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historically , the thick disc of the milky way was first identified through star counts toward the galactic poles as a vertically extended disc component with scale height @xmath5 1 kpc @xcite , in addition to the thin disc with scale height @xmath6 that dominates the disc stars in the immediate solar neighbourhood . therefore , at the outset , the tentative definition of the thick - disc stars was those disc stars with large vertical orbital excursions and large vertical velocities .    later , spectroscopic studies ( e.g. , @xcite ) on kinematically - selected disc stars suggested that stars with large vertical motions ( which are likely to belong to the thick disc ) tend to show lower [ fe / h ] and higher [ @xmath7/fe ] than those with small vertical motions ( thin - disc stars ) . these chemical properties suggest that the thick - disc stars are older than thin - disc stars ( lower [ fe / h ] ) and that the star formation timescale of the thick disc was shorter than that of the thin disc ( higher [ @xmath7/fe ] ) . recently , klaus fuhrmann @xcite investigated a kinematically - unbiased volume - complete sample of solar - type disc and halo stars located within 25 pc of the sun . the distribution of his sample stars in the [ fe / h]-[mg / fe ] space reveals two chemically distinct populations of disc stars ( see figure 15 of @xcite ) . based on the kinematical properties of these populations , he identified the lower-[fe / h ] and higher-[mg / fe ] population to be the thick disc , and the other population to be the thin disc . this clear separation between the thin and thick discs is also confirmed in the nearby ( heliocentric distance @xmath8 100 pc ) kinematically - unbiased sample of @xcite for which [ fe / h ] and [ @xmath7/fe ] are available through high - resolution spectroscopy . these recent observations suggest that the thick disc is better defined by chemical compositions of stars , especially by [ fe / h ] and [ @xmath7/fe ] @xcite .    in the past decades , our understanding of the structure of the thick disc towards the galactic poles has been greatly improved @xcite . the next step forward is to unravel its more global structure , such as the radial dependence of its vertical structure . so far , many attempts have been made to fulfil this goal , and they are broadly categorised into two classes of studies .    the first class of studies are based on nearby samples of disc stars . one such example is @xcite , who fitted the distribution of local disc stars with his distribution function model . he discussed some global structure of the thick disc by looking into his best - fit models . although this kind of studies can predict almost everything if the best - fit models are reasonably correct , one critical problem with these studies is the validity of the functional forms of the assumed distribution functions .    the second class of studies are based on in - situ samples of ( relatively ) distant disc stars . this class can be further categorised into three sub - classes : those studies using ( i ) high - resolution spectroscopic samples ; ( ii ) medium - resolution spectroscopic samples ; or ( iii ) photometric samples . the advantage of sub - class ( i ) studies is that we can define the thick disc purely by means of the stellar chemistry @xcite . however , the number of stars that are currently available is less than a few hundred , and this small sample size makes it hard to obtain some statistical properties of distant disc stars . also , since the errors in distance and proper motion are large , kinematical analyses are difficult for these stars . in the sub - class ( ii ) studies , much larger sample of stars are available than in the sub - class ( i ) studies . a recent example of this sub - class is @xcite , who studied the density distribution of chemically defined disc populations by fitting the sdss / segue data with analytic models of the density profile . however , since the chemical abundances of these samples are less accurate than high - resolution samples , some blending of the thin- and thick - disc stars is inevitable in this kind of studies . most of the sub - class ( iii ) studies are based on the star count method ( e.g. , @xcite ) . photometric samples are advantageous in grasping the global structure of the stellar disc because the available sample size is the largest among these subclasses , and because the sample stars cover a wide range of heliocentric distance . however , since the photometric data lack chemical information for each sample star , it is not possible to separate the sample into stellar components . this inseparability means that one needs to adequately model all the stellar components that contribute to the sample , which is harder than adequately modelling the thick disc only .    by taking into account these problems , in this paper we explore the possibility of constraining the global structure of the thick disc based on a small number of nearby chemically - selected stars , but not relying on analytic modelling of the thick disc . specifically , we reconstruct the distribution function of the thick disc within a certain portion of the action space which is accessible from local observations . this paper is organised in the following manner . first , we describe our sample stars in section [ section : sample ] . in section [ section : theory ] , we introduce the concepts of the observable action space @xmath9 and the observable distribution function @xmath10 . there , we discuss the possibility of inferring some information on the velocity dispersion of the thick disc by using @xmath10 . in section [ section : method ] , we present our method to reconstruct @xmath10 from a local sample of thick - disc stars . in section [ section : analysis ] , we present our main results . the implications from our study are discussed in section [ section : discussion ] , and section [ section : conclusion ] sums up our analyses . in this paper , we assume that the milky way potential is axisymmetric and adopt the usual galactocentric cylindrical coordinate system @xmath11 to describe the three - dimensional positions of stars in the milky way . the three - dimensional velocities along the respective coordinates are denoted by @xmath12 , where @xmath13 is positive outwards , @xmath14 is positive in the direction of galactic rotation , and @xmath15 is positive in the direction of the north galactic pole . we assume that the local standard of rest ( lsr ) is in a circular orbit with a rotation speed of @xmath16 @xcite and that the galactocentric distance of the sun is @xmath17 . we also adopt the peculiar motion of the sun with respect to the lsr determined by @xcite .      in this paper , we assume that the thick disc of the milky way is a single stellar population and define the thick - disc stars to be those stars with @xmath18 } \leq -0.2 $ ] and @xmath19 } \geq 0.2 $ ] , by taking into account the solar neighbour observations of disc stars ( e.g. , @xcite ) . we constructed a kinematically - unbiased sample of 127 thick - disc stars , for which accurate position , velocity , [ fe / h ] , and [ mg / fe ] are available . 84 stars in our sample were taken from @xcite , and the other 43 stars were taken from @xcite . most of our sample stars are distributed within 100 pc from the sun . @xcite published a kinematically - unbiased sample of 1111 stars , for which accurate [ fe / h ] and [ mg / h ] estimated from high - resolution spectroscopic observations as well as the three dimensional cartesian velocity based on hipparcos observations are available . we derived the distances and the line - of - sight velocities for 1103 stars in their sample that were also included in the updated hipparcos catalog of @xcite . in the course of this inversion , we took into account the solar peculiar motion adopted in @xcite and we used the three - dimensional cartesian velocity taken from @xcite and the position on the sky and the proper motion taken from @xcite . we then calculated the three - dimensional velocity @xmath12 in our coordinate system , by using our assumption on @xmath20 . from these 1103 stars , we selected those stars that satisfy both the thick - disc chemical criteria ( @xmath18 } \leq -0.2 $ ] and @xmath19 } \geq 0.2 $ ] ) and the kinematical criteria of @xmath21 and @xmath22 . the latter velocity cuts were applied to minimise possible contamination from halo stars . finally , we discarded those stars that had been flagged as non - single stars , and we obtained 84 stars . @xcite published a catalog of 743 stars compiled from 11 catalogs of nearby stars and listed the chemical abundances of these stars . among these 11 catalogs , those catalogs taken from @xcite , @xcite , @xcite , @xcite , and @xcite are kinematically unbiased , so we first selected those stars that are included in either of these five catalogs . then , we excluded those stars that happened to be included in @xcite . the position and velocity of these stars were then calculated either by using the data from genova - copenhagen survey @xcite or by using the updated hipparcos catalog @xcite and simbad database . finally , as in the previous subsection , we applied the chemical and velocity criteria and obtained 43 thick - disc stars . as described above , we adopted a pair of velocity cuts to exclude halo stars . rigorously speaking , these cuts potentially could have introduced some kinematical bias on our analyses . however , we have confirmed that adopting a looser velocity constraint of @xmath23 ( without cuts on @xmath13 or @xmath15 ) has only negligible effects on our final results . the distribution of our sample stars in the [ fe / h]-[mg / fe ] plane is shown in figure [ fig : feh_mgfe ] . the uncertainties in the chemical abundances are small enough ( typical uncertainties are @xmath24 } = $ ] 0.03 dex and @xmath25 } = $ ] 0.05 dex for the @xcite sample ) compared with the scatter in this diagram . therefore , our sample does not suffer from severe contamination from low-[mg / fe ] stars . figure [ fig : velocity_distribution ] shows the velocity distribution of our sample in the @xmath26 , @xmath27 , @xmath28 and @xmath29 planes . our sample stars show a mean velocity of @xmath30 , and thus lag from the lsr by nearly @xmath31 . the three - dimensional velocity dispersion is @xmath32 .    in figure [ fig : separability ] , we show the value of @xmath3 as functions of @xmath14 and @xmath33 . to make these plots , we first sort the sample in @xmath14 or @xmath33 , evaluate the dispersion @xmath3 for a binned sample of @xmath34 stars , and move through the sample in steps of 4 stars ( so that any pair of adjacent bins share 36 stars ) . the derived value of @xmath3 for each bin is plotted against the median value of @xmath14 or @xmath33 of the bin . although the bin size of 40 stars is rather large compared with the total sample size of 127 stars , the result of this figure suggests that there is no strong trend in @xmath3 as a function of @xmath33 or @xmath14 . this property suggests that the vertical kinematics ( @xmath3 ) is not strongly correlated with the in - plane motion ( @xmath33 or @xmath14 ) . +      +    +     +   + most of the thick - disc stars in the solar neighbourhood are on non - circular orbits . more specifically , the majority of nearby thick - disc stars have guiding radii smaller than the galactocentric distance of the sun ( @xmath35 ) , due to the asymmetric drift @xcite . therefore , if the thick disc is a static stellar system , kinematics of nearby thick - disc stars carry some information on the thick disc outside the solar neighbourhood , especially on the inner part of the thick disc . of course , nearby thick - disc stars do not carry all the information of the thick disc , since these stars are inherently biased in that they represent a portion of the thick - disc stars whose orbits penetrate through the solar neighbourhood . in other words , a nearby sample only covers a certain fraction of action space @xmath9 where the corresponding orbits bring stars to the solar neighbourhood @xcite . hence , the best we can do with nearby samples is to reconstruct the distribution function @xmath36 within this _ observable action space _ @xmath9 , i.e. to reconstruct the _ observable distribution function _ defined by @xmath37 we here note that @xmath38 is different from the as - observed action distribution . the fraction of its orbital period that a given star spends in the solar neighbourhood depends on its orbit , so that the probability of finding the star in the solar neighbourhood depends on its action . in section [ section : method ] , we take into account this probability to reconstruct the observable distribution function @xmath38 .      in order to have a physical insight into @xmath9 , let us consider the velocity space that corresponds to @xmath9 by assuming a potential model of the milky way . figure [ fig : observable ] shows the allowed range of @xmath4 for disc stars located at @xmath39 kpc with zero vertical velocity to reach the solar neighbourhood . since only a portion of the disc stars at @xmath40 kpc can reach the solar neighbourhood , what we can learn from @xmath10 about the thick disc at @xmath40 kpc is the velocity distribution of stars within the ` observable ' region in figure [ fig : observable ] .    with the information on @xmath10 , we can estimate the _ observable vertical velocity dispersion _ defined by @xmath41 at any given radius @xmath42 on the disc plane . this dispersion corresponds to the vertical velocity dispersion of those stars located at @xmath42 whose velocities are within the observable region . thus @xmath43 is conceptually different from the ( ordinary ) vertical velocity dispersion given by @xmath44 which is the vertical velocity dispersion of all the stars located at @xmath42 . -space for disc stars at @xmath45 kpc with @xmath46 to reach the solar neighbourhood , calculated for the potential model in [ section : analysis ] . the boundary curve between the observable region and unobservable region corresponds to @xmath47 ( @xmath48 is the apocentric distance of the orbit ) , and thus the circular orbit at @xmath40 kpc ( red dot ) lies within the unobservable region . when @xmath49 , the boundary curve slightly shifts downwards in this diagram . however , this difference is not significant for a typical value of @xmath50 of the thick - disc stars . in this plot , we assume a realistic potential model of the milky way described in section [ section : analysis ] . ] here we describe how to reconstruct @xmath38 of the thick disc from a nearby sample of thick - disc stars , based on the idea of @xcite . we note that @xcite investigated a rigorous treatment of an as - observed distribution function in which survey selection function is taken into account ( see appendix ) . we do not follow their formulation here since we do not know the exact survey selection function for our sample stars ( but we do know that our sample stars are not kinematically selected ) . although we do not take into account the survey selection function , our approach has an advantage that we do not have to assume the functional form of the distribution function .      in the following , we regard disc stars as test particles moving in the milky way potential . also , we assume that we have a kinematically - unbiased sample of @xmath51 thick - disc stars which happen to be passing near the sun .    under an axisymmetric potential @xmath52 , each star has three actions @xmath53 @xcite . let us denote the actions of the @xmath54th star as @xmath55 . then , the time - averaged distribution function of a single orbit characterised by @xmath55 is expressed as @xmath56 and the corresponding spatial density is given by @xmath57 the value of @xmath58 can be interpreted as the probability density of finding a star with action @xmath55 at position @xmath59 , since @xmath60 @xcite .    by following @xcite , we approximate the observable distribution function @xmath10 to be the weighted sum of single orbit distribution functions given by @xmath61 here , @xmath62 is a selection index that determines whether the @xmath54th star should be included ( @xmath63 ) or not ( @xmath64 ) , which shall be introduced to stabilise our analyses . the factor @xmath65 is the orbit weighting factor to be determined . although @xcite evaluated @xmath66 through bayesian statistics , here we numerically evaluate @xmath67 and simply set @xmath68 . this simple choice of @xmath66 corresponds to a limiting case where all of our sample stars have identical position @xcite . of finding a given star in the solar neighbourhood is equal to the fraction of its orbital period to stay in the solar neighbourhood . thus , if such a star is observed in the solar neighbourhood , it represents @xmath69 stars in the same orbit ( including those with different orbital phases ) . ] since most of our sample stars are located within 100 pc away from the sun , our simple choice of @xmath66 is well justified . once @xmath10 is reconstructed , we can estimate the observable vertical velocity dispersion @xmath43 at a given point on the disc plane , by using equation ( [ eq : def_sigz_fobs ] ) . it can be shown that @xmath70 if the assumed potential is correct and a large enough number of local sample stars are available . thus , we can perform a sanity check of the adopted potential by seeing if these dispersions reasonably agree with each other . we adopt a realistic potential model of the milky way , consisting of halo , bulge , and disc components .    in our model , the halo potential is given by the navarro - frenk - white potential @xcite of the form @xmath71 where @xmath72 is the galactocentric distance , @xmath73 , and @xmath74 . the bulge potential is given by @xmath75 @xcite . we adopt @xmath76 and @xmath77 . we have confirmed that adopting a slightly flattened bulge potential hardly affects our results . for the disc potential , we assume the miyamoto - nagai potential @xcite given by @xmath78 where @xmath79 , @xmath80 , and @xmath81 . the values of parameters are chosen by using the publicly available code of ``  @xcite , while fixing @xmath17 , @xmath82 , and @xmath77 . after setting up the potential models , we derived the weighting factor for the @xmath54th star , @xmath66 , in the following manner . we first integrated the orbit of the @xmath54th star forward in time for a long enough period . we then calculated the fraction @xmath83 of the integration time which the star spent in a volume defined by @xmath84 , @xmath85 , and @xmath86 . here , we denote the observed position of the @xmath54th star on the meridional plane as @xmath87 and we adopted @xmath88 . by taking into account the relation @xmath89 , we derived @xmath90 and set @xmath91 . we set @xmath64 for those stars that satisfy @xmath92 , where @xmath93 and @xmath94 are the mean and the standard deviation of @xmath95 , respectively . this final procedure was included to prevent our results being affected by small number of stars with large @xmath66 . figure [ fig : sigz_data ] shows the reconstructed profile of @xmath43 . here we use an approximation of @xmath96 we adopt @xmath97 and calculate @xmath98 at @xmath99 in steps of @xmath100 . the radial range of @xmath42 is set so that more than 42 stars ( i.e. , more than @xmath101 stars ) have non - zero contribution to the integrals in equation ( [ eq : approximate_sigz_realistic ] ) , i , e . , have positive values of @xmath102 .    in figure [ fig : sigz_data ] , the red dots show the reconstructed values of @xmath103 , and the shaded region represents the uncertainty which is derived from a bootstrap resampling of the contributing stars . the velocity dispersion declines nearly exponentially as a function of @xmath42 , and the best fit exponential profile of @xmath43 is shown with a black straight line . we find that the best - fit scale length defined by @xmath104 is @xmath105 . the random error is the uncertainty in the fit and the systematic error is estimated by performing mock catalog analyses . we interpolate the values of @xmath43 at @xmath106 kpc and @xmath107 kpc to obtain @xmath108 . this value is in agreement with the local value of @xmath109 in our sample , indicating that the adopted potential is reasonable at least in the solar neighbourhood . we vary the parameters @xmath110 within the range of @xmath111 and @xmath112 and find that the nearly exponential profile of @xmath43 is a generic feature independent of the assumed parameters . @xmath113 mildly depends on these parameters , but it only increases from 7.5 kpc to 9.4 kpc when we vary @xmath114 from 2 kpc to 4 kpc ; and from 8.1 kpc to 9.1 kpc when we vary @xmath115 from 0.15 kpc to 0.5 kpc . +      as discussed in section [ subsection : observable_dispersion ] , @xmath116 represents the vertical velocity dispersion of stars located at @xmath117 whose velocity is within the observable region in the velocity space ( figure [ fig : observable ] ) . in order to see the variation of the vertical kinematics within the observable region , we create the following groups of stars from our sample : ( 1 ) those stars with @xmath118 ( @xmath119 stars ) ; and ( 2 ) those stars with @xmath120 ( @xmath121 stars ) here , we define @xmath122 and @xmath48 as the minimum and maximum value of @xmath42 for each star s orbit . these samples happen to be non - overlapping , and the observed vertical velocity dispersion for groups 1 and 2 are respectively @xmath123 and @xmath124 . all of these stars spend some time at @xmath125 , so in figure [ fig : group12](a ) , we show the distribution of @xmath33 and @xmath14 when they pass through this location . the stars in groups 1 and 2 are distributed in distinct areas in this velocity space , reflecting the different orbital eccentricities . in figure [ fig : group12](b ) , we show the vertical velocity dispersion @xmath116 of these groups , calculated by using @xmath126 derived in section [ section : analysis ] . we see that the radial profiles of @xmath116 for these groups are quite similar to each other at @xmath127 . this result suggests that the vertical kinematics of the stars in the observable region is more or less homogeneous at least at @xmath127 .    ) and group 2 ( @xmath120 ) in the @xmath128 plane when they pass through @xmath129 kpc . ( b ) the reconstructed profile of @xmath43 for these groups . as a reference , we show the exponential profile shown in figure [ fig : sigz_data ] ( black solid line ) . , title="fig : " ] +   + as seen in figures [ fig : velocity_distribution](a ) , [ fig : velocity_distribution](b ) , and [ fig : separability ] , the vertical velocity dispersion of the thick disc does not have a strong dependency on the in - plane velocities ( @xmath13 and @xmath14 ) in the solar neighbourhood . in order to have some physical insight into this finding , let us suppose , as a working hypothesis , that this apparent homogeneity of vertical velocity distribution across the @xmath4-space is the case not only near the sun but also outside the solar vicinity . then it is implied that the distribution of the vertical action @xmath130 is not strongly dependent on the radial action ( @xmath131 ) or azimuthal action ( @xmath132 ) . it follows that there is a typical value of vertical action @xmath133 that characterises the vertical motions of the thick - disc stars . on the other hand , the maximum height @xmath134 above the disc plane that a given thick - disc star can reach at galactocentric radius @xmath42 is determined mainly by its vertical action @xmath130 . in realistic potential models of the milky way , @xmath135 is an increasing function of @xmath42 , since the gravitational pull towards the disc plane is weaker at larger @xmath42 . thus , if the distribution of @xmath130 of the thick - disc stars is nearly independent of @xmath136 , then the thick disc needs to flare and the scale height of the thick disc roughly follows the radial behaviour of @xmath137 .    then , under what circumstances does the distribution of @xmath130 become ( approximately ) independent of @xmath136 ? here we consider such an example . let us suppose that the thick disc was formed through radial migration @xcite . some numerical simulations suggest that the radial migration conserves the stellar vertical action @xmath130 as the star changes its in - plane motion @xcite . thus , radial migration redistributes the stars in @xmath136-space while keeping @xmath130 unchanged . this redistribution in the action space is expected to erase the initial dependency of @xmath130-distribution on @xmath136 @xcite , unless the initial dependency is too strong to erase . if we take into account that the thick - disc stars are generally very old ( @xmath138 gyr ; @xcite ) , it may be likely that the radial migration ends up with a homogeneous distribution of @xmath130 as a function of @xmath131 or @xmath132 . it is of interest to note that some numerical simulations predict that the thick disc flares if it was formed though radial migration @xcite . if @xmath3 is really independent of @xmath4 anywhere on the disc plane , we have @xmath139 . in this case we can estimate the thickness @xmath140 of the thick disc at galactocentric radius @xmath42 by using our reconstructed profile of @xmath43 . as an example , we define the thickness @xmath141 by @xmath142 where @xmath52 is the galactic potential . by plugging @xmath43 into @xmath3 in equation ( [ eq : def_h ] ) , we obtain @xmath140 shown in figure [ fig : h_data ] . although the @xmath140 profile mildly depends on the adopted potential , we have confirmed that @xmath141 increases as a function of @xmath42 independent of the potential parameters as long as they are within the range explored in section [ subsection : result_potential_dependece ] . +      as we have discussed in the previous subsection , the radial dependence of the thickness of the thick disc is related to the dependency of the vertical kinematics on the in - plane motions .    by taking into account some observational evidence of the thick - disc component in external galaxies with a constant scale height @xcite , let us consider what we would need to keep the scale height @xmath141 constant . here we concentrate our discussion on the situation at a certain radius @xmath143 in the inner disc ( @xmath144 ) . in order to satisfy @xmath145 , we require @xmath146 so that the vertical velocity distribution is inhomogeneous in the @xmath4 space . since @xmath43 represents the vertical velocity distribution of observable stars ( those stars whose velocity is inside the observable region ) and @xmath3 represents the vertical velocity distribution of all the stars , this requirement implies the following situation : in the inner disc , those thick - disc stars with large @xmath132 ( large guiding centre radii ) need to have smaller vertical action than those thick - disc stars with small @xmath132 and @xmath131 . with small @xmath132 to be unobservable , it has to have small @xmath33 ( see figure [ fig : observable ] ) and thus small @xmath131 . ] this situation may be attained if there are enough low - eccentricity , shell - like orbits with large vertical motions in the inner disc . we have investigated an orbit - based method to reconstruct the observable distribution function @xmath10 , which describes the distribution of action within the portion of action space covered by a local disc sample . by applying our method to 127 chemically - selected thick - disc stars in the solar neighbourhood , we found that the vertical velocity dispersion @xmath43 that corresponds to @xmath10 decreases nearly exponentially as a function of @xmath42 with a scale length of @xmath147 kpc . we also found that the vertical velocity dispersion @xmath3 of the local thick - disc stars shows only weak dependency on the in - plane motions @xmath4 . this apparent homogeneity of @xmath3 in @xmath4-space may be naturally explained if the distribution of the vertical action @xmath130 is only weakly dependent on the radial action @xmath131 and the azimuthal angular momentum @xmath132 . one possible mechanism to produce such action distribution is radial migration , where the thick - 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larsen , j. , & zhen , c.  1990 , mnras , 242 , 10 soubiran , c. , & girard , p.  2005 , a&a , 438 , 139    van der kruit , p.  c. , & searle , l.  1981 , a&a , 95 , 105 van leeuwen , f.  2007 , a&a , 474 , 653    yoshii , y.  1982 , pasj , 34 , 365 yoshii , y.  2013 , planets , stars and stellar systems . volume 5 : galactic structure and stellar populations , 393 in our analyses , we followed the approach of @xcite to relate the observed kinematic data to the underlying distribution function . we note here that our approach owes much to the fact that our sample stars are located at the immediate solar neighbourhood ( @xmath148 ) . let us consider a sample of stars taken from an imaginary magnitude - limited survey in which all the stars with apparent magnitude brighter than a certain value are observed . in this case , those sample stars at large distances are high - luminosity stars , while the sample covers a wider luminosity range for nearby stars . thus , such a sample puts more weight on nearby stars . if the kinematical properties are highly inhomogeneous within the surveyed region , this sample is kinematically biased . in general , if the surveyed region is large , the survey selection function has to be taken into account in order to estimate the kinematic properties of the system ( mcmillan & binney 2012 ) . this means that our methodology is not applicable to a sample that covers a large volume unless the completeness is taken into account , since our method implicitly requires that the sample stars are fair representatives of the stellar population . however , since our sample is distributed in a small region ( @xmath149 pc ) , it is safe to assume that the kinematical properties of the thick - disc stars are homogeneous inside this region ( the thick disc has a scale height of @xmath150 kpc and a scale length of @xmath151-@xmath152 kpc ) . therefore , the effect from the ( poorly - known ) survey selection function is relatively small in our sample .
we explored a method to reconstruct the distribution function of the galactic thick disc within the action space where nearby thick - disc stars are distributed . by applying this method to 127 chemically - selected thick - disc stars in the solar neighbourhood 
 , we found that the vertical velocity dispersion that corresponds to the reconstructed distribution function declines approximately as @xmath0 at @xmath1 , with @xmath2 kpc . 
 also , we found that the vertical velocity dispersion @xmath3 of our local thick - disc stars shows only weak dependency on radial and azimuthal velocities @xmath4 . 
 we discuss possible implications of these results on the global structure of the milky way thick disc . 
 [ firstpage ]     galaxy : disc  galaxy : evolution  galaxy : formation  galaxy : kinematics and dynamics  galaxy : structure  solar neighbourhood