Limits on the microlens mass function of Q2237+0305

Abstract

Gravitational microlensing at cosmological distances is potentially a powerful tool for probing the mass functions of stars and compact objects in other galaxies. In the case of multiply imaged quasars, microlensing data have been used to determine the average microlens mass. However, the measurements have relied on an assumed transverse velocity for the lensing galaxy.

Since the measured mass scales with the square of the transverse velocity, published mass limits are quite uncertain. In the case of Q2237+0305 we have properly constrained this uncertainty. The distribution of light-curve derivatives allows a quantitative treatment of the relative rates of microlensing arising from proper motions of microlenses, the orbital stream motion of microlenses and the bulk galactic transverse velocity. By demanding that the microlensing rate arising from the motions of microlenses is the minimum that should be observed we determine lower limits for the average mass of stars and compact objects in the bulge of Q2237+0305. If microlenses are assumed to move in an orbital stream the lower limit ranges between 0.005 and 0.023 M_⊙, where the systematic dependence arises from the fraction of smooth matter and the size of photometric error assumed for the published monitoring data. However, if the microlenses are assumed to move according to an isotropic velocity dispersion then a larger lower limit of 0.019-0.11 M_⊙ is obtained. A significant contribution of Jupiter-mass compact objects to the mass distribution of the galactic bulge of Q2237+0305 is therefore ruled out unambiguously.

quasars: general, quasars: individual: Q2237+0305, gravitational lensing

1 Introduction

Q2237+0305 comprises a source quasar with a redshift of z=1.695 that is gravitationally lensed by a foreground galaxy at z=0.0394, producing four images with separations of ∼1 arcsec. Each of the four images are observed through the bulge of a galaxy which has an optical depth in stars that is of the order of unity (e.g. Kent & Falco 1988; Schneider et al. 1988; Schmidt, Webster & Lewis 1998). In addition, the proximity of the lensing galaxy means that the effective transverse velocity may be high. The combination of these facts make Q2237+0305 the ideal object from which to study microlensing. Indeed, Q2237+0305 is the only object in which cosmological microlensing has been confirmed (Irwin et al. 1989; Corrigan et al. 1991).

Numerical microlensing simulations (e.g. Wambsganss, Paczynski & Schneider 1990; Witt, Kayser & Refsdal 1993) have shown that the statistics of microlensed high-magnification events (HMEs) obtained from long-term monitoring may provide information on properties of the lens such as the stellar mass function and the percentage of mass in stars for the bulge. However, the monitoring period required is greater than 100 yr (Wambsganss et al. 1990). There are several other unknown quantities in the problem. These include the magnitude and direction of any transverse motion, as well as the source size. In particular, the magnitude and direction of the transverse motion are degenerate with the density of caustics (a function of the mean compact object mass) for a given set of microlensing statistics.

Microlensed fluctuations in the continuum of the quasar result from motion owing both to a galactic transverse velocity and to the random proper motion or stream motion of stars and compact objects. Wyithe, Webster & Turner (2000; hereafter WWTb) define the equivalent transverse velocity as the transverse velocity in a model containing stars with static positions that produces a rate of microlensing most closely resembling that of a model containing stellar proper motions. Foltz et al. (1992) measured the central velocity dispersion of Q2237+0305 to be ∼215 km s⁻¹, and theoretical models (Schmidt et al. 1998) predict a value of ∼165 km s⁻¹. If the dispersion is isotropic then [as discussed in Wyithe, Webster & Turner (1999; hereafter WWTa)] the equivalent transverse velocity calculated from the overall microlensing rate (all four images) of Q2237+0305 is larger than this value. The microlensing rate that results from the line-of-sight velocity dispersion (of an isotropic distribution) is therefore comparable to that of the likely transverse velocity. This means that the often-made assumption that random proper motions provide a negligible contribution to microlensing in Q2237+0305 is incorrect.

While the microlensing rate resulting from random stellar motions is dependent on the mean microlens mass, the equivalent transverse velocity (WWTa; WWTb) that describes the microlensing rate is not. We use this fact to break the degeneracy between the microlensing rate and the mean microlens mass, and hence obtain useful limits on the mass function along the line of sight through the bulge of the lensing galaxy.

This paper is presented in five parts. Section 2 contains a general discussion of microlensing and the mass function. Section 3 discusses the numerical methods used to model microlensing in Q2237+0305 and Section 4 describes a method to place limits on the mass function through consideration of microlensing arising from both transverse velocity and stellar velocity dispersion.

2 Microlensing and the stellar mass function

Stellar mass functions have traditionally been measured through the combination of an observed luminosity function and an empirical mass—luminosity relationship. However, because of the difficulties inherent in the observations of faint stars, these determinations become uncertain as the hydrogen burning limit (∼0.08 M_⊙) is approached and crossed. In contrast to this approach, microlensing uses the mass of a lens to magnify the observed flux of a background source through gravitational deflection of the light bundle. The statistics obtained are therefore free of any bias introduced by the hydrogen burning limit, and so microlensing is a powerful tool for determining the contribution to the mass function of low-mass stars and dark compact objects.

Gravitational microlensing is observed in two very different regimes. Paczynski (1986) suggested surveying millions of stars in the Magellanic Clouds for microlensing-induced flux variation as a way of detecting solar-mass compact objects in the halo of our own Galaxy. Several groups have undertaken such searches (e.g. Alcock et al. 1997a; Renault et al. 1998). In a similar vein, Paczynski (1991) and Griest et al. (1991) suggested microlensing experiments towards the Galactic bulge as a method to probe the masses of stars along the line of sight in the Galactic disc. Microlensing searches towards the Galactic bulge have since found a higher rate of microlensing events than are found towards the Large Magellanic Cloud (LMC) (e.g. Paczynski et al. 1994; Alcock et al. 1997b). In a very different microlensing regime, the observed flux of a background quasar can be altered by gravitational lensing by stars or halo objects in a foreground galaxy. Q2237+0305 is an example of a quasar that lies very close to a galactic line of sight. In such cases multiple imaging occurs, allowing the microlensed variation to be easily separated from intrinsic fluctuations which are observed in all images.

The analyses of microlensing in the Galactic and cosmological regimes are very different. While there are only a few sources (quasar images) in the cosmological case rather than the millions available in Galactic microlensing searches, the optical depth (or equivalently the probability of lensing) is a factor of ∼10⁶ greater. The transverse velocity is a critical parameter for the determination of a mass or masses responsible for a microlensing event, regardless of the microlensing scenario. Unfortunately, it is unknown in both cases. In Galactic microlensing calculations the velocities are inferred from an assumed distribution. However, in the cosmological case the transverse velocities of the lensing objects are not independent. Rather, they are equal except for the contribution of the individual stellar proper motions. Moreover, cosmological microlensing results from the gravitational contribution of a large ensemble of many hundreds of masses rather than a single microlens. Microlensed, multiply imaged quasars are an excellent probe of the mass function of compact objects along the quasar image line of sight because they interact with a large number of microlenses.

Several authors have placed limits on the masses of microlenses responsible for cosmological microlensing. Schmidt & Wambsganss (1998) use the lack of an observed variation in Q0957+561 to place a lower limit on the mass of microlenses in the halo of the lensing galaxy. They rule out a mean mass of halo objects 〈m〉≪10⁻² M_⊙. However, their determination is dependent on the source size and the fraction of smooth matter employed in their calculations. A transverse velocity of υ_t=600 km s⁻¹ is assumed. The uncertainty in this assumed value is the most serious problem for this kind of analysis because the values of mass obtained are Lewis & Irwin (1996) compared the monitoring data of Q2237+0305 with simulations using a structure function to analyse variability. They too assumed a transverse velocity of υ_t=600 km s⁻¹ and concluded that the mean mass of objects in Q2237+0305 is 0.1≲〈m〉≲10 M_⊙. This result is also proportional to the square of the unknown transverse velocity.

While they do not measure precisely the same quantity, the above results are supportive of those of the MACHO experiment (e.g. Alcock et al. 1997a,b). From microlensing observations towards the LMC by the Galactic halo, Alcock et al. (1997a) conclude that the average MACHO masses are 0.08–0.93 M_⊙. The quoted range includes both statistical uncertainties and systematic effects introduced by the assumption of the halo model. In addition, they find evidence for an excess of events over those predicted from stellar lensing alone. Alcock et al. (1997b) find that the range of time-scales towards the Galactic bulge is consistent with microlens masses of 0.1≲〈m〉≲1.0 M_⊙. Table 1 summarizes the results described.

Table 1.

A collection of results from microlensing data for the average mass of compact objects in galactic haloes and/or bulges of the Milky Way and quasar lensing galaxies. The velocity is in units of km s⁻¹.

3 The microlensing models

3.1 Microlensing models for Q2237+0305

Throughout this paper, standard notation for gravitational lensing is used. The Einstein radius of a 1-M_⊙ star in the source plane is denoted by η₀. The normalized shear is denoted by γ and the convergence or optical depth by κ. The model for gravitational microlensing consists of a very large sheet of point masses that simulates the section of galaxy along the image line of sight, together with a shear term that includes the perturbing effect of the mass distribution of the lensing galaxy as a whole. The normalized lens equation for a field of point masses with an applied shear in terms of these quantities is

(1)

where x and y are the normalized image and source positions, respectively, and and m^j are the normalized positions and masses of the microlenses. κ_c is the optical depth in smoothly distributed matter. Equation (1) is solved for the macroimage magnification at many points along a pre-defined source trajectory through the inversion technique of Lewis et al. (1993) and Witt (1993). The region of the lens plane in which image solutions need to be found to ensure that 99 per cent of the total macroimage flux is recovered from a source point was described by Katz, Balbus & Paczynski (1986). The union of areas in the lens plane, corresponding to the flux collection area of each point on the source line, is known as the shooting region; the method for determining the dimensions of this region is described in Lewis & Irwin (1995) and Wyithe & Webster (1999). The radius of the disc of point masses is chosen to be 1.2 times that required to cover this shooting region.

We assume the macroparameters for Q2237+0305 calculated by Schmidt et al. (1998) (the values are shown in Table 2). Two models are considered for the mass distribution: one with no continuously distributed matter and one where smooth matter contributes 50 per cent of the surface mass density. Both the microlensing rate arising from a transverse velocity (Witt et al. 1993; Lewis & Irwin 1996; WWTa) and the corresponding rate due to proper motions (WWTb) are not functions of the details of the microlens mass distribution, but depend only on the mean microlens mass.

Table 2.

Values of the total optical depth and the magnitude of the shear at the position of each of the four images of Q2237+0305. The quoted values are those of Schmidt et al. (1998). κ* and κ_c are the optical depths in stars and in smoothly distributed matter, respectively.

The mean of the microlens mass function can therefore be determined independently of its form. On the other hand, information on the form of the mass function is not obtainable through consideration of the microlensing rate. We limit our attention to models in which all the point masses have identical masses.

Each of our models was computed for a source track of length 10η₀ (corresponding to ∼70 yr for an effective transverse velocity of 600 km s⁻¹). We set 500 to be the minimum number of stars in a model. Two orientations were chosen for the transverse velocity with respect to the galaxy, with the source trajectory being parallel to the A–B or C–D axes. At each orientation, 100 simulations were made for each of the four images of Q2237+0305 in combination with each of the model mass distributions. The two orientations bracket the range of possibilities, and, because the images are positioned approximately orthogonally with respect to the galactic centre, correspond to shear values of γ_A,γ_B<0, γ_C,γ_D>0 and γ_A,γ_B<0,γ_C,γ_D>0, respectively. Table 3 shows the number of stars used for each of these models along with the mean magnification 〈μ〉 and the theoretical magnification 〈μ_th〉 for comparison. The average magnification was found from the combination of the two sets of models and the error calculated from the standard deviation of a subset of six simulations (three per model orientation). Fig. 1 shows the magnification distributions for each image in each model. Simulations having γ<0 and γ>0 are represented by light and dark lines, respectively. The distributions are quantitatively comparable to those presented for similar values of κ and γ in Lewis & Irwin (1995), and demonstrate that the magnification distribution is independent of the source direction as required.

Table 3.

Values for the number of stars in the microlensing models (left-hand column γ>0; right-hand column γ<0), the average model magnifications and the theoretical values for comparison.

Figure 1.

The magnification distributions for images A/B (left), C (centre) and D (right). The top and bottom rows show the distributions for models that have no smooth-matter component, and a 50 per cent smooth-matter component, respectively. The light and dark lines represent distributions that are sampled parallel to and at right angles to the shear.

We assume that microlensing is produced through the combination of a galactic transverse velocity with each of two classes of proper motion for individual stars with respect to the galaxy: an isotropic velocity dispersion and a circular stream motion. Moreover, we assume that the magnitudes of the dispersion or stream motions are the same for each of the four images. We take the theoretical value of σ*∼165 km s⁻¹ for the line-of-sight velocity dispersion of the stars in the galactic bulge. This is lower than the value observed by Foltz et al. (1992), so lower limits placed on the mass are more conservative.

3.2 The effective transverse velocity

We define the effective transverse velocity as the transverse velocity that produces a microlensing rate from a static field model equal to that of the observed light curve. The effective transverse velocity therefore describes the microlensing rate arising from the combination of the effects of a galactic transverse velocity and random microlens proper motion.

Our calculation of effective transverse velocity utilizes the cumulative histogram of derivatives calculated from the six difference light curves (A–B, A–C, A–D, B–C, B–D, C–D). Through computation of the difference light curves, the intrinsic flux variation and the systematic component of the observational uncertainty are removed from the data. WWTa describes a procedure for determining the probability that the effective transverse velocity is less than an assumed value. That paper provides correct upper limits, but does not obtain the cumulative probability function. A modified procedure to determine the cumulative probability for effective transverse velocity from ensembles of mock data sets is described below.

Many mock observations of the six difference light curves were produced from model light curves with a sampling rate and period that are identical to the published monitoring data (Irwin et al. 1989; Corrigan et al. 1991; Østensen et al. 1996). Observational errors were also simulated by applying a random fluctuation to each point on the light curve, distributed according to a Gaussian with a half-width σ describing the published photometric error.

The simulations used two different estimates for the uncertainty in the photometric magnitudes. In the first case a small error was assumed (SE). For images A and B, σ_SE=0.01 mag, and for images C and D σ_SE=0.02 mag. In the second case, a larger error was assumed (LE). For images A and B, σ_LE=0.02 mag and for images C and D σ_LE=0.04 mag. The observational error in Irwin et al. (1989) was 0.02 mag. We note that the adoption of a larger error leads to a smaller bound on the measured upper limit of effective transverse velocity, and subsequently a larger lower bound on the mean mass.

The simulations assume a point source since the value of the effective transverse velocity measured with a model is not a sensitive function of the source size (in the range of interest) assumed (WWTa). The insensitivity arises from the fact that the measurement is derived predominantly from points that are not part of an HME.

We define a set of effective transverse velocities (V_eff) and determine the probability, based on light-curve derivatives, that each of these correctly describes the true value for Q2237+0305 (V_gal). For each model, 5000 mock observations were produced at each effective transverse velocity from the four sets (one per image) of 100 10η₀ simulated light curves. As a result of the finite sampling period, histograms of microlensed light-curve derivatives produced from the individual mock observations describe typical but not average behaviour. Therefore an average histogram was also computed from each set of mock observations. For each mock observation at each pre-defined effective transverse velocity (V_eff), a mock measurement of effective transverse velocity υ_eff was made by minimizing the Kolmogorov—Smirnov (KS) difference between the average histogram for υ_eff and the histogram of the mock observation. Thus we calculate the function representing the likelihood p_lhood(υ_eff|V_eff) for observing υ_eff given an assumption for the true value (V_eff).

Similarly, from the observed histogram of microlensed light-curve derivatives we find the effective transverse velocity (υ_obs) that best describes the observed microlensing rate. Using Bayes' theorem we calculate the posterior probability that the effective galactic transverse velocity V_gal is less than an assumed value V_eff:

(2)

where p_prior(V_eff) is the prior probability for V_eff.

We have assumed two different and physically unmotivated priors. These are, respectively, flat:

(3)

and logarithmic:

(4)

These priors bracket the range and spread of physically plausible priors such as a Gaussian based on observed peculiar velocities (e.g. Mould et al. 1993) or those computed from numerical studies. Our results are insensitive to the choice of these priors, and we conclude that our estimate of probability is dominated by the microlensing observations rather than our assumed prior for V_eff.

P_υ(υ_gal<V_eff|υ_obs) is plotted in Fig. 2 for the models discussed in this paper (〈m〉=1 M_⊙). In these plots the solid and dot—dashed lines correspond to source trajectories along the C–D and A–B image axes, and the dark and light lines correspond to models with 0 and 50 per cent smoothly distributed matter, respectively. The upper and lower plots correspond to the cases where the photometric error was assumed to be σ_SE=0.01 mag in images A/B and σ_SE=0.02 in images C/D, and σ_LE=0.02 mag in images A/B and σ_LE=0.04 in images C/D.

Figure 2.

Plots of the cumulative probability (P_υ) for the effective transverse velocity. The solid and dot—dashed lines correspond to the models with trajectory directions that are aligned with the C–D and A–B image axes, respectively. The dark and light lines represent results from models containing 0 and 50 per cent continuously distributed matter. The upper and lower plots correspond to the cases where the photometric error was assumed to be σ_SE=0.01 mag in images A/B and σ_SE=0.02 in images C/D, and σ_LE=0.02 mag in images A/B and σ_LE=0.04 in images C/D. The model assumed 1-M_⊙ microlenses.

3.3 The contribution to microlensing of stellar proper motions

The average histogram of light-curve derivatives can be computed for the case where microlensing results from a random velocity dispersion rather than a transverse velocity. We define the equivalent transverse velocity as the transverse velocity that in combination with a static microlensing model produces a microlensing rate closest to the rate produced by a random velocity dispersion of the point masses. The microlensing rates are considered to be equivalent at the transverse velocity that produces a cumulative histogram of light-curve derivatives closest (has the minimum possible KS difference) to the corresponding proper motion histogram.

The formalism required for the computation of the cumulative distribution of derivatives resulting from proper motion of stars was developed in WWTb. The upper panel of Table 4 shows the values of equivalent transverse velocity of a Gaussian one-dimensional (1D) velocity dispersion with a half-width of σ_*=165 km s⁻¹. These values were computed from the difference light curves for models of Q2237+0305 (the quoted error describes the range of values obtained over three separate sets of simulations). The two cases shown are for a trajectory that is parallel to the shear in images A and B (γ_A,γ_B>0), and for one that is parallel to the shear in images C and D (γ_C,γ_D>0). In all cases the equivalent transverse velocity is larger than the 1D velocity dispersion.

Table 4.

The equivalent transverse velocities and the 99, 95 and 90 per cent lower limits for the mean microlens mass as well as the most likely value. Values for the mass limits are shown corresponding to the cases where the error was σ_SE=0.01 mag in images A/B, σ_SE=0.02 mag in images C/D, and where the error was σ_LE=0.02 mag in images A/B, σ_LE=0.04 mag in images C/D (in parentheses). Values are shown that correspond to an isotropic velocity dispersion of σ*=165 km s⁻¹ (top table), and a circular stream velocity of υ_stream=233 km s⁻¹ (bottom table).

4 The mass of compact objects in Q2237+0305

4.1 Placing limits on the mass using the effective transverse velocity

In this section we describe a method for determining the lower limit for the average compact object mass. The microlensing rate has a minimum possible value determined by the size of the stellar velocity dispersion, thus there is a minimum mass which can explain the observed rate. This is quantified by noting that the measured effective transverse velocity must be greater than the equivalent transverse velocity.

There is a simple scaling that relates models of the same optical depth but consisting of model stars with a different mass. The dimensionless Einstein radius of a point mass is Therefore, reducing all microlens masses by a factor of a reduces all physical distances (in both the source and lens planes) within dimensionless coordinates by a factor of (e.g. Witt et al. 1993). For a given transverse velocity this results in an increase by a factor of in the gradient of the light curve at all points. Similarly, the rate of change of magnification at a static source point arising from stellar proper motions increases by a factor Since it results from the proper motions the minimum rate is larger if a smaller mean microlens mass is assumed. However, the minimum rate cannot statistically exceed the observed rate. The ratio of the equivalent transverse velocity to the velocity dispersion is independent of the mean microlens mass; however, the measurement of the effective transverse velocity is proportional to This allows limits to be placed on the value of a and therefore the stellar mass from the observed microlensing rate.

4.2 Mass limits from random proper motions

For a model where γ_A,γ_B>0, and the simulated errors are assumed to be σ_SE=0.01 and 0.02 mag in images A/B and C/D, respectively, the effective transverse velocity is less than at the 99 per cent level, where 〈m〉 is the microlens mass assumed. In the absence of a galactic transverse velocity, the minimum level of microlensing is described by the equivalent transverse velocity of the stellar velocity dispersion V_equiv(DISP)=280 km s⁻¹ (see Table 4). The value of 〈m〉 at which V_upper(99 per cent) drops below V_equiv(DISP) is m_low(99 per cent):

(5)

The minimum mass of stars in this model is therefore ∼0.04 M_⊙. This calculation assumes a value for the 1D velocity dispersion of 165 km s⁻¹. However, we note that all lower limits obtained in this section are proportional to the square of this value. The procedure also assumes that the measured effective transverse velocity is independent of the assumed source size. WWTb show that this is approximately true for source sizes smaller than ≈ This value compares favourably with the findings of Wambsganss et al. (1990) that a source smaller than ∼ is required typically to reproduce the observed HME amplitude.

The curves shown in Fig. 2 represent the cumulative probability that the effective transverse velocity V_gal is less than It follows that the cumulative probability that the mean microlens mass (in M_⊙) is less than the value

(6)

is

(7)

The calculation of m_low(99 per cent) performed above assumes that the physical transverse velocity of the galaxy with respect to the observer source line of sight V_tran=0. However, since the galaxy may have a component of transverse motion, V_equiv, which describes the minimum microlensing rate, is a function of V_tran. The dependence of V_equiv on V_tran is determined according to the relationships described in WWTa. Since the galactic transverse velocity is not necessarily zero, U_m(〈m〉<M|V_equiv) must be determined at a series of transverse velocities and combined with prior probabilities for V_tran to obtain estimates for the probability of the mean microlens mass. We have assumed the flat and logarithmic priors employed for the calculation of P_υ(V_gal<V_eff):

(8)

The probability density p_m(〈m〉) for the mean microlens mass is obtained by taking the derivative dU_m/d〈m〉. Note that this distribution does not represent the mass function.

Fig. 3 shows the functions p_m(〈m〉) for the 0 and 50 per cent smooth-matter models for both directions considered. The solid and dotted curves represent functions assuming the photometric error to be σ_SE=0.01 mag in images A/B and σ_SE=0.02 in images C/D, and σ_LE=0.02 mag in images A/B and σ_LE=0.04 in images C/D. The light and dark lines refer to the logarithmic and flat priors for transverse velocity. Results for m_low(99 per cent), m_low(95 per cent), m_low(90 per cent) and the mode of the distribution are shown in the top panel of Table 4. In each column, the values correspond to the two assumptions for the simulated error referred to above. m_low(99 per cent) is >0.02 M_⊙ in all models considered. The mode of the distribution ranges between 0.04 and 0.45 M_⊙. It is interesting to note that these models produce distribution modes that are similar to the mean of a Salpeter distribution with a lower cut-off at the hydrogen burning limit. We also note that the distribution is highly asymmetric, with the mode being of the same order as the lower limit.

Figure 3.

The differential distributions for the mean microlens mass. The solid and dotted curves represent the resulting functions when the photometric error was assumed to be σ_SE=0.01 mag in images A/B and σ_SE=0.02 in images C/D, and σ_LE=0.02 mag in images A/B and σ_LE=0.04 in images C/D. The light and dark lines correspond to the assumptions of logarithmic and flat priors for effective transverse velocity referred to in the text.

4.3 The effect of systematic assumptions

In contrast to the calculation of the assumed prior for V_eff makes a significant difference to the microlens mass limits obtained. In particular, low-mass microlenses are less likely if large transverse velocities are assumed. This dependence on the prior illustrates the limitation of the light-curve data for breaking the degeneracy between the galactic transverse velocity and the microlens mass in the high mass—velocity regime.

The assumption of a larger photometric uncertainty yields a lower estimate of the transverse velocity (WWTa). This in turn means that a larger estimate is made for the mean microlens mass. The effect is readily apparent in Fig. 3 as well as from Table 4.

We find lower microlens mass limits if a non-zero fraction of smooth matter is assumed. This results from the combination of a larger determination of effective transverse velocity and a smaller relative contribution to microlensing of the stellar proper motions resulting in a smaller minimum for the microlensing rate. This is expected since no microlensing will be observed because of a galaxy composed entirely of continuously distributed matter. The plots in Fig. 3 demonstrate the level of dependence on the smooth-matter component.

p_m(〈m〉|V_tran) is extremely sensitive to the level of photometric error assumed at large 〈m〉. In addition, microlenses of arbitrarily high mass can produce the required rate in combination with a correspondingly large transverse velocity. Therefore at large 〈m〉, p_m(〈m〉) is a sensitive function of the prior assumed. For these reasons our method does not place useful upper limits on the mean microlens mass.

4.4 Simultaneous limits on mean microlens mass and transverse velocity

To explore the co-dependence of the determined mean microlens mass and galactic transverse velocity we have computed the two-dimensional distribution

(9)

Fig. 4 shows contour plots of this distribution for the assumed source orientations, smooth-matter fraction and photometric errors. All possible combinations of the two values of each parameter are included, so that eight models are shown. The contours are at 0.1, 1.0, 3.6, 14, 26 and 61 per cent of the peak height, and so the extrema of the inner four contours represent 4σ, 3σ, 2σ and 1σ limits on the single variables. The most important benefit of combining the distributions of V_tran and 〈m〉 is that upper limits can be placed on the mean microlens mass and galactic transverse velocity. The limits on 〈m〉 and V_tran are subject to both random and systematic uncertainties. Assuming that the chosen values for these parameters bracket the likely models, upper and lower limits for 〈m〉, and an upper limit for V_tran can be found by using the most extreme values of the contour limits. This yields the 95 per cent result that 0.01≲〈m〉≲1 M_⊙ and V_tran≲300 km s⁻¹ (95 per cent). The upper limit is lower than that obtained by Lewis & Irwin (1996), although sufficiently high that it does not conflict with any popular models of galactic haloes or bulges. The elongation of the outer contours with a power-law slope of 1/2 illustrates the degeneracy between the microlens mass and transverse velocity at large masses.

Figure 4.

Contours of percentage peak height of the function p_m,υ(〈m〉,V_tran). The contours shown are the 0.1, 1.0, 3.6, 14, 26 and 61 per cent levels. The solid and dotted curves represent the resulting functions when the photometric error was assumed to be σ_SE=0.01 mag in images A/B and σ_SE=0.02 in images C/D, and σ_LE=0.02 mag in images A/B and σ_LE=0.04 in images C/D.

4.5 The effect of trajectory direction

Fig. 5 shows the magnification maps for images A and C in the cases of 0 and 50 per cent continuously distributed matter. The caustic networks in both images are significantly stretched so that a source could travel a significant distance in the direction of caustic clustering and be subject to little microlensed flux variation. The measured effective transverse velocity is dependent on the trajectory direction, with a larger measurement resulting from cases where the source trajectory is perpendicular to the shear in images C and D (γ_C,γ_D<0). The larger measurement quantifies the greater stretching of the caustic network in these images, creating a larger difference between the rate of microlensing for sources moving with and against the shear. The variation with orientation of the measurements of effective transverse velocity is slightly larger when the model contains a component of smooth matter, owing to additional stretching of the caustic network.

Figure 5.

The magnification maps for images A (top) and C (bottom), in the cases of 0 per cent smooth matter (left) and 50 per cent smooth matter (right). The point masses each had a mass of 1 M_⊙.

A similar argument applies to the determination of the equivalent transverse velocity of a stellar velocity dispersion. The rate of microlensing due to stellar proper motions at a series of fixed source points is independent of the direction in which those points are sampled, regardless of the contributing fraction of smoothly distributed matter. However, microlensing that results from a transverse velocity has a rate that varies more with direction in images C and D. The equivalent transverse velocity is therefore a function of direction. The effect is more pronounced in models that include a smoothly distributed mass component.

Thus there are two separate effects. First, if the source trajectory is aligned with the image C–D axis, the measured effective transverse velocity increases. This effect is magnified by the inclusion of a component of smooth matter. In addition, the equivalent transverse velocity of the stellar proper motions is also increased under the same circumstances. Interestingly, during our calculation of a lower mass limit these effects tend to have a cancelling effect, rendering the limits reasonably insensitive to the source trajectory direction assumed.

4.6 Mass limits from stellar stream motions

The calculations in Sections 4.2 and 4.4 assumed that microlensing results from the combination of stellar proper motions and galactic transverse velocity. However, the bulge dynamics may be dominated by rotational motion resulting in stream motions of the starfield. For an isothermal sphere model of the bulge, the rotational velocity is Kundic, Witt & Chang (1993) obtained the expression for the caustic velocity υ_caust resulting from a stream motion υ_stream=(υ_stream,0):

(m10)

If the rotational motion is assumed to be circular then the stream motions are perpendicular to the image—galactic centre axis, and therefore parallel to the shear vector at each image. We choose the stream motions to be in the x₁-direction, making the shear positive for all four images. An equivalent transverse velocity was defined for the stream motion and determined in analogy to that for the stellar proper motions. For a circular stream motion of υ_stream=233 km s⁻¹, equivalent transverse velocities [V_equiv(STREAM)] were calculated for the two transverse directions discussed previously. These values are presented in Table 4. When combined with the equivalent velocities for an isotropic velocity dispersion, the values describe the contribution to the microlensing rate of the extreme cases for stellar motions in the galactic bulge.

The lower panel in Table 4 shows results for models with stream motions. Values are shown for m_low(99 per cent), m_low (95 per cent), m_low(90 per cent) and the mode for all models considered. m_low(99 per cent) is >0.005 M_⊙ in all models. The equivalent transverse velocities presented in Table 4 show that an orbital stream motion produces a much lower microlensing rate than a transverse velocity of equal magnitude (independent of its direction). The reason for this behaviour is apparent from equation (10); a positive shear reduces the caustic velocity resulting from stream motions. In addition, the caustic velocity and hence microlensing rate are also decreased when there is a component of smooth matter. The lower microlensing rate translates to an equivalent transverse velocity that is smaller than the one obtained from a stellar velocity dispersion. This in turn means a lower estimate of m_low(P). In the case discussed, the mass limit obtained is only ∼0.2 times that where the proper motions are assumed to be isotropic.

The last row of Table 1 summarizes the range of values for the mean microlens mass obtained by this study. These compare favourably with the results of related studies, none of which finds that Jupiter-mass compact objects are the dominant component of mass in the bulges and or haloes of the spiral galaxies studied.

5 Conclusion

Calculations of the transverse velocity from published monitoring data of Q2237+0305, as well as complementary characterizations of caustic clustering from previous studies, show that the caustic networks produced by populations of microlenses have a structure independent of the mass spectrum, but a scalelength proportional to the square root of the mean microlens mass. Previous determinations of average mass from microlensing data have therefore been proportional to the square of the unknown transverse velocity. However, a minimum rate of microlensing is produced by the proper motions of microlenses in the bulge of the lensing galaxy. We have used limits on the effective transverse velocity obtained from published monitoring data for Q2237+0305 in combination with calculations of the microlensing effect of isotropic stellar proper motions to calculate the probability for the mean microlens mass. We find that the most likely value for the mean microlens mass is 0.04–0.45 M_⊙. The lower limits on the mean microlens mass responsible for the observed microlensing are 〈m〉>0.02–0.24 M_⊙ (99 per cent level). A similar calculation assuming an orbital stream motion of stars rather than a velocity dispersion produces a most likely value of 0.01–0.05 M_⊙ and a lower limit of 〈m〉>0.005–0.071 M_⊙ (99 per cent level). Through joint consideration of the probability for mean microlens mass and transverse velocity under the assumption of an isotropic velocity dispersion, we obtain a mass between 0.01 and 1.0 M_⊙ (95 per cent) and a galactic transverse velocity of less than 300 km s⁻¹ (95 per cent).

Our result that very low-mass objects do not comprise a significant mass fraction of microlenses in Q2237+0305 supports the results of Schmidt & Wambsganss (1998) who rule out microlenses in Q0957+561 having 〈m〉≪0.01[υ_t/(600 km s⁻¹)]². The most likely mean microlens mass of ∼0.2 M_⊙ supports both the conclusions of Lewis & Irwin (1996) who found that microlenses in Q2237+0305 have 0.1≲〈m〉[υ_t/(600 km s⁻¹)]²≲10 M_⊙, and the conclusion of the MACHO collaboration (Alcock et al. 1997a,b) who find from microlensing in the Milky Way that 〈m〉≳0.05 M_⊙ in the halo and 0.1≲〈m〉≲1.0 M_⊙ towards the bulge.

While our limits depend on the correctness of the published macrolensing parameters, as well as on the fraction of smoothly distributed matter assumed, they do not depend on other unknowns present in the problem, including the source size, source intensity profile, effective transverse velocity and direction of the source trajectory. We have investigated models having smooth-matter fractions of 0 and 0.5, and find that smaller masses can explain the observed microlensing rate if a non-zero smooth-matter fraction is assumed. Our results depend on the Bayesian prior chosen for galactic transverse velocity. However, we have chosen priors to bracket physically reasonable choices and find that the contribution of the prior to the systematic uncertainty is smaller than that of the assumption for the smooth-matter fraction.

Our results unambiguously rule out a significant contribution of Jupiter-mass compact objects to the mass distribution of the bulge whether the microlenses move predominantly in orbital or random motions. However, if the microlens proper motions are distributed according to an isotropic velocity dispersion then the lower limit obtained is of the same order as the hydrogen burning limit.

Acknowledgments

The authors would like to thank Chris Fluke and Daniel Mortlock for helpful discussions. This work was supported in part by NSF grant AST98-02802. JSBW acknowledges the support of an Australian Postgraduate award and a Melbourne University Overseas Research Experience Award.