}{\left

^2} \EE in their notation. This works out to \BE t_0= \frac{\pi}{4}\frac{\sum_i(\hat{t}_i/\epsilon_i)^2}{\sum_i \hat{t}_i/\epsilon_i} \EE which completely specifies our likelihood function. The beauty of this likelihood function approach is in its flexibility. We can use it to directly compare the models to the data without first calculating an ``observed'' optical depth, or we can use it to explore what the data say about the optical depth. We will discuss later our construction of a primitive map of optical depth from the presently observed data. This approach also shines in the analysis of observations of a disparate collection of lines of sight, especially if some have low optical depth and produce no events. One simply uses a density function which is non-simply connected and the null data is taken into account automatically. Overlapping fields are also easy to handle in this formalism. Areas in which one has an overlap are simply counted as having twice the density. We use our likelihood to rank the models we will discuss in the next section. It is important to note that we will be able to compute only relative likelihoods. The true structure of the Galaxy almost certainly does not fall exactly into the classes of models we consider. By considering a range of models that have passed a variety of other tests we hope to include at least some models that approximate reality. \subsection{Models} Our Galaxy can be described loosely as consisting of three parts: a disk, a dark halo, and a central bulge. In the following section, we describe the models we use in our calculations of optical depths. For each component, we indicate a range for its parameters that we feel is reasonable based on the literature. Because the values of these parameters are so uncertain we felt that it would indicate a false level of certainty to use a Gaussian prior for the parameters of our models. Accordingly, we have taken flat priors over the parameter ranges indicated. Since we expect the amount of microlensing due to objects in the halo to be negligible \cite{halocont} and almost constant over the small range of directions examined, we do not model the halo. The Galactic density enters the equations for the optical depth in two distinct ways: source and lens. One must therefore be very careful to keep the two roles separate. If both the bulge and the disk have the same lens to source ratio (or equivalently the same mass to light ratio), then the distinction becomes meaningless and can be dropped. It is not clear that this is the case. We handle the issue in the following manner. The visible content of the disk, that which could be seen as sources, is relatively well known. On the other hand, the ratio of bulge to disk source stars in the MACHO fields has been estimated by the MACHO collaboration as around 20\% based on the fraction of the 2.2 micron flux contributed by the disk\cite{MACHO1st}. We therefore model the disk with a source disk and a lens disk, and adjust the bulge source fraction to give approximately an 80\% contribution along our line of sight. \subsubsection{Bulge} Models of the Galactic bulge are very uncertain. The difficulties are both practical (most fields towards the bulge suffer extremely high extinction) and theoretical (it is extremely difficult to invert gas and stellar dynamics to obtain potentials). Nevertheless, a number of models have emerged as standards\cite{Dwek}. These models have widely ranging functional forms. We consider two representative forms, Dwek et al's G2 and E2. Their densities can be expressed \begin{eqnarray} ~~~~~\rho_{G2} & = & 1.2172\frac{M_{BAR}}{8\pi x_0y_0z_0} e^{-{r_s^2}/{2}} \\ ~~~~~\rho_{E2} & = & \frac{M_{BAR}}{8\pi x_0 y_0 z_0} e^{-r} \end{eqnarray} \begin{eqnarray} ~~~~~r_s & = & {\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}}\nonumber\\ ~~~~~r & = & {\left\{\left(\frac{x}{x_0}\right)^2+\left(\frac{y}{y_0}\right)^2+\left(\frac{z}{z_0}\right)^2\right\}^{1/2}}\nonumber \end{eqnarray} with the major axis inclined towards us at an angle $\theta_B$, which we will take to be between $0\deg$ and $45\deg$, with the near side in the first Galactic quadrant. The G2 model is the best fit to the DIRBE infrared maps \cite{Dwek}, while the E2 is favored by an analysis of the distribution of red clump giants in the OGLE fields\cite{OGLEfields}. At the location of the MACHO fields we will be analyzing, G2 models tend to have high optical depths while E2 models produce optical depth considerably less efficiently \cite{lateZhao}. By considering both models with a wide range of parameters we hope to cover a large portion of possible bulge models. Estimates of bulge masses cover a wide range. More recently, however, estimates have fallen approximately in the range $(2.0 - 3.0)\times10^{10}\Msol$\cite{lateZhao}. To be conservative we consider the range $1.0 - 4.0\times10^{10}\Msol$. The bulge scale factors are less well known. The best known of these are the two vertical scale heights. We follow the results of Stanek et al.\cite{OGLEfields} and fix these at $0.43\kpc$ for the G2 models and $0.25\kpc$ for the E2 models. We consider the ranges $0.3 - 2.7\kpc$ (G2) and $0.2 - 1.8\kpc$ (E2) for both $x_0$ and $y_0$. Following the review paper by Kerr \& Lynden-Bell \cite{R0} we fix the distance to the bulge to be the IAU recommended value of 8.5kpc. \subsubsection{Disk} As discussed above the structure of the luminous component of the disk is relatively better known than a possible dark component \cite{morelumdisk}. Thus we consider two separate disks. The first is a thin luminous double exponential disk, \begin{equation} \rho_{lum} = \frac{\Sigma_{lum}}{2 r_z}\exp\left({-\frac{z}{r_z}}\right)\exp\left({\frac{r_0-r}{r_d}}\right), \end{equation} composed of luminous stars that can serve as sources. The parameters for this disk are fixed: $\Sigma_{lum}=15\Msol \pc2$, $r_z=0.3 \kpc$, and $r_d=3.5\kpc$.\cite{GWK,morelumdisk} Not all stars in the disk are bright enough to been seen, however, and in fact there is evidence that the disk contains considerable mass beyond that which is visible, perhaps distributed somewhat differently from the luminous mass \cite{sigma0,sigma1,sigma2}. Hence we consider also a second double exponential disk, \begin{equation} \rho = \frac{\Sigma_0}{2 r_z}\exp\left({-\frac{z}{r_z}}\right)\exp\left({\frac{r_0-r}{r_d}}\right), \end{equation} whose parameters are not fixed. This component is allowed to serve as lenses. Analysis of the vertical velocity distributions of stars in the vicinity of the sun, gives $2\sigma$ upper limits on the total surface density within $1\kpc$ of the disk plane of at most $85\Msol \pc2$\cite{sigma0,sigma1,sigma2}. Of this about $30\Msol \pc2$ is in the form of bright stars and gas, while perhaps $10\Msol \pc2$ is in the form of M dwarfs which could serve as lenses. About $10\Msol \pc2$ is contributed by the halo, more in flattened models. This leaves a maximum of $35\Msol \pc2$ for a possible dark disk. Adding in the M dwarf lenses, the surface density of lenses should be in the range $10-45\Msol \pc2$. To be conservative we choose the range $10-55\Msol \pc2$. The scale height for the dark component of the disk is unknown. We therefore consider the range $0.2-1.5\kpc$, which corresponds to populations tighter than the visible stars at the low end, to a significantly heated population at the high end. We fix the disk scale length at a value of 3.5kpc\cite{scalelength}. \section{Data} \subsection{Events \& Fields} The basic requirement for microlensing searches is a very large sample of distant stars. For studies of the halo the obvious choices of fields are towards the Large Magellanic Cloud and the Small Magellanic Cloud. Searches towards the bulge are complicated by the extinction from intervening dust obscuring the Galactic Center. In the red and blue bands used by the MACHO collaboration, there are only a relatively few fields with a large enough density of stars. One of these ``holes in the dust'' is Baade's Window at Galactic longitude (l) and latitude (b) (1$\deg$,-4$\deg$), towards which the first-year observations are clustered. We show in Fig.~1 a plot of the 24 fields reported on by the MACHO collaboration. For reference, also included are some of the OGLE fields, which are also of relatively high density. The MACHO collaboration observed 12.6 million stars for a period of 190 days during the 1993 season in these 24 fields\cite{MACHO1st}. The 41 events that pass their criteria for microlensing are plotted on Fig.~1 as dots with radii proportional to their duration. The location and duration of each event can be found in Alcock et al. Table 1\cite{MACHO1st}. In addition to the location and duration of the events we also need a variety of other numbers. Information on the location, size and orientation of the 24 fields can be found at the MACHO website, http://wwwmacho.anu.edu.au. % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure1.eps}}} \caption[$20\deg \times 20\deg$ region centered at the galactic center.]{$20\deg \times 20\deg$ region centered at the galactic center. Light boxes are the 24 MACHO 1st year fields. Smaller heavy boxes are OGLE fields. The 41 events we analyse are shown as dots with radii proportional to the event duration.} \end{figure} % The two quantities that pose the greatest problems are the observational efficiencies and the density of the stars in each field. As of this writing, the MACHO collaboration has not completed its analysis of the full blending efficiencies for the bulge fields. As a reasonable approximation, they suggest using their sampling efficiencies with a 0.75 factor correction. Accordingly, we following this prescription and use their standard cut sampling efficiencies from Fig.~5 of Alcock et al\cite{MACHO1st}. Since the fields under consideration are similar and all are crowding limited, we assume that the efficiencies are uniform across the sample. The reader is warned however, that this is a major source of uncertainty. It is possible that the efficiency is not only a function of duration but also of position \cite{HanGould} which would introduce spurious spatial structure into the microlensing distribution. The final quantity we need for our analysis is the density of observed sources. In view of the fact that the fields are crowding limited and in the absence of better data, we assume a uniform density of source stars across the fields. Accounting for overlap we get $\sigma= 1.06\times 10^6 {\rm deg}^{-2}$. We note an encouraging point. The efficiencies are likely to get better as the fields get less crowded since blending effects are smaller. On the other hand, less crowded fields mean a smaller source density. Hence the errors due to our assumptions of constant efficiencies and source densities should be in opposite directions and are likely to at least partly cancel. \subsection{Structure of the Data} Before attempting to extract information about the structure of the Galaxy from the data, we first look at what we can learn about the structure of the data itself. We apply our likelihood formalism to the very simplest model of the optical depth possible: \begin{equation} \tau = \tau_0, \end{equation} a flat optical depth over the entire set of fields. We obtain the result $\tau_0 = 1.93\pm0.39\times 10^{-6}$ where we quote 68\% confidence limits. The MACHO collaboration's reported value of $2.4\pm0.5\times10^{-6}$ for the same events includes a ``correction'' factor of $1/0.8$ introduced to adjust for contamination by source stars in the disk. Undoing this ``correction'', we obtain $1.9\pm0.4\times10^{-6}$, in agreement with our result. Note that our errors are from our analytic calculations and not the result of Monte Carlo simulations. \subsection{Gradients} We next apply our formalism to a slightly more complicated set of models: those with an optical depth gradient in the $b$ and $l$ directions. We consider models with the form, \begin{equation} \tau = \tau_0 + \frac{d\tau}{db} (b-b_0) + \frac{d\tau}{dl} (l-l_0), \end{equation} where $b_0=-4$ and $l_0=2$. Fig~.2 shows contours of the likelihood, marginalized over $\tau_0$, in the $\frac{d\tau}{db}$ -- $\frac{d\tau}{dl}$ plane. A gradient in the optical depth in latitude is clearly indicated. The case for a slope in longitude is less clear-cut. Marginalizing over the remaining parameter in each case we obtain $\frac{d\tau}{db}=1.12\pm0.37\times10^{-6}/{\rm deg}$ and $\frac{d\tau}{dl}=-1.71\pm1.19\times10^{-7}/{\rm deg}$, again with 68\% confidence limits. % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure2.eps}}} \caption[Likelihood contours for $l$ and $b$ gradients in the optical depth.]{Likelihood contours for $l$ and $b$ gradients in the optical depth. Solid lines are the 68\% confidence contours. Dotted lines denote 95\% confidence. Dashed lines denote 99\% confidence. Also included, to guide the eye are long dashed lines for 38\% confidence.} \end{figure} % Our latitude gradient can be compared with an estimate based on the MACHO clump giant optical depths reported for fields above and below $b=-3.5\deg$\cite{MACHO1st}. Scaling the calculated slope, $s=(6.32-1.57)\times10^{-6}/1.38\deg$, to the full sample and undoing their $1/0.80$ disk correction, we obtain an estimate of $\frac{d\tau}{db}=1.7\times10^{-6}/{\rm deg}$ with likely errors of at least $1.0\times10^{-6}/{\rm deg}$. This is fully consistent with our results. \subsection{Maps} Just how much information about the structure of the event distribution can we extract? Our likelihood method lends itself nicely to the construction of the model independent ``most likely'' map of the microlensing optical depth. Consider a general function $\tau(b,l)$ giving the optical depth. We would like to find the positive definite function $\tau(b,l)$ which maximizes the likelihood or equivalently minimize the negative log likelihood, \begin{equation} LL = \int\int \left[ \frac{T}{t_0}\tau(b,l) \sigma(b,l) - Q(b,l) \log{(\tau(b,l))} \right] db dl \end{equation} where \BE Q=\frac{\pi}{4 t_0} \sum_{events}\hat{t}_i \delta(l-l_i,b-b_i). \EE Taken alone, however, this condition is insufficient. The solutions to this equation turn out to be delta functions at the event locations, a clearly unphysical situation. What is missing is that the optical depth should be a smooth function of position on the sky. Thus we must add a smoothing term to the log likelihood to be minimized. Our smoothing term must discourage structure unmotivated by the data, yet at the same time not penalize legitimate gradients such as we have seen in the data. We choose to minimize the extrinsic curvature, given by \BE K= \frac{d^2\tau}{db^2}+\frac{d^2\tau}{dl^2}+2\frac{d^2\tau}{dldb}. \EE Thus we require a minimum of \begin{equation} LL = \int\int \left[ \frac{T}{t_0} \tau(b,l) \sigma(b,l) - Q(b,l) \log{(\tau(b,l))} + \lambda K^2\right] db dl \end{equation} where $\lambda$ controls how much smoothing we require. We solve for $\tau$ by an iterative scheme, starting with a flat $\tau(l,b)$. Setting $\lambda$ to provide a reasonable smoothness we produce the map shown in Fig.~3. Our generated map contains few surprises. We note a pronounced tilt in galactic latitude and a small one in galactic longitude just as we saw in the earlier sections. A slight bending of the contours to wrap around the Galactic center is also present. It is important to remember that although the generated map is smooth and does not appear ``noisy'', this is an artifact of the way it is % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure3.eps}}} \caption[``Most likely'' map of the optical depth based on the first year events.]{``Most likely'' map of the optical depth based on the first year events. The solid lines show contours of $4.0, 3.0, 2.0, 1.0, {\rm and} 0.5\times10^{-6}$ from the top down.} \end{figure} % created: the smoothing term ensures that the resulting map is fairly smooth. The significance of the present map is low, due to the small number of events. A considerably larger data set would be needed before the map could be used to yield detailed information. \section{Results} With the results from our look at the data in mind, we now consider the more realistic models of the Galaxy discussed above. For each type of bulge, G2 and E2, we calculate the likelihood as a function of the various Galactic parameters with the stated ranges. Since the functional form of the bulge is so poorly constrained, we resist the temptation to make a direct comparison between the two bulge models. Given the number of parameters in our models and the present number of events, any such comparison would be of marginal significance. Instead, we focus on the parameters which have meaning independent of the functional form such as the mass of the bar, $M_B$, the inclination angle of the bar, $\theta_B$ and the bar axis ratio, $r=\frac{x_0}{y_0}$. In this section we will discuss the limits we can put on bulges of only these functional forms. \subsection{Bar} Due to the large number of parameters in our full model and limits on computational power, in our exploration of the implication of the microlensing events for bulge parameters we fix the disk parameters to reasonable values: $\Sigma_0 = 30.0, r_z=0.3\kpc,$ and $r_d=3.5\kpc$, while varying $M_B$, $\theta_B$, $x_0$ and $y_0$. The bulge quantities we are most interested in, $M_B$, $\theta_B$ and $r$, should be most strongly influenced by the magnitude and longitudinal gradient of the observed optical depth. Of these, only the magnitude of the optical depth is sensitive to the disk parameters. We expect that as we increase the disk surface density, the inferred mass will decrease. We have checked our results and find that this effect amounts to a less than $0.3\times10^{10}\Msol$ shift even when we increase the disk to $55\Msol \pc2.$ Our limits on other bulge parameters are unaffected. \subsubsection{Bar Mass and Orientation} Our major results concerning the bar mass and orientation are summarized in Figs.~4\&5. These figures show contours of likelihood as a function of the mass of the bar and orientation angle away from our line of sight. The two horizontal scale lengths, $x_0$ and $y_0$ were marginalized. Fig.~4 shows results for G2 models, while Fig.~5 presents E2 models. We discuss the G2 case first. % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure4.eps}}} \caption[Contours of likelihood in the $\theta_B - M_B$ plane for G2 models.]{Contours of likelihood in the $\theta_B - M_B$ plane for G2 models with $x_0$ and $y_0$ marginalized. Disk values where held fixed at $\Sigma_0=30\Msol pc^-2$ and $r_z=0.3\kpc$. Contours are as in Fig.~2.} \end{figure} \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure5.eps}}} \caption{Same as Fig.~5. but for E2 models.} \end{figure} One feature of Fig.~4 is immediately apparent: mass can be traded off for angle. A ridge in the likelihood lies on the line $M_{BAR}(10^{10}\Msol)-0.11\theta_B({\rm deg})=1.4$. There are limits to this trade-off, however. If the mass of the bar is much beyond $4.0\times10^{10}\Msol$, too high an optical depth will be produced to fit the data even if the angle is increased dramatically. There is also a sharp cutoff when the mass drops to below about 1.7. At such low masses, decreasing the angle no longer helps but rather hurts since the maximum optical depth is at a non-zero angle.\cite{lateZhao} The situation is much the same for the E2 models (Fig.~5), except shifted by about $1.1\times10^{10}\Msol$ in bulge mass. The E2 models drop off too rapidly to be efficient at producing microlensing optical depth even when optimally aligned, and hence need considerably higher bar masses\cite{lateZhao}. Below a mass of about $2.0\times10^{10}\Msol$ it becomes difficult to produce the high optical depths required by the data. We show in Fig.~6 the likelihood for the bar mass now marginalized over the bar orientation as well. The 90\% confidence limit is at $1.75\times10^{10}\Msol$ for the G2 bulge and $2.6\times10^{10}\Msol$ for the E2 bulge. Taking into account the uncertainty in the disk normalization, we arrive at lower bounds on the bulge mass of $1.5\times10^{10}\Msol$ (G2) and $2.3\times10^{10}\Msol$ (E2). The most likely values are around $3.5\times10^{10}\Msol$ for G2 models and beyond $4.0\times10^{10}\Msol$ for E2 models. % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure6.eps}}} \caption[Likelihoods from Figs.~4\&5 now with $\theta_B$ marginalized.]{Likelihoods from Figs.~4\&5 now with $\theta_B$ marginalized. Solid line for G2 models, dashed line for E2 models.} \end{figure} \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure7.eps}}} \caption[Same as Fig.~6, but with $M_B$ marginalized.]{Same as Fig.~6, but with $M_B$ marginalized.} \end{figure} In Fig.~7 we plot the marginalized likelihood versus bulge inclination angle. Low inclination angles are clearly favored. The likelihood has dropped off strongly by $30\deg$ and $20\deg$ for the G2 and E2 models respectively. Beyond these angles, even bar masses as high as $4.0\times10^{10}\Msol$ cannot produce enough microlensing to be compatible with the experimental results. The $90\%$ confidence limits are $30\deg$ (G2) and $21\deg$ (E2). Our results, using only a single year of microlensing data, fit very nicely with attempts to constraint the bulge mass and orientation angle by other means. Analysis of the stellar motions in the bulge gives a range of mass estimates from slightly below $2.0\times10^{10}\Msol$ to almost $3.0\times10^{10}\Msol$.\cite{Kent,zhao,blum} Several authors have derived bulge masses around $2.2\times10^{10}\Msol$ using a variety of methods based on modeling of the gas content of the bulge.\cite{Binney,ZRS96} A simple argument by Zhao et al. gives this as an upper limit\cite{lateZhao}. Our limit on the bulge mass is consistent with the reported values for the bulge mass. The OGLE collaboration has reported an analysis of the distribution of the bulge red clump giants within their fields shown in Fig.~1. They report a bulge orientation of between 20 and $30\deg$ almost independent of the bulge model.\cite{OGLEfields} Dwek et al. analyze the DIRBE maps of the infrared emmision from the bulge and conclude that the orientation angle lies in the range $10-40\deg$. Our range of $0-30\deg$ is also consistent with the value $16\pm2\deg$ suggested by Binney et al.'s analysis of gas dynamics\cite{Binney}. Perhaps more important than our limits on $M_B$ and $\theta_B$ separately, are the full contours of likelihood in the $M_B$ - $\theta_B$ plane showing the correlation between high bulge mass and high orientation angle. As we discuss later, increases in the number of events at Baade's window do little to break the $M_B$ - $\theta_B$ degeneracy, but make the ridge considerably narrower. It requires more information to uniquely pick out a bulge mass. An analysis using the tensor virial theorem for the bulge by Blum \cite{blum} gives exactly the opposite degeneracy: high mass is correlated with low angle. Thus the results of microlensing and dynamics arguments are complementary and may be able to break the mass-angle degeneracy for either alone. \subsubsection{Bar Axis Ratios} The other bulge quantity we look at is the bar axis ratio $r=\frac{x_0}{y_0}$. Although we do not explicitly have the axis ratio as an input quantity for our bulge models, microlensing puts limits on the ratio of the scale lengths. In Figs.~8\&9 we have marginalized $\theta_B$ and the $x_0$-$y_0$ pair subject to the constraint $r=\frac{x_0}{y_0}$. One can immediately see a trade-off between the axis ratio and the mass. A higher axis ratio concentrates the mass where it will do the most microlensing. Hence, in conjunction with a low orientation angle, this allows a lower mass. Since a bar-like configuration seems to be favored by % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure8.eps}}} \caption[Contours of likelihood in the $M_B$ - $r$(axis ratio) plane.]{Contours of likelihood in the $M_B$ - $r$(axis ratio) plane. $\theta_B$ and one degree of freedom from $x_0$, $y_0$ have been marginalized. As in Fig.~4, disk parameters have been fixed. Contours are as in Fig.~2.} \end{figure} % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure9.eps}}} \caption[Same as Fig.~8 but for E2 models.]{Same as Fig.~8 but for E2 models.} \end{figure} % the most recent data on the bulge it is interesting to note that our results do not completely rule out axisymmetric models. The favored axisymmetric models have very high masses. On the other hand, measurements of velocity dispersions in the bulge give low constraints on masses of axisymmetric models. The strongest microlensing evidence against axisymmetric models comes from a consideration of the more limited bulge giant subsample of the events, which probe the optical depth for sources distributed throughout the bar. An axisymmetric model can not produce enough microlensing to account for the high optical depth implied by these events\cite{long}. Our most likely values, r=3.5 (G2) and r=2.5 (E2), are consistent with the axis ratios determined by other means. Dwek et al. give a range $2.5-5.0$ for their models\cite{Dwek}. Stanek et al. obtain a value in the range $2.0-2.5$ again independent of model choice\cite{OGLEfields}. The distribution of bulge Mira variables gives an axis ratio of 3.9.\cite{whitelock}. \subsection{Disk-Bulge Discrimination} Since we hope to discriminate between the bulge and disk on the basis of the latitude gradient information, the parameters $M_B$, $\theta_B$, $r_z$, and $\Sigma_0$ are most relevant. Thus we vary these quantities while holding the two horizontal bulge scale lengths constant at $x_0=1.58\kpc$ and $y_0=0.62\kpc$ for G2 models. The results for the E2 models are similar to those for the G2 models, except for a shift in the bulge mass as noted above. Since we are interested only in the disk parameters we discuss only the results for the G2 models. \subsubsection{Surface Density} Fig.~10 shows the marginalized likelihood as a function of the surface density of the disk versus mass of the bulge. It is immediately clear that we cannot uniquely fix the surface density of the disk. Rather, as was the case with the orientation angle of the bulge, $\Sigma_0$ shows a linear degeneracy with $M_{BAR}$, with the ridge of the likelihood at $M_{B}(10^{10}\Msol)+0.0088\Sigma_0(\Msol/pc^2)= 2.88.$ A heavy disk can add as much as $0.7\times10^{-6}$ to the optical depth, allowing the bar to be less massive. The widening of the likelihood contours towards the top shows a slight tendency towards a more massive disk. This tendency, although not significant, is due to the slope of the optical depth in latitude. % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure10.eps}}} \caption[Contours of likelihood in the $M_B$ - $\Sigma_0$ plane.]{Contours of likelihood in the $M_B$ - $\Sigma_0$ plane. $\theta_B$ and $r_z$ have been marginalized. Bulge parameters $x_0$ and $y_0$ have been held fixed at $x_0=1.58\kpc$ and $y_0=0.62\kpc$. Contours are as in Fig.~2.} \end{figure} % % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure11.eps}}} \caption[Contours of likelihood in the $M_B$ - $r_z$ plane.]{Contours of likelihood in the $M_B$ - $r_z$ plane. $\theta_B$ and $\Sigma_0$ have been marginalized. Bulge parameters $x_0$ and $y_0$ have been held fixed at $x_0=1.58\kpc$ and $y_0=0.62\kpc$. Contours are as in Fig.~2.} \end{figure} % \subsubsection{Scale Height} We show in Fig.~11 the likelihood as a function of the scale height of the disk and the bar mass. The scale height is only very weakly correlated with the bar mass and is basically unconstrained. There is a spreading of the likelihood contours for low scale heights possibly favoring scale heights below about $0.6\kpc$. At low scale lengths the gradient in optical depth of the disk contribution is high. As the scale length increases, the disk gradient decreases. Hence low values of $r_z$ are favored, with high values, where the gradient is low, suppressed. % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure12.eps}}} \caption[Slope in latitude for the optical depth as a function of $\Sigma_0$.]{Slope in latitude for the optical depth as a function of $\Sigma_0$, the disk surface density. The total optical depth has been held constant by varying $M_B$ as $\Sigma_0$ varies. The lines represent: solid ($r_z=0.3\kpc$), dashed ($r_z=0.5\kpc$), long dashed ($r_z=1.0\kpc$), and dot-dashed ($r_z=1.5\kpc$). Total variation across our range of disk surface densities is less than 10\%. } \end{figure} % \subsubsection{Disk Bulge Degeneracy} Our results for disk parameters are something of a disappointment. With the present data we can say virtually nothing about the disk structure. Part of the problem is the trade-off that occurs between $M_B$ and $\Sigma_0$, keeping the optical depth constant and producing the ridge in the $M_B$ versus $\Sigma_0$ likelihood plot. We had hoped, however, that gradient information would allow us to break the degeneracy between $M_B$ and $\Sigma_0$. The reason that this does not happen is not that the area that the MACHO fields span is too small to show strong structure with these few events; slopes in $b$ and $l$ are indicated. The problem lies with the position of Baade's window and the details of the Galactic models. Firstly, because the longitudes of the bulk of the MACHO fields are low, the longitude slope expected in this region is small for virtually any model. This makes the slope in longitude a poor diagnostic. Secondly, at the location of Baade's window the $M_B$ - $\Sigma_0$ trade-off also keeps the latitude gradient fairly constant over a large range. We show this in Fig.~12, where the latitude gradient is shown as a function of $\Sigma_0$, with the optical depth kept constant by varying the $M_B$. The different curves show various values of the scale height of the disk. Since the $b$ slope in the region of Baade's window is constant for a wide range of models it is very difficult to discriminate among them. This is why we see only hints of structure in our likelihood plots. \section{Future Directions} Over the next year, two collaborations, OGLE and EROS, will be moving to dedicated telescopes and fully automated analysis systems. Together with the currently running MACHO system, these groups have the potential for producing many times the data we have used here. How will such a wealth of data effect our results? To explore this question we have synthesized 4 years of observations and rerun our analysis. Our synthetic observations were constructed assuming a G2 model with $M_B=3.0\times10^{10}\Msol, x_0=1.58\kpc, y_0=0.62\kpc, \theta_B=15\deg, \Sigma_0=30.0\Msol \pc2, r_z=0.3/kpc$. The total number of events expected was calculated as \BE N_{exp}= \frac{T}{\langle t \rangle}\int \sigma\tau(l,b) dl db ~~~~~~~~{\rm w}/ \langle t \rangle=\frac{\pi}{4}\sum_i\frac{\hat{t}_i}{\epsilon_i}/\sum_i\frac{1}{\epsilon_i}, \EE where the average was done over the present data set and the integral is over the MACHO first-year fields. From this expected number, the actual number was picked assuming Poisson statistics. Since the rate is proportional to the optical depth (assuming constant average duration), the events were laid down randomly according to the optical depth. A duration was picked out of the efficiency weighted set of observed durations. The proposed event was then ``observed'' or not based on the efficiency for that duration. The final set of ``observed'' events was run through our analysis. The results are shown in Fig.~13. We note a number of features of the results. First is that the confidence regions have tightened somewhat as expected. Second, the basic degeneracy between $M_B$ and $\theta_B$ is uneffected. Although $\theta_B$ now seems to be quite well constrained, $M_B$ still varies over a wide range. As we discussed earlier, part of the problem is in the distribution of our present lines of sight. Baade's window is a poor location for determining longitude slope, and the latitude slope is correlated with the optical depth, making it less useful as a diagnostic tool. If we wish to determine the bulge parameters solely from microlensing, simply collecting more data in the same fields will not easily break the degeneracies between parameters. We must look to expanding our range of fields. % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure13.eps}}} \caption[Likelihood in the $M_B$ - $\theta_B$ plane for a simulated 4 seasons of data.]{Likelihood in the $M_B$ - $\theta_B$ plane for a simulated 4 seasons of data based on a G2 model with $M_B=3.0\times10^{10}\Msol, x_0=1.58\kpc, y_0=0.62\kpc, \theta_B=15\deg, \Sigma_0=30.0\Msol \pc2, r_z=0.3/kpc$. Contours same as Fig.~2. } \end{figure} % So, what is the best strategy to use? Microlensing searches are costly and very time-consuming: we would like to find a strategy that maximizes the scientific return. We attempt to address this issue by analyzing a set of very stylized strategies that will allow us some insight into real searches. Let us assume that we observe four identical fields centered at $(1.0,-3.0)$, $(1.0+\Delta l,-3.0)$, $(1.0,-3.0-\Delta b)$ and $(1.0+\Delta l,-3.0-\Delta b)$. Further, we take the limit as the size of the fields goes to zero but the exposure for each field, $E$, stays constant. In this way the only gradient information will come from the field separation, and not from the distribution of events within the fields. Each choice of $\Delta l$, $\Delta b$ will be a distinct strategy. Let $\hat{t}_i^j$ represent the durations of the events observed in the $j$-th field. Then the likelihood for a given model, $\tau(l,b)$ is \BE L=\prod_j e^{-\frac{E}{t_0}(\tau_j)}(\sigma\tau_j)^{\frac{\pi}{4t_0}\sum_i\frac{\hat{t}_i^j}{\epsilon_i}} \EE where $\tau_j$ is the predicted optical depth at the $j$-th field. Let $\tau^0(l,b)$ represent the optical depth of the underlying model. Then on average $\frac{\pi}{4t_0}\sum_i\frac{\hat{t}_i^j}{\epsilon_i}=\frac{E}{t_0}\tau^0_j$ and we will have \BE L=\prod_j e^{-\frac{E}{t_0}(\tau_j)}(\sigma\tau_j)^{\frac{E}{t_0}\tau^0_j}. \EE Using this likelihood we can now determine how well a given strategy can recover our underlying model. Figs.~14,15\&16 show the magnitude of the 68\% confidence intervals in various quantities as a function of the longitude and latitude separation of the fields assuming twice the exposure of present experiments. We used the same model as in the previous section. The first thing one notices is that the varying strategies don't make as much difference as might be hoped. As the separation of fields is increased, our lever arm for making determinations of the quantities increases. However, at the same time the outer fields are in regions with low $\tau$ and hence few events and poor statistics. These two effects tend to cancel out, making dramatic improvements difficult. Nevertheless, what can we learn from these graphs? First of all, the present data correspond to roughly $\Delta l=3.0$, $\Delta b=2.0$. This is very close to the worst possible region for determination of all three parameters. % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure14.eps}}} \caption[Contours of the size of the 68\% confidence regions in the determination of $M_B$.]{Contours of the size of the 68\% confidence regions in the determination of $M_B$. Values are calculated as a function of the latitude and longitude spacing of the observed fields. Solid: 1.46, Dotted: 1.53, Dashed: 1.61, Long Dash: 1.69$\times10^{10}\Msol$ } \end{figure} % % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure15.eps}}} \caption[Same as Fig.~14 but for $\theta_B$.]{Same as Fig.~14 but for $\theta_B$. Solid: 16$\deg$, Dotted: 19$\deg$, Dashed: 23$\deg$, Long Dash: 26$\deg$} \end{figure} % % \begin{figure}[htb] \epsfysize=13.5cm \centerline{ \rotate[r]{\epsfbox{figure16.eps}}} \caption[Same as Fig.~14, but for $\Sigma_0$.]{Same as Fig.~14, but for $\Sigma_0$. Solid: 29, Dotted: 35, Dashed: 42, Long Dash: 48$\Msol \pc2$.} \end{figure} % If we wish to improve our determination of $M_B$ the results suggest a relatively large $\Delta b$ and a smallish $\Delta l$. For $\Sigma_0$, on the other hand, we should have a high $\Delta l$ and $\Delta b$ is irrelevant. Finally, for $\theta_B$, a moderate $\Delta l$ is required and again $\Delta b$ is mostly unimportant. A single search strategy to fix the parameters would need to probe the entire range of scales in longitude. Coverage in latitude appears to be less important; only the large scales need to be probed. The best strategy would seem to be one which includes many fields scattered over the entire bulge instead of concentrated in one region. Although the optical depth at any given location would be less well defined, better limits on the global parameters would be obtained. \section{Conclusions} In this paper we have developed a novel likelihood technique for the analysis of the microlensing data towards the bulge. We construct a likelihood that is sensitive to the spatial distribution of the events. Our technique is both more flexible than calculations that have been done before, and allows for a direct comparison of the data to models of the mass distribution for the Galaxy. It is particularly good for dealing with data from more than one line of sight, field overlaps and variations in the density of stars observed. Its sensitivity to the position of events makes it ideal for determining gradients in the optical depth. Applying this technique to the first year MACHO data we have confirmed the strong slope in latitude found by the MACHO collaboration, and found hints of one in longitude. We have for the first time, given quantitative measure of these slopes, $\frac{d\tau}{dl}=-1.71\pm1.19\times10^{-7}/{\rm deg}$ and $\frac{d\tau}{db}=1.12\pm0.37\times10^{-6}/{\rm deg}$. We also apply our analysis to constructing a crude ``most likely'' map of the microlensing optical depth over the observed region. We confront a set of Galactic models consisting of a bulge with either G2 or E2 functional form and an exponential disk, with the data. With only one season of microlensing data, we can already set meaningful limits on various bulge parameters. We find that $M_B>1.5 (2.0)\times10^{10}\Msol$, for the G2 (E2) based models. Most likely values for the bulge mass are much higher: $M_B=3.5 (>4.0)\times10^{10}\Msol$. Previous work has shown that such high bulge masses imply low halo MACHO fractions\cite{long}. A massive bulge puts tight constraints on the contribution of the disk to the rotation curve at small radii. A small disk, however, leaves more room for the halo in the outer rotation curve, implying a massive halo. Since microlensing results towards the LMC fix the MACHO content of the halo, a massive halo implies a smaller MACHO fraction. We also constrain the inclination angle of the bulge finding that $\theta_B < 30\deg (21\deg)$, consistent with other measurements. Our most likely values for the axis ratio of the bulge, $r=\frac{x_0}{y_0} = 3.5 (2.5)$, are consistent with determinations by other methods. We note that axisymmetric bulge models are not entirely ruled out with the full sample of events. Such models, however, typically need very high ($\approx4.0\times10^{10}\Msol)$ bulge masses. Such high bulge masses are unlikely for axisymmetric bulge models\cite{kentpriv}. No limits could be set on the disk component due to a degeneracy in the latitude slope of the optical depth between the bulge and disk contributions. We discuss what can be expected with an increase in the number of seasons of data. Finally, we have attempted to quantitatively discuss various strategies for microlensing searches and conclude that a strategy observing many fields well scattered in longitude offers the best return in determining bulge and disk parameters. The distribution of fields in latitude is less important. Despite the difficulties, this is perhaps more important than simply getting the optical depth more accurately at one location as it will allow a better determination of the relative contributions of the bar and the disk. Ideally, the optical depth can be mapped over a wide range in both latitude and longitude, yielding detailed information about the mass distribution in the inner Galaxy. The field of microlensing promises to be an eventful one for the foreseeable future! \section*{Acknowledgments} This work was supported by the DOE (at Chicago and Fermilab) and the NASA (at Fermilab through grant NAG 5-2788). I would like to thank Craig J. Copi for many stimulating conversations, and for his gracious work on the hMgo package. I would also like to thank Michael Turner, Evalyn Gates and Robert Nichol. \begin{thebibliography} {std} \bibitem{galstudy} {The first investigations might be considered to be those of Galileo when he first resolved the Milky Way into stars in 1610. Pioneering studies by W. Herschel paved the way for more modern investigation of the Milky Way by such luminaries as Kapteyn, Shapley, Lindblad and Oort. For more recent discussion see D. Milhalas and J. Binney, ``Galactic Astronomy: Structure and Kinematics.'',2nd edition, W.H. Freeman and Company, New York (1981). Also see {\it Back to the Galaxy} (AIP Conference Proceedings 278), eds. 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