ASTRONOMY AND ASFROPHYSICS of low mass companions to stars

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904 T EStepinski,D C Black Statistics of low mass companions. Minimum mass of brown dwarf This argument stems 2 Data adjustments. from the alleged lower limit to the mass of brown dwarfs based. LMC data considered in this paper come from several differ. upon the concept of opacity limited fragmentation Estimates. ent surveys This fact puts in question the representativeness. of this mass limit yield values of about 10 Mj Rees 1976 Silk. of the overall LMC sample and thus its suitability for statistical. 1977 although lower estimates are possible if fragmentation. analysis To alleviate this problem some adjustments are needed. occurs in a disk Boss 1998 Such a limit could provide theo. when joining LMC data from different surveys into a single set. retical support to the notion of duality of the LMC population. In the context of low mass and stellar mass companions such. provided it falts near the purported mass cutoff However this. adjustments are discussed by Mazeh et al 1998 The correc. limit must be considered as highly uncertain quantitatively. tions take into account the effects due to instrumental precision. Additionally the minimum mass argument ignores the pos. and number of stars examined in the various radial velocity sur. sibility that evolutionary effects such as mass exchange have. veys In addition Mazeh et al 1998 correct for the sin i factor. altered the masses of the closer companions The strength of. because they are ititerested in a distribution of an actual mass. this argument as support for a heterogeneous LMC population. of companions rather than a distribution of a projected mass. is marginal at best, We collect our LMCs sample from numerous surveys but. Mass eccentricity relation This argument arises from. it is only necessary to consider two distinct categories objects. plotting LMC projected masses versus their eccentricities Sup. obtained from relatively low precision 300 m sec survey. porters of the dual character of LMCs have pointed out that. of 570 stars by Mayor et al 1997 and objects obtained from. LMC below a certain mass have low eccentricities and those. relatively high precision 10 raJsec surveys of about 300. above that mass have high eccentricities again revealing a dis. stars see Marcy et al 1999 and references therein, continuity that suggests the existence of two sub populations. A correction protocol described by Mazeh et al 1998 is. Mayor et al 1998 This argument seems to be a historical foot. valid assuming that low and high precision surveys are compat. note as new data do not conform to the alleged mass eccentricity. ible statistically independent and unbiased However due to. differences in target selection criteria different surveys are not. Metallicity The fourth argument given for a dual charac. entirely compatible and are likely not to be statistically indepen. ter of LMCs comes from the metallicities of stars with LMCs. dent Therefore it is not clear what adjustment protocol if any. Stars with LMC designated as EP are metal rich compared to. is the most appropriate Given these uncertainties we use both. field stars Gonzalez 1998 Gonzalez et al 1999 However as. unadjusted and adjusted LMCs data to infer the distribution of. no direct comparison of metallicities between parent stars of M sin i We do not correct the data for the sin i factor because we. designated EP and parent stars of designated BD has been pub. restrict ourselves to studying the distribution of projected mass. lished the metallicity argument does not at present contribute. only This is dictated by the small size of LMCs sample Thus. to the question of homogeneity or heterogeneity of the LMC. the names EP and BD have to be taken with caution inasmuch. population, as M sin i is used as a surrogate for an actual mass Finally only. The case for the existence of two distinct species in the. unadjusted data is used to infer distributions of LMCs orbital. population of LMC is deserving of a more extensive treatment. parameters, than it has received in the literature to date In this paper we.
concentrate on evaluating two of the above arguments using. statistical analysis of data retating to all 27 LMC Our goal is to 3 Statistical model. estimate from available data the underlying PDFs for projected. We look for the underlying PDFs for projected masses periods. masses periods and eccentricities of LMC and eccentricities using the MLE Such an estimation maxi. Our approach will be to employ a parametric statistical mizes the probability of drawing a particular datum that was in. model in which the data is assumed to be drawn from a mixture. fact obtained This approach requires specifying the functional. of two PDFs one describing putative EP and the second describ. form of the PDF and estimating the values of free parameters. ing putative BD each having a specific form inferred from the The form of the PDF can be deduced from the empirical CDF. empirical CDF but undetermined parameters In addition the constructed for LMC quantities Let Yl Y2 YN be a. parameter describing the relative contribution of two compo list of either projected masses periods or eccentricities for N. nents to the overall mixture is undetermined MLE is used to. LMCs and assume that has been already sorted by size in. determine all unknown parameters This approach distinguishes increasing order Then the empirical CDF denoted by F y is. our work from that of Heacox 1999 who employed a nonpara. defined by, metric statistical model to analyze distributions of various LMC. quantities, Sect 2 discusses data adjustments and Sect 3 contains a de F v. scription of our statistical analysis In Sect 4 we present re 1 yg y. suits pertaining to projected masses Separately in Sect 5 we The estimation process is complicated by the fact that we. present results pertaining to periods and eccentricities Finally have to allow for the existence of two sub populations in the. in Sect 6 we present conclusions and discussion overall population of LMC We assume that is drawn from a. mixture of two PDFs fl YI01 which describes the distribution. T F Stepinski,Statistics,oflow mass,companions 905. ofquantityy for planets and f2 g102 which describes the form f y P with the index p 1 Convex departures from. distribution of quantity y for brown dwarfs where 81 and 02 the straight line indicate PDF in the form of the power law with. are lists of parameters characterizing respective PDFs Thus the p 1 whereas concave departures from the straight line indi. PDF for the entire LMC population can be expressed as follow cate power law PDF withp 1 Similarly PDFs in other forms. for example normal distribution log normal distribution etc. f ylaz 02 c 1 s fl ylOz Ef2 y 02 2 have their own characteristic CDF signatures In the case when. the gradient of the empirical CDF changes abruptly a mixture. where 0 1 is a mixture parameter Drawing observing. of two PDFs is indicated, say projected masses M sin i from a total LMC population dis. tributed according to 2 can be interpreted as a two step process The empirical CDF for projected masses of LMCs regard. First a Bernoulli random variable b is drawn taking a value of less of considered adjustments can be best characterized as. either a single smooth curve quite close to a straight line or. 1 with probability 1 s or value 2 with probability E Ac. a piecewise smooth curve with two component curves Thus. cording to the value of b M sin i is then drawn from one of. we infer from the data that the PDF of projected masses has a. the two sub populations with PDFs fl y101 and f2 y 02 We. functional form that is either a single power law or a mixture. assume that the allocation variable b is not directly observed. of two power laws Of course the empirical CDF constructed. This means that we don t a priori put any labels on the data The. from only 27 data points cannot be used to unambiguously in. labels if any can be put a posteriori if indicated by statistical. fer the underlying PDF and it is conceivable that the data came. analysis The complete data is thus zl z2 zjv where. from a distribution having functional form different from what. zj yj bj The PDF given by 2 can be interpreted as 9 zl0. we have inferred Newertheless a power law provides the least. with 0 01 02 The log likelihood function formed from. the data is structured candidate for the underlying PDE Therefore we adopt. the following form for the PDF of LMC projected masses. log1 log Ig zjl0 logg z 10 3 f M sinilO 1 e Ai M sini m H1. cA2 M sin i 2 2 4, A MLE is a value of 0 denoted by 0 that maximizes log L where H1 and 2 are cut offs defined in terms of the Heaviside.
In genera obtaining 0 is a nontdvial undertaking because 0 step function H. is a vector of potentially many dimensions and 9 z O can be. Hx H Msini,Msmz e mira H Msln p max Msini l, a complicated function We use the Expectation Maximization. EM numerical algorithm Dempster et al 1977 to find a MLE H2 H Msini MsmZ b d. rain H Msin b max d Msini, Note that this estimate contains the mixture parameter If the. estimation of e is close to zero a homogeneous population is. indicated In other words the PDF consists of two components the EP. component which is a power law with the index Pa and valid. 4 Projected masses for projected masses between Msm p and Msmz p. and the BD component given by a power law with the in. We carried out calculations for several cases set apart by differ dex P2 and valid for projected masses between Msm b d. ent adjustments to the LMC data no adjustments adjustment for and Msm b d Values of Msmz p 03Mj and. sample size and adjustments for both sample size and precision Msini a 70Mj are set by observations but there are. Adjustments are achieved by enlarging the population of objects. no a priori assumptions about values of Msini and, in a certain projected mass range by an appropriate factor To. M sin z bd, rain the distributions may in principle overlap con. correct for sample size we enlarged the population of EP by the. nect or there may be a gap between them The parameter. factor of 2 Following Mazeh et al 1998 we correct fo r instru. mental precision by further enlarging the population of planets list 0 has five components Pl P2 M sm. with A sin 1Mj and BD with 10Mj Msini 30 Mj and e because we decided to fix values of M sin z emin. by another factor of 2 It should be noted that Mazeh et al use M sin i Constants A1 and A2 are to assure that contribut. a 2a where a is a measurement error criterion for establish ing PDF s integrate to 1 They are expressible in terms of already. ing the minimum detectable EP signal 4a peak to peak This defined parameters. is in contrast to the 4a semi amplitude criterion suggested by Our goal is to determine the MLE of 0 We set up our calcula. Marcy Butler 1998 and used by us later in this paper Use tions as follows We allow both M sin i ep and M sin i bmdn. of this more stringent detection criterion would yield a modest to be any value from 5Mj to 50Mj in steps of 2 5 M j Thus. increase in the correction factor for the low end of the EP data there are altogether 192 361 possible PDFs under considera. set but it would not alter the conclusion tion For each possible PDF with the pre defined mass domain. The first step is to calculate an empirical CDF for LMC pro we employ the EM algorithm which finds the MLE of Pl P2. jected masses Displaying a CDF on the log linear scale makes and e and record the corresponding maximized value of log L. an identification of the underlying PDF easier In such a scal Note that in principle the EM algorithm should be able to find. ing any straight line corresponds to a PDF having a power law the MLE of the entire 0 without auxiliary cycling through two. 9O6 T EStepinski,Statistics,oflow mass,companions,10 20 30 40 20 30 40 50 10 20 30 40 50.
M sin MJ M sin Mj M sin Mj, Fig 1 The summary of testing the hypothesis that the PDF of M sin i for LMCs are given by a single power law against the alternative hypothesis. rltl X mill, that it is given by a mixture of two power laws Possible mixture PDFs are indexed by M sm t e p and M sm t bd The single power law. hypothesis is accepted over the mixture hypothesis for mixtures in the white region The mixture hypothesis is accepted at the significance level. s 1 for mixtures in the gray region The black subset of the gray region contains mixtures accepted at s 0 05 These best fit models can. be grouped into several types indicated by arrows The panels from left to right are for the unadjusted LMC data data adjusted for sample. size and data adjusted for both sample size and instrumental precision. of its components However due to the special character of The first result is that in all considered cases our formal. these parameters they define cut offs of PDFs we find such procedure locates some mixture models that are better fits than. a straightforward application of the EM algorithm difficult to the best single power law model However there is no unique. implement We also calculate the MLE of 80 p and record the best fit mixture instead in all cases the best mixture fits can be. maximized value of log L for a PDF given by a single power grouped into several types set apart by their overall character. law f M sini p over the entire domain of masses The best single power law fit to the unadjusted LMC data. Astror Astrophys 356 903 912 2000 ASTRONOMY AND ASFROPHYSICS Statistics of low mass companions to stars Implications for their origin T E Stepinski and D C Black

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