ce p r o b a b ili t y Thin disk (1894 stars) Thick disk (114 stars)
Age_D = 2.84 +0.32-0.26
GyrAge_TD = 10.42 +1.82-0.73
Gyr
Figure 17.
Distributions of the kinematic age t (Gyr) for1,894 planet host stars in the thin disk (blue) and 114 planethost stars in the thick disk (red). divided the planet host sample into 10 bins with ap-proximately equal size ( ∼ σ U , σ V , σ W , and σ tot .Finally, we compare these kinematic ages in Figure 16.As can be seen, kinematic ages derived using differentvelocities are well consistent with each other. To furthersee the differences quantitatively, we fit the relations be-tween age derived from σ tot and ages from σ U , σ V , and σ W . The results are: t σ U = 0 . +0 . − . × t σ tot ,t σ V = 0 . +0 . − . × t σ tot ,t σ W = 0 . +0 . − . × t σ tot . (17)As can be seen, the relative differences between agesfrom different velocities are ∼ − -2 -1 T D/D t ( G y r ) Figure 18.
The kinematic age t (Gyr) vs. the relativeprobability, T D/D ( thick-disk-over-thin-disk) for our planethost sample. unless otherwise specified, the kinematic age refers tothe one derived using the dispersion of total velocity,i.e., σ tot .4.4.2. kinematic ages of planet host stars in the Galacticdisk Besides the element abundances and Galactic veloci-ties, age is one of the main differences of different Galac-tic components. It has been known that thin disk starsare generally younger than thick disk stars with a divid-ing age of ∼ ∼ −
12 Gyr (Jofr´e & Weiss 2011; Kalirai 2012; Guoet al. 2016, 2019). For Hercules stream, it has threesubstructures. The age distribution is peaked at 4 Gyrand extend to very old age for Hercules a and Hercules9b; and Hercules c has a more uniform age distributionfrom ∼ −
10 Gyr (Torres et al. 2019).As mentioned in section 3.2, there is no clear trend be-tween velocity dispersions and ages for stars in Herculesstream and halo. Thus here we only calculate the kine-matic age of planet host stars in Galactic disks, whichcontains the majority (97.9%) of our sample. This willbe useful in future study on the characteristics and evo-lution of planetary systems related to the Galactic com-ponents and age of host star.We obtain the kinematic age and uncertainty for starsin thin disk (1,894 stars) and thick disk (114 stars) withthe methods described at the beginning of section 4.3.The typical ages are 2 . +0 . − . Gyr, 10 . +1 . − . for thinand thick disk respectively. As shown in Figure 17, theage distribution of stars in the thick disk is generallylarger than 8 Gyr, while the thin disk is populated byyounger stars. This is well consistent with previous stud-ies (Fuhrmann 1998; Bensby et al. 2003; Haywood et al.2013; Bensby et al. 2014).To explore the relation between T D/D and kinematicage, we first sorted the plant host sample by the rel-ative probabilities for the thick-disk-to-thin-disk, i.e.,
T D/D . Next, according to
T D/D , we divided theplanet host sample into 10 bins with approximatelyequal size. Then, for each bin, we calculated theirkinematic ages. As shown in Figure 18, the kinematicage generally increase with
T D/D , demonstrating that
T D/D is a indicator of age for stars in the Galactic disk. DISCUSSIONS5.1.
Systematic Differences in Radial Velocity fromVarious Sources
As mentioned in section 4.1.3, we obtained the RVfrom five sources: APOGEE, LAMOST, RAVE, Gaia,and EA. As the RV is one of the basic parameters to cal-culate the Galactic velocity, the systematic differencesin RV will induce differences in the Galactic velocityand then effect the classification of Galactic components.Therefore, it is necessary to calibrate the systematic dif-ference in RV from various sources.Here we choose the APOGEE RV data as standard ref-erence because it has the highest resolution ( R ∼ RV are1.21, 0.05, 0.06 km s − for LAMOST, Gaia, and EA re-spectively. For RAVE, there is only 2 common stars andtoo few to make analysis. Here we refer to Huang et al. (2018), which presents a new catalog of RV standardstars selected from the APOGEE data and find that thesystematic offset is only 0.17 km s − for RAVE. The sys-tematic differences in RV between APOGEE and othersources are all smaller than the typical uncertainties inRV of our sample 1.58 km s − and thus have no signifi-cant influence in the calculation of Galactic velocity andthe classification of Galactic components.5.2. Kinematic Age vs. Asteroseismic Age vs.Isochrone Age
In order to access the reliability of kinematic ages de-rived in this work, we compare them to ages derivedfrom asteroseismology and isochrone. The Kepler as-teroseismic LEGACY sample (Silva Aguirre et al. 2017)provides a well characterized sample of 66 Kepler planethosts with asteroseismic age estimates whose averageuncertainy is ∼ N ( t, σ t )). The distribution of kinematic ages were con-structed with AVR from Equation 15 using a MonteCarlo method which took into account the uncertaintiesof AVR parameters (Table 4) and velocity dispersions.As can be seen in Figure 20, the results are 3 . +0 . − . Gyr, 3 . +0 . − . Gyr and 3 . +0 . − . Gyr for asteroseismicage, kinematic age, and isochrone age, respectively. En-couragingly, our kinematic ages match better with theasteroseismic ages, though the three kinds of ages arenot inconsistent with each other within their errorbars.5.3.
Future Studies
For the clarity and simplicity of the paper, here weapply the revised kinematic methods (section 2 and 3)only to the planet host sample. Next, in our secondpaper of the PAST project (Chen et al. in prep.), wewill apply the revised kinematic methods to the wholeKepler star sample, enabling us to further connect stellarkinematic properties to stellar rotations and activities.The planet host catalog provides stellar parameters,spatial position, Galactic velocity and component clas-sification for 2,174 stars, which hosts 2,872 planets. Fur-thermore, using AVR as we show in section 3.6, one can0 -100 -50 0 50
LAMOST RV (km s − ) -100-50050 A P O G EE R V ( k m s − ) APOGEE-LAMOST (292 stars) -100 -50 0 50
Gaia RV (km s − )APOGEE-Gaia (181 stars) -100 -50 0 50 EA RV (km s − )APOGEE-EA (146 stars) (a) (b) (c) Δ RV = 0.06 km s -1 Δ RV = 0.05 km s -1 Δ RV = 1.21 km s -1 Figure 19.
The comparison of radial velocities (RV) from different sources: (a). LAMOST RV vs. APOGEE RV; (b).Gaia RV vs. APOGEE RV; (c) EA RV vs. APOGEE RV. The black lines in the figure indicate where the horizontal andvertical coordinates are equal. ∆ RV represents the systematic offset in RV comparing to APOGEE measurements. t (Gyr) F r e q u e n c y p r o b a b ili t y Asteroseismic ageKinematic ageIsochrone age
Asteroseismic age: 3.06 +0.13 -0.13
GyrKinematic age: 3.30 +0.32 -0.28
GyrIsochrone age: 3.79 +0.74 -0.49
Gyr
Figure 20.
Distributions of the asteroseismic age (blue),kinematic age (red) and isochrone age (green) of 54 Keplerstars. obtain the kinematic age for a group of stars with respec-tive properties. As shown in Figure 13 and Table 7,these planet hosts spread over different Galactic compo-nents in a wide range of distance up to 1 ,
500 pc. Withsuch rich stellar information, future studies are allowedto explore and answer some fundamental questions onexoplanets, such as, what are the differences in variousproperties of planetary systems at different positions inthe Galaxy with different ages? Specifically, in a subse-quent paper of the PAST project (Yang et al. in prep),we will study whether/how planetary occurrence and ar-chitecture change with the Galactic environment. Theanswers of these questions will be crucial in constrainingvarious models and theories of planetary formation andevolution. GUIDELINES FOR USING THE METHODS ANDCATALOGIn this section, we provide the guidelines, cautionsand limitations to utilize the revised kinematic methods(section 2 and 3) and planet host catalog (Table 10).To classify stars into different Galactic compo-nents, the key is to calculate the relative prob-abilities between two different components (i.e.
T D/D, T D/H, Herc/D, Herc/D ) with Equation(4) and (5), which rely on the X factor and velocityellipsoid (i.e. σ U , σ V , σ W , and V asym ) of differentGalactic components. Here, we suggest two ways toobtain these kinematic parameters for a given Galacticposition ( R, Z ). The most easy way is to use our fittingformulae (e.g., Equation 7 with coefficients in Table 2for σ U , σ V and σ W , and Equation 8 for V asym ). Al-ternatively, one can conduct interpolation based on ourrevised characteristics in Table 9.To derive the kinematic age for a group of stars, onecan use the AVR (Equation 15) with the revised coef-ficients in Table 4. The typical age uncertainty can beestimated from Equation 16, which is ∼ −
20% here.For the purpose of avoiding potential spacial biases, werecommend to adopt the total velocity dispersion (Equa-tion 13) when using the AVR (Equation 15).Applying the above revised methods to our planet hostsample, we provide a catalog with kinematic properties(e.g. Galactic position, velocity and
T D/D ) and otherbasic parameters (e.g. T eff and element abundances).With this catalog, one can divide planet hosts into binsaccording to respective properties (e.g., planetary pe-riod, multiplicity etc.) and calculate the kinematic agefor each bin, which will be practical and useful for sta-1tistical studies of age effects on planetary systems. Al-though the kinematic method can not directly measurethe ages for individual stars, some of the derived kine-matic properties (e.g., T D/D ) could serve as good agetracers given their significant correlations (e.g., Figure18).However, there are some notable cautions and limita-tions, which are listed as follows:(1). The revised velocity ellipsoid and AVR arestrictly applicable for stars within the region calibra-tion sample covers, i.e, within 1.54 kpc to the Sun(corresponding to R = 7 . − . | θ | <
10 deg& | Z | = 0 − . θ (e.g. Williams et al. 2013). Thusthe criterion | θ | <
10 deg are not necessary for theGalactic disk stars. Besides, with Equation 7 and Ta-ble 2, the velocity dispersion can be extrapolated to theregion outer to 1.54 kpc. However, this extrapolationshould be adopted with caution.(2). Since the additional motions caused by binaryorbits could affect the stellar kinematic, binaries shouldbe applied with caution.(3). Due to the small numbers of Halo and Herculesstream stars in our calibration sample, we adopt thevelocity dispersion values derived from stars in the so-lar neighborhood as in Bensby et al. (2014). As thevelocity dispersions change with the Galactic position(Williams et al. 2013), there might be some deviationsin the classification of Halo and Hercules stream starswhen utilizing the characteristic parameters in Table 9and the planet host catalog in Table 10. It will be morereliable to take other parameters (e.g. velocity, elementabundance, angular momentum Bensby et al. 2007; Leeet al. 2011; Bonaca et al. 2017; Kushniruk & Bensby2019) into consideration.(4). There is no clear trend between velocity disper-sions and ages for stars belong to Hercules stream andhalo in our calibration sample. Therefore, the methodto derive kinematic age is only suitable for stars in theGalactic disk.(5). The kinematic ages and uncertainties derivedfrom Equation 15 and 16 are the typical (median/mean)values for a group of stars. SUMMARYSince 1995, the discovered exoplanet population hasexpanded significantly from the solar neighborhood toa much larger area in the Galaxy (Figure 1). We aretherefore entering a new era to study exoplanets in abig context of the Galaxy. In the Galactic context, the relations between the properties of planetary systemsand the kinematics as well as the ages of planet hoststars have yet to be explored. To answer these ques-tions, we perform a series of studies in a project dubbedas PAST (Planets Across Space and Time). In this pa-per, as the Paper I and the basis of the PAST series, werevisit the kinematic methods for classification of Galac-tic components (section 2) and estimation of kinematicages (section 3) and apply them to planet host stars(section 4).For classification of Galactic components (section 2),we adopt the well-used kinematic approach as in Bensbyet al. (2003, 2014). However, so far, the kinematic char-acteristics of this method has been applied only to theSolar neighborhood within ∼ −
200 pc. For this rea-son, using a calibration sample based on the GAIA andLAMOST data (section 2.3.1), we extend the kinematiccharacteristics to ∼ ,
500 pc (section 2.3, Table 9) tocover the majority of planet hosts (Figure 1 and 13).For estimation of kinematic ages, we refit the Age-Velocity dispersion Relation (AVR) with the calibrationsample (section 3, Figure 8). Our AVR is consistent withthose in previous studies (e.g., Holmberg et al. (2009))but with much smaller internal uncertainties (Table 4)thanks to the large and high quality calibration sample.Based on this refined AVR, we are able to derive kine-matic age with an uncertainty of 10-20% (section 3.6),which is a factor of ∼ α/ Fe] and the kinematic age generally increasewith the relative probabilities for the thick-disk-to-thin-disk, i.e.,
T D/D , while [Fe / H] decreases with
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Revised characteristics at different Galactic radii ( R ) and heights ( Z ) for different Galactic components using thecalibration sample. | Z | R σ DU σ DV σ DW V Dasym σ TDU σ TDV σ TDW V TDasym X D X TD X H X Herc (kpc) (kpc) ————————— (km s − ) —————————0 − . . − . −
16 63 37 31 -41 0.81 0.13 0.0010 0.068 . − . −
14 65 39 35 −
44 0.84 0.10 0.0013 0.068 . − . −
15 70 37 33 −
51 0.85 0.10 0.0013 0.059 . − . −
12 70 35 34 −
51 0.87 0.09 0.0014 0.049 . − . −
11 68 37 31 −
48 0.89 0.08 0.0016 0.030 . − . . − . −
13 69 37 39 −
43 0.78 0.16 0.0011 0.068 . − . −
15 66 42 36 -45 0.79 0.14 0.0010 0.078 . − . −
15 71 41 36 -52 0.81 0.13 0.0014 0.069 . − . −
13 71 42 34 −
52 0.85 0.12 0.0012 0.049 . − . −
11 70 40 34 −
50 0.87 0.10 0.0019 0.030 . − . . − . −
16 67 39 37 −
46 0.75 0.20 0.0016 0.078 . − . −
17 68 42 40 −
47 0.76 0.17 0.0015 0.078 . − . −
17 72 42 37 −
54 0.78 0.16 0.0017 0.069 . − . −
14 71 40 36 −
52 0.81 0.14 0.0016 0.059 . − . −
11 67 39 35 −
48 0.84 0.13 0.0140 0.030 . − . . − . −
18 68 41 38 -47 0.71 0.24 0.0016 0.058 . − . −
19 68 41 41 −
47 0.71 0.21 0.0020 0.088 . − . −
17 71 41 38 −
52 0.75 0.19 0.0020 0.069 . − . −
15 71 42 37 −
52 0.77 0.17 0.0020 0.069 . − . −
14 71 41 35 −
52 0.80 0.16 0.0025 0.040 . − .
55 7 . − . −
21 71 42 40 -52 0.65 0.29 0.0021 0.068 . − . −
21 70 41 42 −
50 0.66 0.26 0.0026 0.088 . − . −
19 71 42 38 −
52 0.70 0.23 0.0022 0.079 . − . −
17 71 41 38 −
52 0.73 0.21 0.0024 0.069 . − . −
16 70 40 35 −
50 0.75 0.19 0.0025 0.060 . − .
75 7 . − . −
21 71 45 41 -52 0.55 0.37 0.0042 0.088 . − . −
23 72 43 42 −
54 0.58 0.34 0.0035 0.088 . − . −
20 72 43 40 −
54 0.61 0.31 0.0026 0.089 . − . −
18 72 43 40 −
52 0.65 0.29 0.0026 0.069 . − . −
16 71 40 40 −
52 0.68 0.26 0.0029 0.060 . − . . − . −
24 71 43 44 -52 0.43 0.48 0.0055 0.088 . − . −
22 73 43 43 −
55 0.46 0.45 0.0053 0.088 . − . −
21 73 45 41 −
55 0.50 0.42 0.0039 0.099 . − . −
19 72 43 40 −
54 0.56 0.36 0.0026 0.081 . − . . − . −
25 72 45 46 -54 0.25 0.63 0.0129 0.108 . − . −
22 73 45 44 −
55 0.28 0.61 0.0100 0.108 . − . −
24 73 45 43 −
55 0.31 0.59 0.0103 0.099 . − . −
23 74 45 40 −
57 0.34 0.56 0.0056 0.09 Table 10.
The catalogue of kinematic properties and other basic properties for the combined planet host stars
Column Name Format Units descriptionParameters obtained from Gaia, APOGEE, RAVE, LAMOST and NASA exoplanet archive (EA)1 Gaia ID Long Unique Gaia source identifier2 LAMOST ID string LAMOST unique spectral ID3 APOGEE ID string APOGEE unique spectral ID4 RAVE ID string RAVE unique spectral ID5 pl hostname string NASA Exoplanet archive unique planet host name6 Kepler ID integer Kepler Input Catalog (KIC) ID7 Gaia RA Double deg Barycentric right ascension8 Gaia Dec Double deg Barycentric Declination9 Gaia parallax Double mas Absolute stellar parallax10 Gaia e parallax Double mas Standard error of parallax11 Gaia pmra Double mas yr − Proper motion in right ascension direction12 Gaia e pmra Double mas yr − Standard error of proper motion in right ascension direction13 Gaia pmdec Double mas yr − Proper motion in declination direction14 Gaia e pmdec Double mas yr − Standard error of proper motion in declination direction15 Gaia G mag Double mag
Gaia G band apparent magnitude16 T eff Float K Effective temperature from RAVE, LAMOST, APOGEE, Gaia, EA17 flag T integer flag represents which source each value is collected from18 log g Float Surface gravity from RAVE, LAMOST, APOGEE, Gaia, EA19 flag log g integer flag represents which source each value is collected from20 [Fe / H] Float dex Metallicity from RAVE, LAMOST, APOGEE, Gaia, EA21 flag [Fe / H] integer flag represents which source each value is collected from22 [ α/ Fe] Float dex α elements abundance from RAVE, LAMOST, APOGEE, Gaia, EA23 flag [ α/ Fe] integer flag represents which source each value is collected from24 rv Double km s − Radial velocity from APOGEE, RAVE, Gaia, LAMOST, EA25 e rv Double km s − Error of radial velocity26 flag rv integer flag represents which source each value is collected fromParameters derived in this work27 R Double kpc Galactocentric Cylindrical radial distance28 θ Double deg Galactocentric Cylindrical azimuth angle29 Z Double kpc Galactocentric Cylindrical vertical height30 V R Double km s − Galactocentric Cylindrical R velocities31 V θ Double km s − Galactocentric Cylindrical θ velocities32 V Z Double km s − Galactocentric Cylindrical Z velocities33 U LSR
Double km s − Cartesian Galactocentric X velocity to the LSR34 e U LSR
Double km s − error of Cartesian Galactocentric X velocity to the LSR35 V LSR
Double km s − Cartesian Galactocentric Y velocity to the LSR36 e V LSR
Double km s − error of Cartesian Galactocentric Y velocity to the LSR37 W LSR
Double km s − Cartesian Galactocentric Z velocity to the LSR38 e W LSR
Double km s − error of Cartesian Galactocentric Z velocity to the LSR39 T D/D
Double Thick disc to thin disc membership probability ratio40
T D/H
Double hick disc to halo membership probability ratio41
Her/D
Double Hercules stream to thin disc membership probability ratio42
Herc/T D