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it is now generally accepted that the galactic bulge is a tri - axial , bar - like structure . observational evidence for a bar has arisen from several sources , such as the study of gas kinematics ( e.g. @xcite ) , surface brightness ( e.g. @xcite ) , star counts ( e.g. @xcite ) and microlensing ( e.g. @xcite ) ; see @xcite for a review . observational data have been used to constrain dynamical models of the galaxy . @xcite used the cobe - dirbe multi - wavelength observations of the galactic centre @xcite to constrain several analytic bar models . @xcite used optical observations of red clump giant ( rcg ) stars to constrain theoretical bar models . similarly , @xcite and @xcite traced the bulge rcg population in the infrared . this work uses a sample of stars 30 times larger than that of @xcite , with a greater number of fields distributed across a larger area of the galactic bulge , thus allowing finer constraints to be placed on the bar parameters than those determined by @xcite . our current understanding of the galactic bar is that it is orientated at about @xmath6 to the sun  galactic centre line - of - sight , with the near end in the first galactic longitude quadrant . the bar length is around 3.1  3.5 with axis ratio approximately @xmath7 @xcite . the above bar parameters are generally accepted as a working model , however they are not well determined . our understanding of the complete structure of the inner galactic regions is similarly incomplete . for example , recent infra - red star counts collected by the spitzer space telescope for galactic longitudes @xmath8 = @xmath9  @xmath10 are best explained assuming a long thin bar oriented at an angle of @xmath11 to the sun  galactic centre line @xcite while most previous studies ( performed at @xmath12 ) prefer a short bar with an opening angle of @xmath13 . recently , @xcite report that nir observations of rcgs support the hypothesis that a long thin bar oriented at @xmath14 co - exists with a distinct short tri - axial bulge structure oriented at @xmath15 . in addition , there may be some fine features , such as a ring in the galactic bulge @xcite , or a secondary bar @xcite , that are not yet firmly established . it is therefore crucial to obtain as many constraints as possible in order to better understand the structure of the inner galaxy .    in this paper we present an analysis of rcg stars observed in the galactic bulge fields during the second phase of the ogle microlensing project @xcite . these stars are bright and they are approximately standard candles , hence their magnitudes can be taken as an approximate measure of their distances . number counts in 34 central bulge fields with @xmath16 and @xmath17 are used to constrain analytic tri - axial bar models , and thereby obtain estimates on bar parameters . we repeat the analysis with 44 fields with @xmath18 . we find the fitted bar parameters support the general orientation and shape of the bar reported by other groups . this paper is organised as follows : in section  2 we describe the ogle microlensing experiment and photometry catalogue and we illustrate how rcg stars can be used as approximate distance indicators ; in section  3 we detail how rcgs in the ogle - ii proper motion catalogue are selected ; in section  4 we compute the distance modulus to the red clump in 45 ogle - ii fields and thereby trace the central mass density of the galaxy ; in section  5 we describe how rcg star count histograms for each field can be used to constrain analytic bar models of the inner galaxy ; our results and their comparison to previous works is given in section  6 and in section  7 we discuss the implications and limitations of these results . the ogle @xcite and moa @xcite microlensing collaborations currently make routine observations of crowded stellar fields towards the galactic bulge , and issue alerts when a microlensing event is detected . a result of this intense monitoring is the creation of massive photometry databases for stars in the galactic bulge fields . such databases are extremely useful for kinematic and population studies of the central regions of the galaxy . @xcite obtained the proper motions for millions of stars in the ogle - ii database for a large area of the sky . [ fig : fields ] shows the ogle - ii fields towards the galactic bulge . in this paper we focus on the population of red clump giant stars at the galactic centre . red clump giants are metal - rich horizontal branch stars ( * ? ? ? * and references therein ) . theoretically , one expects their magnitudes to have ( small ) variations with metallicity , age and initial stellar mass @xcite . empirically they appear to be reasonable standard candles in the @xmath19-band with little dependence on metallicities @xcite . @xcite used rcg stars in 12 fields ( see fig .  [ fig : fields ] ) observed during the first phase of the ogle microlensing experiment , ogle - i , to constrain several analytic models of the galactic bar density distribution . @xcite , @xcite and @xcite similarly used ir observations of rcgs to trace the bulge stellar density . we follow similar procedures to extract rcg stars from the ogle - ii galactic bulge fields and to constrain analytic models . we compute the reddening - independent magnitude for all stars in each of the 45 ogle - ii fields : @xmath20 where @xmath21 and @xmath22 are the extinctions in the @xmath19 and @xmath23 bands determined by @xcite . we select stars which have @xmath24 , where @xmath25 is a constant chosen for each field that excludes the main - sequence dwarf stars , and @xmath26 , which corresponds to the magnitude of rcg stars closer than 15 kpc and @xmath27 respectively , with @xmath28 . ] . [ fig : cmd ] shows the sample of stars selected from the @xmath29 cmd for ogle - ii field 1 . reddening - independent magnitude vs colour diagram for ogle - ii field 1 . the red clump is clearly visible . stars are selected ( black dots ) using the criteria @xmath24 , where @xmath25 is a constant chosen for each field , and @xmath26 ( solid lines , see text ) . ] the selected stars are then collected in @xmath30 magnitude bins , see fig . [ fig : hist ] . a function comprised of quadratic and gaussian components is used to model this number count histogram in each of the ogle - ii fields :    @xmath31 \label{eq : fitfun}\ ] ]    where is the spread of the red clump giant magnitudes , is the mean apparent magnitude of the red clump , and @xmath32 , @xmath33 and @xmath34 are coefficients for the gaussian and quadratic components respectively . these six parameters are allowed to vary for each of the ogle - ii fields and the best - fitting values obtained by minimising @xmath35^{2}$ ] where the sum is taken over all @xmath36 histogram bins which cover the range @xmath37 . the error on histogram number counts is assumed to be @xmath38 , i.e. poissonian . number count histogram for selected stars in ogle - ii field 1 . the heavy solid line is the best - fitting model of the form given by equation  ( [ eq : fitfun ] ) . the last histogram bin is generally not included in the fitting due to incompleteness effects near the limiting magnitude of @xmath26 . ]    the errors on the mean magnitude and distribution width of the red clump stars , and respectively , are determined using a maximum - likelihood analysis ( see e.g. @xcite ) . we determine the distance modulus to the red clump in each field as :    @xmath39    where @xmath40 is the distance to the red clump measured in parsecs , is the fitted peak reddening - independent magnitude of the red clump , @xmath41 is the mean value of @xmath42 for each field . @xmath43 @xcite and @xmath44 @xcite are the mean absolute magnitude and colour of the population of local red clump giant stars . we assume that the properties of the local population of red clump giant stars are the same as the galactic bulge population , however it is likely that population effects are significant . we discuss the effects of red clump population effects in section  [ sec : popeffects ] . table  [ tab : distmod ] lists the fitted parameters @xmath45 and @xmath46 , along with the mean value of @xmath41 for each field . the distance modulus , @xmath47 , computed from equation  ( [ eq : mu ] ) showed that there was a significant offset : the fitted mean red clump giant magnitudes were uniformly too faint compared to that expected of typical local rcg stars at 8 , resulting in overly large distance moduli . @xcite found that the ogle - ii rcgs are approximately 0.3 mag fainter than expected when assuming that the population of rcgs in the bulge is the same as local . for this reason , we apply an offset of 0.3 mag to the distance moduli computed above . the shifted distance moduli @xmath48 are given in table  [ tab : distmod ] . the true line - of - sight dispersion , , can be approximated by @xmath49 where is the gaussian dispersion fitted using equation  ( [ eq : fitfun ] ) , @xmath50 is the intrinsic dispersion of the rcg luminosity function and @xmath51 is an estimate of the photometric errors @xcite . we use @xmath52 ( see section  [ sec : data ] ) and @xmath53 , along with the tabulated values of to determine . [ fig : distmod ] shows the mean distance to the red clump stars in each of the 45 ogle - ii fields listed in table  [ tab : distmod ] . the fields with galactic longitude @xmath16 show clear evidence of a bar , with a major axis oriented at @xmath54 to the sun - galactic centre line - of - sight . for fields with @xmath55 and @xmath56 the mean position of the red clump stars do not continue to trace the major axis of a linear structure . @xcite find similar evidence that the position of the red clump stars is not predicted by a bar for @xmath57 , suggesting that this is a detection of the end of the bar , or the beginning of a ring - like structure . we investigate these possibilities in section  [ sec : ring ] . the uncertainties on the mean position of the red clump are large ; the largest term in the error expression for arises from the relatively large uncertainty in the intrinsic colour of the rcgs . the true line - of - sight dispersions , , are consistent with a wide range of bar position angles , but the mean position of the red clump in each direction strongly suggest a bar oriented along a direction consistent with that determined by the previous work referred to in section  [ sec : intro ] . ccccccccc field & @xmath8 & @xmath33 & & & @xmath41 & @xmath58 & & @xmath59 +    structure of the inner region of the milky way as traced by red clump giant stars extracted from the ogle - ii microlensing survey data . the mean position of the red clump for 45 ogle - ii fields is indicated by black dots with errors shown by black lines . the grey lines along each line of sight indicate the 1-@xmath60 spread in distances obtained from fitting equation  ( [ eq : fitfun ] ) to the red clump data in each field and correcting for intrinsic red clump luminosity dispersion and photometric errors ; see text . a position angle of 25is shown by the thin solid line . lines - of - sight at @xmath61 and @xmath62 are indicated along the top axis . ] @xcite found that the extinction - corrected magnitudes of rcg stars in the ogle - ii bulge fields were 0.3 mag higher than that of a rcg with an intrinsic magnitude equal to those of local rcgs , placed at a distance of 8 . @xcite notes that the cause of this offset is uncertain but may be resolved when detailed v - band ogle - ii photometry of rr lyrae stars will allow improved extinction zero - point estimations for all fields . there is evidence that the properties of bulge rcgs are different to local rcg stars , in particular age and metallicity , resulting in a different average absolute magnitude ( see e.g. @xcite ) . this could in part explain the observed offset of 0.3 mag between local and bulge rcg stars . the distance modulus plotted in fig . [ fig : distmod ] is as for equation  ( [ eq : mu ] ) : @xmath63 where @xmath64 mag is the offset between the observed bulge rcg population and a fiducial local rcg placed at 8 . including the difference in intrinsic magnitude between local and bulge rcg populations we have : @xmath65 where is the intrinsic rcg magnitude difference between local and bulge populations and @xmath66 is a magnitude offset arising from effects other than population differences . the theoretical population models of @xcite estimate @xmath67 . the remaining magnitude offset @xmath68 can be accounted for by decreasing the assumed value of the galactocentric distance from 8 to 7.3 . this assumption of is in agreement with the value of @xmath69 ( stat ) @xmath70 ( sys ) determined by @xcite , and that of @xmath71 determined by @xcite . we note that as long as the stellar population is uniform in the galactic bulge ( an assumption largely consistent with the data ) , then this unknown offset only affects the zero point ( and hence the distance to the galactic centre ) , and the absolute bar scale lengths . however the ratios between bar scale lengths are very robust , which we demonstrate in section  [ sec : changer0 ] where we determine the bar parameters for several values of . a metallicity gradient across the galactic bulge would have an effect on the intrinsic colour and absolute magnitude of bulge rcg stars as a function of galactic longitude . @xcite find that there is a connection between metallicity gradient and structural features in a sample of 6 barred galaxies . @xcite measured metallicities for k giants in two fields at 1.5 and 1.7 from the galactic centre . the average metallicity was [ fe / h ] = @xmath72 , lower than that of k giants in baade s window . however , no such metallicity gradient was reported for galactocentric distance range 500 pc  3.5 by @xcite . similarly , determine that there is no metallicity gradient within angles @xmath73 to @xmath74 of the galactic centre ( corresponding to 300  800 pc from the gc assuming a sun - gc distance of 8 ) . we continue to investigate the structure of the galactic bar by fitting analytic models of the stellar density to the observed rcg data in the ogle - ii fields . following @xcite we use the analytic models of @xcite to fit the observed data . three model families ( gaussian , exponential and power - law ) are used :    @xmath75 \rho_{\rm e1 } & =   \rho_{0 } \exp(-r_{\rm e } ) \\ \rho_{\rm e2 } & =   \rho_{0 } \exp(-r )   \label{eq : e2 } \\ \rho_{\rm e3 } & =   \rho_{0 } k_{0}(r_{\rm s } ) \\[20pt ] \rho_{\rm p1 } & =   \rho_{0 } \left(\frac{1}{1+r}\right)^{4}\\ \rho_{\rm p2 } & =   \rho_{0 } \left(\frac{1}{r(1+r)^{3}}\right ) \\ \rho_{\rm p3 } & =   \rho_{0 } \left(\frac{1}{1+r^{2}}\right)^{2 } \label{eq : p3 } \\[10pt ]   \notag\end{aligned}\ ] ] where @xmath76 is the modified bessel function of the second kind and    @xmath77^{1/2 } \\ r_{\rm e } & \equiv   \left[\frac{|x|}{x_{0 } } +                           \frac{|y|}{y_{0 } } +                           \frac{|z|}{z_{0 } } \right ] \\ r_{\rm s } & \equiv \left[\left[\left(\frac{x}{x_{0}}\right)^{2 } +                                 \left(\frac{y}{y_{0}}\right)^{2}\right]^{2 } +                          \left(\frac{z}{z_{0}}\right)^{4 } \right]^{1/4}\end{aligned}\ ] ]    the co - ordinate system has the origin at the galactic centre , with the @xmath78 plane defining the mid - plane of the galaxy and the @xmath79 direction parallel to the direction of the galactic poles . the @xmath80 direction defines the semi - major axis of the bar . the functions are rotated by an angle @xmath81 around the @xmath79-axis . an angle of @xmath82 corresponds to the major axis of the bar pointing towards the sun . the functions can also be rotated by an angle @xmath83 around the @xmath84 axis , corresponding to the sun s position away from the mid - plane of the galaxy . we aim to fit the observed number count histograms for each field as a function of magnitude . given a magnitude , the number of stars with this magnitude is @xcite : @xmath85 where the integration is taken over distance @xmath86 . we perform the integration over this range of @xmath87 as we do not expect the tri - axial bulge density structure to exceed these limits . the constant @xmath88 is dependent on the solid angle subtended around each line of sight . the luminosity @xmath89 is given by @xmath90 and @xmath91 is a constant . the luminosity function @xmath92 is @xmath93\ ] ] where @xmath94 is the luminosity of the red clump and @xmath95 is the intrinsic spread in red clump giant luminosity and is held constant . there are ten parameters to be determined in the above equations : the three bar scale lengths @xmath96 , @xmath97 , @xmath3 ; the bar orientation and tilt angles @xmath81 and @xmath83 ; the luminosity function parameters @xmath98 , @xmath99 , @xmath100 and @xmath94 and the density function parameter @xmath101 . there is evidence that the centroid of the bar is offset from the centre of mass of the galaxy , a feature commonly observed in external galaxies ( @xcite and references therein ) . we include an additional parameter in the modelling process , @xmath102 , which allows for this possible centroid offset . theoretical number counts are computed for the @xmath103 fields at longitudes @xmath104 as @xmath105 where the offset parameter @xmath102 is determined over all fields for a given density model . we apply an exponential cut - off to the density functions similar to that of @xcite :    @xmath106    where @xmath107 is in kpc , @xmath108 and @xmath109 is a cut - off radius . the theoretical constraint on the maximum radius of stable stellar orbits is the co - rotation radius , @xmath109 . we adopt the co - rotation value of @xmath110 determined by @xcite in equation  ( [ eq : cutoff ] ) . we later repeat the modelling process without this theoretical cut - off , see section  [ sec : changer0 ] . @xcite included a further fitting parameter to account for a possible metallicity gradient across the bulge , but found that this did not significantly affect the bar parameters . the discussion in section  [ sec : popeffects ] suggests that there is no appreciable metallicity gradient over the bulge region investigated here . we therefore do not include an additional model parameter corresponding to a metallicity gradient in the model fitting analysis . the model fitting was performed using a standard non - linear neadler - mead minimisation algorithm . for each of the nine density profile models we minimise @xmath111 : @xmath112 where the summations are taken over each of the 26 histogram bins in @xmath113 ogle - ii fields , @xmath114 is the observed histogram data for field @xmath25 and @xmath115 is the model number count histogram from equation  ( [ eq : numcount ] ) . the error in @xmath116 was taken to be @xmath117 . the 11 parameters @xmath96 , @xmath97 , @xmath3 , @xmath81 , @xmath83 , @xmath118 , @xmath98 , @xmath99 , @xmath100 , @xmath94 and @xmath102 were fitted for each of the nine density profiles given in equations . ( [ eq : g1 ]  [ eq : p3 ] ) for the 34 ogle - ii fields which have @xmath16 . a naive interpretation of fig . [ fig : distmod ] is that fitting all fields including those with @xmath55 and @xmath56 with a single bar - like structure would not be successful and indeed initial modelling using data from all fields resulted in bar angles of @xmath119 . as seen in fig . [ fig : distmod ] this result is consistent with the mean position angle of all fields , but clearly does not describe the bar correctly . we therefore begin modelling the bar using data from the central 34 fields which have @xmath16 . in section  [ sec : ring ] we proceed by including the data from fields with @xmath55 and @xmath56 in the modelling process and in section  [ sec : changer0 ] we consider the effect of changing . the range of field latitudes is small and therefore the information available in the latitude direction from the total dataset unlikely to be able to constrain @xmath83 strongly . initial modelling runs held @xmath83 constant at @xmath120 . once a minimum was determined for each model holding @xmath121 , this parameter was allowed to vary with the ten other parameters . figure [ fig : matrixe2 ] shows the number count histograms of observed red clump giants in the 34 ogle - ii fields with @xmath16 , along with the best - fitting model ( equation  ( [ eq : numcount ] ) ) using the e2 density profile given in equation  ( [ eq : e2 ] ) as an example  [ eq : p3 ] ) appear in the on - line supplementary material . each set of axes is arranged roughly in order of decreasing galactic longitude . the magnitude of the red clump peak increases with decreasing longitude , consistent with a bar potential . the observed number count histograms are reasonably well - fitted by all models in most fields . the most obvious exceptions are fields 6 , 7 , 14 , 15 and 43 . these fields are at the most extreme latitudes represented in the data . in the case of fields 14 and 15 the algorithm fails to fit satisfactorily both observed histograms for a given model . upon closer inspection of the observed histogram data for these two fields , it is noted that the total number of stars in the field 15 histogram is significantly less than that for field 14 . the models preferred by all fields clearly can not reproduce the decline in total star count between fields 14 and 15 . upon inspection of the ccd pixel position of stars for field 15 it was found that there are no stars recorded in a region along one edge of the field . the lack of stars in this strip is due to missing @xmath23-band data for these stars . similarly , there is a lack of stars in field 15 in the lower right corner , due to heavy dust extinction in this area . all models predict the peak of the red clump to be at a lower magnitude than that observed for field 43 . a similar offset is seen in the other high latitude fields 14 and 15 . these fields are all at latitude @xmath122 and the observed offset in all three fields is likely to be related to this common position in latitude . the cause of this effect is currently unknown . the observed offset between the maximum density position predicted by the tri - axial bar models and that observed for some fields may imply some asymmetry of the bar in the latitude direction , as suggested by @xcite and references therein . fields 6 and 7 , both at @xmath123 , are also poorly fitted by all models . the histograms of observed red clump stars in these fields show a curious double peak . clearly the density functions equations  ( [ eq : g1 ]  [ eq : p3 ] ) are inadequate for describing such a feature . this double - peaked structure may be the result of another population of stars lying along the line of sight , at a distance different to the main bulge population . the best - fitting model curves typically have a peak at magnitudes corresponding to the highest peak in the observed histogram , thereby suggesting that the second population , if real , and composed of a significant fraction of rcgs , exists between us and the bulge population . a population of asymptotic giant branch ( agb ) stars @xcite may also be the cause of the secondary peak in the number count histograms for these and similar fields ( e.g. field 10 , fig .  [ fig : histmatrixwide_g ] ) . closer investigation of the stellar characteristics and kinematics of stars in these fields may help in describing the origin of the second number count peak . the best - fitting parameter values for all the nine density models are presented in table  [ tab : paramvals ] . two sets of parameter values are given , one where the tilt angle , @xmath83 , was held at @xmath124 , and the other where @xmath83 was allowed to vary . the values of are well in excess of the number of degrees of freedom : @xmath125 , and can therefore not be used to set reliable errors on the parameters values in the usual way . it is likely that the errors on the histogram data are underestimated , resulting in extreme values of . we assumed that the errors on the number count data would be poissonian and we have not added any error in quadrature that might arise from any systematic effects . furthermore , as mentioned above , the extreme latitude fields 6 , 7 , 14 , 15 and 43 are poorly fit by every model . the cause for this is unknown , however the effect is to add a relatively high contribution to the total value of compared to other fields . the values of can only be used to differentiate between the various models in a qualitative manner . in terms of relative performance , model e3 best reproduces the observed number count histograms for the fields tested here , and model g2 provides the worst fit to the data . cccccccc & & & & axis ratios + model & @xmath83 & @xmath126 & @xmath96 & @xmath97 & @xmath3 & & @xmath127 +    the position angle of the bar semi - major axis with respect to the sun - galactic centre line - of - sight is @xmath126 and is related to the rotation angle @xmath81 by @xmath128 . we find @xmath129  @xmath130 . the addition of @xmath83 as another variable parameter does not result in a significant improvement in . the lack of information in the latitude direction means that the data have little power in constraining parameters such as @xmath83 . the absolute values of the scale lengths in table  [ tab : paramvals ] can not be directly compared between models . the axis ratios @xmath131 and @xmath132 can be compared however . [ fig : axisratios ] shows the axis ratios @xmath131 and @xmath132 for the nine models tested . the ratio of the major bar axis scale length to the minor bar axis in the plane of the galaxy , @xmath131 , has values @xmath133 @xmath134 , with the exception of that for model e1 , for which @xmath135 . @xcite found @xmath136  @xmath137 , with the exception of model e1 for which @xmath138 . it is interesting to note that our values of @xmath131 have a similar range to that of @xcite , when we exclude the outlying result from the same model ( e1 ) . our results do suggest a slimmer bar , i.e. higher values of @xmath131 compared to @xcite . both these results and the results of @xcite are consistent with the value of @xmath139 reported by @xcite . the ratio of the major bar axis scale length to the vertical axis scale length @xmath132 was found to be @xmath140 @xmath141 , with the same exception of model e1 , which has an outlying value of @xmath142 . again , the range of ratio values is comparable to that of @xcite who found @xmath143 @xmath144 , with the exception of model e1 again , which had @xmath145 . scale length ratios @xmath131 ( squares ) and @xmath132 ( triangles ) for all models . solid and open symbols show the ratios for the best - fitting models using the 34 fields with @xmath16 , and all 44 fields ( see section  [ sec : ring ] ) respectively . ] the mean axis ratios are @xmath127 are @xmath146 : @xmath147 : @xmath148 . excluding the outlying results from the e1 model , the ratios are @xmath146 : @xmath149 : @xmath150 . these results suggest a bar more prolate than the general working model with @xmath7 @xcite . [ fig : distmod ] shows the mean position of the bulge red clump stars . at longitudes @xmath16 the bulge red clump stars follow the main axis of the bar . at greater angular distances , the mean positions of the red clump stars are clearly separated from the main bar axis . @xcite find similar evidence and suggest that this could indicate either the end of the bar or the existence of a ring - like structure . we used the ogle - ii data to investigate these possibilities . the ogle - ii data from fields with longitude @xmath151 and @xmath152 were excluded from the original bar modelling based on the findings illustrated by fig . [ fig : distmod ] , and because initial modelling efforts using all data from all fields simultaneously failed to produce satisfactory results . using the best - fitting model ( e3 ) determined using the central ogle - ii fields , fig . [ fig : histmatrixwide ] shows the predicted number count densities for fields with @xmath151 and @xmath152 , overlaid on the observed number count histograms . it is clear from the left hand column of fig . [ fig : histmatrixwide ] that the mean position of the red clump in fields with @xmath152 are significantly removed from that predicted by the e3 linear bar model . however , the right hand column of fig . [ fig : histmatrixwide ] shows that the observed position of the red clump is in rough agreement with that predicted from the model for fields with @xmath151 . we test this further , by taking lines of sight through the best - fitting e3 model and computing the number count profile using equation  ( [ eq : numcount ] ) . the mean magnitude of the red clump is converted to a distance for each line of sight . [ fig : midplanescan ] shows the density contours of the analytic e3 model overlaid with the mean position of the red clump determined in this way . mean position of the red clump determined using the best - fitting analytic bar model ( e3 ) . gray lines indicate 11 lines - of - sight through the e3 model , with @xmath153 and @xmath154 and contours indicating the density profile of the model . the dashed line shows the orientation of the bar major axis , with the sun positioned at @xmath155 . number count profiles are generated using equation  ( [ eq : numcount ] ) . the mean red clump magnitude for each line - of - sight was converted to a distance , and plotted as solid circles . black circles indicate the mean position of the red clump when the exponential cut - off , equation  ( [ eq : cutoff ] ) is applied . grey circles indicate the mean rcg position when this cut - off is not imposed . the observed features in fig .  [ fig : distmod ] at longitudes @xmath156 are more clearly reproduced when the cut - off is applied to the analytic model . ] the features of fig . [ fig : distmod ] at longitudes @xmath156 are qualitatively reproduced in fig . [ fig : midplanescan ] . this suggests that the observed data are likely to be consistent with a single bar - like structure , rather than requiring additional structure elements . we note however , that the abrupt departure of the location of the density peak in fig . [ fig : distmod ] at longitudes @xmath157 is best reproduced in fig . [ fig : midplanescan ] by the analytical density model when the exponential cut - off of equation  ( [ eq : cutoff ] ) is imposed . fields with @xmath151 and @xmath152 should be included in the modelling of the bar , as they are likely to provide further constraints on the final model . initial attempts to model the bar using all fields failed because the fitting algorithm found a model with a bar angle of @xmath158 , consistent with the observed data , but not consistent with the current understanding of the bar . [ fig : midplanescan ] shows that the morphology indicated by the observed data in fig . [ fig : distmod ] can be explained using a bar structure oriented at @xmath159 to the sun - gc line - of - sight . @xcite also note that the line - of - sight density for tri - axial bulges reaches a maximum where the line - of - sight is tangential to the ellipsoidal density contours . these authors also quantify the difference between the positions of maximum density and the intersection of the line - of - sight with the major axis of the bar . the fitting procedure for all nine tri - axial models was repeated , including now the 10 ogle - ii fields with @xmath151 or @xmath152 . figure  [ fig : histmatrixwide_g ] shows the best - fitting number count profiles for the ten non - central fields for the ` g ' type of analytic tri - axial model . the best - fitting number count profiles for fields with @xmath16 are not significantly different to those shown in figure  [ fig : matrixe2 ] . the main features of the observed number count profiles are reproduced by the best - fitting analytic models . there are however instances where details of the observed number counts are not reproduced . magnitude bins with @xmath160 are poorly fitted by the g - type models in positive longitude fields . two of the e - type models ( e2 , e3 ) similarly fail to trace these data . the number count profile for model e1 shows a flattened peak , resulting from the pronounced box - like nature of the density profile . all the p - type models appear to reproduce the data at these magnitudes with similar profiles . all models predict a rcg peak at magnitudes greater than that observed in fields 28 and 29 . this offset between predicted and observed rcg peak location is also seen in fields 26 and 27 but to a lesser degree . the best - fitting parameters are shown in table  [ tab : paramvalswide ] . the model which results in the lowest value of is still e3 . model g2 also remains the worst - fitting model to the data . cccccccc & & & & axis ratios + model & @xmath83 & @xmath126 & @xmath96 & @xmath97 & @xmath3 & & @xmath127 +    we now consider the change to the best - fitting model parameters when data from all fields are used in the analysis . the position angle of the bar semi - major axis with respect to the sun - galactic centre line - of - sight is now @xmath161  @xmath1 . the bar scale lengths @xmath96 , @xmath97 and @xmath3 also changed ; fig . [ fig : paramchange ] shows the ratio of the best - fitting scale lengths determined using all 44 fields to those found using only the central 34 fields . on average , upon including the data from wide longitude fields , the semi - major axis scale length , @xmath96 , decreased by @xmath162 ; the semi - minor axis scale length , @xmath97 , increased by @xmath163 and the vertical scale length , @xmath3 , remained essentially unchanged . the relatively large change in the semi - minor axis scale length is intuitively understandable , as a bar position angle of @xmath164 with respect to the sun - gc line - of - sight mean the semi - minor axis direction has a large vector component in the galactic longitude direction . the additional constraints of the data from fields at extended galactic longitudes can therefore have a pronounced effect on the fitted values of @xmath97 . similarly , we expect some weak correlation between the semi - major and semi - minor axis scale lengths , evidenced by the slight decrease on average for the @xmath96 values upon adding the data from wide longitude fields . it is unsurprising that the values of @xmath3 are unchanged , as adding data from the wide longitude fields has little power in further constraining model elements in the direction perpendicular to the galactic midplane . ratio of bar scale lengths from best - fitting models using data from all 44 ogle - ii fields to those using data from the central 34 ogle - ii fields . ]    as above , we consider the ratio of the axis scale lengths for each model ( see fig .  [ fig : axisratios ] ) . the ratio @xmath131 for all nine models using all field data has values 2.7  3.0 . this range is now completely consistent with the range 2.5  3.3 reported by @xcite . the ratio @xmath132 now lies in the range 3.6  3.9 , with the exception of that for model g1 which has @xmath165 . the range @xmath166 3.6  3.9 is narrower than that found previously , 3.4  4.2 , yet still higher on average than 2.8  3.8 reported by @xcite . the scale length ratios are now @xmath4 , which compared to those determined using only the central ogle - ii fields ( @xmath167 ) are closer to the working model proposed by @xcite which has @xmath5 . there are features in the number count histograms that are not faithfully reproduced by the analytic tri - axial bar models for wide longitude fields . specifically , the predicted number count dispersions around the rcg peak magnitudes for fields with @xmath168 are too small compared to that observed , see fig . [ fig : histmatrixwide_g ] . while the location of the observed maximum line - of - sight density can be reproduced using just a bar , it is possible that the reason why the analytic bar models underestimate the observed number count dispersions for fields with @xmath168 is because these models do not include elements which correspond to extended aggregations of stars at or near the ends of the bar . clumps of stars at the ends of the bar would increase the line - of - sight density dispersion at wide longitudes . such extra aggregations of stars have been observed in galaxies with boxy or peanut - shaped bulges @xcite . their origin may be due to the superposition of members of the @xmath169 family of orbits @xcite many of which have loops near the end of the bar . alternatively , the aggregations of stars near the ends of the bar may be the edge - on projection of an inner ring ( @xcite and references therein ) . the presence of another density structure such as a long thin bar oriented at @xmath170 @xcite might also contribute to the relatively large line - of - sight density dispersion . the effects of such a structure would be most pronounced at wide longitudes . the presence of a spiral arm might similarly contribute to the large line - of - sight density dispersion for these fields . we consider in this section the effect on the fitted model parameters of changing the galactocentric distance @xmath171 . it was noted in section  [ sec : popeffects ] that there is an offset between the peak magnitude of rcgs observed in the ogle - ii data and that expected for a rcg , with absolute magnitude similar to that of local rcgs , placed at a distance of 8.0 . the observed rcgs are systematically 0.3 mag fainter than the fiducial local rcg at 8.0 . there are two possible reasons for this magnitude offset . firstly , the adopted galactocentric distance of @xmath172 may be incorrect ; secondly , there may be population variations between local and bulge rcgs , resulting in different intrinsic magnitudes in the two populations . in section  [ sec : popeffects ] we noted that the offset predicted from the theoretical population models of @xcite estimate @xmath67 . we postulated that if the remaining magnitude offset is completely accounted for by a change in the galactocentric distance , this would mean that @xmath173 . conversely , if we adopted the value of @xmath174 determined by @xcite , the magnitude offset would become @xmath175 mag , where including the population effect predicted by @xcite would leave an unaccounted - for offset of @xmath176 mag . in this section we present the results of further modelling where we apply magnitude offsets which correspond to different values of @xmath171 .    in section  [ sec : results ] , we note that for the ogle - ii fields 6 , 7 , 14 , 15 and 43 , which have latitudes removed from the majority of ogle - ii fields , there are systematic offsets between the rcg peak in observed number count histograms and the predictions of all nine tri - axial models tested . for this reason , we limited the fields used to those which have galactic latitude @xmath177 . we also noted in section  [ sec : results ] that due to the lack of field coverage in the latitude direction the data are not effective at constraining the bar tilt angle @xmath83 . for this reason , @xmath83 was held constant at @xmath178 in the following modelling analysis . during the previous modelling efforts of section  [ sec : results ] it was also found that applying the exponential cut - off of equation  ( [ eq : cutoff ] ) had little effect on determining the best - fitting bar parameters . this cut - off was not applied in the following analysis . the best - fitting bar parameters were determined for the nine tri - axial bar models for the ogle - ii fields occupying the latitude strip @xmath177 using magnitude offsets corresponding to @xmath171 values of 8.0 , 7.6 and 7.3 . the resulting best - fitting parameters are listed in table  [ tab : changer0 ] . cccccccc & & & axis ratios + model & @xmath171 ( ) & @xmath81 ( ) & @xmath96 & @xmath97 & @xmath3 & & @xmath127 +    from the results in table  [ tab : changer0 ] we see that the fitted bar opening angle @xmath81 increases for all models as @xmath171 decreases . the fitted scale lengths @xmath96 , @xmath97 and @xmath3 all decrease linearly as @xmath171 decreases . the values of in table  [ tab : changer0 ] indicate that a smaller value of @xmath171 is favoured by all models . the ratio between the scale heights remains remarkably constant with varying @xmath171 , indicating that while the orientation of the bar changes slightly with varying @xmath171 , the overall shape of the bar does not change . the mean values of the scale length ratios for all models , and all values of @xmath171 are @xmath179 . this result is very close to that determined using data from all 44 ogle - ii fields , see section  [ sec : results ] above . red clump giant stars in the ogle - ii microlensing survey catalogue can be used as tracers of the bulge density over a large region towards the galactic centre . nine analytic tri - axial bar models were initially fitted to the number count histograms of red clump stars observed in 34 ogle - ii fields which have @xmath16 . the models all have the major axis of the bar oriented at @xmath180 @xmath130 to the sun - galactic centre line - of - sight . this orientation is in agreement with the results of @xcite and @xcite which give a bar angle of @xmath180 @xmath181 , and is marginally in agreement with the value of @xmath182 from @xcite . @xcite obtain best - fitting non - parametric models to the cobe / dirbe l band map of the inner galaxy with bar angles of @xmath180  @xmath183 . we find the ratio of the bar major axis scale length to minor axis scale length in the plane of the galaxy to be @xmath184  @xmath134 , higher than the value of 2.0  2.4 reported by @xcite , but consistent with the upper end of the range 2.5  3.3 found by @xcite . the ratio of bar major axis scale length to vertical axis scale length was found to be @xmath185  4.2 , again higher on average than that reported by @xcite who found @xmath186  3.8 , and higher than the value of @xmath187 generally adopted @xcite . the working model proposed by @xcite gives the scale length ratios as @xmath5 . our results suggest a more prolate model with @xmath167 . the observed separation of the mean position of red clump giants from the bar major axis at @xmath156 was shown to be a geometric effect , rather than evidence of a more complicated structure such as a ring . the observed data from these fields were used to further constrain the bar models . the resulting bar position angles was found to be @xmath0  @xmath1 . this narrower range is consistent with the several values of the bar position angle found by previous studies . the bar scale length ratios were determined to be @xmath4 , which are closer to the working model proposed by @xcite than those values found using data from only the central 34 ogle - ii fields . reasons for the difference between the bar axis ratios determined here and the general working model of @xcite may include rcg population effects noted above in section  [ sec : popeffects ] . the intrinsic luminosity of bulge rcg stars was assumed to be the same as the local population , however it was found that an offset of @xmath188 mag had to be applied to the computed distance moduli in order to obtain results consistent with standard bar models . @xcite found the mean magnitude of observed bulge clump stars is found to be 0.3 mag fainter to that expected assuming ( i ) the properties of bulge rcgs are the same as the local population and ( ii ) the distance to the galactic centre is 8 . a possible implication is that the adopted distance to the galactic centre of 8 may be incorrect . the bar modelling procedure was repeated for three values of the sun - galactic centre distance @xmath171 : 7.3 , 7.6 and 8.0 , using data from ogle - ii fields which have @xmath177 . some fields with latitudes outside this range were found to have number count histograms which were not reproducible by any linear tri - axial model of the bar tested in this work , and were therefore excluded . in addition , most of the ogle - ii fields latitudes @xmath189 were excluded . the low amount of information in the latitude direction means that the data have little leverage in determining some of the model parameters . without more data in these regions , model fitting algorithms may not be able to refine some model parameters such as the tilt angle @xmath83 . the three values of @xmath171 used are the ` default ' value of 8.0 ; 7.6 , as determined by @xcite ; and 7.3 , which corresponds to the value of @xmath171 consistent with the observed mean rcg magnitudes in the ogle - ii bulge fields assuming the population effects of @xcite . the fitted scale lengths @xmath96 , @xmath97 and @xmath3 were found to increase linearly with increasing values of @xmath171 , while the bar opening angle @xmath81 decreased slightly with increasing @xmath171 . the shape of the bar , as quantified by the ratio of axis scale lengths , was found to be insensitive to different values of @xmath171 . the mean ratio of scale lengths over all models and values of @xmath171 was found to be @xmath190 , which is slightly closer to the working model of @xcite for the galactic bar than that determined using all 44 ogle - ii fields . the goodness - of - fit measure decreased for all models as the value of @xmath171 was lowered . improved modelling for the galactic bar may be possible through the addition of further elements to the methods presented here . the possibility of a metallicity gradient across the bulge was not accounted for in this work . including spiral terms ( see e.g. @xcite ) in the analytic density profiles may similarly result in a closer reproduction of the observed number count profiles . the observed maximum line - of - sight density can be reproduced using just a bar , without requiring additional structure elements . however , the finer details of the number count histograms , especially at wide longitudes , were not reproduced by the tri - axial bar models used here . the predicted number count dispersions around the peak rcg magnitude were underestimated by the analytical models , particularly for fields with @xmath168 . it is possible that these finer features can be reproduced using models which include extra stellar agglomerations at the ends of the bar . the additional stellar densities could arise from specific stellar orbits aligned with the bar , or due to the projection effect of an inner ring . a long thin bar as postulated by @xcite might similarly increase the line - of - sight density dispersion , particularly for wide longitude fields , as might the presence of a spiral arm . modelling the bar using non - parametric methods ( see e.g. @xcite ) may provide valuable insight into the existence and nature of such additional features . some preliminary work applying these methods has begun . data from current and future surveys of the galactic bulge region will be useful for refining the constraints on the bar parameters . the third evolution of the ogle microlensing experiment , ogle - iii , is currently in progress , covering a larger region of the central galactic region than ogle - ii . infra - red observations of the bulge have the advantage of lower extinction effects due to dust , compared to optical observations . current catalogues which would be suitable for investigating the structure of the galactic bar include the point source catalogue from the 2mass all sky data release @xcite , the galactic plane survey from the ukirt infra - red deep sky survey @xcite and data from the isogal , spitzer / glimpse @xcite and akari @xcite space telescope surveys . proposed infra - red surveys towards the galactic centre include the galactic bar infra - red time - domain ( gabarit ) survey which aims to monitor the galactic bar region in the k band for microlensing events ( kerins 2006 , private communication ) . the analysis of red clump giant star counts in these surveys may result in tighter constraints on the properties of the bar , and can be combined with constrains on stellar kinematics from proper motion surveys @xcite in order to develop dynamical models of the inner galaxy . we thank d. faria ,  . wyrzykowski , r. james and w. evans for helpful discussions . njr acknowledges financial support by a pparc pdra fellowship and the lkbf . njr thanks the kapteyn astronomical institute , rug , for visitor support . this work was partially supported by the european community s sixth framework marie curie research training network programme , contract no . mrtn - ct-2004 - 505183 ` angles ' .
red clump giant stars can be used as distance indicators to trace the mass distribution of the galactic bar . 
 we use rcg stars from 44 bulge fields from the ogle - ii microlensing collaboration database to constrain analytic tri - axial models for the galactic bar . 
 we find the bar major axis is oriented at an angle of @xmath0 
 @xmath1 to the sun - galactic centre line - of - sight . 
 the ratio of semi - major and semi - minor bar axis scale lengths in the galactic plane @xmath2 , and vertical bar scale length @xmath3 , is @xmath4 , suggesting a slightly more prolate bar structure than the working model of @xcite which gives the scale length ratios as @xmath5 . 
 galaxy : bulge - galaxy : centre - galaxy : structure