------------------------------------------------------------------------ From: Ortwin Gerhard Ortwin.Gerhard@unibas.ch Subject: replaced texfile Binney etal To: gcnews@aoc.nrao.edu MIME-version: 1.0 Content-transfer-encoding: 7bit %astro-ph/0003330 \documentclass[preprint]{aastex} \shorttitle{Microlensing and Galactic Structure} \shortauthors{Binney, Bissantz and Gerhard} \begin{document} \title{Is Galactic Structure Compatible with Microlensing Data?} \author{James Binney} \affil{Oxford University, Theoretical Physics, Keble Road, Oxford, OX1 3NP, U.K.} \author{Nicolai Bissantz and Ortwin Gerhard} \affil{Astronomical Institute of the University of Basel, Venusstrasse 7, CH-4102 Binningen, Switzerland} %\pubyear{2000}\maketitle \begin{abstract} We generalize to elliptical models the argument of Kuijken (1997), which connects the microlensing optical depth towards the Galactic bulge to the Galactic rotation curve. When applied to the latest value from the MACHO collaboration for the optical depth for microlensing of bulge sources, the argument implies that the Galactic bar cannot plausibly reconcile the measured values of the optical depth, the rotation curve and the local mass density. Either there is a problem with the interpretation of the microlensing data, or our line of sight to the Galactic centre is highly atypical in that it passes through a massive structure that wraps only a small distance around the Galactic centre. \end{abstract} \section{Introduction} Searches for gravitational microlensing events, over 500 of which have now been observed, provide an important constraint on the mass in the inner Galaxy. Deriving a mass from an observed optical depth is not straightforward because one usually has only limited knowledge of the distances to the stars that are lensed, and even less information about the distances to the deflecting objects. Kuijken (1997) showed that the {\it minimum\/} mass required to generate a given optical depth towards the Galactic centre from an axisymmetric distribution of matter can be determined without any knowledge of the location of the lenses provided one knows the optical depth to a source on the Galaxy's symmetry axis. He showed, further, that the minimum mass required by the then available microlensing data was barely compatible with the measured rotation curve and local mass density. Here we generalize Kuijken's argument to the case of elliptical distributions of matter, which require less matter for a given optical depth. The apparent magnitude of a red-clump star at the Galactic centre is relatively bright, so the microlensing optical depth to such objects can be determined without significant uncertainty due to blending. Moreover, the red-clump stars must follow the general distribution of near infra-red light quite closely, because they are part of the population of evolved stars that are responsible for most of the Galaxy's near-IR luminosity (McWilliam \& Rich, 1994). Given this, it proves possible to deduce from the measured mean optical depth of the clump giants the optical depth to a source on the Galaxy's axis. >From 13 lensing events Alcock et al.\ (1997) inferred $\tau = 3.9^{+1.8}_{-1.2}\times 10^{-6}$ for clump giants centred on $(l,b) = (2.55\deg, -3.64\deg)$. From a difference image analysis of observations that include 99 events, Alcock et al.\ (2000b) recently measured $\tau=2.9^{+0.47}_{-0.45}\times10^{-6}$ for a mix of stars centred on $(l,b)=(2.68\deg,-3.35\deg)$, and from this measurement deduced for the same direction $\tau=(3.88\pm0.6)\times10^{-6}$ for bulge sources after making allowance for the lower optical depth for lensing of disk stars. Here we use our results to show that such large values of the optical depth for microlensing of bulge sources cannot be reconciled with measurements of the rotation curve and local mass density, even by elliptical Galaxy models. The paper is organized as follows. In \S2 we rederive Kuijken's result, generalize it to elliptical systems, and demonstrate that it is applicable to the measured clump-giant optical depth. In \S3 we show that axisymmetric models are very clearly excluded. In \S4 we show that barred models can be excluded, albeit with somewhat less confidence. \section{Lower limits on the Galactic mass} We determine the minimum mass in stellar objects that is required to generate a given optical depth $\tau$ towards sources that lie on the Galaxy's symmetry axis a distance $h\ll R_0$ from the Galactic plane. The optical depth to microlensing of a stellar object at distance $s_0$ is \begin{equation}\label{basiceq} \tau={4\pi G\over c^2}\int_0^{s_0}\d s\,\rho_*\hat s, \end{equation} where $s$ is the distance to the lens and \begin{equation} \hat s=\Big({1\over s_0-s}+{1\over s}\Big)^{-1}. \end{equation} Consider the contribution to $\tau$ from a circular band of mass $M$ and radius $r$ around the Galactic centre. If we assume that the band's surface density never increases with distance from the plane, then its mass will be minimized for a given optical depth when its surface density is constant and the line-of-sight to the source just cuts its edge. So we take the band's half-width to be $h(R_0-r)/R_0$, which makes the band's surface density \begin{equation}\label{sigmaMeq} \Sigma={M\over4\pi rh}\Big(1-{r\over R_0}\Big)^{-1}. \end{equation} Substituting this into equation (\ref{basiceq}) we find the band's optical depth to be (Kuijken, 1997) \begin{equation}\label{axieq} \tau={GM\over c^2 h} \end{equation} independent of radius. Realistically we must assume that the surface density of the band falls off with distance from the plane, and if this decline is exponential with the optimal scale-height ($h[1-r/R_0]$), the band's mass must be e times that given by equation (\ref{axieq}) for a given optical depth, while if the vertical density profile is Gaussian with optimal scale-height ($h[1-r/R_0]$) equation (\ref{axieq}) underestimates the band's mass by a factor $\sqrt{\pi\e/2}\simeq2.066$. Given the implausibility of assuming that the band's scale height is optimal, we can safely conclude that the minimum stellar mass required in a circular band if the observations are to be met is \begin{equation}\label{axieq2} M_{\rm a}\equiv{2c^2h\over G}\tau=4.2\times\Big({h\over100\pc}\Big) \Big({\tau\over10^{-6}}\Big)\times10^9\msun. \end{equation} Note that the minimum mass estimate (\ref{axieq2}) holds also if the mass is widely distributed in radius rather than concentrated in a single band, because we can imagine a radially continuous mass distribution to be made up of a large number of bands, and we have shown that the optical depth contributed by each band depends only on its mass when the band is optimally configured. Can one achieve a higher optical depth within a give mass budget by making the bands elliptical rather than circular? Imagine deforming an initially circular band into an elliptical shape while holding constant the radius $r$ at which the line of sight to our sources cuts the band. It is straightforward to show that if the column density through the band to the sources is to be independent of the band's eccentricity $e$, its mass $M(e)$ must satisfy \begin{equation} M(e)=M(0){1-e^2\cos^2\phi\over\sqrt{1-e^2}}, \end{equation} where $\phi$ is the angle between the band's major axis and the Sun--centre line. For $\phi<\pi/4$, $M(e)$ is a minimum with respect to $e$ at \begin{equation}\label{emineq} e_{\rm min}=\sqrt{2-\sec^2\phi}. \end{equation} For $\phi=20\deg$, a value favoured by Binney, Gerhard \& Spergel (1997), the optimal axis ratio is $q_{\rm min}=0.36$ and $M(e)/M(0)=0.64$; for $\phi=15\deg$, we find $q_{\rm min}=0.27$ and $M(e)/M(0)=0.50$. Hence, making the bands elliptical realistically reduces the mass required to generate a given optical depth by at most $50\%$ to \begin{equation}\label{ellipeq} M_\e\equiv{c^2h\over G}\tau=2.1\times\Big({h\over100\pc}\Big) \Big({\tau\over10^{-6}}\Big)\times10^9\msun. \end{equation} Physically, there is an optimum eccentricity because the major axis of the ellipse grows with $e$, and for large $e$ this growth in scale overwhelms the reduction in the mass per unit length around the ellipse that is possible because the line of sight intersects the ellipse ever more obliquely as $e$ is increased. \subsection{Applicability to sources not on the axis} We wish to apply these formulae to sources that are broadly distributed around the Galactic centre, rather than lying precisely on the Galaxy's symmetry axis. How much error will we incur by so doing? Suppose there is a thin circular band of lenses of radius $r$ and imagine moving two sources along the line of sight away from the axis. One source moves toward and one away from the observer, with $x$ being the distance of each source from the axis. One can easily calculate as a function of $x$ the average, $\tau(x)$, of the optical depths for each source to be microlensed by the ring. We then evaluate the ratio \begin{equation}\label{taueq} {\tau(x,r)\over\tau(0,r)}=\cases{ {\displaystyle {R_0(R_0r-x^2)\over r(R_0^2-x^2)}}&for $xr$ } \end{equation} For $xr$, when the sources are outside the ring, the mean optical depth rises steadily with $x$ because there is nothing to offset the gain in optical depth for the further source. To estimate the error involved in assuming that all sources lie on the axis, we calculate the ratio \begin{equation}\label{defsfeq} f(x)={\int_{r=0}^{r=R_0}\d\Sigma(r)\tau(x,r)\over \int_{r=0}^{r=R_0}\d\Sigma(r)\tau(0,r)} \end{equation} of the total mean optical depth due to a series of rings when the sources lie distance $x$ either side of the axis, and when they lie on axis. In this calculation we assume that the surface density of each band, $\d\Sigma$, is given by equation (\ref{sigmaMeq}) with the mass of each band proportional to $r\d r\,e^{-r/R_\d}$, where $R_\d=2.5\kpc$. Fig.~\ref{spread2} shows the ratio $f$ as a function of $x$. Both the apparent-magnitude distribution of clump giants reported by Stanek et al.\ (1994) and the COBE/DIRBE near-infrared photometry of the Galaxy imply that $\sim95\%$ of the clump giants seen towards a typical MACHO field lie within $2\kpc$ of the point at which the line of sight passes closest to the Galactic centre (Bissantz \& Gerhard, 2000).\footnote{Moreover, the apparent-magnitude distribution of the clump giants that have suffered lensing is consistent with these sources lying within the bulge (Alcock et al., 1997).} From Fig.~\ref{spread2} it now follows that the maximum error made in $\tau$ by placing any of these sources on the axis is $\sim16\%$. For the $50\%$ of sources that lie within $\sim700\pc$ of the point of closest approach the maximum error is $\sim2\%$. Whereas the non-zero distribution in depth of the clump giants causes us to slightly underestimate the mean optical depth to them when we place them on the Galactic axis, the fact that many of the MACHO fields lie at $l\neq0$ gives rise to an error of the opposite sign. Maps of microlensing optical depth from models based on the COBE/DIRBE data (e.g., Fig.~7 of Bissantz et al., 1997) show that the errors from this source are also very small. \section{Lensing by an axisymmetric Galaxy} We have seen that the mean microlensing optical depth towards bulge sources at $(l,b)\simeq(2.68\deg,-3.35\deg)$ gives a reasonable approximation to the optical depth $\tau$ in the formulae above with $h=470\pc$. Setting $\tau$ equal to the value $(3.88\pm0.6)\times10^{-6}$ inferred by Alcock et al.\ (2000b) we find \begin{equation}\label{micros} M_{\rm a}=(7.6\pm1.2)\times10^{10}\msun\qquad M_{\rm e}=(3.8\pm0.6)\times10^{10}\msun. \end{equation} A crude estimate of the mass required to generate the circular speed at the Sun is \begin{equation}\label{naive} M_0=(220\kms)^2\times8\kpc/G\simeq8.9\times10^{10}\msun, \end{equation} which is barely more than the minimum stellar mass for an axisymmetric body from microlensing observations. The naive estimate (\ref{naive}) is for a spherical mass distribution, whereas the mass estimates of equation (\ref{micros}) are based on strongly flattened mass distributions, and $v_c(r)$ for a flattened body can differ significantly from $v_c(r)$ for the spherical body that has the same cumulative mass function $M(r)$. Consider therefore Fig.~\ref{vcaxi}, which shows $v_c(r)$ for two axisymmetric mass distributions in which the vertical density profile is a Gaussian in $z$ with the scale-height $\sigma$ chosen to maximize the microlensing optical depth to $(R,z)=(0,470\pc)$ subject to the auxiliary condition $\sigma\ge30\pc$. The masses of the two bodies are both $M=7.6\times10^{10}\msun$ to $R=R_0$, so they both produce optical depth $\tau\sim3.9\times10^{-6}$ to a source at $(R,z)=(0,470\pc)$. One body has a surface density $\Sigma(R)\propto\e^{-R/R_\d}$, with $R_\d=2.5\kpc$, whereas the other has $\Sigma(R)\sim R^{-0.8}$. The rotation curve of the exponential model clearly exceeds the approximate fit to the measured circular speed that is given by the short-dashed curve: $v_c(R)\sim 220(R/R_0)^{0.1}\kms$ (Binney et al., 1991). By contrast, the circular-speed curve of the power-law model lies well below the measured value of $v_c$ even though it corresponds to the same mass profile $M\propto r^{1.2}$. The reason for this is quite subtle: in this power-law disk there is at any radius $R$ a substantial outward pull from rings at $R'>R$, and this outward pull more than compensates for the fact that matter at $R'R_0$. Hence, if we taper the disk near $R_0$ to avoid conflict with the observed local densities, we will immediately generate a violation of the limit on the circular speed. A little numerical experimentation suffices to convince one that there is no way of generating the required optical depth to $(R,z)=(0,470\pc)$ with a circular disk without violating either the constraint on the circular speed or that on the local mass density. There is, moreover, negligible probability that the Galaxy will be structured vertically so as to maximize $\tau$ towards $(R,z)=(0,470\pc)$. Hence, we may be pretty sure that the measured optical depth is not produced by an axisymmetric Galaxy (Kuijken 1997). \section{Lensing by a barred Galaxy} Since it is now generally accepted that the Galaxy contains a bar, the conclusion we have just reached may not be surprising. Equations (\ref{micros}) show that non-axisymmetry can in principle reduce the requirement for stellar mass to half that required in the stellar case. It is not in practice possible to reduce the required stellar mass by so large a factor, however, because the structure of the Galaxy's stellar bar is strongly constrained by both near-IR photometry (Blitz \& Spergel, 1991; Bissantz et al., 1997) and radio-frequency observations of gas that flows in the Galactic plane (Englmaier \& Gerhard, 1999; Fux, 1999). We estimate the reduction in mass that can be achieved as follows. We suppose that the surface density of the disk projected along $z$ is exponential, but that at $R\lta R_0/2$ the material is arranged on elliptical rings; at every radius the vertical density profile is Gaussian with the dispersion that maximizes the optical depth for given mass subject to the condition $\sigma\ge30\pc$. Then, the circular speed of the disk will be the same as that of an axisymmetric disk of the same mass, while the optical depth will be given by \begin{equation}\label{Mtoteq} \tau={GM_{\rm tot}\over2c^2h}\biggl( {\sqrt{1-e^2}\over1-e^2\cos^2\phi}\,\alpha+(1-\alpha)\biggr), \end{equation} where $e$ is determined from equation (\ref{emineq}) and \begin{equation} \alpha={1-(1+R_0/2R_\d)\e^{-R_0/2R_\d}\over1-(1+R_0/R_\d)\e^{-R_0/R_\d}} \end{equation} is the fraction of the disk's mass, $M_{\rm tot}$, that lies inside $R_0/2$. We determine $M_{\rm tot}$ by setting $\tau=3.9\times10^{-6}$ in equation (\ref{Mtoteq}) and evaluate the resulting curve $v_c(r)$. The long-dashed curve in Fig.~\ref{vcaxi} shows the result for $R_\d=2.5\kpc$ and $\phi=20\deg$. Now the constraint on $v_c$ is not violated, but that on the local surface density is: the model has $72\msun\pc^{-2}$ at $R_0$. Our assumption that the central Galaxy is barred makes mass placed inside $R_0/2$ roughly twice as effective at generating optical depth as mass placed at $R>R_0/2$. Can we then evade both constraints by concentrating mass in the bar? Fig.~\ref{vcbulge} shows an attempt along these lines. The model is made up of two exponential components. The larger has a scale length $R_\d=3\kpc$ and its mass is determined by requiring that at $R_0$ its surface density is $35\msun\pc^{-2}$. At $R400\pc$ from the plane, where it can generate useful optical depth. To raise $v_c$ at $R<500\pc$ much of this matter will have to be moved down, closer to the Galactic centre, thus reducing the model's optical depth. Moreover, near-infrared photometry confirms that at $R<500\pc$ most of the bulge's mass does lie below $z=400\pc$. \item Not all mass causes microlensing. About $5\%$ of the mass of the inner Galaxy is made up of gas. At $R\gta R_0$ most mass is contained in a dark halo, most of which does not cause microlensing (Alcock \etal 2000a). The dark halo is thought to make a significant contribution to the local circular speed, and its density surely increases inwards. Hence, the peak circular speed of whatever component causes microlensing should be strictly less than the measured value of $v_c$; it is not satisfactory to conclude that it may be no larger than the measured value. \item Most importantly, it is not plausible that the inner Galaxy is structured so as to maximize the optical depth in a particular direction from a particular star. In fact mass models obtained by deprojecting the near-infrared photometry and assuming that mass follows light are able to reproduce many kinematic properties of the inner galaxy (Englmaier \& Gerhard, 1999; H\"afner et al., 2000). If mass even approximately follows light, the photometry implies that the inner Galaxy is not structured so as to maximize the optical depth produced by a given mass. In fact the recent model of Bissantz \& Gerhard (2000) yields $\tau\simeq1.4\times10^{-6}$ in Baade's window. \end{enumerate} Thus, we conclude that the marginal satisfaction of the three principal constraints [optical depth, $v_c$, $\Sigma(R_0)$] that Fig.~\ref{vcbulge} implies cannot be considered satisfactory and that even realistic barred models cannot simultaneously satisfy all the observational constraints. \begin{thebibliography}{99} \bibitem{A+97} Alcock, C., et al., 1997, \apj 479 119 \bibitem{A+00} Alcock, C., et al., 2000a, submitted to ApJ, astro-ph/0001272 \bibitem{A+00b} Alcock, C., et al., 2000b, submitted to ApJ, astro-ph/0002510 \bibitem{BGSBU} Binney, J.J., Gerhard, O.E., Stark, A.A., Bally, J., Uchida, K.I., 1991, \mn 252 210 \bibitem{BGS} Binney, J.J., Gerhard, O.E., Spergel, D.N., 1997, \mn 288 365 \bibitem{BEBG} Bissantz, N., Englmaier, P., Binney, J.J. \& Gerhard, O.E., 1997, \mn {289} {651} \bibitem{BG} Bissantz, N., Gerhard, O.E., 2000, in preparation \bibitem{BS} Blitz, L., Spergel, D.N., 1991, \apj 379 631 \bibitem{creze} Cr\'ez\'e, M., Chereul, E., Bienaym\'e, O. \& Pichon, C., 1998, \aa 329 920 \bibitem{EG} Englmaier, P. \& Gerhard, O.E., 1999, \mn 304 512 \bibitem{FX} Fux, R., 1999, \aa 345 787 \bibitem{HEDB} H\"afner, R.M.,Evans, N.W., Dehnen, W., Binney, J.J., 2000, MNRAS, in press, astro-ph/9905086 \bibitem{HolmF} Holmberg J., Flynn C., 2000, \mn 313 209 \bibitem{Kuijken} Kuijken K., 1997, \apj 486 L19 \bibitem{KuijkenG} Kuijken, K. \& Gilmore, G., 1991, \apj 367 L9 \bibitem{McWill} McWilliam A., Rich R.M., 1994, \apjsupp 91 749 \bibitem{Stanek+} Stanek, K. Z., Mateo, M., Udalski, A., Szymanski, M., Kaluzny, J., Kubiak, M., 1994, \apj 429 L73 \end{thebibliography} \begin{figure} % spread2.ps = figure1.ps \centerline{\psfig{file=figure1.ps,width=.8\hsize}} \caption{The ratio (\ref{defsfeq}) of optical depths. The abscissa $x$ is the distance of the sources from the axis. The lenses are distributed such that their mass density projected along $z$ is proportional to $\e^{-R/R_\d}$ with $R_\d=2.5\kpc$.\label{spread2}} \end{figure} \begin{figure} % vcaxi.ps = figure2.ps \centerline{\psfig{file=figure2.ps,width=.8\hsize}} \caption{Full curves: the circular speeds of axisymmetric bodies that generate $\tau\sim3.9\times10^{-6} $ towards $(R,z)=(0,470\pc)$ with the minimum mass, $7.6\times10^{10}\msun$ interior to $R_0$. The curve that is lower at $R_0$ is for a surface mass density $\Sigma\sim R^{-0.8}$, while the upper curve is for $\Sigma\sim\e^{-R/R_\d}$ with $R_\d=2.5\kpc$. The short-dashed curve shows the approximate analytic form $v_c(r)\lta 220(R/R_0)^{0.1}$ (Binney et al., 1991). The long-dashed curve shows the circular speed of the barred exponential galaxy described in the text. \label{vcaxi}} \end{figure} \begin{figure} % vcbulge.ps = figure3.ps \centerline{\psfig{file=figure3.ps,width=.8\hsize}} \caption{Full curve: the circular speed of the two-exponential elliptical model described in the text. The dashed curve is the same analytic estimate of $v_c$ that is shown in Fig.~\ref{vcaxi}.\label{vcbulge}} \end{figure} \end{document} ------------- End Forwarded Message ------------- ------------- End Forwarded Message ------------- ------------- End Forwarded Message ------------- ------------- End Forwarded Message -------------