A&A 616, A37 (2018) https://doi.org/10.1051/0004-6361/201833456 Astronomy c ESO 2018 & Astrophysics
New stellar encounters discovered in the second Gaia data release?
C. A. L. Bailer-Jones, J. Rybizki, R. Andrae, and M. Fouesneau
Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany e-mail: [email protected] Received 19 May 2018 / Accepted 19 June 2018
ABSTRACT
Passing stars may play an important role in the evolution of our solar system. We search for close stellar encounters to the Sun among all 7.2 million stars in Gaia DR2 that have six-dimensional phase space data. We characterize encounters by integrating their orbits through a Galactic potential and propagating the correlated uncertainties via a Monte Carlo resampling. After filtering to remove spurious data, we find 694 stars that have median (over uncertainties) closest encounter distances within 5 pc, all occurring within 15 Myr from now. 26 of these have at least a 50% chance of coming closer than 1 pc (and 7 within 0.5 pc), all but one of which are newly discovered here. We confirm some and refute several other previously-identified encounters, confirming suspicions about their data. The closest encounter in the sample is Gl 710, which has a 95% probability of coming closer than 0.08 pc (17 000 AU). Taking mass estimates obtained from Gaia astrometry and multiband photometry for essentially all encounters, we find that Gl 710 also has the largest impulse on the Oort cloud. Using a Galaxy model, we compute the completeness of the Gaia DR2 encountering sample as a function of perihelion time and distance. Only 15% of encounters within 5 pc occurring within 5 Myr of now have been identified, mostly due to the lack of radial velocities for faint and/or cool stars. Accounting for the incompleteness,± we infer the present rate of encounters within 1 pc to be 19.7 2.2 per Myr, a quantity expected to scale quadratically with the encounter distance out to at least several pc. Spuriously large parallaxes± in our sample from imperfect filtering would tend to inflate both the number of encounters found and this inferred rate. The magnitude of this effect is hard to quantify. Key words. Oort Cloud – methods: analytical – methods: statistical – astrometry – solar neighborhood – surveys
1. Introduction 7.2 million stars. This is 22 times larger than our previous encounter study using TGAS in the first Gaia data release. Fur- The first convincing evidence for relative stellar motion thermore, TGAS contained no radial velocities, so these had to came from Edmund Halley in 1718. Since then – if not be obtained by cross matching to external catalogues. well before – people have wondered how close other stars Here we report on results of looking for close encounters may come to our own. Several studies over the past quar- in Gaia DR2. Our approach follows very closely that taken ter century have used proper motions and radial velocities by Bailer-Jones (2015a; hereafter paper 1) for HIPPARCOS and to answer this question (Matthews 1994; Mülläri & Orlov Bailer-Jones (2018; hereafter paper 2) for TGAS. The main dif- 1996; García-Sánchez et al. 1999, 2001; Dybczynski´ 2006; ferences here are: Bobylev 2010a,b; Jiménez-Torres et al. 2011; Bailer-Jones – We use only Gaia data. This avoids the complications of 2015a; Dybczynski´ & Berski 2015; Mamajek et al. 2015; having to obtain radial velocities from several different cata- Berski & Dybczynski´ 2016; Bobylev & Bajkova 2017; Bobylev logues, which resulted in some data heterogeneity and dupli- 2017; Bailer-Jones 2018). cate sources. Other than being interesting in their own right, close encoun- – Using mass estimates from multiband photometry and ters may have played a significant role in the evolution of our Gaia DR2 astrometry in Fouesneau et al. (in prep), we es- solar system, in particular of the Oort cloud. This may also have timate the momentum transfer from an encountering star to had implications for the development of life, since a strong per- Oort cloud comets for essentially every encounter. turbation of the Oort cloud by an encountering star could push – We account for the incompleteness of our encounter sam- comets into the inner solar system. An ensuing collision with the ple by sampling from a self-consistent spatial and kinematic Earth could be catastrophic enough to cause a mass extinction. Galaxy model. This is more realistic that the analytic model Such a fate probably befell the dinosaurs 65 Myr ago. Studies of developed in paper 2. stellar encounters and comet impacts on the Earth can also be The main limitation in the number of encounter candidates in used to learn about the general hazards for life on exoplanets. this study is still the availability of radial velocities. While The recent publication of the second Gaia data release Gaia DR2 contains five-parameter astrometry for 1.33 billion (Gaia DR2; Gaia Collaboration 2018) is a boon to encounter stars, only 7.2 million have published radial velocities. These are studies. It contains six-dimensional (6D) kinematic data – predominantly brighter than G = 14 mag, and are also limited to position, parallax, proper motion, and radial velocity – for the approximate Teff range of 3550 < Teff/K < 6900 (Katz et al. ? Full Tables2 and3 are only available at the CDS via 2018). This is simply the temperature range of the templates anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via used. Teff estimates in Gaia DR2 are described in Andrae et al. http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/616/A37 2018.
Article published by EDP Sciences A34, page 1 of 13 A&A 616, A37 (2018)
In Sect.2 we describe how we select our sample and infer measurements to perihelion parameters, neglecting the full PDF the distribution of the encounter parameters for each candidate. can lead to erroneous results (and not just erroneous uncertain- We analyse the results in Sect.3, discussing new (and dubious) ties), in particular for stars with long travel times. cases, and highlighting disagreement with earlier results. Issues of spurious data and imperfect filtering we discuss in subsec- tions 2.2 and 3.2. In Sect.4 we introduce the new complete- 2.2. Filtering on astrometric solution quality metrics ness model and use this to derive the completeness-corrected There are many reasons why astrometric solutions in Gaia DR2 encounter rate. We conclude in Sect.5 with a brief discussion. may be “wrong” for some stars, in the sense that the reported uncertainties may not be representative of the true uncertainties. The main reasons are: neglect of accelerated motions (i.e. unseen 2. Identification and characterization of close companions); cross-matching errors leading to the inclusion of encounters observations of other sources (spurious data); a poor correction of the so-called “DOF bug” (see Appendix A of Lindegren et al. 2.1. Initial selection and orbital integration 2018). These can lead to erroneous estimates of the quantities or We searched the Gaia archive for all stars which would approach their uncertainties. within 10 pc of the Sun under the assumption that they move on Various metrics on the astrometric solution are reported unaccelerated paths relative to the solar system (the so-called in Gaia DR2 to help identify good solutions. Lindegren et al. “linear motion approximation”, LMA, defined in paper 1). The (2018) discuss some of these and give an example of a set of archive ADQL query for this is in AppendixA. This yielded cuts which may be used to define a conservative sample, i.e. 3865 encounter candidates, which we call the unfiltered sample. an agressive removal of poor solutions (e.g. in their Fig. C.2). In both this selection and the orbital integration used later, we es- This is not appropriate for our work, however, because we timate distance using inverse parallax rather than doing a proper are looking to determine the encounter rate, not just find the inference (Bailer-Jones 2015b; Bailer-Jones et al. 2018). This is most reliable encounters. While quantiles on the various metrics acceptable here, because for the filtered sample we define later, are easily measured, there is no good model for the expected dis- 95% have fractional parallax uncertainties below 0.08 (99% be- tribution of these for only non-spurious results. Concepts like low 0.14; the largest is 0.35), meaning the simple inversion is a “the reduced χ2 should be about one” are simplistic at best (and reasonably good approximation. statistically questionable), and also don’t tell us whether devi- The Gaia data are used as they are, other than we added ations from this are “wrong” or just have mildly deviant as- 0.029 mas to the parallaxes to accommodate the global paral- trometry or slightly underestimated uncertainties. Even a highly lax zeropoint Lindegren et al.(2018). (Neglecting this would significant astrometric excess noise of a few mas may be of remove 83 sources from the unfiltered sample.) There is no evi- little consequence if the parallax and proper motion are large. dence for a proper motion zeropoint offset (Arenou et al. 2018). It is therefore difficult to make a reliable cut. The main qual- While there may be a systematic difference between the Gaia ra- ity metric of interest here is the “unit weight error”, defined in 1 2 1/2 dial velocities and other catalogues of up to 0.5 km s− , the ori- Appendix A of Lindegren et al.(2018) as u = (χ /ν) , where gin of this is unclear (Katz et al. 2018). This is of the order of χ2 is the metric astrometric_chi2_al and ν is the de- the gravitational redshifts which are also not corrected for. None grees of freedom, equal to astrometric_n_good_obs_al 5. of the Gaia uncertainties have been adjusted to accommodate For our unfiltered sample, u correlates quite strongly with− the a possible over- or underestimate of their values (e.g. missing astrometric_excess_noise and reasonably well, but less RMS systematics). tightly, with astrometric_excess_noise_sig. There is no To get more precise encounter parameters for the set of can- correlation between u and visibility_periods_used. Figure didates, we integrated the orbits of the unfiltered sample through C.2 of Lindegren et al.(2018) plots u against G. We see u a smooth Galactic potential forward and backwards in time. increasing for brighter sources, albeit it with a lot of scat- (It was shown in paper 1 that the deviation of an orbit due ter. The full range of u in our unfiltered sample is 0.63 to to perturbations by individual stars can be neglected.) We use 122, and the brighest star in our unfiltered sample (Gaia DR2 the same procedure and model as described in paper 2. The 5698015743040715264 = rho Puppis) has G = 2.67 mag. This Galactic potential, described in detail in paper 1, is a three- corresponds to u < 35 for the selection in Figure C.2, so component axisymmetric model. The bar and spiral arms are not we only retain sources with u < 35. This moderately lib- included, partly because their properties are not well determined, eral cut reduces the sample size to 3465. Rho Puppis it- but mostly because using the same potential as in our previous self has u = 29, astrometric_excess_noise = 2.4 mas, and studies eases comparison of results. As the orbit segments up to astrometric_excess_noise_sig = 2458, yet its parallax and encounter are generally short compared to the scale lengths in proper motion in Gaia DR2 agree with those from HIPPARCOS-2 the model, the exact choice of potential will have only a small (van Leeuwen 2007) to within about 2%. A more agressive cut impact on the encounter parameters for most stars. would clearly be removing genuine solutions. To ensure that the In order to accommodate and propagate the uncertainties in astrometric solutions are reasonably overdetermined, we further the data, we draw 2000 samples from the 6D covariant proba- require that visibility_periods_used is at least 8. Accord- bility density function (PDF) over the data – position, parallax, ing to Arenou et al.(2018), this should help to remove the most proper motion, and radial velocity – for each star and integrate spurious proper motions. This cuts the sample down to 3379. the orbits of each of these “surrogates” through the potential. Large uncertainties in the data, provided they are represen- The distribution of the perihelion time, distance, and speed over tative of the true uncertainty, are not a problem per se because the surrogates for each star is used to characterize the encounter they are accommodated by the resampling we used to map the (in the next section). Comparisons of encounter parameters com- PDF of the perihelion parameters. We will also accommodate puted with the LMA, orbital integration of the nominal data, and this PDF when we compute the completeness and encounter rate orbital integration of the surrogates were shown in paper 1 and in Sect.4. We therefore do not filter out sources due to large paper 2. Due to the nonlinear transformation from astrometric uncertainties.
A34, page 2 of 13 Bailer-JonesBailer-Jones et et al.: al.: Close Close encounters encounters to to the the Sun Sun in in Gaia-DR2Gaia-DR2 placeOne them should all nearer be very than careful Proxima about filteringCentauri) out. However, extreme val- as theyues of all the have dataG just> 19 because mag none they of are them extreme. appearGaia in anyDR2 of is our known se- lections.to include Other, sources less with extreme, implausibly spurious large values parallaxes. are impossible For exam- to 0.10 identifyple, there without are 21 using sources additional with $ > information.1000 mas (which Spuriously would large place ) / pc 0.08 properthem all motions nearer are than less Proxima of a problem, Centauri). because However, close asencounter they all h p starshave generallyG > 19 mag have none relatively of them small appear proper in any motions of our (unless selections. they d are currently very close to encounter). In contrast, stars which
Other, less extreme, spurious values are impossible to identify 0.06 comewithout close using tend additional to be those information. that currently Spuriously have large large radial proper ve- locitiesmotions (compared are less of to a problem, their transverse because velocities). close encounter The radial stars gen- ve- locityerally processing have relatively for Gaia small DR2 proper may motions have produced (unless they spuriously are cur- 0.04 largerently radial very velocities close to encounter). (the pipeline In contrast,can produce stars values which up come to perihelion distance ( −1 ±close1000 tend km s to).Katz be those et al.that (2018) currently report have they large were radial hard velocities to ver- 0.02 ify(compared due to the to their absence transverse of observations velocities). of The standards radial velocity with radial pro- velocities larger than 550 kms. They further state that the preci- cessing for Gaia DR2 may have produced spuriously large radial 0.00 −1 1.15 1.20 1.25 1.30 1.35 1.40 1.45 sions are lower for stars with |vr| > 175 km s , but not that the1 velocities (the pipeline can produce values up to 1000 km s− ). perihelion time (tph) / Myr radialKatz etvelocities al.(2018 themselves) report they are were problematic. hard to verify Moreover,± due to they the ab- vi- | | −1 suallysence inspected of observations all results of standards with vr > with500 radial km s velocitiesand removed larger Fig.Fig. 1. 1. TheDistribution distribution of the of the perihelion perihelion parameters parameters for forthe the2000 2000 sur- 1 suspiciousthan 550 km results. s− . They At this further point state in that the thefiltering precisions we have are lower only surrogatesrogates used used to to characterize characterize the the encounter encounter of of Gl Gl 710 710 = GaiaGaia DR2DR2 1 −1 sevenfor stars stars with withvr |v>r|175> km500 s− km, but s , not six that of thewhich radial were velocities deter- 4270814637616488064.4270814637616488064. minedthemselves by just are| two problematic.| focal plane Moreover, transits (rv_nb_transits they visually inspected= 2; 1 theall resultsminimum with forvr inclusion> 500 km in s− Gaiaand DR2) removed whereas suspicious the median results. | | TableTable 1. 1. Number of stars in the filtered sample found by the orbit inte- numberAt this for point this in sample the filtering is 8. They we do have not have only particularly seven stars large with The numbermed ofmax stars in the filteredmed sample found by the orbit
) / pc ● ● ● ● ●●● ● ● ● ● orbitd integrations result in much larger median perihelion dis- ● ● e ●● ● ● ● ● ●● ● h ● ● ● ● ● ● ters from papers 1 and 2 not found in the present study. m p ● ● ● ● ● ● ● tances.d Similarly, our● ● list● will● be missing those stars which would ● ● ● ● ●● ● ● ●● 3. Close encounters found in Gaia DR2 30 ●● ● ●● ● ● ● ● ● med ● ● ● ●●●● ● ● ● ● ● ● have had d ● < 10 pc had● they been subject to the orbital in- ● ●● ● ● ph ● ● ● ●●● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ●●● ● We first look at the overall results, then discuss individual stars ● ● ● ●● ● ● > tegration, but were never selected● because they●● had dph 10 pc 3.1. Overall results ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ●● ● found approaching within 1 pc, and finally those close encoun- 20 ● ●● ● ● ●● ● ● from the LMA. This latter●● omission● will● in●●●● principle● lead to an ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ●●● ● ● ● ●●●●●● ● ● ● ● ●● ● ●●● ●● The perihelion for each star is described by the distribution of ● ●● ● ●●● ● ters from papers 1 and 2 not found in the present study. ● ●● ●●●● ●● ●●● ● ● ●● ● underestimate of the● derived● ● ● ● encounter●● rate.●●●●● ● However,● as shown ●● ● ●● ● ● ●●● ●● ● ●● ● ● ● ●● ● ●●●●●●● ● ●●●●●●●●●● ● ● ● ● ●●●●●● ● ● ●●●●●● ● ●●●● ●● ● ● ● ●●● ● ●● ● ●●● ●●●● ● ● the 2000 surrogates in the perihelion time, t , distance, d , and ● ●● ●●●●●●●●●●●● ● ● ●●●●●● ●●●●●●●●●● ph ph ● ●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●● ●●● ● ●●● inperihelion distance ( both papers 1 and 2, this● ●●●● ● mostly● ●●● ● ●●● ● aff● ●ects● stars● that● will en- ● ● ●● ●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●● ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●● ● ● 10 ●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● speed v . The distribution for one particular star is shown in ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ph ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ● counter further in the past●● /●future●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● and●●●●●●●●●●●●●●●●●●●●●/●●●or●●●●●●●●●●● nearer to the edge of ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● 3.1. Overall results ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●● ● ● Figure 1. As in papers 1 and 2 we summarize these using the ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● the 10 pc distance limit.● As● ● ● stars●●●●●●●●●●●●●●●●●●●●●●●● at●●●●●●●●●● such● ●●● ● large● distances can ● ●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●● ●●● ●●●● ● ● ● ●● ●●● ●● ●●●●●●●●●● ●●● ● ●● ● ● ●●●●● ● ● median,The perihelion and characterize for each their star isuncertainty described using by the the distribution 5th and 95th of hardly0 be considered encountering, we will only be interested in percentiles (which together form a 90% confidence interval, CI). −20 −10 0 10 20 the 2000 surrogates in the perihelion time, tph, distance, dph, and encounters within 5 pc from now on.me Moreover,d when we later perihelion time (tph ) / Myr Thespeed numbervph. The of stars distribution coming within for one various particular perihelion star is distances shown in compute the (completeness-corrected) encounter rate, we will isFig. shown1. As in in Table papers 1 (see 1 and also 2 Figure we summarize 10 later). these Summary using perihel- the me- limit the sample to a narrower time window. med Fig. 2. Perihelion times and distances computed by orbit integration for ion data for those stars with d < 1 pc are shown in Table 2 med < dian, and characterize their uncertaintyph using the 5th and 95th the filteredThe encounters sample (278with pointsd lieph outside5 pc the are plotting shown range). in Fig. Open3. cir- We (thepercentiles online table (which at CDS together reports form higher a 90% numerical confidence precisions interval,CI). for clessee show a strong the median drop in of the perihelion density of time encounters and distance with distributions. increasing theThe whole number filtered of stars sample). coming Negative within timesvarious indicate perihelion past distances encoun- Thetph error. This bars is show primarily the limits a consequence of the 5% and of 95% the percentiles. magnitude limit in ters.is shown The magnitude, in Table1 (see colour, also and Fig. quality 10 later). metrics Summary from perihelionGaia DR2 the| | sample (95% brighter than G = 13.4 mag): Encounters that are shown in Table 3. Somemed of these encounters we have cause data for those stars with dph < 1 pc are shown in Table2 (the would occur further in the past/future generally correspond to toonline disregard table (discussed at CDS reports later). higher numerical precisions for the stars that are currently more distant, and so more likely to be Figure 2 plots the perihelion times and distances. (This and tances. Similarly, our list will be missingff those stars which would whole filtered sample). Negative times indicate past encounters. below themed limiting magnitude. The e ective time limit of this otherThe magnitude, plots do not colour, remove and bogus quality cases.) metrics The starsfrom wereGaia selectedDR2 are havestudy had is 5–10dph Myr.< 10 Comparing pc had they this been with subject Fig. 3 to of the paper orbital 2, we inte- see byshown the LMA in Table to come3. Some within of these 10 pc encounters of the Sun, we but have some cause of the to gration,that while butGaia wereDR2 never has selected found many because more they encounters had dph > than10 ourpc orbitdisregard integrations (discussed result later). in much larger median perihelion dis- fromTGAS the study LMA. (by This a factor latter of omission about seven willwithin in principle 5 pc), leadGaia toDR2 an
Article number, pageA34, 3 of page 13page.13 3 of 13 A&A 616, A37 (2018)
med Table 2. Perihelion parameters for all stars with a median perihelion distance (median of the samples; dph ) below 1 pc, sorted by this value.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 Gaia DR2 source ID tph / kyr dph / pc vph / km s− $ σ($) µ σ(µ) vr σ(vr ) M
1 1 med 5% 95% med 5% 95% med 5% 95% mas mas yr− km s− M
4270814637616488064 1281 1221 1352 0.068 0.052 0.084 14.5 13.8 15.2 52.55 0.05 0.46 0.11 14.5 0.4 0.68 − 955098506408767360 737 817 672 0.151 0.043 0.279 38.5 34.7 42.1 34.53 0.61 0.82 1.21 38.5 2.1 1.26 − − − 5571232118090082816 1162 1173 1151 0.232 0.199 0.264 82.3 81.5 83.1 10.23 0.02 0.42 0.05 82.2 0.5 0.82 − − − 2946037094755244800 908 1054 795 0.338 0.101 0.662 42.1 36.9 47.2 25.66 1.12 1.34 1.82 42.1 3.2 NA − − − 4071528700531704704 438 268 1109 0.374 0.213 1.037 44.2 16.9 71.7 50.43 0.89 8.91 1.92 44.5 17.1 1.00 − 510911618569239040 2788 2853 2731 0.429 0.368 0.494 26.5 25.9 27.1 13.23 0.04 0.56 0.05 26.4 0.3 1.07 − − − 154460050601558656 372 332 418 0.444 0.225 0.734 233.5 218.2 247.8 11.29 0.67 2.16 0.99 233.1 8.9 NA − † 6608946489396474752 2757 2826 2694 0.491 0.301 0.681 45.3 44.3 46.3 7.89 0.05 0.67 0.12 44.2 0.6 0.82 − − − 3376241909848155520 450 516 396 0.508 0.276 0.773 79.9 70.8 88.9 27.18 1.09 5.96 2.56 79.9 5.6 1.04 − − − 1791617849154434688 1509 1533 1487 0.579 0.503 0.654 56.4 55.6 57.2 11.49 0.04 0.88 0.09 56.3 0.5 0.80 − − − 4265426029901799552 655 686 627 0.580 0.368 0.791 46.6 46.3 46.9 32.05 0.88 5.93 1.95 46.6 0.2 0.49 − − − 5261593808165974784 897 916 879 0.636 0.608 0.664 71.1 69.7 72.6 15.32 0.02 2.33 0.06 71.0 0.9 0.55 − − − 5896469620419457536 6680 5880 7550 0.657 0.250 1.088 16.8 14.9 19.1 8.69 0.03 0.75 0.08 16.9 1.3 0.62 − 4252068750338781824 886 452 3177 0.668 0.352 3.257 27.6 5.8 51.7 38.86 0.61 5.74 1.24 27.7 14.2 0.89 − 1949388868571283200 710 738 685 0.673 0.591 0.762 347.4 336.5 357.9 3.96 0.04 0.80 0.05 347.3 6.5 NA − − − † 1802650932953918976 5341 4770 6033 0.740 0.213 1.370 53.0 47.1 59.2 3.46 0.04 0.76 0.07 52.6 3.9 0.98 − 3105694081553243008 715 790 656 0.760 0.516 0.990 38.3 35.1 41.4 35.72 0.97 7.44 2.01 38.4 1.9 0.75 − − − 5231593594752514304 89 89 89 0.815 0.807 0.822 715.9 714.1 717.6 15.35 0.03 29.87 0.07 715.8 1.0 0.67 − † 4472507190884080000 1836 1210 3382 0.819 0.214 2.371 52.0 28.6 77.1 10.37 0.61 0.51 1.13 52.2 15.2 0.96 − 3996137902634436480 641 578 717 0.820 0.547 1.122 38.5 34.7 42.1 39.71 1.07 10.51 2.73 38.4 2.3 0.95 − 3260079227925564160 909 890 928 0.824 0.798 0.849 33.4 32.7 34.1 32.19 0.06 6.00 0.12 33.4 0.4 0.47 − 5700273723303646464 1636 1846 1464 0.836 0.227 1.768 38.1 36.6 39.5 15.70 1.08 0.26 1.98 38.0 0.9 0.95 − − − † 5551538941421122304 3815 4023 3626 0.866 0.772 0.955 30.4 28.8 31.9 8.47 0.01 1.03 0.04 30.0 1.0 0.65 − − − 2924378502398307840 1841 1880 1803 0.880 0.803 0.957 87.1 85.5 88.8 6.10 0.03 0.74 0.06 87.0 1.0 0.75 − − − 6724929671747826816 1045 986 1111 0.884 0.512 1.324 54.8 53.0 56.7 17.07 0.49 3.13 0.99 54.8 1.1 0.72 − 3972130276695660288 511 530 493 0.888 0.855 0.921 31.9 30.7 33.0 59.97 0.05 22.01 0.14 31.8 0.7 0.58 − − − 5163343815632946432 4965 5499 4523 0.896 0.484 1.361 37.1 33.7 40.7 5.39 0.04 1.45 0.09 35.4 2.3 0.76 − − − 2926732831673735168 1680 1698 1664 0.917 0.820 1.014 66.5 66.1 66.9 8.75 0.04 0.99 0.08 66.5 0.3 1.15 − − − 2929487348818749824 5364 5645 5108 0.926 0.239 2.004 70.0 67.9 72.3 2.61 0.06 0.42 0.10 69.9 1.4 1.34 − − − 939821616976287104 90 91 90 0.989 0.974 1.002 568.4 567.1 569.6 19.05 0.07 45.71 0.11 568.2 0.8 NA − − − † 3458393840965496960 866 1419 600 0.996 0.396 2.104 86.4 52.8 120.9 13.20 1.05 2.79 2.06 86.6 19.9 1.17 − − − med med med Notes. The first column is the Gaia DR2 source ID. Columns 2, 5, and 8 are tph , dph , and vph respectively. The columns labelled 5% and 95% are the bounds of corresponding confidence intervals. Columns 11–16 list the parallax ($, plus the 0.029 mas zeropoint offset), total proper motion (µ), and radial velocity (vr) along with their 1-sigma uncertainties. Column 17 is the estimated mass of the star from Fouesneau et al. (in preparation); NA indicates it is missing. The formal 1-sigma uncertainties in the masses are a few percent (systematics are likely to be higher). Those encounter results we consider bogus are marked with the dagger symbol in the final column. These are discussed in the text. Other may also be dubious: quality metrics from Gaia DR2 can be found in Table3. The full table is available at the CDS and includes all 3379 stars in the filtered sample and reports some columns to a higher numerical precision. does not allow us to probe much further into the past/future, be- the entire filtered sample are 0.53, 0.91, and 1.64 M respec- cause of the similar magnitude limits on the samples. tively. The lack of massive stars in the sample is a conse- We also see in Fig.3 a slight reduction in the density of en- quence of the Teff filtering on radial velocities published in counters very close to the present time. (The effect is not quite Gaia DR2. as strong as first appears, as it is partly an illusion produced by Close encounters are interesting not least because they the shorter error bars.) This was also seen with in paper 2, where perturb the Oort cloud. The degree of perturbation, or rather the we argued this was due to two things: missing bright stars in the impulse which comets can receive, depends not only on the dis- Gaia catalogue; and the ever smaller volume available for en- tance of the encountering star, but also on its speed and the mass. counters to occur at arbitrarily near times. Both of these apply According to the simple impulse approximation (Öpik 1932; for Gaia DR2, although we argue later that this is also a conse- Oort 1950; Rickman 1976; Dybczynski 1994) the impulse trans- 1 α quence of the limited Teff range for stars with radial velocities in fer is given by Mvph− dph− where α is 1 for very close encounters Gaia DR2. (on the order of the comet–Sun separation), and 2 otherwise. The The uncertainties in tph and dph are correlated. This can be impulse of the encountering stars is visualized in Figs.6 and7 seen in Fig.4, where we plot the individual surrogates instead for α equal to 1 and 2 respectively. Regardless of which impulse of the summary median and axis-parallel error bars. Each star approximation we use, it is the closest encounters which have is generally represented as an ellipsoidal shape pointing roughly the greatest impact. towards tph = 0, dph = 0. med The perihelion speeds for encounters with dph < 5 pc are med 1 3.2. Colour-magnitude diagram shown in Fig.5. 90% have vph < 100 km s− . The very fast en- counters are almost entirely due to stars with large radial veloci- Figure8 shows the colour-absolute magnitude diagram (CMD) ties. for the close encounters on the assumption of zero interstellar The masses of the encountering stars are listed in Table2. extinction. The upper panel shows the 26 non-bogus encounters med The 5th, 50th, and 95th quantiles of the mass distribution over with dph < 1 pc from Table2. Eleven of these have unit weight
A34, page 4 of 13 Bailer-Jones et al.: Close encounters to the Sun in Gaia-DR2
Table 3. Additional data from Gaia DR2 for the close encounters listed in Table2.
Gaia DR2 source ID G BP-RP u No. Astrometric Astrometric No. l b mag mag visibility excess excess RVS deg deg periods noise noise sig transits 4270814637616488064 9.06 1.70 1.22 10 0.00 0.00 2 27 6 955098506408767360 12.41 0.76 16.19 10 2.49 1661.28 5 176 10 5571232118090082816 11.79 1.50 1.25 15 0.00 0.00 10 249 25 2946037094755244800 12.34 1.49 27.41 10 3.12 4225.87 22 228 − 7 4071528700531704704 12.44 0.78 23.52 10 3.92 3138.25 3 7 −11 510911618569239040 8.88 0.77 1.57 17 0.00 0.00 6 126− 0 154460050601558656 15.37 NA 6.47 10 1.85 359.52 3 174 11 6608946489396474752 12.28 1.44 0.89 10 0.00 0.00 6 23 −61 3376241909848155520 12.52 0.78 28.53 11 5.46 8824.25 8 190− 5 1791617849154434688 11.00 1.09 1.49 11 0.00 0.00 9 70 18 4265426029901799552 12.20 2.07 18.88 10 3.44 2533.02 3 32− 0 5261593808165974784 12.69 2.02 1.39 18 0.00 0.00 14 285 27 5896469620419457536 13.55 1.98 1.06 11 0.03 0.23 5 313− 6 4252068750338781824 12.10 0.92 20.42 9 2.53 1453.10 5 26 3 1949388868571283200 13.12 NA 1.34 11 0.13 5.81 2 86 −13 1802650932953918976 12.69 1.02 0.99 14 0.00 0.00 11 52 −11 3105694081553243008 12.31 1.24 25.05 9 4.78 6025.61 6 215 − 2 5231593594752514304 12.03 1.87 1.22 16 0.00 0.00 2 293 −9 4472507190884080000 12.90 0.87 13.00 9 2.17 1206.88 5 29− 15 3996137902634436480 11.74 0.87 32.65 9 5.07 7026.13 4 212 64 3260079227925564160 11.73 2.13 1.98 13 0.12 8.12 7 188 33 5700273723303646464 11.96 0.84 32.64 15 5.26 8991.52 8 242− 7 5551538941421122304 13.10 1.71 1.23 16 0.06 1.31 11 257 21 2924378502398307840 12.62 1.25 1.10 14 0.00 0.00 14 232 −16 6724929671747826816 11.97 1.31 13.55 9 1.87 1105.66 2 352 −11 3972130276695660288 9.88 2.18 1.58 8 0.00 0.00 3 227− 65 5163343815632946432 12.92 1.25 1.21 10 0.00 0.00 5 194 50 2926732831673735168 9.56 0.72 1.41 14 0.00 0.00 9 230 −12 2929487348818749824 11.21 0.78 1.54 12 0.00 0.00 4 233 − 5 939821616976287104 9.91 1.43 1.50 8 0.00 0.00 2 181− 18 3458393840965496960 12.07 0.63 11.71 9 3.07 2046.73 2 172 9
Notes. All data are taken directly from the catalogue, except for the astrometric “unit weight error” u which is calculated as sqrt[astrometric_chi2_al/(astrometric_n_good_obs_al-5)]. The full table available at the CDS includes all 3379 stars in the filtered sample. error u > 10 (the others have u < 2) and lie some way below and there is in fact a decreased tendency for encounters to be below left of the main sequence (MS), yet well above the white dwarf the MS compared to nearby stars. This comparison is not ideal, (WD) sequence. The dimensionless quantity u measures how however, because some of these nearby stars are WDs – essen- well the five-parameter solution describes the astrometric data tially absent from the encountering sample due to the magnitude (see Sect. 2.2). If we assumed that all the encounters were sin- limit – and the encounters are drawn from a much larger volume. gle main sequence stars, then one interpretation of this diagram We noted in Sect. 2.2 that a large value of u does not mean is that the parallaxes of these 11 sources are wrong; specifically the parallax is wrong. Indeed, a relatively poor fit of the five- that they are all overestimated by one or two orders of magni- parameter solution is in principle expressed by larger values tude (i.e. the sources should be much further away). This seems of the astrometric uncertainties (and these are quite correlated unlikely, although it is the case that a search for encounters will with u). Here, the 11 sources in the upper panel of Fig.8 have preferentially include sources with spuriously large parallaxes σ($) between 0.5 and 1.1 mas, whereas the other 15 have rather than sources with spuriously small parallaxes (because the σ($) < 0.06 mas. We have already accommodated the uncertain- former are more likely to have apparent close encounters). As we ties in the identification of the close encounters. Furthermore, see a smaller proportion of sources between the MS and WDs although the parallax uncertainties are larger, they produce un- among nearby stars (blue points in Fig.8), it at first seems un- certainties in the absolute magnitude of no more than 0.08 mag, likely that we would get such a large proportion of such sources which is smaller than the size of the points plotted in Fig.8. Con- among the closest encounters (upper panel). The magnitude of tamination of the astrometric solution by nearby (sub-arcsecond) this selection effect is impossible to estimate without knowing sources is possible, and although most of the sources are near which parallaxes are actually spurious. On the other hand, if we to the Galactic plane (as are unquestioned sources), they are look instead at all encounters (lower panel), then we find that just all bright, so less likely to be affected by crowding. Visual in- 14% of all encounters within the plotted range lie below the MS, spection of non-Gaia images of these targets shows no obvious whereas this figure is 24% for the nearby stars. In other words, contaminants. They do generally have fewer visibility periods in
A34, page 5 of 13 Bailer-Jones et al.: Close encounters to the Sun in Gaia-DR2 place them all nearer than Proxima Centauri). However, as they all have G > 19 mag none of them appear in any of our se- Bailer-Jones et al.: Close encounters to the Sun in Gaia-DR2 lections. Other, less extreme, spurious values are impossible to 0.10 identify without using additional information. Spuriously large Table 3. Additional data from Gaia DR2 for the close encounters listed in Table 2. They are all taken directly from the catalogue, except for the ) / pc 0.08 astrometric “unit weight error” u which is calculated as sqrt[astrometric_chi2_al/(astrometric_n_good_obs_al-5)]. The online table
proper motions are less of a problem, because close encounter h p stars generally have relatively small proper motions (unless they d at CDS includes all 3379 stars in the filtered sample. are currently very close to encounter). In contrast, stars which come close tend to be those that currently have large radial ve- 0.06 Gaia DR2 source ID G BP-RP u No. astrometric astrometric No. l b locities (compared to their transverse velocities). The radial ve- mag mag visibility excess excess RVS deg deg locity processing for Gaia DR2 may have produced spuriously 0.04 periods noise noise sig transits 4270814637616488064 9.06 1.70 1.22 10 0.00 0.00 2 27 6 large radial velocities (the pipeline can produce values up to perihelion distance ( −1 955098506408767360 12.41 0.76 16.19 10 2.49 1661.28 5 176 10 ±1000 km s ).Katz et al. (2018) report they were hard to ver- 0.02 5571232118090082816 11.79 1.50 1.25 15 0.00 0.00 10 249 -25 ify due to the absence of observations of standards with radial 2946037094755244800 12.34 1.49 27.41 10 3.12 4225.87 22 228 -7 velocities larger than 550 kms. They further state that the preci- 4071528700531704704 12.44 0.78 23.52 10 3.92 3138.25 3 7 -11 0.00 sions are lower for stars with |v | > 175 km s−1, but not that the 1.15 1.20 1.25 1.30 1.35 1.40 1.45 510911618569239040 8.88 0.77 1.57 17 0.00 0.00 6 126 0 r perihelion time (t ) / Myr radial velocities themselves are problematic. Moreover, they vi- ph 154460050601558656 15.37 NA 6.47 10 1.85 359.52 3 174 -11 | | −1 A&A proofs: manuscript no. stellar_encounters_gdr26608946489396474752 12.28 1.44 0.89 10 0.00 0.00 6 23 -61 sually inspected all results with vr > 500 km s and removed Fig. 1. The distribution of the perihelion parameters for the 2000 3376241909848155520 12.52 0.78 28.53 11 5.46 8824.25 8 190 5 suspicious results. At this point in the filtering we have only surrogates used to characterize the encounter of Gl 710 = Gaia DR2 1791617849154434688 11.00 1.09 1.49 11 0.00 0.00 9 70 -18 −1 Table 2. Perihelion parameters for all stars with a median perihelion distance (median of the samples; dmed) below 1 pc, sorted by this value. The seven stars with |vr| > 500 km s , six of which were deter- 4270814637616488064. 4265426029901799552ph 12.20 2.07 18.88 10 3.44 2533.02 3 32 0 med med med 5261593808165974784 12.69 2.02 1.39 18 0.00 0.00 14 285 -27 mined by just two focal plane transits (rv_nb_transits = 2; first column is the Gaia DR2 source ID. Columns 2, 5, and 8 are tph , dph , and vph respectively. The columns labelled 5% and 95% are the the minimum for inclusion in Gaia DR2) whereas the median bounds of corresponding confidence intervals. Columns 11–16 list the parallax ($, plus the 0.0295896469620419457536 mas zeropoint offset), 13.55total proper 1.98 motion 1.06 (µ), 11 0.03 0.23 5 313 6 4252068750338781824 12.10 0.92 20.42 9 2.53 1453.10 5 26 -3 number for this sample is 8. They do not have particularly large Tableand radial 1. The velocity number (vr of) along stars with in the their filtered 1-sigma sample uncertainties. found by the Column orbit 17 is the estimated mass of the star from Fouesneau et al. (in preparation); med max Bailer-Jonesmed et al.: Close encounters to the Sun in Gaia-DR21949388868571283200 13.12 NA 1.34 11 0.13 5.81 2 86 -13 uncertainties, because Gaia DR2 only includes stars for which integrationNA indicates to it have is missing.dph < Thedph formal(for any 1-sigmatph ). uncertainties Stars with potentially in the masses are a few percent (systematics1802650932953918976 are likely to be higher). 12.69 Those 1.02 encounter 0.99 14 0.00 0.00 11 52 -11 −1 results we consider bogus are marked with the dagger symbol in the final column. These are discussed in the text. Other may also be dubious: |σ(vr)| < 20 km s . It is tempting to filter out stars with a small problematic data have not been excluded. 3105694081553243008 12.31 1.24 25.05 9 4.78 6025.61 6 215 -2 qualityTable 3.metricsAdditional from Gaiadata fromDR2 canGaia be DR2 found for in the Table close 3. encounters The online listed table in at Table CDS 2. includes They are all all 3379 taken5231593594752514304 stars directly in the from filtered the sample 12.03catalogue, and 1.87 exceptreports 1.22 for some the 16 0.00 0.00 2 293 -9 number of transits, but this also removes valid measurements, astrometric “unit weight error” u which is calculated as sqrt[astrometric_chi2_al/(astrometric_n_good_obs_al-5)]. The online table such as that for the closest known encounter, Gl 710, which has columns to a higher numericalmax precision. 4472507190884080000 12.90 0.87 13.00 9 2.17 1206.88 5 29 15 at CDS includes all 3379dph stars/ pc in the No. filtered stars sample. 3996137902634436480 11.74 0.87 32.65 9 5.07 7026.13 4 212 64 two transits and a radial velocity consistent with non-Gaia mea- 3260079227925564160 11.73 2.13 1.98 13 0.12 8.12 7 188 -33 1∞ 2 33379 4 5 6 7 8 9 10 11 12 13 14 15 16 17 surements. We do not filter on rv_nb_transits. Gaia DR2 source ID G BP-RP u No. astrometric astrometric5700273723303646464 No. l b11.96 0.84 32.64 15 5.26 8991.52 8 242 7 10 2548 −1 5551538941421122304 13.10 1.71 1.23 16 0.06 1.31 11 257 -21 We define the remaning set of 3379 stars as the filtered Gaia DR2 source ID tph / kyr magdph mag/ pc visibility vph / km s excess$ excess σ($) RVSµ σ deg(µ) deg vr σ(vr) M 2924378502398307840−1 12.62−1 1.25 1.10 14 0.00 0.00 14 232 -16 sample. It comprises all stars that approach within 10 pc of med5 5% 694 95% med 5% 95% medperiods 5% 95% noise noise mas sig transits mas yr km s M 3 283 6724929671747826816 11.97 1.31 13.55 9 1.87 1105.66 2 352 -11 the Sun according to the LMA (but not necessarily the or- 42708146376164880644270814637616488064 1281 1221 1352 0.068 9.06 0.052 1.70 0.084 1.22 14.5 10 13.8 15.2 0.00 52.553972130276695660288 0.00 0.05 0.46 2 0.11 27 -14.5 6 9.88 0.4 2.18 0.68 1.58 8 0.00 0.00 3 227 65 bital integration) and which satisfy the filters u < 35 and 955098506408767360955098506408767360 -7372 -817 129 -672 12.41 0.151 0.043 0.76 0.279 16.19 38.5 10 34.7 42.1 2.49 34.53 1661.285163343815632946432 0.61 0.82 5 1.21176 10 12.9238.5 2.1 1.25 1.26 1.21 10 0.00 0.00 5 194 -50 visibility_periods_used ≥ 8. When it comes to consider- 55712321180900828165571232118090082816 -11621 -1173 -1151 31 11.79 0.232 0.199 1.50 0.264 1.25 82.3 15 81.5 83.1 0.00 10.232926732831673735168 0.00 0.02 0.42 10 0.05249 -25 82.2 9.56 0.5 0.72 0.82 1.41 14 0.00 0.00 9 230 -12 ing the completeness-corrected encounter rate (section 4.3), we 29460370947552448002946037094755244800 -9080.5 -1054 -795 8 12.34 0.338 0.101 1.49 0.662 27.41 42.1 10 36.9 47.2 3.12 25.66 4225.872929487348818749824 1.12 1.34 22 1.82228 -7 11.2142.1 3.2 0.78 NA 1.54 12 0.00 0.00 4 233 -5 40715287005317047044071528700531704704 4380.25 268 1109 3 12.44 0.374 0.213 0.78 1.037 23.52 44.2 10 16.9 71.7 3.92 50.43 3138.25939821616976287104 0.89 8.91 3 1.92 7 -11 -44.5 9.91 17.1 1.43 1.00 1.50 8 0.00 0.00 2 181 18 shall further limit this sample to stars with G < 12.5 mag, which 510911618569239040510911618569239040 -2788 -2853 -2731 0.429 8.88 0.368 0.77 0.494 1.57 26.5 17 25.9 27.1 0.00 13.233458393840965496960 0.00 0.04 0.56 6 0.05126 012.0726.4 0.3 0.63 1.07 11.71 9 3.07 2046.73 2 172 9 contains 2522 stars. 154460050601558656154460050601558656 372 332 418 15.37 0.444 0.225 NAA&A 0.734 6.47 616, 233.5 A37 10 218.2 (2018) 247.8 1.85 11.29 359.52 0.67 2.16 3 0.99174 -11-233.1 8.9 NA † 66089464893964747526608946489396474752 -2757 -2826 -2694 12.28 0.491 0.301 1.44 0.681 0.89 45.3 10 44.3 46.3 0.00 7.89 0.00 0.05 0.67 6 0.12 23 -61 44.2 0.6 0.82 ● ● ● ●● 3376241909848155520● -450 -516 -396 0.508 0.276● 0.773 79.9 70.8 88.9 27.18 1.09 5.96 2.56 79.9 5.6 1.04 ● ● 3376241909848155520 12.52 0.78 28.53 11 5.46 8824.25 8 190 5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●
● 5 5 ● 50 ● ● ● ● ●● ● ● ● ● ● 1791617849154434688 -1509 -1533 -1487 0.579● 0.503 0.654 56.4 55.6 57.2 11.49 0.04 0.88 0.09 56.3 0.5 0.80 ●● ● ●●● ● ● ● ● ● 1791617849154434688● 11.00● ● 1.09 1.49 11 0.00 0.00 9 70 -18 ● ● ● ● ●●●●● ● ●●●●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ●● ● ● 3. Close encounters found in Gaia DR2 ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ●●●● ● ● ● ●●●● ● ● 42654260299017995524265426029901799552 -655 -686 -627 12.20 0.580● ● 0.368 2.07 0.791 18.88 46.6 10 46.3 46.9 3.44 32.05 2533.02 0.88 5.93 3 1.95 32 046.6 0.2 0.49 ● ● ●●●● ●●● ●●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ●●● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 5261593808165974784● -897 -916 -879 0.636 0.608 0.664 71.1 69.7 72.6 15.32 0.02 2.33 0.06 71.0 0.9 0.55 ● ● ● ●●● ● ● ● ●● ● ● 5261593808165974784● 12.69 2.02● 1.39 18 0.00 0.00 14 285 -27 ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●●●●● ●● ● ● ●● ● We first look at the overall results, then discuss individual stars ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●
● 4 4 589646962041945753640 ●● ● 6680● 5880 7550 0.657 0.250 1.088 16.8 14.9 19.1 8.69 0.03 0.75 0.08 -16.9 1.3 0.62 ● ●●● ● ●5896469620419457536● ● 13.55● ● 1.98 1.06● 11 0.03 0.23 5 313 6 ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ●● ● found approaching within 1 pc, and finally those close encoun- 4252068750338781824 886 452 3177 0.668● 0.352 3.257 27.6 5.8 51.7 38.86 0.61 5.74 1.24 -27.7 14.2 0.89 ● ● ●●●●●● ● ● ● ● ● ●● ● ) / pc ● ● ● ● ● 4252068750338781824●●● ● ● ● 12.10 0.92 20.42 9 2.53 1453.10 5 26 -3 ● ●● ● ●● ● ● ● ) / pc ● ● ● d ● ● ●● ● ●●● ) / pc ●● ● ● ● d ● ● ● e ●● ● ● ● ● ●● ● ● ●● ●●● ● ● ● ● ●
h ● h ● ● ● ● ● ● ● † e ● ● ● ● ● ● ● ● ● 1949388868571283200● -710 -738 -685 0.673 0.591 0.762 347.4 336.5 357.9 3.96 0.04 0.80 0.05 347.3 6.5 NA h ● ● p m p ● ters from papers 1 and 2 not found in the present study. ● 1949388868571283200● ● 13.12 NA 1.34 11 0.13 5.81 2 86 -13 ● ● ● ●● ●● ●● ● m p ● ● ● ● ● ● ●● ●● ●● ● d d ● ● ● ● ● ● ●● ● ●● ● ● ● ● d ● ● ● ● ● ● ● 1802650932953918976● 5341●● ● 4770 6033 0.740● ●● 0.213 1.370 53.0 47.1 59.2 3.46 0.04 0.76 0.07 -52.6 3.9 0.98 ● ● ●● ● ● ● 3 3 30 ● ● ● ● 1802650932953918976● 12.69● ● 1.02 0.99 14 0.00 0.00 11 52 -11 ● ● ● ●● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ●●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 3105694081553243008● ● -715● -790 -656●●● 0.760● ● ●● 0.516● 0.990 38.3 35.1 41.4 35.72 0.97 7.44 2.01 38.4 1.9 0.75 ●● ● ● ● 3105694081553243008● ● 12.31● 1.24● 25.05 9 4.78 6025.61 6 215 -2 ● ●● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ●● ●● ● ● ● ●● ● ●●● ● ● ● ● ●● ● ●●● ● 5231593594752514304 89● 89 89● 0.815●●● ● 0.807 0.822 715.9 714.1 717.6 15.35 0.03 29.87 0.07 -715.8 1.0 0.67 † ● ● ● ●●● ● ● 5231593594752514304● ● 12.03● ● ● 1.87 1.22 16 0.00 0.00 2 293 -9 ● ● ●● ● ●●● 3.1. Overall results ● ● ● ●●●● ●● ● ●● ●● ● ●● ● ●● ● ● ● ●● ●● ● ● ● ● ●●●● ●●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● 4472507190884080000● 1836● 1210 3382●● ● 0.819● ● 0.214 2.371 52.0 28.6 77.1 10.37 0.61 0.51 1.13 -52.2 15.2 0.96 ● ● ●●● ●● ●● ● 2 2 20 4472507190884080000● ● ● 12.90● ●● ● ● 0.87 13.00 9 2.17 1206.88 5 29 15 ● ●● ● ●● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ●● ●●●●● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ●● ●● 39961379026344364803996137902634436480● ● 641●●●●●● 578 717● 11.74 0.820● ● 0.547 0.87● 1.122 32.65 38.5 9 34.7 42.1 5.07 39.71 7026.13 1.07 10.51 4 2.73212 64 -38.4 2.3 0.95 ● ● ● ● ● ●● ● ●●● ●● ● ● ● ● ● The perihelion for each star is described by the distribution of ● ●● ● ●●● ● ● ● ● ●● ●●●● ●● ●●● ● ● ●● ● ● ●● ●● ●● ● ● ● ● ● ● ● ●● ●●●●● ● ● ● ●● ● ● ● ● 3260079227925564160●● 909● ●● ● ● 890 928●●● 0.824●● ● ●● 0.798● 0.849 33.4 32.7 34.1 32.19 0.06 6.00 0.12 -33.4 0.4 0.47 ● ● 3260079227925564160● ●● ● ●●●●●●● ● ●●●●●●●●● 11.73● ● ● ● 2.13 1.98 13 0.12 8.12 7 188 -33 ●● ● ● ● ●●●●●● ● ● ●●●●●● ● ●●●● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● the 2000 surrogates in the perihelion time, t , distance, d , and ● ●● ●●●●●●●●●●● ● ● ●●●●●●●● ●●●●●●●● ● ● ● ● ● ph ph ● ●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●● ●● ● ●●●● ● ● ● ● ● perihelion distance ( ● perihelion distance ( ● ● ● ● ● ● 5700273723303646464● ● -1636●● ●●●●● -1846●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● -1464●●●●●●●●●●●●● ●●●●● ●● 0.836● 0.227 1.768 38.1 36.6 39.5 15.70 1.08 0.26 1.98 38.0 0.9 0.95 † ● ● ● ● ● ● 5700273723303646464● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● 11.96●●● ● ● 0.84 32.64 15 5.26 8991.52 8 242 7 perihelion distance ( ● ● ● ● ● ● 1 1 10 ●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● speed v . The distribution for one particular star is shown in ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ph 5551538941421122304 -3815●●●●●● ●●●● -4023●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● -3626●●●●●●●●●●●●●●●● ●●●●● 0.866● 0.772 0.955 30.4 28.8 31.9 8.47 0.01 1.03 0.04 30.0 1.0 0.65 ● ● 5551538941421122304●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 13.10● 1.71 1.23 16 0.06 1.31 11 257 -21 ● ● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ● ● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●● ● ● ● 2924378502398307840 -1841● ●●●● -1880●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● -1803●●●●● ●●●● ●●●●● ●● 0.880● ● ● 0.803 0.957 87.1 85.5 88.8 6.10 0.03 0.74 0.06 87.0 1.0 0.75 ● ● Figure 1. As in papers 1 and 2 we summarize these using the 2924378502398307840● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ● 12.62 1.25 1.10 14 0.00 0.00 14 232 -16 ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ● ● ●● ●● ●●●●●●●●●●●●●●●●●●●●● ●●● ●●●● ● ● ●● ●●● ●●●●●●●●●●●● ●●● ● ● ● ● 6724929671747826816 1045● 986●●●● ● ● 1111 0.884 0.512 1.324 54.8 53.0 56.7 17.07 0.49 3.13 0.99 -54.8 1.1 0.72 ● median, and characterize their uncertainty using the 5th and 95th 0 6724929671747826816 11.97 1.31 13.55 9 1.87 1105.66 2 352 -11 3972130276695660288 -511 -530 -493 0.888 0.855 0.921 31.9 30.70 33.0 59.97 0.05 22.01 0.14 31.8 0.7 0.58 0 percentiles (which together form a 90% confidence interval, CI). −20 3972130276695660288−10 0 9.8810 2.1820 1.58 8−15 0.00−10 0.00−5 30 227 65 5 10 15 5 10 20 50 100 200 500 51633438156329464325163343815632946432 -4965perihelion -5499 time ( -4523tmed) / Myr 12.92 0.896 0.484 1.25 1.361 1.21 37.1 10 33.7 40.7 0.00 5.39 0.00 0.04 1.45 5 0.09194 -50 35.4 2.3 0.76 med −1 ph perihelion time (tph) / Myr perihelion speed (vph ) / km s The number of stars coming within various perihelion distances 29267328316737351682926732831673735168 -1680 -1698 -1664 0.917 9.56 0.820 0.72 1.014 1.41 66.5 14 66.1 66.9 0.00 8.75 0.00 0.04 0.99 9 0.08230 -12 66.5 0.3 1.15 is shown in Table 1 (see also Figure 10 later). Summary perihel- Fig.2929487348818749824 2. Perihelion times2929487348818749824 and -5364 distances -5645 computed -5108 11.21 0.926 by orbit 0.239 0.78 integration 2.004 1.54 for 70.0 12 67.9 72.3 0.00 2.61 0.00 0.06 0.42 4 0.10233 -5 69.9 1.4 1.34 Fig. 2. Fig. 4. As Fig.3, but now showing the individual surrogates from the ion data for those stars with dmed < 1 pc are shown in Table 2 the939821616976287104 filteredPerihelion sample times (278 and points939821616976287104 distances -90 lie outside -91 computed the -90 plotting by 0.989 9.91 orbit range). 0.974 integration 1.43 Open1.002 1.50 for cir- 568.4Fig. 8567.1 4. As 569.6 Figure 0.00 19.05 3, but now 0.00 0.07 showing 45.71 2 the 0.11181 individual 18568.2 surrogates 0.8 NA from† the Fig. 5. Median perihelion velocities from the orbit integrations for those ph orbital integrations (but just plotting 100 per star rather than all 2000 per thecles3458393840965496960 filtered show sample the median (278 of points3458393840965496960 the -866 perihelion lie outside-1419 time the -600 plotting and 12.07 0.996 distance range). 0.396 0.63 distributions. Open 2.104 11.71 cir- 86.4orbital 9 52.8 integrations 120.9 3.07 13.20 (but 2046.73 just 1.05 plotting 2.79 2 100 2.06172 per star 986.6 rather 19.9 than all 1.17 2000 per encounters shown in Figure 3 The velocity axis is a logarithmic scale. (the online table at CDS reports higher numerical precisions for star). Surrogates from stars with median perihelion parameters outside clesThe show error the bars median show theof the limits perihelion of the 5% time and and 95% distance percentiles. distributions. star). Surrogates from stars with median perihelion parameters outside the whole filtered sample). Negative times indicate past encoun- The error bars show the limits of the 5% and 95% percentiles. thethe plotting plotting range range (and (and so so not not shown shown in in Figure Fig.3) 3)areareshownshown here. here. ters. The magnitude, colour, and quality metrics from Gaia DR2 ●● ●● ● ● ● ● 5 ● ● ●● ● ● ● ● ●● ●● ● counter further in the past/●future● ● ● ● and●● /or● nearer to the edge of 5 ●●● ●● ● ●● ● ● 5 ● ●● ● ●●● ● ● ● ● ● ● ● ●● ●●● ● ●●●● ●● ● ●● ● ●● ● ● ● ● ●● ●●●● ● ● ● are shown in Table 3. Some of these encounters we have cause ● ● ● ● ● ●●● ● ●●● ●● ● ● ●● ● ●● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ●● ●● ●●●●● ● ● ● ● ● ● ● ●●●● ●●● ● ● ● ● ●● ●● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● is generally represented as an ellipsoidal shape pointing roughly ● ●● ● ● ● ●● the 10 pc distance limit.● ●● As● stars●●● ●● ● at●● ● such● ● ● large● distances can ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ●●● ●● ●●● ● ● ● ● to disregard (discussed later). ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ●● ● ● ● ● ● ●●● ●●● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ●● ● ● towards t = 0, d = 0. ● ● ● ●●● ● ● ● hardly be considered encountering,● ● ● ● ●● ● ● ● we will only be interested in ph ph ● ● ●● ● ●●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● tances. Similarly, our● list will● ● be missing●● ● ● those● ● stars● which would ● ● ●●●●● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● Figure 2 plots the perihelion times and distances. (This and 4 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● 4 ● ●● ● ● ● ● ● ●● ● counters4 to occur at arbitrarily●● ● near●●● ● times.● Both of these apply ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● med med ● ● ● ● ● ●● ● encounters within● 5 pc● from now●● on.●●●● Moreover,● ● when● we later ●● ● ● ●●●● ●● ● ● ●● ● ●●●● The perihelion speeds for encounters with d < 5 pc are ●● ● ● ●●● ● ● ● ●●● ●● ● ● have had d < 10 pc had● they● ● been● ● ● ● subject●● ● to● the● orbital● inte- ● ● ● ●● ●● ● ●●● ● ph other plots do not remove bogus cases.) The stars were selected ● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ) / pc ph ● ● ●●● ● ● ● ● ● for Gaia DR2, although we argue● ●● ● later●● ● that● this is also a conse- ●● ● ● ) / pc ●● ●●● ●●● ● d ● ● ● ● ● ● ) / pc ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● computed the (completeness-corrected)● ● ● ● ● encounter rate,● we will e ● ● ● ●● ● ● ● ●● ● ● ● ● med −1 h ● ● ● ● ● ● ●
h ● ● ● ● ●● ● e ● ● ● ● ● ● ● ● ● ● ● h ● p ● ● ● m p ● ●● ● ● ●●● ●● ● ● ● ●● ●● ●● shown in Figure 5. 90% have v < 100 km s . The very fast by the LMA to come within 10 pc of the Sun, but some of the gration, but were never selected because● ●● they had d > 10 pc m p ● ● ● ● ● ● ph T ● ● ● ●● ●● ●● ● d ● ● ● quence of the limited range for stars with radial velocities in d ● ● ●● ● ● eff ● ● ●● ● ●● ● ph ● d ● ● ● ● ● ● ● limit the sample to a narrower● ● time● ● window.● ● ● 3 ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● 3 ●●●● ● ● ● ● ● 3 ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ●● ●● ● orbit integrations result in much larger median perihelion dis- from the LMA. This latter omission● ●● ●● ● will● in● principle● lead to an ● ● ●● ● ● ● ● ● ● ● ● Gaia DR2. ●●● ● ● ● ● ● ● ●●● encounters are almost entirely due to stars with large radial ve- ● ● ●●● ● ● ● ● ● ● ● ● ●● med ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● The encounters with d● ● ● < 5 pc●● ● are● ● shown in● Figure● 3. We ● ●● ● ● ●● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ph ● ● ●●● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● locities. ●● ●● ● ●● ● The uncertainties● in ●t●ph and● dph●● are● correlated. This can be ● ● ● ● ● ● ●● ● ● ● ●●●● ●●● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ●●●● ● ●● ● ● ● ● ● ● Article● ●● number,● page 3 of 13page.13 see a strong drop in the density of● ● encounters with increasing ●● ●● ●● ● ● ● ● ● ● ● ●● ● 2 ● ● ● ● ● ● ● ●●● ● ● ● 2 ● ● ● ● ● ● ● seen in2 Figure 4, where we plot the● ●● individual● surrogates instead The masses of the encountering stars are listed in Table 2. ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● |t |. This is primarily● a● consequence● ● ●● of●● the magnitude limit in ●● ●● ● ph ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● of the summary median● and axis-parallel● ● ● error bars. Each star The 5th, 50th, and 95th quantiles of the mass distribution over ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● the sample (95% brighter● than ●G = 13.4● ● mag): Encounters that ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ●● ● perihelion distance ( ● ● ●● perihelion distance ( ● ●● ● ● ● ● ● ● ● 1 ● perihelion distance ( ● ● ● ● 1 ● ● ●● ● ● would1 occur further in the past/future●● generally correspond to ● ●● ● ● ● ● ● ● ● ● ● ● ● Article number, page 5 of 13page.13 ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● stars that are currently more distant,● ● and so more likely to be be- ● ● ● ● ● ● ● ● ●● ● ● ● low the limiting magnitude. The● effective time limit of this study ● ● 0 0 is 5–100 Myr. Comparing this with Figure 3 of paper 2, we see −15−15 −10−10 −5−5 00 55 1010 1515 5 10 20 50 100 200 500 perihelion time (tmed) / Myr med −1 perihelion time p(htph) / Myr that while Gaia DR2 hasperihelion found manyspeed (v moreph ) / km encounters s than our
med TGAS study (by a factor of about seven within 5 pc), Gaia DR2 Fig. 3. As Fig.2, but just showing the 694 stars with dmedph < 5 pc. The Fig.Fig. 3. 4.AsAs Figure Figure 2, 3, but but just now showing showing the the 694 individual stars with surrogatesdph < 5 frompc. The the doesFig. 5. notMedian allow perihelion us to probe velocities much fromfurther the into orbit the integrations past/future, for those be- timeorbitaltime axis axis integrations is is scaled scaled to(but to show show just plotting all all encounters encounters 100 per in star in this this rather perihelion perihelion than all distance 2000 distance per causeencounters of the shown similar in FigureFig. magnitude3 The 3 The velocity limits velocity axis on axis the is a is samples. logarithmic a logarithmic scale. scale. range. range.star). Surrogates from stars with median perihelion parameters outside We also see in Figure 3 a slight reduction in the density of the plotting range (and so not shown in Figure 3) are shown here. encounters very close to the present time. (The effect is not quite the solutions, however: 9, 10, or 11, as opposed to 8–18 for the asevidence strong as to first claim appears, that these as it stars is partly have spuriousan illusion solutions produced or byun- sources lying near the main sequence. theisaccounted generally shorter error for represented errors, bars.) and This as it wasan is ellipsoidal quite also possibleseen withshape that in pointing paper some 2, or roughly where all are underestimate of the derived encounter rate. However, as shown wetowardsMS-WD arguedtph binaries. this= 0, wasdph Spectroscopydue= 0. to two things: may missing help identify bright the stars nature in the of countersGiven to their occur location at arbitrarily in the near CMD, times. it is Both possible of these that apply some inor both all of papers these 111 and sources 2, this mostly are subdwarfs affects stars or MS-WD that will bina- en- GaiatheseThe catalogue; sources, perihelion and and astrometry speeds the ever for smaller over encounters a longer volume with timebase availabledmed < (e.g.5 for pc from en- are for Gaia DR2, although we argue later that this is also a conse- Gaia DR3) should help to determine if the astrometricph solutions ries. The latter, of which many have been discovered (e.g. shown in Figure 5. 90% have vmed < 100 km s−1. The very fast Articlequence number, of the pagelimited 4 ofT 13page.13eff range for stars with radial velocities in are good. ph GaiaRebassa-Mansergas DR2. et al. 2016), can in principle lie in much of encounters are almost entirely due to stars with large radial ve- the region between the MS and WD sequence. Such binarity The uncertainties in t and d are correlated. This can be locities. could potentially explainph elevatedph values of u, although only if seen in Figure 4, where we plot the individual surrogates instead 3.3.The Individual masses encounters of the encountering found to bestars coming are listed within in 1 Table pc 2. the periods were short enough and the amplitudes large enough of the summary median and axis-parallel error bars. Each star The 5th, 50th, and 95th quantilesmed of the mass distribution over to be detectable over Gaia DR2’s 22-month baseline. Even then We find 31 stars with dph < 1 pc. All have been visually the parallax should be reasonably accurate, because the different checked against 2MASS, DSS,Article PS1, number, and AllWISE page 5 of images 13page.13 in astrometric displacements at different times should not create a Aladin and all are confirmed as real sources. Some have mod- large bias (Lennart Lindegren, private communication). Further- erately high values of u, astrometric_excess_noise and/or more, four of the 11 sources have much larger radial velocity astrometric_excess_noise_sig, as can be seen in Table3. 1 uncertainties than the other encounters in Table2( >14 km s− Only one of the encounters has been discovered by the previous 1 as opposed to <6 km s− ). As this uncertainty is computed from studies listed in the introduction. 21 do not have HIPPARCOS or the standard deviation of measurements at different epochs Tycho IDs, so could not have been discovered in papers 1 or 2. (Eq. (1) of Katz et al. 2018), this could be an indication of bi- Of the remaining nine stars, seven have obvious Tycho matches narity. but were not found in paper 2 for reasons discussed below. The In conclusion, the correlation of the CMD location with u is last two stars are problematic and are also discussed below. suspicious, and would be consistent with a selection effect that The closest approaching star, and the only one to have preferentially includes spuriously large parallaxes in a search for been found already, is Gaia DR2 4270814637616488064, bet- stellar encounters. On the other hand, we do not have specific ter known as Gl 710 (Tyc 5102-100-1, Hip 89825). This is a
A34, page 6 of 13 A&A proofs: manuscript no. stellar_encounters_gdr2 A&A proofs: manuscript no. stellar_encounters_gdr2 Bailer-JonesA&A proofs: et al.:manuscript Close encounters no. stellar_encounters_gdr2 to the Sun in Gaia-DR2
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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● 5 ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● 35 ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● 5 ● ● ● ● ● 2 ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 2 ● ● ● ● 35 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● 35 ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●●● ● ● ●● ● ● ●● 4 ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● 4 ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 30 ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●●● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● 4 ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● 4 ●● ● ● ● ● ● ● ● ● ● ● ● ● 30 ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ●● ● ●●●● ● ● ●● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● 4 ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● 4 ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ) / pc ● ● ● 30 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●●● ● ● ● d ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● e ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● 25 ● ● ● ●● ● ● ● ● ● ● ● ● ● h ● ● ● ● ●● ● ● ● ) / pc ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● 6 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● m p ● ●● ● ● ● ● ● ● ● ● ● ● ● ● d ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ●● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● e ● ●● ● ● ● ● ● ●● ● ● ●● ● ) / pc ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●●● ● ● ● ● 25 ● ● ● d ● ●● ● ●● ● ● ● ● ● ● ● ●
h ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●● ● ●●● ● ● ● ●● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● d ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● 6 ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● m p ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●
3 ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● e ● ● ● ● ● ●●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●● ●● ● 25 ● ● ● ● ●● ● ● ● ● ● ● ● h ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● d ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● 6 ● ●● ● ● ● ● ● ● ● ● ● m p ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ●
3 ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ●●●●● ●● ● ● ● ●● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ●● ●● ● ● ●● ●●● ● d ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● + 5 [mag] ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ●●●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●●●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● 20 ● ● ●
● 8 ● ● ● ● ● ● ● ● ● ●
● ϖ ● ● ● ● ● ● u ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ●● ● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● + 5 [mag] ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● 20 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● 8 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ϖ ● ● ● ● ● ● ● ● ● ● u ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ●●● ●● ● ● ● ● ● ● + 5 [mag] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● g ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●●● ● ●●● ● ● ● 15 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 8 ● ● ● ● ● ● ● ● ● ● ● ● ϖ ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● u ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●
● o ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ●●● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● l ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ●●● ● ● ● ● ● g ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ●● ● ●● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●
● 10 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● 15 ●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ●● ● o ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ●●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● l ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● g ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●●● ● ● ●●●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ●●● 15●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ●● ● ● o ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● ●● ● ● ●●● ● ●● ● ● ● ●● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● l ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●●●● ● ●● 2 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●●●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●●● ●● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●● ● ●● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ●● ●●● ● ● ●●●● ●● ●● ● ● ●●● ● ●●●● ●●● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ●● ●● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ●● ● ●●●●● ●●●●● ●●● ● ● ●● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●●● ●● ● ● ●● ●● ●● ●●●● ●●● ● ●●● ●●● ●● ●● ●●● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ●● ● ● ● ● ● ● ● ● ● ● G + 5 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●●● ●● ● ●● ● ● ●●● ● ●● ● ● ●● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ●● ●● ● ● ● ●● ●●● ● ●●●● ●● ● ●●● ●● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ●●●●● ●● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●● ●●●●●●●● ●●● ●●●● ●●● ●●●●●●● ●● ● ● ●●● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●●● ●●● ●●● ●●●●● ●● ● ●●●●● ● ●●● ●●● ●●●●● ● ●● ●● ●● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●●● ● ● ● ●● ●●●● ● ● ●●●● ● ● ● ● ● ● ● ●● ●●●● ● ● ●● ● ● ● ● ● ● ● ●●● ● ●● ● ●●●●● ●● ●●●●●● ●● ●●● ● ● ● ● ● ● ● ● 10● ● ●● ● ● ● ● ● ● ●● ● ● ● 10 ●● ●● ●● ● ●●● ● ● ●●● ● ● ●● ● ●● ● ● ● ● ●● ● ● ●● ●● ●●● ● ● ●●●● ●●●●●● ●●●●●● ● ● ● ● ●● ● ● ●● ● ● ● ●●● ● ● ●● ● ●● ● ● ● ●● ● ● ●●●● ● ● ●● ●●●●●●● ● ●●● ●● ●●● ● ● ● ●● ● ● ● ● ● ● G + 5 ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●●●●●●● ●●● ●●●● ●● ●●●●● ●● ● ●● ● ●●●● ●●● ●●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ●● ●●●●● ●● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ●●● ●●●● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ●●● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ●● ●●●●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ●●●● ●● ●●● ●● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ●● ● ●●● ● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ●●●● ● ● ●● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ●●● ●● ●● ●●● ● ● ●●● ● ●●● ●●●● ●● ●●● ● ● ● perihelion distance ( ● ● ● ● ●● ● ● ● ●● ●●● ● ●●● ● ●●●●●● ● ●● ● ● ●● ● ● ● ● ● ● ●●● ● ● ●●● ● ●● ● ●●●●● ●● ●● ●●● ● ● ● ● ● ●●●● ● ●●●● ● ● ●● ● ● ● ● ● G + 5 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ●● ● 1 ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ●●● ● ●● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ●● ● ●● ●●●● ●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ●●● ●●●● ●●●● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ●●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 12 ● ● ● ● ● ● ● ●●●● ●●●●● ●● ●●● ● ● ●● ● ● ● ●●● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●●
perihelion distance ( ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● 1 ● ● ● ● ●● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●●● ●● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● 12 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● perihelion distance ( ● ● ● 5● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
1 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 12 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 5 ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● 0 ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 14 ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ●
0 ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● 14 ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ●●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● 0 ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● 14 ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● −15 −10 −5 0 5 10 15 0.0 0.5 ● 1.0● ● ● ●● ● 1.5● 2.0 2.5 ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● med ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● −15 −10 −5 0 5 10 15 0.0 0.5 ● 1.0● ● ● ●● ● 1.5● 2.0 2.5 ● ● ● ● ● ● perihelion time (t ) / Myr ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ph ● ●● ● ● ● ● ●● ● med ● ● ●● ● BP−RP● [mag] ● ● ● ● ● −15 −10 −5 0 5 10 15 0.0 0.5 1.0 ● ● ● ● 1.5● 2.0 2.5 ● ● ● ● ● ●● ● ● ● ● ● ● perihelion time (t ) / Myr ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● pmhed ● ● ● ● ●● ●●● ● ● ● BP−RP●●●● ●[mag] ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● perihelion time (t ) / Myr ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ph ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● BP−RP● ●●●●● [mag]● ●●●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●●● ●● ● ● ●● ● ● ● ●● ● ● ●●● ●● ● ● ● ● ●● ●●● ●●●● ● ● ● ● ●●● ● ● ●●● ●● ●●● ●●● ●● ● ●● ● ●● ● ● ● ●●●● ● ● ●●●● ● ● ● ● ● ● ●●●●●●●●●● ● ●●● ● ● ● ● ●●● ● ●● ●● ● ● 2 ● ●● ● ●●● ● As Figure 3, but now plotting each star as a circle, the area of ●● ● ● ● ●●● ● ● ● ●●● ● ● ● Fig. 6. ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●●● ● ● ●●●●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ●●●●● ●● ●●●●●●● ●● ●●●●●● ●●● ● 35 ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●
2 ● ● ● ● med med ● ● ● ●● ● ● ● ● ●● ● ● As Figure 3, but now plotting each star as a circle, the area of ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Fig. 6. ● ● ● ● ●●●● ● ● ●● ● ● ● ● ● ● ●●●● ● ●● ● ● As Fig.3, but now plotting each star as a circle, the area of which ● ●● ● ● ● ● ●● ● ● ● ● ●●● ● ● ●● ● ● ● Fig. 6. ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● / ● ●●●● ●●●●●●●●● ● ●● ● ● M d ● ● ● ● which is proportional to (v ). ● ●●●●● ●●● ●●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● Fig. 6. ●●● ● ●●●● ● ●●● ● ● ● ● ● ● 2 ● As Figure 3, but now plotting each star as a circle, the area of ● ●●● ●●●●●●●●● ●●● ●● ● ● ● ● 35 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●●● ● ph ph ● ● ●●●●●● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● med med ● ● ●●●●● ● ● ● ● ●●● ●● ●●● ● ●●● ● ●●● ● ●● ● ● ● ● ● ●● ●● ●●●● ● ●● ● ● ●● ● ● ●●●● ●● ● ● ● ●●● ●●● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● med med ● ● ● ●●●●●●● ●● ● ●●● ● ● ●● ● ● ●● ●● ● ● ● ● ●●●●● ●●●●● ●● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ●●●●●●● ● ● ● ● ●● ●●● ●● ● ● ● ●● ●●● ●●●●●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ●●●●●●●● ●●● ● ●● ● ● ● ●● ● / ● ●● ●●●●●● ● ● ● ● ● ●●●●●●● ● ● ●● ● ● ● ● ● which is proportional to M (v d ). ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ● ●●● ● ●● ● 35 ● ● ● ● ● ● ● / ●●● ● ●● ● ● ● ● ●● ● med med ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●●●●● ● ● M v d ●● ●●●●● ●●●● ●●●● ● ● ● is proportional to ( ). ●●●●●●● ● ● ● ● ● ● ● ● ● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●●●●●●●●●●●● ●● ● ● ● ●● ● ● ph ph ● ●●●● ● ● ●● ● ● ● ● ● ●● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●● ● ●● ● ● ●● ● ● ●● ●●● ● ● ●● ● ● ●●● ● ●● ● ● ● ●● ●●●●● ● ● ● ● ● ● ● ● ●●●●● ● ●●● ●●● ● ● ● ● ● ●● ● ●●●●● ● ● ● ●● ● ● ● ● ●●● ● ●● ● ●● ●● ●● ● ph ph ● ● ●●● ● ● ●●● ●●●●●●●● ● ●●●●●●●●●●●●● ● ● ● ● ●● ●● ● ● ● ●● ● ●●●●●●●● ●●●●● ● ● ●● ●● ●●●●● ●●● ● ● ●● ● ● ● ●●● ● ● ● ●● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● M/ d 4 ● ●●●●●●●●●●●●● ● ●● ●● ● which is proportional to (v ). ● ● ● ●● ● ● ● ● ●●●●●●●●●●●●● ● ● ●● ● ●● ●● ● ●● ●● ●● ●● ● ● ● ● ● ●● ● ● ● ●●●● ●● ●● ●●●●● ● ● ● ● ● ● ● ●●●●●● ●●● ● ●● ● ● ● ●● ● ●● ●●●●● ● ●● ●● ● ● ● ● ● ●● ●●●●● ●● ●● ● ● ●●●● ●●●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ●●●● ● ● ● ● ● ● ●●●●●● ●● ●● ● ● 30 ● ● ● ●●● ● ● ● ● ●● ●● ● ● ●●●●●●●●●●●●●●●●●●●●●● ● ● ●● ● ●● ●● ● ●●● ●●●●●● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ●●●●●●●●●● ● ● ● ●● ●●● ph ph ● ●● ● ● ● ●●● ●● ●●● ●●● ● ● ● ● ● ●●●● ● ● ● ● ● ●●● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ●● ●● ● ●● ● ● ●●●●● ●●●●●●●●●●●● ● ●● ● ●●● ●● ●● ● ● ● ● ●● ● ● ● ● ● ●●●● ●● ● ● ●● ● ● ● ● ●●● ● ●● ●●● ● ● ● ● ● ●● ● ●● ●●●●● ●●●●●●●● ● ●●● ●● ● ● ●●● ●●●●●●●● ●●●●● ●● ● ●● ● ● ●●●●●●● ● ● ● ● ● ● ● ●●● ●●●●● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●● ● ● ● ●● ● ● ● ●●● ● ● ● ● ●●●●●● ● ● ● ● ●●●● ● ●● ● ● ● ● ● ●● ● ●● ●●● ● ● ●● ● ●● ●●● ● ● ●●●● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ●● ●●●● ●●●● ● ●● ●●● ● ● ● ●● ●●●● ●●● 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6 ● ● ●● ●● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ●●●● ●●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●● ●●●●● ●●●● ● ●●● ●●● ● ●●●●● ● ● ● ●●●●● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●●●●● ●● ● ●● ●● ●● ● ● ● ● ● ● ●● ● ●●●●● ●●● ● ● ● ● ● ●●●●●●●● ● ●●●●●●●●●● ● ● ● ● ● ● ●●●● ●● ●●● ● ● ●●●● ● ●●●●●●● ●●● ● ● ● ●●● ●●● ●● ●● ● ●● ● ●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ●●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ●●● ●●● ● ●● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●●●● ● ● ●●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● 25 ● ●● ● ● ● ● ● ●● ● ●● ● ●●● ● ●● ●● ● ● ●●●●● ●●●●● ●● ● ●● ●● ● ● ●●●●● ●●● 5 ● ● ● ●● ● ●● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●●● ●● ● ● ● ● ●● ●●●●●●● ●●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ●●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ●●●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●●● ●●●●●●●● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ●● ●● ● ● ●● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● 6 ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●●●●●● ● ●● ● ● ● ● ●● ●●● ● ●●●●● ● ● ● ●●●● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ●●● ● ●● ● ● ● ● ● ● ●● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ●● ●●●● ● ● ● ● ● ● ● ● ● ●● ●●●●● ●●●●● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ●●●● ●●● ● ●● ● ● ●●●●● ●●●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●●●● ● ●● ●●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● + 5 [mag] ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●● ●● ● ● ●● ●● ● ● ● ● ● ●● ● ●● ● ● ●● ● ●● ● ● ●●● ●● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●●● ● ● ●● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●●● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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●●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ϖ ● ● ● ● ● ●● ● ●
● u ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ●● ● ● ● ●●●●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● + 5 [mag] ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ●●●● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ●●● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ● ●●● ●● ● ● ● ● ●● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 8 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ϖ ● ● ● ●●● ● ● ● ● ● ●● ● u ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● + 5 [mag] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● g ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●
4 ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●●● ● ● ● ●●● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●●●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● 15 ● ●● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●
10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 8 ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ϖ ● ●● ● ● ● ● ●● ● ● ● u ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● o ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● l ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ● ●● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● g ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● 4 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● 10 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 15 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● o ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
l ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● g ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ) / pc ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 15 ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● o ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● d ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
l ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● e ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●●●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ●● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● h ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ) / pc ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 10 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ●●● ●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ●● ● ●● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
10 ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● G + 5 ●● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● m p ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● d ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●●● ● ●● ●●● ● ● ●● ● ●● ●●● ●● ● ● ● ● ● ● ● ●● ●●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●●●● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● e ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ●● ●● ● ● ● ● ●● ) / pc ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● d ● ● ● ●● ● ●● ● ● ● ●● ●●● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● h ● ● ● ● ● ● ● ● ● ● ● ●●●● ●● ●● ● ●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● 10● ● ● ●● ● ● ●● ● ● 10 ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●●●● ● ● ●●●● ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ●● ●●● ●●● ● ●●● ● ● ●●● ●●●● ●●● ●● ●●● ●● ● ● ● ● ●●● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● d ● ● ● ● ● ● ●● ● ● ● ● G + 5 ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ●● ● ●● ● m p ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3 ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● e ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●●●●● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ●● ● ● ● ●● ●●●●●●● ●● ●●● ● ●●● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ●●●● ● ●●●●●● ● ● ● ● ● ● ● ● ● h ● ●● ● ● ● ●● ● ●● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● 10 ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ●● ● ●●● ●● ●● ● d ● ● ● ● ●●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ●●●●●●●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ●● ●● ● ● ● ● ● ● ●●●● ● ●●● ● ● ●● ● ● ●
G + 5 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ●● ●● ● ● ● ●● ●●● ●● m p ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ●● ● ●● ●●● ●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ● ● ● 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ●●●●● ● ●●●● ● ● ●●●● ● ●● ●●● ● ● ● ● ● ●● ● ● ● ● 12 ● ● ● ● ● ● ● ●●●● ●●●●● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ●● d ● ● ● ● ● ●●● ● ● ● ● ● 5 ● ● ●● ● ● ● ● ● ● ● ●● ●●●● ● ●●● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● 12 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● 5● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 12 ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● 5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● 2 ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 14 ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● 14 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● 0 ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 14 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● 0.0 0.5 1.0● 1.5● 2.0 2.5 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● 0.0 0.5 ● 1.0● ● ● ● ● 1.5● 2.0 2.5 ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● BP−RP● [mag] ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.5 1.0 ● ● ● 1.5● 2.0 2.5 ● ● ● ● ● ● perihelion distance ( ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● BP−RP [mag] ● ● ● ● ● ● ● ● ● ● ● ● ●● ● perihelion distance ( ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● BP−RP● [mag] ● ● ● ● ● ●●● ● ● perihelion distance ( ● ●● ● ● ● ● ●● ● ● ● ● ● 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ●● ● ● ● ● Fig. 8. Colour–magnitude diagram for stellar encounters coloured ac- ● ● ●●● ●● Fig.Fig. 8. 8. Colour–magnitudeColour-magnitude diagram for stellar encounters coloured coloured ac- ac-
● ● 0 ● ● Fig.cording 8. Colour–magnitude to the unit weight error diagramu in Table for stellar 3. The encounters vertical axis coloured equals M ac-G ● cording to the unit weight error u in Table3. The vertical axis equals MG
0 cording to the unit weight error u in Table 3. The vertical axis equals M −15 −10 −5 0 5 10 15 cordingassuming to extinction the unit weight is zero. error Foru in orientation, Table 3. The the vertical grey lines axis equalsare unred-MG 0 G −15 −10 −5perihelion time0 (tmed) / Myr5 10 15 assumingassuming extinction extinction is is zero. zero. For For orientation, orientation, the the grey grey lines lines are are unred- unred- mphed dened solar metallicity PARSEC isochrones for 1 Gyr (solid) and 10 Gyr −15 −10 −5perihelion time0 (t ) / Myr5 10 15 assuming extinction is zero. For orientation, the grey lines are unred- pmhed deneddened solar solar metallicity metallicity PARSEC PARSEC isochrones isochrones for for 1 1 Gyr Gyr (solid) (solid) and and 10 10 Gyr Gyr perihelion time (tph ) / Myr dened(dashed) solar from metallicity Marigo PARSEC et al. (2017), isochrones the grey for points1 Gyr (solid) are white and 10dwarfs Gyr (dashed)(dashed) from from MarigoMarigo et et al.(2017 (2017),), the the grey grey points points are are white white dwarfs dwarfs Fig. 7. As Fig.3, but now plotting each star as a circle, the area of which (dashed)within 20 from pc of Marigo the Sun et identified al. (2017), by the Hollands grey points et al. are(2018) white (many dwarfs lie Fig. 7. As Figure 3, butmed nowmed plotting2 each star as a circle, the area of within 20 pc of the Sun identified by Hollands et al.(2018) (many lie is proportionalAs Figure to M 3,/ but(v now(d plottingmed) ). med each2 star as a circle, the area of within 20 pc of the Sun identified by Hollands et al. (2018) (many lie Fig.which 7. isAs proportional Figure 3, but tophM now/(vph plotting(d ) each). star as a circle, the area of withinoutside 20 the pc range of the of theSun plot), identified and the by small Hollands blue dots et al. show (2018) all sources(many lie in which is proportional to M/(vmedph (dmedph )2). outsideoutside the the range range of of the the plot), plot), and and the the small small blue blue dots dots show show all all sources sources in in M/ phmed dphmed 2 outsideGaia DR2 the with range$ of > the50 masplot), (plotted and the on small the bottom blue dots layer, show so all most sources are ob- in which is proportional to (vph ( ph ) ). GaiaGaia DR2 DR2 with with$$ > > 50 mas (plotted on the bottom layer, so most most are are ob- ob- Gaiascured DR2 by withcoloured$ > and50 mas grey (plotted points, on especially the bottom in the layer, lower so most panel). are The ob- scuredscured by by coloured coloured and and grey grey points, points, especially especially in in the the lower lower panel). panel).med The The scuredupper panel by coloured shows all and reliable grey points, (non-bogus) especially encounters in the lower with panel).d med< 1 The pc K7 dwarf known from many previous studies to be a very close upperupper panel panel shows shows all all reliable reliable (non-bogus) (non-bogus) encounters encounters with withddmedph <<11 pc pc upper(frompanel Table shows 2). For all comparison, reliable (non-bogus) the lower encounters panel shows with alld encountersphmedph < 1 pc theencounter. entire filtered In paper sample 1 we found are 0.53, a median 0.91, encounter and 1.64 M distancerespec- of (from(from Table Table2 2).). For For comparison, comparison, the the lower lower panel panel shows shows all all encounters encountersph the entire filtered sample are 0.53, 0.91, and 1.64 M respec- (fromin the filteredTable 2). sample For comparison, (some lie outside the lower the plottingpanel shows range). all Encounters encounters thetively.0.267 entire pc The (90% filtered lack CI of0.101–0.444 sample massive are stars 0.53, pc). in the A 0.91, much sample and smaller is 1.64 a consequence M – andrespec- more inin the the filtered filtered sample sample (some (some lie lie outside outside the the plotting plotting range). range). Encounters Encounters tively. The lack of massive stars in the sample is a consequence inhave the been filtered plotted sample with (some larger lie values outside of u theon plotting higher layers range). so Encounters as to better tively.of the T Theeff filtering lack of massiveon radial stars velocities in the published sample is in a consequence Gaia DR2. havehave been been plotted plotted with with larger larger values values of ofuuonon higher higher layers layers so so as as to to better better ofprecise the T – properfiltering motion on radial measured velocities by published TGAS gave in Gaia a closer DR2. en- havelocate been larger plotted values with of largeru. The values colour of baru on spans higher the layers full range so as of to values better of theCloseTeff encountersfiltering on are radial interesting velocities not published least because in Gaia they DR2. per- locatelocate larger larger values values of of uu.. The The colour colour bar bar spans spans the the full full range range of of values values counter,eff with a median distance of 0.076 pc (90% CI 0.048– locatein both larger panels. values of u. The colour bar spans the full range of values turbClose the Oort encounters cloud. The are degree interesting of perturbation, not least because or rather they the per- im- inin both both panels. panels. turb0.104 the pc) Oort (paper cloud. 2 and The Berski degree & of Dybczy perturbation,nski´ 2016 or rather). Gaia theDR2 im- in both panels. turbpulsehasslightly the which Oort comets decreased cloud.can The this receive, degree distance of depends perturbation, – 0.0676 not only pc or on (13 rather the 900 distance the AU) im- – pulseof the which encountering comets can star, receive, but also depends on its speed not only and on the the mass. distance Ac- ofand the has encountering narrowed the star, uncertainty but also on – its 90% speed CI and 0.0519–0.0842 the mass. Ac- pc ofcording(10 the 700–17 encountering to the 400 simple AU) star, – impulse although but also approximation on this its is speed statistically and (Öpik the 1932;mass. consistent Oort Ac- terstellarthey used extinction. a very different The potential upper panel from shows ours. Notethe 26 that non-bogus ignoring cording1950; Rickman to the simple 1976; impulse Dybczynski approximation 1994) the impulse (Öpik 1932; transfer Oort is terstellar extinction.med The upper panel shows the 26 non-bogus with the TGAS result. This is well within the Oort cloud. terstellarencountersthe parallax extinction. with zeropointdmedph The< and1 upperpc/or from ignoring panel Table shows the 2. Eleven astrometric the 26 of non-bogusthese correla- have 1950; Rickman1 1976;α Dybczynski 1994) the impulse transfer is encounters with dmed < 1 pc from Table 2. Eleven of these have 1950;givenDue to byRickman theMv slightlyph−1 d 1976;ph−α where more Dybczynski negativeα is 1 for 1994) radial very the velocity close impulse encounters published transfer (on inis encounterstions changes with thedph estimate< 1 pc by from less Table than 2. 0.0001 Eleven pc, of so these these have are given by Mv− d− where α is 1 for very close encounters (on unit weight error uph> 10 (the others have u < 2) and lie some way giventheGaia orderDR2 by M of thanvph the1dph used comet–Sunα where in theα earlieris separation), 1 for studies, very and close the 2 otherwise.encounterencounters time The (on unitnot theweight cause error of theu > di10fference. (the others have u < 2) and lie some way the order of theph ph comet–Sun separation), and 2 otherwise. The unitbelow weight and lefterror ofu the> 10 main (the sequenceothers have (MS),u < 2) yet and well lie someabove way the theimpulseis slightly order of of earlier the the encountering comet–Sun (but no more stars separation), precisely is visualized determined). and 2 in otherwise. Figures We 6 note, The and belowThe and second left of the closest main approach sequence (MS), in Table yet2 well, Gaia aboveDR2 the impulse of the encountering stars is visualized in Figures 6 and belowwhite dwarf and left (WD) of the sequence. main sequence The dimensionless (MS), yet quantity well aboveu mea- the impulseFigureshowever, 7 of forthat theα theequal encountering radial to 1 velocityand stars 2 respectively. in isGaia visualizedDR2 Regardless incomes Figures from of which 6 andjust white955098506408767360, dwarf (WD) sequence. is also The the dimensionless one with the quantity secondu largestmea- Figures 7 for α equal to 1 and 2 respectively. Regardless of which whitesures howdwarf well (WD) the sequence. five-parameter The dimensionless solution describes quantity theu astro-mea- Figuresimpulsetwo Gaia 7 approximation forfocalα equal plane to transits,1 we and use, 2 respectively. it the is the minimum closest Regardless encountersfor a value of which to be suresimpulse. how It well is theone five-parameter of the mostsolution massive describes of the closest the astro- en- impulse approximation we use, it is the closest encounters which suresmetric how data well (see the section five-parameter 2.2). If we solutionassumed describes that all the the encoun- astro- impulsehavereported the approximation greatest in the catalogue. impact. we Despite use, it is its the low closest mass, encounters Gl 710 imparts which metricterscounters, were data single with (see a section main perihelion sequence 2.2). inIf we the stars, assumed relatively then one that recent interpretation all the past. encoun- Four of have the greatest impact. ters were single main sequence stars, then one interpretation of havethe biggest the greatest impulse impact. according to either impact approximation tersthisfainter diagramwere 2MASS single is that main sources the sequence parallaxes lie within stars, of these 8 then00, but 11 one sources there interpretation is are no wrong; good of (Figs.6 and7), in part because of its low velocity. Using the thisreason diagram to think is that these the parallaxes have interfered of these with 11 sources the Gaia aresolution. wrong; 3.2. Colour–magnitude diagram thisspecifically diagram that is that they the are parallaxes all overestimated of these 11 by sources one or aretwo wrong; orders 3.2.same Colour–magnitude data, de la Fuente diagram Marcos & de la Fuente Marcos(2018) specificallyofThe magnitude most recent that (i.e. they reliable the are sources all encounter overestimated should in be the bymuch table one further or is twoGaiaaway). ordersDR2 3.2. Colour–magnitude diagram of magnitude (i.e. the sources should be much further away). *****get a slightly New subsection closer mean describing (and median) and discussing distance of Figure 0.052 pc 8 ofThis3376241909848155520, magnitude seems unlikely, (i.e. the although sources which it should is has the a case be 50% much that chance a further search of foraway).having en- This seems unlikely, although it is the case that a search for en- *****(with a New symmetric subsection standard describing deviation and of discussing0.010 pc). Their Figure esti- 8 Thiscounterspassed seems within will unlikely, preferentially 0.5 pc, although around include 450 it is kyr thesources ago. case with that spuriouslya search for large en- *****mateFigure is close 8 showsto what the one colour–absolute gets for a constant magnitude gravitational diagram po- countersparallaxesAmong will rather the preferentially 21 than sources sources include without with sources spuriously Tycho with matches, small spuriously mostparallaxes large have (CMD)tential,Figure namely for 8 the shows close 0.0555 theencounters pc colour–absolute (computed on the by assumption propagating magnitude of the diagram zero set in- of parallaxes(becausematches to the rather 2MASS former than are(and sources more a couple likelywith ofspuriously to others have apparent to small other parallaxes closesurveys). en- (CMD)surrogates for using the close the LMA encounters and taking on the the assumption mean). This of suggests zero in- (becauseA few of the these former are in are very more crowded likely regions to have (e.g apparent at low close Galactic en- Article number, page 6 of 13page.13 Article number, page 6 of 13page.13 Article number, page 6 of 13page.13 A34, page 7 of 13 A&A 616, A37 (2018) latitude) and have close companions, but in all but one case these Hip 85605. This was the closest encounter found in paper 1 med companions are much fainter, and there is no specific evidence at dph = 0.103 pc (90% CI 0.041–0.205 pc). It was deemed du- to suggest a problem with the Gaia measurements. The one ex- bious due to the inconsistency of its magnitude and HIPPARCOS ception is Gaia DR2 5700273723303646464, which has a com- parallax with its apparent spectral type. This suspicion is con- panion 400 away that is just 1.6 mag fainter in the J-band (the best firmed with Gaia DR2, which has a much smaller parallax of match to this in Gaia DR2 is 0.7 mag fainter in the G band, and 1.822 0.027 mas compared to 146.84 29.81 in HIPPARCOS-2 has a completely different parallax and proper motion). This is a (the proper± motions and radial velocities± are consistent within borderline case. Its measurements are not suspicious, but some 1σ). This now places the closest approach at 420 pc. It was con- of the quality metrics make it questionable. We decide to flag jectured in paper 1 that the cause for the incorrect parallax is a this as bogus. binary companion at a very problematic separation for the There are seven encounters with obvious Tycho matches that HIPPARCOS detectors. were not found as encounters in paper 2, either because they Hip63721.Itsbestmatch,Gaia DR21459521872495435904, lacked a radial velocity, or because their astrometry has changed has a parallax of 5.99 0.04 mas, compared to 216.62 ± ± substantially from TGAS to Gaia DR2. These are as follows 56.53 mas in HIPPARCOS. This HIPPARCOS solution was doubted (with notes on the problematic cases). in paper 1 on the grounds of inconsistency with spectral type and – Gaia DR2 510911618569239040 = TYC 4034-1077-1 . it being a double star. – Gaia DR2 154460050601558656 = TYC 1839-310-1. This Hip 91012. The parallax and proper motion of its best is rather faint, G=15.37 mag, and has a rather high radial ve- match, Gaia DR2 4258121978471892864, are consistent with locity based on just 3 transits. It also has no BP-RP colour HIPPARCOS, but the Gaia DR2 radial velocity is now much 1 and a large astrometric_excess_noise_sig of 360, so smaller, at 16.27 0.61 km s− , compared to the RAVE value we consider this encounter to be bogus. of 364.1 −22.4 adopted± in paper 1. – Gaia DR2 1791617849154434688 = TYC 1662-1962-1. −Hip± 23311. This matches to Gaia DR2 – Gaia DR2 1949388868571283200 = TYC 2730-1701-1. 3211461469444773376. It is a high proper motion 1 The large radial velocity is based on just two transits, so we (>1000 mas yr− ) nearby (113 mas) star – Gaia DR2 and consider this to be bogus. HIPPARCOS agree – but the Gaia DR2 radial velocity is much 1 – Gaia DR2 3972130276695660288 = TYC 1439-2125-1 = smaller: 21.38 0.15 km s− compared to the implausibly large GJ 3649. value of 813.7± 48.6 from RAVE that led it to being flagged – Gaia DR2 2926732831673735168 = TYC 5960-2077-1. as dubious− in paper± 1. – Gaia DR2 2929487348818749824 = TYC 5972-2542-1. Hip 85661. The parallax and radial velocity used The final two sources have ambiguous matches. Simbad in paper 1 agree with the best match star, Gaia DR2 lists four objects within 500 of Gaia DR2 5231593594752514304 4362793767434602752, but the Gaia DR2 proper motion 1 1 (G = 12.03 mag), although in reality these four are probably is much larger (8.5 0.2 mas yr− versus 0.5 0.6 mas yr− ), just two or three unique sources. Gaia DR2 itself identifies meaning it no longer comes± near to the Sun. ± one of these as Gaia DR2 5231593599052768896, which with Hip 55606. The proper motion of its best match, Gaia DR2 G = 9.12 mag is almost certainly Hip 53534 with V = 9.21 mag. 3590767623539652864, is consistent with HIPPARCOS (parallax Whichever other star our close encounter is, its RVS spectrum is less so), but the Gaia DR2 radial velocity is now much smaller, 1 surely contaminated by this much brighter star just a few arcsec- at 18.12 0.70 km s− , compared to the implausibly large value onds away, suggesting that the whoppingly large radial velocity of− 921.0± 91.9 from RAVE that led it to being flagged as du- of 715 km s 1 measured by RVS is in fact spurious. We there- − ± − − bious in paper 1. fore consider this encounter to be bogus. Hip 75159. Its counterpart, Gaia DR2 Finally, Gaia DR2 939821616976287104 would appear to 6251546901899531776, has no radial velocity in Gaia DR2. match with TYC 2450-1618-1. However the 2MASS image The Gaia DR2 astrometry agrees within 2σ of the HIPPARCOS clearly shows this to be a double star, and this surely has contam- astrometry (which was rather imprecise, with a proper motion inated the RVS spectrum. In that case we do not trust the high consistent with zero). Using the (large) RAVE radial velocity 1 1 radial velocity reported (568 km s− ) and conservatively consider of 368 km s− from paper 1 with the Gaia DR2 astrometry, the this to be a bogus encounter too. LMA gives a closest approach of 2.1 pc 660 kyr in the past. Hip 103738 (gamma Microscopii). This G6 giant was the po- tentially most massive encounter coming within 1 pc in paper 1, 3.4. Individual encounters coming within 1 pc in papers 1 and med found to have dph = 0.83 pc (90% CI 0.35–1.34 pc) based on the 2, but not found in the present study 1 HIPPARCOS-2 proper motion of 1.78 0.35 mas yr− . (A second med ± In paper 2 we found two stars with dph < 1 pc, one of which was entry in the catalogue based on the larger Tycho-2 proper motion Gl 710, discussed above. The other was Tyc 4744-1394-1 which put it at 3.73 pc; 90% CI 2.28–5.22 pc) This star is Gaia DR2 is in Gaia DR2 but without a radial velocity. The astrometry is 6781898461559620480 with G = 4.37 mag. The proper motion 1 consistent with that from TGAS, confirming the reality of that in Gaia DR2 is ten times (17σ) larger, 17.7 0.9 mas yr− , with ± encounter also. no suggestion from the quality metrics that this is a poor solu- In paper 1 we found 14 stars coming within 1 pc. One was tion. (Both the parallaxes and radial velocities agree to within Gl 710. The other 13 are not found in the present study for var- 1σ.) Either there is a significant, unaccounted for acceleration, ious reasons. We briefly discuss each of these cases. The match or the proper motion in HIPPARCOS-2 or Gaia DR2 (or both) is in Gaia DR2 was found by searching for sources within about wrong. Using the Gaia DR2 data we find the encounter to be at 30 using Simbad, and identifying the one with the closest mag- tmed = 3440 kyr (90% CI 3630 to 3260 kyr), dmed = 20.3 pc 00 ph − − − ph nitude. In all cases but one the match was unambiguous, and in (90% CI 18.0–22.9 pc). none of these is there anything to suggest the Gaia DR2 data are Hip 71683 (α Centauri A). This is not in Gaia DR2 on ac- problematic. count of its brightness.
A34, page 8 of 13 Bailer-Jones et al.: Close encounters to the Sun in Gaia-DR2 Bailer-Jones et al.: Close encounters to the Sun in Gaia-DR2
using the new astrometry together with the old radial velocity 5 Hip 71681 (α Centauri B). This is not in Gaia DR2 on ac- are consistent with the result in paper 1 (now more precise). count of its brightness. Hip 3829 (van Maanen’s star). This is Gaia DR2 Hip 70890 (Proxima Centauri) Gaia
. This is DR2 4 2552928187080872832. It has no radial velocity in Gaia DR2, 5853498713160606720. It has no radial velocity in Gaia DR2, ) / pc
but the parallax and proper motion agree with Hipparcos h but the parallax and proper motion agree with HIPPARCOS to p to better than 1σ. Given their high precision, the perihelion d within 2σ. Given their high precision, the perihelion parameters 3 parameters using the new astrometry with the old radial velocity using the new astrometry together with the old radial velocity are consistent with the result in paper 1 (now more precise). are consistent with the result in paper 1 (now more precise). Hip 42525. Two Gaia DR2 sources with the same par- 2 Hip 3829 (van Maanen’s star). This is Gaia DR2 allax, proper motion, and radial velocity as each other 2552928187080872832. It has no radial velocity in Gaia DR2,
(within their uncertainties) match this. They are Gaia DR2 perihelion distance ( but the parallax and proper motion agree with HIPPARCOS to bet- 1 913394034663258752 and Gaia DR2 913394034663259008. ter than 1σ. Given their high precision, the perihelion parameters The proper motion and radial velocity agree with the data used using the new astrometry with the old radial velocity are consis- in paper 1, but the Gaia DR2 parallax of 6.1±0.8 mas is quite dif- 0 tent with the result in paper 1 (now more precise). ferent from the Hipparcos-2 value of 68.5 ± 15.5 mas adopted in −15 −10 −5 0 5 10 15 Hip 42525. Two Gaia DR2 sources with the same par- perihelion time (tph) / Myr paper 1. It was noted in that paper, however, that the Hipparcos- allax, proper motion, and radial velocity as each other 2 parallax was almost certainly corrupted by a close companion, (within their uncertainties) match this. They are Gaia DR2 Fig. 9. Perihelion times and distances for star samples from the mock and that the Hipparcos-1 solution of 5.1±4.3 mas was more plau- Fig. 9. Perihelion times and distances for star samples from the mock 913394034663258752 and Gaia DR2 913394034663259008. full Galaxy (no Gaia selection). Each star is represented by a single sible. This value is in fact consistent with the new Gaia DR2 one, full Galaxy (no Gaia selection). Each star is represented by a single The proper motion and radial velocity agree with the data sample. which is five times more precise. It is (they are) no longer a close sample. used in paper 1, but the Gaia DR2 parallax of 6.1 0.8 mas is encounter. ± quite different from the HIPPARCOS-2 value of 68.5 15.5 mas adopted in paper 1. It was noted in that paper, however,± that the computedvelocities the above perihelion 200 km parameters s 1 would reach of all the stars Sun using within the 15 LMA. Myr. HIPPARCOS-2 parallax was almost certainly corrupted by a close − 4. Completeness correction and the encounter rate Although(Our final the completeness-corrected LMA is not an accurate encounter model for rate long only paths uses inthe companion, and that the HIPPARCOS-1 solution of 5.1 4.3 mas the Galaxy, the inaccuracy introduced by this for the distribu- 4.1.was Principle more plausible. of the completeness This value is in model fact consistent with± the new model out to 5 Myr, so only mock stars beyond 3 kpc with radial tion as a whole will be small.1 The impact is further diminished Gaia DR2 one, which is five times more precise. It is (they are) velocities above 600 km s− will be neglected.) We checked on a Gaia DR2 does not identify all encounters, primarily because of given that we are ultimately interested only in the ratio of the no longer a close encounter. subsample of the data that increasing this limit to 10 kpc made its faint-end magnitude limit for radial velocities. To compute twonegligible encounter diff distributionserence to the from distribution these two of models. encounters. We then an encounter rate over a specified time and distance window we computedFigure 9 the shows perihelion the positions parameters in perihelion of all stars parameter using the space LMA. must4. Completeness correct for this. This correction was done and in paper the encounter 2 using a simple rate ofAlthough individual the stars LMA from is the not sampled an accurate mock model full Galaxy. for long The paths den- in analytic model. Although insightful and easy to work with, that sitythe of Galaxy, encounters the inaccuracy shows essentially introduced no dependence by this for the on t distribu-ph over model4.1. Principle made severe of the assumptions completeness of isotropic model spatial and homoge- thetion range as a shown. whole willThis be is di small.fferent The from impact what is was further found diminished from the neous velocity distributions. Here we take the same conceptual modelgiven in that paper we are 2 which ultimately showed interested a drop only off to in large the ratio|t |.of This the Gaia DR2 does not identify all encounters, primarily because of ph approach to building a completeness map, but replace the ana- wastwo in encounter part due to distributions the spatial distributionfrom these two adopted models. (and its incon- its faint-end magnitude limit for radial velocities. To compute lytic model with a more realistic Galaxy simulation, in which the sistencyFigure with9 shows the velocity the positions distribution in perihelion adopted). parameter In the present space an encounter rate over a specified time and distance window we spatial and kinematic distributions are also self-consistent. We modelof individual we also starsdo not from see the dropsampled in density mock towardsfull Galaxy.t = The0. The den- must correct for this. This was done in paper 2 using a simple ph first simulate the positions and velocities of all stars in the nearby encountersity of encounters density in shows Figure essentially 9 does show no a dependence strong dependency on t over on analytic model. Although insightful and easy to work with, that ph Galaxy (“mock full Galaxy”), and trace the orbits backwards and dthe. Closer range shown. inspection This shows is different that the from number what was of encounters found from per the model made severe assumptions of isotropic spatial and homoge- ph forwards in time (with LMA) to determine the distribution of unitmodel distance in paper varies 2 linearlywhich showed with d a, dropi.e. the off numberto large of encoun-t . This neous velocity distributions. Here we take the same conceptual ph | ph| encounters in perihelion coordinates. Call this Fmod(tph, dph), the was in part due to the spatial2 distribution adopted (and its incon- approach to building a completeness map, but replace the ana- ters within dph varies as dph. This is what the simple model in pa- number of encounters per unit perihelion time and distance. We sistency with the velocity distribution adopted). In the present lytic model with a more realistic Galaxy simulation, in which the per 2 predicts (see section 4.2 of that paper, which also explains then repeat this including only the stars which would have full model we also do not3 see the drop in density towards t = 0. spatial and kinematic distributions are also self-consistent. We why it does not vary as d , as one may initially expect). Toph con- 6D kinematic information in Gaia DR2 (“mock Gaia-observed The encounter density inph Fig.9 does show a strong dependency first simulate the positions and velocities of all stars in the nearby struct the completeness map we therefore replace the 2D distri- Galaxy”), then trace orbits to give F (t , d ). The ratio of on d . Closer inspection shows that the number of encounters Galaxy (“mock full Galaxy”), and traceexp theph orbitsph backwards and butionph in Figure 9 with a 1D distribution F (t , d ) = a d , these two distributions gives the completeness map, C(t , d ), per unit distance varies linearly with d , i.e.exp theph numberph ofph en- forwards in time (with LMA) to determine the distributionph ph of where the constant a is fit from the simulatedph data. The distribu- which can be interpreted as the fraction of encounters at a spe- 2 encounters in perihelion coordinates. Call this F (t , d ), the tioncounters of the within real encountersdph varies also as d showsph. This this is whatlinear the variation simple out model to cific perihelion time and distance that will actuallymod beph observedph number of encounters per unit perihelion time and distance. We severalin paper pc, 2 as predicts shownin (see Figure Sect. 10. 4.2 of that paper, which also ex- (in our sample). 3 then repeat this including only the stars which would have full plainsTo produce why it does the notmock vary Gaia-observed as dph, as one Galaxy may initially we query expect). the 6D kinematic information in Gaia DR2 (“mock Gaia-observed mockTo construct catalogue the of completeness Rybizki et mapal. (2018) we therefore using the replace following the 2D 4.2.Galaxy”), The model then trace orbits to give Fexp(tph, dph). The ratio of ADQLdistribution query in Fig.9 with a 1D distribution Fexp(tph, dph) = a dph, these two distributions gives the completeness map, C(tph, dph), where the constant a is fit from the simulated data. The distribu- Towhich generate can beour interpreted mock full Galaxy as the fraction we use Galaxia of encounters (Sharma at aet spe- al. SELECTtion of parallax, the real encounters pmra, pmdec, also radial_velocity shows this linear variation out to 2011)cific perihelion to sample time stars and following distance a that Besancon-like will actually (Robin be observed et al. FROMseveral gdr2mock.main pc, as shown in Fig. 10. 2003)(in our model sample). distribution of the Galaxy, in a similar way to how WHERETo phot_g_mean_mag produce the mock <= 12.5Gaia-observed Galaxy we query the we generated the mock Gaia DR2 catalogue described in Rybizki ANDmock teff_val catalogue > 3550 of ANDRybizki teff_val et al. <(2018 6900 ) using the following et al. (2018). The main difference is that we now apply no mag- ADQL query nitude4.2. The cut, model but instead a distance cut, to keep the computations The Teff limit simulates the limit on the radial velocities pub- manageable. We use 3 kpc, as only stars with radial velocities lished in Gaia DR2 (Katz et al. 2018), and the magnitude limit is To generate our mock full Galaxy we use Galaxia (Sharma et al. SELECT parallax, pmra, pmdec, radial_velocity above 200 km s−1 would reach the Sun within 15 Myr. (Our fi- the selection we will apply to the observed encounter sample to 2011) to sample stars following a Besancon-like (Robin et al. FROM gdr2mock.main nal completeness-corrected encounter rate only uses the model compute the completeness-corrected encounter rate below. We 2003) model distribution of the Galaxy, in a similar way to WHERE phot_g_mean_mag <= 12.5 out to 5 Myr, so only mock stars beyond 3 kpc with radial ve- do not specifically account for the incompleteness at the bright how we generated the mock Gaia DR2 catalogue described in AND teff_val > 3550 AND teff_val < 6900 locities above 600 km s−1 will be neglected.) We checked on a end in Gaia DR2 because it is poorly defined. There are so few Rybizki et al.(2018). The main di fference is that we now apply subsample of the data that increasing this limit to 10 kpc made bright stars in the model that this is anyway a minor source of no magnitude cut, but instead a distance cut, to keep the com- The T limit simulates the limit on the radial velocities pub- negligible difference to the distribution of encounters. We then error. Thiseff above query delivers 4.4 million stars. (As a compar- putations manageable. We use 3 kpc, as only stars with radial lished in Gaia DR2 (Katz et al. 2018), and the magnitude limit Article number, page 9 of 13page.13 A34, page 9 of 13 A&A proofs: manuscriptA&A 616, no. A37 stellar_encounters_gdr2 (2018) 5 5
500 ●
500 ● ● ●
● ● 4 ● ● ●● ●● 4 ● ● ● ● ●● ● 400 ● ● ● ● ● ● ● ● ● ● ) / pc 400 ● ● ● ● ● ● ● ● ● ● h
● ● ) / pc ● ● ● ● p ● ● ●● ● ● h
● d ● ● ● ● ● ● p ● ● ● d ● ● ● ● ● ● 3 ● ● 3
300 ● ●
300 ● ● ●● ● ● ● ● ●
● 2 ● ● ●● ● ● ● ● ● 2 200 ● ● ● ● ● ● 200 ● ● ● ●● ● ● ● ● ● ●● ● ● ● perihelion distance ( number of encounters per pc number ● 1
●● perihelion distance ( number of encounters per pc number ● ● ● ● 1 100 ●● ● ●
100 ● ● ● ●● ●● ●● ● ● ● 0 ● 0 0 0 2 4 6 8 10 0 −15 −10 −5 0 5 10 15 0 2 4 6 8 10 −15 −10 −5 0 5 10 15 perihelion distance (dph) / pc perihelion time (tph) / Myr perihelion distance (dph) / pc perihelion time (tph) / Myr
Fig. 10. The variation of the number of observed encounters per unit Fig.Fig. 11. 11.PerihelionPerihelion times times and and distances distances for for star star samples samples from from the the mock mock Fig.Fig. 10. 10.TheVariation variation of the of numberthe number of observed of observed encounters encounters per unit per peri-unit Fig.Gaia 11. Perihelion times and distances for star samples from the mock perihelion distance as a function of perihelion distance (for the filtered Gaia-observed-observed Galaxy. Galaxy. Each Each star star is is represented represented by by 100 100 samples. samples. This This is is perihelionhelion distance distance as a as function a function of perihelion of perihelion distance distance (for the(for filtered the filtered sam- Gaia-observedjust a window onGalaxy. a much Each larger star set is of represented encounters, by so 100 includes samples. samples This is of sample for all tph). The open circles count the encountering stars dis- just a window on a much larger set of encounters, so includes samples of sampleple for forall t allph).t Theph). The open open circles circles count count the encountering the encountering stars discretely, stars dis- juststars a windowwhich are on predominantly a much larger set outside of encounters, the range shownso includes (e.g. samples the median of cretely, using dmed as the distance estimate for each star. The filled cir- stars which are predominantly outside the range shown (e.g. the median cretely,using d usingmed asd themedph distanceas the distance estimate estimate for each for star. each The star. filled The circles filled count cir- starsperihelion which position are predominantly is outside). outside the range shown (e.g. the median ph ph perihelion position is outside). cleseach count surrogate each for surrogate each encountering for each encountering star separately. star The separately. green line The is perihelion position is outside). green line is the distribution we expect under the assumption that the greenthe distribution line is the we distribution expect under we expect the assumption under the−1 that assumption the distribution that the is distribution is linear. It has a gradient1 of 50.0 pc−1 . (The number of en- causes for this. One possibility is differing velocity distributions: distributionlinear. It has is a linear. gradient It has of 50.0 a gradient pc− . (The of 50.0 number pc . of (The encounters number2 of within en- counters within some distance is the integral2 of this, i.e. ∝ d2ph.) The if stars5 in the mock catalogue were generally slower, then the counterssome distance within is some the integral distance of is this, the integral i.e. d of.) this, The i.e. data∝ dodph not.) The fol- ∝ ph 5 datalow do this not near follow to the this limit near (at to 10 the pc) limit because (at 10 not pc) allbecause stars foundnot all in stars the most distant visible stars would arrive at larger perihelion times. foundLMA-based in the LMA-based selection actually selection come actually within come 10 pc within after doing 10 pc the after orbital do- An identical reproduction of the observational distribution is not
ing the orbital integration. The error bars show the Poisson noise com- 4 0.4 ingintegration. the orbital The integration. error bars The show error the bars Poisson show noise the Poisson computed noise from com- the necessary,4 however, because the completeness is dependent0.4 pri- puted from the theoretical relation (only attached to the open points to putedtheoretical from relationthe theoretical (only attached relation to (only the open attached points to to the avoid open crowding). points to ) / pc
marilyh on the change from the mock full Galaxy to the mock ) / pc p avoid crowding). h d avoid crowding). p 0.3 Gaiad -observed Galaxy. 3 0.3 As3 an aside, we can use the mock Gaia-observed simula- is the selection we will apply to the observed encounter sample tion to try to explain features in the observed perihelion distri- 0.2
2 0.2
to compute the completeness-corrected encounter rate below. We bution2 (e.g. Fig.3). In particular, the gap at small t might ison, the number of stars in Gaia DR2 with complete 6D kine- | ph| ison,do not the specifically number of account stars in for Gaia the DR2 incompleteness with complete at the6Dbright kine- be explained by the Teff cut. This cut removes cool M dwarfs, matic data brighter than G = 12.5 mag is 3.6 million.) 0.1 maticend in dataGaia brighterDR2 because than G = it12 is. poorly5 mag isdefined. 3.6 million.) There are so few whichperihelion distance ( are also intrinsically faint. To be observable now, they0.1 1 perihelion distance ( brightGiven stars the in nonlinear the model relationship that this is between anyway thea minor 6D kinematic source of must1 therefore be relatively near, in which case they will all data and the perihelion parameters for a star, symmetric (Gaus- dataerror. and This the above perihelion query parameters delivers 4.4 for million a star, stars. symmetric (As a compar- (Gaus- encounter relatively soon (past or future). M dwarfs are highly0.0 0.0 sian)ison, errors the number in the of former stars incanGaia leadDR2 to asymmetric with complete distributions 6D kine- abundant,0 so would dominate the encounters at small t . Yet sian) errors in the former can lead to asymmetric distributions 0 ph in the latter. In other words, measurement errors in the real Gaia −15 −10 −5 0 5 10 15| | inmatic the latter. data brighter In other than words,G = measurement12.5 mag is 3.6 errors million.) in the real Gaia they have−15 been−10 removed−5 by the0 Teff cut.5 A similar10 argument15 in perihelion time (tph) / Myr observationsGiven the can nonlinear lead to stars relationship being preferentially between thescattered 6D kine- into principle also applies toperihelion the brightest time (tph) / Myr stars – many of which are ormatic out of data some and part the of perihelion the perihelion parameters parameter for a space. star, symmet- We ac- bright because they are close by, and thus near to encounter now Fig. 12. The completeness map, C(tph, dph), which can be interpreted commodateric (Gaussian) this in errors our model in the by former replacing can each lead star to in asymmetric the mock Fig.– which 12. The are completeness also missing map, in GaiaC(tph,DR2,dph), which but these can beare interpreted fewer, so Gaia-observed Galaxy with a set of 100 noise-perturbed sam- as the probability of observing an encounter in the selected Gaia DR2 Gaia-observeddistributions in Galaxy the latter. with aIn set other of 100 words, noise-perturbed measurement sam- er- sampletheir net as a contribution function of its is perihelion smaller. time and distance. “Selected” here plesrors generated in the real using Gaia a simple observations error model. can For lead this to we stars use a being 4D- sampleWe as can a function now build of its perihelion the completeness time and distance. map. We “Selected” bin the here dis- ples generated using a simple error model. For this we use a 4D- means for G < 12.5 mag in the filtered sample. Gaussianpreferentially PDF scattered (the Gaia-uncertainties into or out of on some RA partand Declinationof the peri- tribution in Fig. 11, and divide by the number of samples, to arehelion negligible), parameter with space. the following We accommodate standard deviations, this in our obtained model get a binned distribution of the (fractional) number of encoun- fromby replacing inspection each of the star Gaia in DR2 the uncertainties mock Gaia-observed shown in Linde-Galaxy tering stars per bin. We then divide this by the expected (con- from inspection of the Gaia DR2 uncertainties shown in Linde-−1 grenwith et a al. set (2018): of 100σ($ noise-perturbed) = 0.068 mas; samplesσ(µα∗) = generated0.059 mas yrusing−1; marilytinuous) on distribution the change from the mock mock full full Galaxy Galaxy, to which the mock was gren et al. (2018): σ($−1) = 0.068 mas; σ(µ−α∗1) = 0.059 mas yr ; marily on the change from the mock full Galaxy to the mock σ(µδ) = 0.041 mas yr−1; σ(vr) = 0.8 km s−1. We neglect corre- Gaia-observed Galaxy. σa(µ simpleδ) = 0. error041 mas model. yr ; Forσ(vr this) = 0 we.8 km use s a.4D-Gaussian We neglect corre- PDF Gaia-observedjust Fexp(tph, d Galaxy.ph) = a dph, to give our completeness map. This lations as there is no reliable model for these. Just as was done lations(the Gaia as there-uncertainties is no reliable on RA model and Declinationfor these. Just are as negligible), was done is shownAs anaside, in Fig. we 12. can The use map the was mock computed Gaia-observed and fitted simula- over with the surrogates for the real data, the orbits of each of the withwith the the surrogates following forstandard the real deviations, data, the obtained orbits of from each inspec- of the tionthe rangeto try to15 explain to +15 features Myr and in 0 the to observed 10 pc to improve perihelion the distri- fit to samples are traced (but here using LMA rather than a potential). − samplestion of are the tracedGaia DR2 (but here uncertainties using LMA shown rather in than Lindegren a potential). et al. butionFexp(tph (e.g., dph Figure) (by reducing 3). In particular, the shot the noise), gap at but small is only|tph| shownmight 1 bution (e.g. Figure 3). In particular, the gap at small ph might (2018Figure): σ 11($ shows) = 0.068 the mas; resultingσ(µ positionsα ) = 0.059 in mas perihelion yr− ; param-σ(µδ) = beover explained a smaller byd theph Trangeeff cut. (and This will cut only removes be used cool over M dwarfs, a nar- Figure 11 shows1 the resulting positions1 ∗ in perihelion param- be explained by the eff cut. This cut removes cool M dwarfs, eter0.041 space mas of yr− the; samplesσ(vr) = 0 from.8 km the s− mock. WeGaia-observed neglect correlations Galaxy. as whichrower aretph range also intrinsically too). The completeness faint. To be ranges observable between now, nearly they Inthere principle is no reliable this is a model model for for these. the observations Just as was shown done with in Fig- the mustzero andtherefore 0.48, with be relatively an average near, value in (overwhich the case bins) they of will0.09 all for uresurrogates 4. They for are the not real the data, same the due orbits to shot of each noise, of imperfect the samples mod- are encountert < 10 Myr relatively and 0.14 soon for (pastt or< 5 future). Myr. The M dwarfsbinning are of highlyFig. 11 ure 4. They are not the same due to shot noise, imperfect mod- encounter| ph| relatively soon (past| ph| or future). M dwarfs are highly ellingtraced of (but the here (complex) using LMA Gaia DR2 rather selection than a potential). function, and in par- abundant,resulted in so some would cells dominate with no the encounters, encounters which at small would|tph lead|. Yet to elling of the (complex) Gaia DR2 selection function, and in par- abundant, so would dominate the encounters at small |tph|. Yet ticularFigure the fact 11 shows that Galaxia the resulting is not an positions exact model in perihelion of our Galaxy. param- theyzero have completeness been removed (and thus by the a divide-by-zeroTeff cut. A similar when argument we come in to ticular the fact that Galaxia is not an exact model of our Galaxy. they have been removed by the Teff cut. A similar argument in Aeter particular space of di thefference samples is fromthat the the mock mock catalogueGaia-observed shows Galaxy. more principleuse it). We also overcame applies thisto the by brightest replacing stars zero-valued – many ofcells which with are the encountersIn principle at this larger is a perihelion model for times. the observations There are several shown possiblein Fig.4. brightmean because of their they neighbours. are close We by, experimented and thus near to with encounter producing now a causesThey are for this.not the One same possibility due to is shot differing noise, velocity imperfect distributions: modelling –smoother which are map also via missing kernel density in Gaia estimation DR2, but these or fitting are afewer, smooth- so ifof stars the (complex) in the mockGaia catalogueDR2 selection were generally function, slower, and in then particu- the theiring spline, net contribution but the results is smaller. were too sensitive to the fitting param- mostlar the distant factvisible that Galaxia stars would is not arrive an exact at larger model perihelion of our Galaxy. times. etersWe and can su nowffered build from the edge completeness effects (which map. could We bebin mitigated the dis- AnA particular identical reproduction difference is of that the the observational mock catalogue distribution shows is more not tributionby expanding in Figure the grid, 11, and butatdivide the expense by the number of sensitivity of samples, to low necessary,encounters however, at larger because perihelion the times. completeness There are is several dependent possible pri- tocompleteness get a binned values distribution at larger of times). the (fractional) number of en-
ArticleA34, page number, 10 of page 13 10 of 13page.13 A&A proofs: manuscript no. stellar_encounters_gdr2 5
500 ● ●
● ● 4 ● ● ● ● ● ● ● ● ● 400 ● ● ● ●
● ● ● ) / pc ● ● ● ● ● ● h ● ● ● p
● d ● ● ● ● ● ● 3
300 ● ● ● ● ● ● ●
● 2 ● ● ● ● ● ● 200 ● ● ● ● ● ● ● ● ● ●
● perihelion distance ( number of encounters per pc number
● 1 ● ● ● 100 ● ● ● ● ● ● ● ● 0 0 0 2 4 6 8 10 −15 −10 −5 0 5 10 15 perihelion distance (dph) / pc perihelion time (tph) / Myr
Fig. 10. The variation of the number of observed encounters per unit Fig. 11. Perihelion times and distances for star samples from the mock perihelion distance as a function of perihelion distance (for the filtered Gaia-observed Galaxy. Each star is represented by 100 samples. This is sample for all tph). The open circles count the encountering stars dis- just a window on a much larger set of encounters, so includes samples of med stars which are predominantly outside the range shown (e.g. the median cretely, using dph as the distance estimate for each star. The filled cir- cles count each surrogate for each encountering star separately. The perihelion position is outside). green line is the distribution we expect under the assumption that the Bailer-Jones et al.: Close encounters to the Sun in Gaia-DR2 distribution is linear. It has a gradient of 50.0 pc−1. (The number of en- 2 counters within some distance is the integral of this, i.e. ∝ dph.) The data do not follow this near to the limit (at 10 pc) because not all stars 5 build the completeness model. We do not remove bogus encoun- found in the LMA-based selection actually come within 10 pc after do- ters, as their number is small. (Identifying them is a laborious,
ing the orbital integration. The error bars show the Poisson noise com- 4 0.4 manual process that we have only done for the 31 encounters puted from the theoretical relation (only attached to the open points to out to 1 pc.) Within the window tph < 5 Myr, dph < 5 pc we have ) / pc
h | | avoid crowding). p n = 463.4 (the sum in Eq.1 is over 926 736 surrogates), which
d enc 0.3 3 is an uncorrected rate of 46 encounters per Myr within 5 pc. med med This compares to 639 stars with tph and dph in this range (i.e. 0.2
2 neglecting the uncertainties gives an uncorrected rate of 64 stars ison, the number of stars in Gaia DR2 with complete 6D kine- per Myr within 5 pc.) Applying the completeness correction as matic data brighter than G = 12.5 mag is 3.6 million.) 0.1 outlined above yields ncor = 4914 542 encounters, which cor- perihelion distance (
1 ± Given the nonlinear relationship between the 6D kinematic responds to 491 54 encounters per Myr within 5 pc. We could ± data and the perihelion parameters for a star, symmetric (Gaus- calculate the corrected rates for smaller upper limits on dph, 0.0
sian) errors in the former can lead to asymmetric distributions 0 but such results are sensitive to the use of fewer bins in the in the latter. In other words, measurement errors in the real Gaia −15 −10 −5 0 5 10 15 completeness map and are more affected by the scattering of observations can lead to stars being preferentially scattered into perihelion time (tph) / Myr the surrogates. We instead scale the value found for 5 pc using or out of some part of the perihelion parameter space. We ac- the expectation that the number of encounters within some dis- , commodate this in our model by replacing each star in the mock Fig.Fig. 12. 12.TheThe completeness completeness map, map,CC(t(phtph, ddphph),), which which can can be be interpreted interpreted tance grows quadratically with distance. (Figure 10 shows that Gaia-observed Galaxy with a set of 100 noise-perturbed sam- asas the the probability probability of of observing observing an an encounter encounter in in the the selected selected GaiaGaia DR2DR2 out to 5 pc the scaling is as expected, unaffected by the drop-off samplesample as as a a function function of of its its perihelion perihelion time time and and distance. distance. “Selected” “Selected” here here ples generated using a simple error model. For this we use a 4D- near to 10 pc.) This gives encounter rates of 78.6 8.7 per Myr meansmeans for forGG<<1212..55 mag mag in in the the filtered filtered sample. sample. ± Gaussian PDF (the Gaia-uncertainties on RA and Declination within 2 pc, and 19.7 2.2 per Myr within 1 pc. We adopt these ± are negligible), with the following standard deviations, obtained as our final encounter rates. from inspection of the Gaia DR2 uncertainties shown in Linde- 4.3. An estimate of the encounter rate In paper 2, using TGAS and an analytic completeness cor- gren et al. (2018): σ($) = 0.068 mas; σ(µ ) = 0.059 mas yr−1; rection, we derived an encounter rate within 5 pc of 545 59 α∗ marilyIf a star on is the observed change to from be encountering the mock full atGalaxy position tot the, d mock, then ± σ(µ ) = 0.041 mas yr−1; σ(v ) = 0.8 km s−1. We neglect corre- ph ph per Myr (which scales to 21.8 2.4 per Myr within 1 pc). δ r Gaia-observedthe completeness-corrected Galaxy. (fractional) number of encounters ± lations as there is no reliable model for these. Just as was done This is consistent with what we find here. García-Sánchez et al. correspondingAs an aside, to we this can is use simply the mock1/C(t Gaia-observed, d ). When each simula- star with the surrogates for the real data, the orbits of each of the ph ph (2001) obtained a rate of 11.7 1.3 per Myr within 1 pc using tionis instead to try torepresented explain features by a set in of theN observedsurrogates, perihelion the complete- distri- ± sur HIPPARCOS. Possible reasons for this discrepancy are discussed samples are traced (but here using LMA rather than a potential). bution (e.g. Figure 3). In particular, the gap at small |tph| might nesses at which are Ci , the completeness-corrected number of in paper 2. Figure 11 shows the resulting positions in perihelion param- beencounters explained is by the T{ eff} cut. This cut removes cool M dwarfs, eter space of the samples from the mock Gaia-observed Galaxy. which are also intrinsically faint. To be observable now, they Choosing a smaller time window, tph < 2.5 Myr and X | | In principle this is a model for the observations shown in Fig- must therefore1 be1 relatively near, in which case they will all dph < 5 pc, we have nenc = 319.4 and a completeness-corrected ncor = (1) encounter rate of 373 44 per Myr within < 5 pc. This is 1.5σ ure 4. They are not the same due to shot noise, imperfect mod- encounterNsur relativelyCi · soon (past or future). M dwarfs are highly i times smaller than that± obtained with the larger time window. elling of the (complex) Gaia DR2 selection function, and in par- abundant, so would dominate the encounters at small |tph|. Yet This may suggest some time variability in the encounter rate, ticular the fact that Galaxia is not an exact model of our Galaxy. theyThis have same been expression removed applies by the whenTeff wecut. have A similar a set of argument stars repre- in ffi A particular difference is that the mock catalogue shows more principlesented by also surrogates applies overto the some brightest range stars (“window”) – many of whichtph and aredph although this is hard to distinguish given the di culty of propa- encounters at larger perihelion times. There are several possible brightwith corresponding because they are completenesses close by, and thusCi . nearN tois stillencounter the number now gating all the uncertainties. It is also clear that Eq.1 is rather { } sur causes for this. One possibility is differing velocity distributions: –of which surrogates are also per missing stars. ncor in Gaiais then DR2, the but completeness-corrected these are fewer, so sensitive to small values of the completeness. Over the win- if stars in the mock catalogue were generally slower, then the theirnumber net contribution of encounters is in smaller. the window. dow tph < 5 Myr and dph < 5 pc, 3462 of the 926736 surrogates | | < . most distant visible stars would arrive at larger perihelion times. WeRandom can now errors build in n thecor come completeness from two map. main Wesources: bin the (i) Pois- dis- (0.4%) have Ci 0 01. It is an unfortunate and unavoidable fact An identical reproduction of the observational distribution is not tributionson noise in from Figure the 11, finite and number divide by ofencounters the number observed; of samples, (ii) that it is the low completeness regions which contribute the necessary, however, because the completeness is dependent pri- tonoise get ain binned the completeness distribution map. of the The (fractional) first source number is easily of en- ac- largest uncertainty to any attempt to correct the encounter rate. commodated. The fractional number of encounters observed in a Article number, page 10 of 13page.13 window, nenc, is just the sum of the fraction of all surrogates per star which lie in that window. The signal-to-noise ratio in this is 5. Conclusions √nenc, so the standard deviation in ncor due to this alone would be ncor/ √nenc. This does not include any noise due arising from We have identified the closest stellar encounters to the Sun from the original sample selection (e.g. from changes in the filtering). among the 7.2 million stars in Gaia DR2 that have 6D phase The second source can be treated with a first order propagation space information. Encounters were found by integrating their of errors in Eq. (1) orbits in a smooth gravitational potential. The correlated uncer- tainties were accounted for by a Monte Carlo resampling of the 1 X 1 δCi 6D likelihood distribution of the data and integrating a swarm δncor = (2) Nsur Ci Ci · of surrogate particles. The resulting distributions over perihel- i ion time, distance, and speed are generally asymmetric, and are In practice it is difficult to determine δCi. We have experimented summarized by their 5th, 50th, and 95th percentiles. with varying the approach to constructing the completeness map. We find 31, 8, and 3 stars which have come – or which will We approximate the uncertainties arising thereby as a constant come – within 1 pc, 0.5 pc, and 0.25 pc of the Sun, respectively. fractional uncertainty of fc = δCi/Ci = 0.1. In that case Eq. (2) can These numbers drop to 26, 7, and 3 when we remove likely incor- be written δncor = fcncor. Combining this with the Poisson term for rect results (“bogus encounters”) following a subjective analysis (i) we may approximate the total random uncertainty in ncor as (including visual inspection of images). Quality metrics in the Gaia catalogue are not calibrated and are hard to use to iden- !1/2 1 2 tify sources with reliable data. Thus some of the stars in our σ(ncor) = ncor + fc (3) encounter list are sure to be bogus, and we are sure to have nenc · missed some others for the same reason. In particular, a num- To compute nenc we use just the filtered encounter sample ber of encounters with unexpected positions in the CMD have with G < 12.5 mag – 2522 stars – as this is what was also used to large values of the astrometric unit weight error. These are not
A34, page 11 of 13 A&A 616, A37 (2018) necessarily poor astrometric solutions. At least some could be parameters to be rather similar (tmed = 860 kyr, dmed = 1.43 pc). ph − ph main sequence-white dwarf binaries. Sirius, Altair, and Algol are not in Gaia DR2. The closest encounter found is Gl 710, long known to be There are no doubt many more close – and probably closer – a close encounter, now found to come slightly closer and with encounters to be discovered in future Gaia data releases. slightly better determined perihelion parameters. Most of the other encounters found are discovered here, including 25 within Acknowledgements. This work is based on data from the European Space 1 pc. Using newly available masses for 98% of the sample com- Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), pro- puted by Fouesneau et al. (in preparation) from Gaia astrome- cessed by the Gaia Data Processing and Analysis Consortium (DPAC, https: try and multiband photometry, we compute the impulse transfer //www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the to the Oort cloud using the impulse approximation. For both the DPAC has been provided by national institutions, in particular the institutions 1/d or 1/d2 dependencies, Gl 710 induces the largest impulse. participating in the Gaia Multilateral Agreement. We thank Lennart Lindegren ph ph for discussions about the Gaia astrometric solution and Ted von Hippel for dis- Berski & Dybczynski´ (2016) studied the impact of this star on cussions about white dwarfs. This work was funded in part by the DLR (German the Oort cloud in some detail using the encounter parameters space agency) via grant 50 QG 1403. We are grateful for the availability of the from the first Gaia data release. They found that it would inject Simbad object database and Aladin sky atlas, both developed and provided by a large flux of Oort cloud comets toward the inner solar sys- CDS, Strasbourg. tem. Given that the encounter parameters are not significantly changed in Gaia DR2, this conclusion still holds. It remains to References be studied what the cumulative effect is of the many more en- counters found in our study. Andrae, R., Fouesneau, M., Creevey, O., et al. 2018, A&A, 616, A8 Arenou, F., Luri, X., Babusiaux, C., et al. 2018, A&A, 616, A17 The main factors limiting the accuracy of our resulting peri- Bailer-Jones, C. A. L. 2015a, A&A, 575, A35 helion parameters are: for most stars, the accuracy of the radial Bailer-Jones, C. A. L. 2015b, PASP, 127, 994 velocities; for distant stars, the accuracy of our potential model; Bailer-Jones, C. A. L. 2018, A&A, 609, A8 and for some stars, the neglect of possible binarity. Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Mantelet, G., & Andrae, R. Our sample is not complete. The main limitation is the 2018, AJ, 156, 58 Berski, F., & Dybczynski,´ P. A. 2016, A&A, 595, L10 availability of radial velocities in Gaia DR2. 99% of the stars Bobylev, V. V. 2010a, Astron. Lett., 36, 220 (whether in the unfiltered or filtered sample) are brighter than Bobylev, V. V. 2010b, Astron. Lett., 36, 816 G = 13.8 mag. Published radial velocities are also limited to stars Bobylev, V. V. 2017, Astron. Rep., 61, 883 Bobylev, V. V. & Bajkova, A. T. 2017, Astron. Lett., 43, 559 in the approximate T ff range 3550–6900 K, thereby limiting the e de la Fuente Marcos, R. & de la Fuente Marcos, C. 2018, Res. Notes AAS, 2, 30 number of (numerous) late-type stars as well as massive stars. Dybczynski, P. A. 1994, Celest. Mech. Dyn. Astron., 58, 139 Both Teff and G should be extended in subsequent Gaia data re- Dybczynski,´ P. A. 2006, A&A, 449, 1233 leases. Correcting for this incompleteness using a model based Dybczynski,´ P. A. & Berski, F. 2015, MNRAS, 449, 2459 on the Galaxia simulation, we infer the encounter rate averaged Gaia Collaboration (Brown, A. G. A., et al.). 2018, A&A, 616, A1 over the past/future 5 Myr to be 491 54 Myr 1 within 5 pc. García-Sánchez, J., Preston, R. A., Jones, D. L., et al. 1999, AJ, 117, − 1042 When scaled to encounters within 1 pc,± this is 19.7 2.2 Myr 1. ± − García-Sánchez, J., Weissman, P. R., Preston, R. A., et al. 2001, A&A, We caution, however, that the accuracy of this rate is limited 379, 634 by the completeness model assumptions and fitting the result- Hollands, M. A., Tremblay, P. E., Gaensicke, B. T., Gentile-Fusillo, N. P., & ing completeness map, as well as the distribution of the actual Toonen, S. 2018, MNRAS, submitted [arXiv:1805.12590] Jiménez-Torres, J. J., Pichardo, B., Lake, G., & Throop, H. 2011, MNRAS, 418, encounters. It may also be overestimated if there is a large frac- 1272 tion of sources with spuriously large parallax values that are not Katz, D., Sartoretti, P., Cropper, M., et al. 2018, A&A, submitted accounted for by their formal uncertainties. [arXiv:1804.09372] The other source of incompleteness is missing bright stars Lindegren, L., Hernandez, J., Bombrun, A., et al. 2018, A&A, 616, A2 in Gaia DR2. Bright stars saturate even with the shortest CCD Mamajek, E. E., Barenfeld, S. A., Ivanov, V. D., et al. 2015, ApJ, 800, L17 Marigo, P., Girardi, L., Bressan, A., et al. 2017, ApJ, 835, 77 gate on Gaia, making them harder to calibrate. Astrometry for Matthews, R. A. J. 1994, QJRAS, 35, 1 these will only be provided in later data releases. Although few Mülläri, A. A. & Orlov, V. V. 1996, Earth Moon Planet., 72, 19 in number (and therefore acceptably neglected by our complete- Oort, J. H. 1950, Bull. Astron. Inst. Neth., 11, 91 ness correction), some bright stars are currently nearby and/or Öpik, E. 1932, Proc. Am. Acad. Arts Sci., 67, 169 Rebassa-Mansergas, A., Ren, J. J., Parsons, S. G., et al. 2016, MNRAS, 458, massive, so encounter parameters more accurate than those ob- 3808 tained with HIPPARCOS could ultimately reveal some important Rickman, H. 1976, Bull. Astr. Inst. Czechosl., 27, 92 encounters. The case of gamma Microscopii – no longer a close Robin, A. C., Reylé, C., Derrière, S., & Picaud, S. 2003, A&A, 409, 523 encounter in Gaia DR2 – has already been discussed. Another Rybizki, J., Demleitner, M., Fouesneau, M., et al. 2018, PASP, 130, 074101 case is the A2 dwarf zeta Leporis, computed in paper 1 to have Sharma, S., Bland-Hawthorn, J., Johnston, K. V., & Binney, J. 2011, ApJ, med 730, 3 dph = 1.30 pc 850 kyr ago. It is in Gaia DR2 but without a ra- van Leeuwen, F., ed. 2007, Hipparcos, the New Reduction of the Raw Data, dial velocity. 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Appendix A: Gaia archive query astrometric_excess_noise_sig, rv_nb_transits, l, b FROM gaiadr2.gaia_source Below is the ADQL query used to select stars from Gaia DR2 WHERE (parallax IS NOT NULL) AND (parallax > -0.029) which have perihelion distances less than 10 pc according to the AND (radial_velocity IS NOT NULL) AND linear motion approximation (specified by equation 4 in paper 1). ( ( ( 1000*4.74047*sqrt(power(pmra,2)+power(pmdec,2)) SELECT source_id, ra, dec, parallax, pmra, pmdec, / power(parallax+0.029,2) radial_velocity, ra_error, dec_error, parallax_error, ) pmra_error, pmdec_error, radial_velocity_error, / sqrt( (power(pmra,2)+power(pmdec,2)) ra_dec_corr, ra_parallax_corr, ra_pmra_corr, * power(4.74047/(parallax+0.029),2) ra_pmdec_corr, dec_parallax_corr, dec_pmra_corr, + power(radial_velocity,2) dec_pmdec_corr, parallax_pmra_corr, parallax_pmdec_corr, ) pmra_pmdec_corr, visibility_periods_used, ) < 10 phot_g_mean_mag, bp_rp, astrometric_chi2_al, ) astrometric_n_good_obs_al, astrometric_excess_noise,
A34, page 13 of 13