Weighing a galaxy bar in the lens Q2237+=0305

© 1998 RAS

Mon. Not. R. Astron. Soc. 295, 488–496 (1998)

Weighing a galaxy bar in the lens Q2237+++0305

Robert Schmidt,1w† Rachel L. Webster1 and Geraint F. Lewis2‡1School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia2SUNY at Stony Brook, Stony Brook, NY 11794-2100, USA

Accepted 1997 November 17. Received 1997 July 22; in original form 1997 March 25

A B STR ACTIn the gravitational lens system Q2237+0305 the cruciform quasar image geometryis twisted by 10° by the lens effect of a bar in the lensing galaxy. This effect can beused to measure the mass of the bar. We construct a new lensing model for thissystem with a power-law elliptical bulge and a Ferrers bar. The observed ellipticityof the optical isophotes of the galaxy leads to a nearly isothermal elliptical profile forthe bulge, with a total quasar magnification of 16+5

Ð4. We measure a bar mass of(7.5¹1.5)Å108 hÐ1

75 M> in the region inside the quasar images.

Key words: galaxies: fundamental parameters – galaxies: individual: 2237+0305 –galaxies: spiral – gravitational lensing.

1 INTRODUCTION

Gravitational lensing provides a unique way to weighobjects at cosmological distances without any assumptionabout the connection between light and dark matter. Sincethe discovery of the first gravitational lens (Walsh, Carswell& Weymann 1979) several gravitational lenses have beenfound, and this method has been used many times toexplore the mass distribution of galaxies. In this paper, wemodel the lens system Q2237+0305 in order to weigh thebar in the lensing galaxy.

The quasar Q2237+0305 (zq\1.695) was found byHuchra et al. (1985) at the centre of an SBb spiral galaxy(z\0.0394) that is situated in the outskirts of the Pegasus IIcluster. The quasar was later resolved into four images thatare situated around the core of the galaxy within a radius of1 arcsec (Schneider et al. 1988; Yee 1988).

Two fundamentally different approaches have been usedto model the lensing galaxy. One was to fit a parametricmass profile with several free parameters to the observedquasar image configuration (Kent & Falco 1988); the otherwas to use a model of the light distribution of the galaxy, andto fit for the mass-to-light ratio as the single free parameter(Schneider et al. 1988; Rix, Schneider & Bahcall 1992). Theformer approach was naturally more precise in the repro-duction of the observed image geometry due to the greaternumber of free parameters.

The length-scale over which the lensing galaxy influencesa light bundle from the quasar is small compared to thecosmological distances between observer, lens and source.The lens can therefore be treated as a mass sheet at theposition of the galaxy. Since the galaxy disc of 2237+0305 isinclined with respect to the sky, elliptical surface mass dis-tributions must be used in the models of this system.

Interestingly, the position angle, counted counterclock-wise from north, of the major axis of the elliptical lensmodels found by Kent & Falco (1988) was about 67°. This isalmost parallel to the axis through images C and D, and justbetween the angle of the inclination axis of the galaxy (77°;Yee 1988) and the angle of the bar (39°; also Yee 1988).This situation is shown in Fig. 1. This was also found byKochanek (1991) and Wambsganss & Paczynski (1994),who used simple circular mass distributions with an addi-tional quadrupole perturbation. When fitted to theobserved image geometry, the direction of the perturbationturned out to be close to the one Kent & Falco (1988) foundfor their model major axis. More recent investigations byKassiola & Kovner (1995), Witt, Mao & Schechter (1995)and Witt (1996) obtained the same position angle for theperturbation or major model axis.

On the other hand, the bar shows up prominently in CCDimages of the galaxy. It has been noted (Tyson & Goren-steinn 1985; Yee 1988; Foltz et al. 1992) that it might contri-bute significantly to the lensing in the system – initially thiswas actually an aid to explain the lensing effect when onlythe images A, B and C were known (Tyson & Gorenstein1985).

The motivation for this paper is the idea that theapparent misalignment of the predicted model major axis

wPresent address: Astrophysikalisches Institut Potsdam, An derSternwarte 16, 14482 Potsdam, Germany.†E-mail: [email protected] (RS)‡ Visiting Astronomer at the University of Melbourne.

and observed galaxy inclination axis as shown in Fig. 1 is dueto the lensing influence of the bar. In Section 2 we constructand analyse a lensing model that includes the bar compo-nent and takes the observed position angle for the inclina-tion axis of the galaxy into account. Section 3 deals with theimplications of this model for the bar. In Section 4 we finallydiscuss our results. We use a cosmological model withH0\75 h75 km sÐ1 MpcÐ1, W\1 and L\0.

2 THEOR ETICA L MODEL

The lensing model we construct has two components, eachwith several free parameters. In this section we introducethe components and determine the values for these thatprovide the best description of the observations.

2.1 The bulge

Yee (1988) identified three components in the inner part ofthe galaxy: bulge, disc and bar. In the lensing models for thissystem, bulge and disc have been represented by just oneeffective component, since there are only limited observa-tional data from the quasar image and galaxy positions toconstrain free parameters of the model. For simplicity, wecall this composite component ‘bulge’.

Moreover, Kochanek (1991) and Wambsganss & Paczyn-ski (1994) found that the quasar image positions and thegalaxy position in this system do not constrain the param-eters even of simple lens models. In particular, Wambsganss& Paczynski (1994) showed that for a circular power-lawmass distribution with an external shear there is a wholefamily of models that fit the observations; they discoveredthat for this family there is a linear relation between themagnitude of the external shear and the exponent of themass profile for a vast range of exponents. The shear and

the mass exponent are degenerate, and one needs moreinformation than only the positions of the quasar imagesand the galaxy to break this degeneracy. If one uses a two-component galaxy model, the shear contributions from thetwo components will also be degenerate, since the resultingshear is degenerate.

One way to get around the shear degeneracy is to use anelliptical mass distribution instead of a circular profile. Inthis case, the ellipticity parameter of the mass distributionreplaces the shear parameter as a free parameter of themodel. An analogous degeneracy in the ellipticity can thenbe broken by using the observed ellipticity of the isophotesof the galaxy.

Generalizing the approach by Wambsganss & Paczynski(1994), we accordingly modelled the bulge with an ellipticalpower-law mass distribution with a major-axis positionangle of 77°. There is some disagreement in the literature onthe value of this position angle (e.g. Rix et al. 1992). Fitte &Adam (1994) showed that this is because the position angleof elliptical isophotes is increasingly twisted towards the barwith increasing distance from the galaxy centre. The valuewe adopted from Yee (1988) is identical with the positionangle determined from the galaxy continuum map within aradius of 1 arcsec from the galaxy centre (Fitte & Adam1994) where most of the lensing mass is situated. Let e bethe elliptical parameter, so that the ratio b/a of minor andmajor axis of concentric elliptical shells is

b

a\

1Ðe

1+e. (1)

It is useful to express surface mass densities in units of thecritical lensing density (Schneider, Ehlers & Falco 1992)Scrit\c2Ds/4pGDdDds, where Ds, Dd and Dds are the angularsize distances between observer and source, observer anddeflector (lens), as well as deflector and source. The surfacemass density of k of a power-law elliptical mass distributionin units of Scrit is given by

k (y1, y2)\E0

2y ve

. (2)

y1 and y2 are the coordinates on the sky as measured in acoordinate system oriented with the observed major andminor axis of the bulge. ye is the elliptical radius,

ye\X y 21

(1+e)2+

y 22

(1Ðe)2, (3)

v is the power-law exponent of the elliptical mass distribu-tion, and E0 is a constant. In the analysis of our results weuse the ellipticity e\1Ðb/a\2e /(1+e), since this is thevalue that is usually used in the observations. In Appendix Adeflection potential and angles for this mass distribution aredescribed.

2.2 The bar

The light distribution of bars has a well-defined elongatedshape, and is non-singular and centrally condensed (Sell-wood & Wilkinson 1993). It is not straightforward to deter-

Weighing a galaxy bar in the lens Q2237+0305 489

© 1998 RAS, MNRAS 295, 488–496

Figure 1. Illustration of the image geomtry of Q2237+0305, moti-vated by fig. 1 in Kent & Falco (1988). The images are labelledusing the convention by Yee (1988), with positions from Crane etal. (1991). The relative areas of the circles correspond to the radioflux ratios from Falco et al. (1996). The position of the galaxycentre is indicated with a cross. The long arrows indicate the direc-tions of the galaxy inclination axis and the bar, as well as theposition angle of 67°. North is up, and east to the left.

mine the true form of bar mass distributions from this, sothat we have to assume a model. Very simple models withthese properties of the bar light are the Ferrers profiles(Ferrers 1877). They were used in dynamical studies of bars(Freeman 1966a,b,c; Martinet & de Zeeuw 1988; Sellwood& Wilkinson 1993), since they can be treated analytically.To model the surface mass distribution of the bar of2237+0305, we used two-dimensional Ferrers profiles ofthe form

kc21Ðy 2

1

a 2Ð

y 22

b23l

if y 2

1

a 2+

y 22

b2R1

k (y1, y2)\ (4)

0 otherwise.8

In this equation, y1 and y2 are the coordinates on the sky asmeasured in a coordinate system oriented with the observedmajor and minor axis of the bar. l is a real number, kc is thecentral surface density in units of the critical lensing densityScrit defined in Section 2.1, and a and b are the semimajorand semiminor axes of the bar respectively.

In the analysis we restricted ourselves to moderate expo-nents l\0.5, 1 and 2. The deflection potential and anglesfor integer values of l can be calculated analytically, asdescribed in Appendix B. For l\0.5 the profiles were con-structed through the numerical superposition of many ellip-tical slices of constant density (l\0) and different size(Schramm 1994).

2.3 The effect of shear

The lensing influence of the bar can be understood by con-sidering a system with two shear tensors with shear direc-tions as shown in Fig. 1 for galaxy inclination axis and bar(see Schneider et al. 1992 for the definition of the sheartensor). The resulting shear tensor can be found by addingup the single tensors, and the resulting shear direction isdetermined by the shear ratio of the two components.

A source almost directly behind the core of the lensinggalaxy appears lensed with four of the five images in a crossformation aligned with the axes parallel and perpendicularto the resulting shear direction, while the fifth is seen in thecentre (Schneider et al. 1992, p. 252). In the case of2237+0305, the major axis position angle as found by theone-component lens models (67°) can be interpreted to bethis resulting shear direction, which almost coincides withthe axis through images C and D. The axis through imagesC and D has effectively been twisted away from the galaxyinclination axis by the bar.

In Fig. 2 this twisting is illustrated by plotting the criticallines, caustics and image positions of two barred lenses withidentical source positions. The caustics are the lines in thesource plane that separate regions of different image multi-plicity. The critical lines are the corresponding lines in thelens plane where pairs of images are created or destroyed(Schneider et al. 1992).

In this figure the first lens has a weak bar that barelychanges the elliptical shape of the bulge’s critical line or thecorresponding diamond shape of the caustic. The other baris significantly more massive; it warps the shape of thesestructures and shifts the image positions.

2.4 Detailed modelling

Our barred galaxy model has 10 adjustable parameters.These are the positions of the galaxy and the source plus theconstant E0, the ellipticity e and the exponent v for the bulgeas well as the bar mass normalization kc, the semiminor axisb and the exponent l for the bar. This number can bereduced by two if the observed ellipticity of the bulge andonly fixed values for l are used. For the length of the semi-major axis of the bar we used the observed value of a29arcsec (taken from the figures of Yee 1988 or Irwin et al.1989). The lens effect is insensitive to the precise value of a,because a is much larger than the radius of the ring ofimages (21 arcsec). The length of the semiminor axis must,however, remain a free parameter of the model, since it iscomparable to this radius, but not known well enough(b21–2 arcsec; see Section 3).

There are 10 observational constraints that the systemimposes upon theoretical models. These are the coordinatesof the four observed quasar images and the galaxy centre.The positions for the images and galaxy centre were takenfrom Crane et al. (1991). These positions have been deter-mined from Hubble Space Telescope (HST) observations andhave quoted measurement errors of 0.005 arcsec. Ingeneral, the ratios of the fluxes of the different images of agravitational lens also provide good constraints for a model.Unfortunately, in the case of Q2237+0305 the light curves

490 R. Schmidt, R. L. Webster and G. F. Lewis

© 1998 RAS, MNRAS 295, 488–496

Figure 2. Illustration of the effect of a strong bar. Assuming identi-cal source positions, the diamond-shaped caustics, the correspond-ing critical curves and the image positions are plotted for twodifferent lenses. The smaller caustic and the almost elliptical criti-cal curve belong to a model with a v\1 power-law bulge and a l\2Ferrers bar that was fitted to the observed parameters ofQ2237+0305. The positions of the images created by this modelare labelled with unprimed letters as in Fig. 1. In this model, thebulge mass inside the ring of images is about 20 times larger thanthe bar mass (see Table 1). The caustic and the critical curvetransform into the two elongated curves if the mass of the barinside the ring of images is increased to half the mass of the bulge,while the bulge mass is kept fixed. The images are shifted by themore massive bar to the positions labelled with the primedletters.

from Corrigan et al. (1991) or Østensen et al. (1996) clearlyshow that all optical image fluxes are subject to flux varia-tions due to microlensing of the quasar light from the starsin the lensing galaxy. In addition, the light from the quasaris non-uniformly dust-reddened during the passage throughthe galaxy. The fluxes were therefore neglected in themodelling procedure. We will, however, compare the modelpredictions with the recently measured radio flux densitiesby Falco et al. (1996).

Let hk, hg be the positions on the sky the model predictsfor quasar images and the galaxy centre, and hko, hgo theobserved positions with their positional uncertainties sk, sg.To find the best-fitting model, the expression

x 2\ +4

k\1

(hkÐhko)2

s 2k

+(hgÐhgo)

2

s 2g

(5)

(Wambsganss & Paczynski 1994) was minimized throughvariation of the model parameters using a multidimensionalminimization routine (direction set or downhill simplexmethods according to Press et al. 1992).

In order to find an estimator for the separations hkÐhko

between modelled and observed images for the first term onthe right-hand side of equation (5), we used the method byKochanek (1991); the separations between an optimallyweighted source position and the positions in the sourceplane where the observed image positions are mapped to bya given lens model are propagated back into the lensplane.

2.5 Analysis

For a nearly circularly symmetric lens with a source almostin the origin, it follows from Newton’s theorem in twodimensions (Foltz et al. 1992; Schramm 1994) that the massinside the circle of images is approximately given by theseparation Dy of the images at opposite ends of the crossvia

M\c 2

16G

DdDs

Dds

(Dy)2 (6)

(see, e.g., Narayan & Bartelmann 1996). The galaxy2237+0305 is situated relatively close to us at an angularsize distance Dd\0.15 hÐ1

75 Gpc, so that Dds/Ds21. Theseparations of the quasar images are Dy21.8 arcsec, so thatwe get M21.5Å1010 hÐ1

75 M>. This value was also found inprevious models for this system (Rix et al. 1992:1.44¹0.03Å1010 hÐ1

75 M>, Wambsganss & Paczynski 1994:1.48¹0.01Å1010 hÐ1

75 M>). The other value theoreticalmodels for 2237+0305 agreed on was the resulting sheardirection of 267°.

In our barred lens model, the bulge acts as the mainlensing mass, and the bar as a perturbation; for a givenellipticity e or exponent v, the bulge parameter E0 andhence the bulge mass do not change very much for differentl-bar models. In fact, experiments with different bar massesas in Fig. 2 showed that for similar masses of bulge and barinside the quasar images the cruciform image symmetrybecomes skewed. Models with a strong bar thus cannotreproduce a symmetric image geometry as in Q2237+0305.

In order to explore the effect of l on the models, we usedfixed values l\0.5, 1 and 2.

Besides E0 and l, the parameter space of e, v, kc and bhad to be examined. We first scanned the parameter spaceof e and v while leaving kc and b as free parameters. In Fig.3 a contour plot of the confidence regions that contain 68.3,95.4 and 99.7 per cent of normally distributed modelsaround the minimum of x2 (Press et al. 1992) in the param-eter space of v and e is shown. Only the plot for l\2 ispresented; the cases with l\0.5 and 1 are very similar. Theparameter space was scanned with a step size of 0.02 for vand 0.01 for e. For every point the best parameters weredetermined through minimization, and a x2-value was com-puted. The best-fitting models lie in a long valley thatextends up to v21.25. The unclear structure at the lowerend of the valley for vR0.5 and eR0.1 is due to numericaleffects. Detailed investigation shows that the valley con-tinues towards smaller v, becoming shallower. The modelsin this region, with low e and v, are similar to circular discswith constant surface mass density, and are not examinedhere because they do not represent realistic galaxymodels.

The external-shear models by Wambsganss & Paczynski(1994) yielded no constraints on their circular mass distribu-tions for the whole range of parameters from v\0.07 to 2.The smaller allowed range for v in Fig. 3 shows that theellipticity does not allow the same kind of freedom as theshear. For high ellipticities of the power-law bulges, thequasar image geometry canot be reproduced anymore.


© 1998 RAS, MNRAS 295, 488–496

Figure 3. Contour plot of confidence regions in the parameterspace of bulge exponent v and ellipticity e. The indicated contourscontain 68.3, 95.4 and 99.7 per cent of normally distributed modelsaround the minimum of x2.

In Fig. 4 the total mass, the bulge mass, and the bar massinside a circle of 0.9 arcsec, as well as the semiminor axis band the total magnification m along the valley of best fitsaround the v\1 profile, are shown for 0.75RvR1.3 and thethree values of l. The plots for different l differ noticeablyonly in their semiminor axis predictions. Beginning withv21.2, the minimization produces numerical noise at theupper end of the valley from Fig. 3, since the fits becomeworse.

It can be seen that the total mass of the model inside acircle of 0.9 arcsec is almost constant. The constituent

masses of the bar and the bulge exhibit a dependence on v.It can be interpreted that the bar mass increases with v inorder to counter the increased contribution of the bulge tothe resulting shear due to the increase of e, and hence thebulge mass decreases in order to conserve the mass inside0.9 arcsec given by equation (6). The magnification dropsstrongly with increasing v; this was also observed by Wambs-ganss & Paczynski (1994).

As motivated in the introduction, we chose the modelfrom the family of best models in Fig. 3 that exhibits theobserved ellipticity of the bulge. The ellipticity has beenmeasured by Racine (1991) to be e\0.31¹0.02, in agree-ment with the results from the continuum map by Fitte &Adam (1994). This range of ellipticities is encompassed bythe range v\1.05¹0.1 of bulge exponents. We use thisregion of allowed bulge models to determine the uncertain-ties of the predictions of models with different bars.

From all possible values for v and the three bar models weobtain an estimate of the total mass inside a circle of 0.9arcsec of (1.49¹0.01)Å1010 hÐ1

75 M>. This is consistent withthe estimate and the results from the literature given at thebeginning of this section. In a review, Narayan & Bartel-mann (1996) mention only two cases of gravitational lensesin which it has been possible to constrain the radial distribu-tion of the lens galaxy. Both models are based on singularelliptical or nearly elliptical profiles. For MG 1654+134,Kochanek (1995) obtained v\1.0¹0.1, and for QSO0957+561, Grogin & Narayan (1996) obtainedv\1.1¹0.1. The predictions of our model for the bulge arein good agreement with these values.

In Table 1, the model parameters for the different valuesof l and the uncertainties due to the uncertainty in v areshown. The x2-values from equation (5) have been dividedby the number of degrees of freedom, giving a measure ofthe quality of the fit. The number of degrees of freedom isthe number of constraints minus the number of free param-eters; here we have three degrees of freedom since e, v andl are fixed.

The value for the total magnification predicted from thev\1.05 model is about half the value found by Wambsganss& Paczynski (1994) for their v\1 circular power-law profilewith an external shear. This illustrates that the use of anelliptical mass distribution drastically changes the predic-tions of the model; a similar discrepancy is apparent for thetime-delays. Note that m total23/e\16.7, with e\0.18(corresponding to e\0.31; see Section 2.1) as derived for av\1 power-law lens with small e and a source in the originby Kassiola & Kovner (1993). In contrast to the non-singu-lar mass models by Kent & Falco (1988), our lens models donot produce a fifth image in the centre due to the centralsingularity of the mass distribution. The predicted sourcepositions are very similar to the ones given by Kent &Falco.

The remaining parameter space of the bar models is illus-trated in Fig. 5. As in Fig. 3, the contours of the confidenceregions of normally distributed models around the mini-mum of x2 in the parameter space of kc and b are shown forthe model with v\1.05, e\0.31 and l\2. The parameterspace was scanned with a step size of 0.003 in kc and 0.025arcsec in b. The bar parameters are well constrained at thebottom of a steep boomerang-shaped valley.

The surface mass density k in units of the critical densityand the local shear g at the positions of the images are givenin Table 2. The values for v\0.95 and 1.05 are similar towhat one gets for a circular v\1 power-law mass distribu-tion (singular isothermal sphere) with a mixture of internaland external shear (Kochanek 1991; Witt & Mao 1994). Itcan be taken from Tables 1 and 2 that a change of l does notcause a measurable change of observable quantities exceptfor the semiminor axis of the bar; the uncertainty of themodel parameters is dominated by the uncertainty of thebulge ellipticity e /exponent v.

3 PROPERTIES OF THE BA R

The width of the bar has not been measured previously. Infig. 1 of their paper, Irwin et al. (1989) present a contourplot of the galaxy, where the bar has been separated fromthe disc. In this plot the bar appears about 18 arcsec long,but only 22–4 arcsec wide. It is just the less well-knownminor axis that enters the lensing model, since the quasarimages are situated in the centre of the galaxy. In theirimage analysis Irwin et al. subtracted structure from theirimage that is smooth on scales of 25–10 arcsec. This pro-cedure removed the galaxy disc very efficiently, but, beinglong as well as thin, the bar shown in their fig. 1 could alsobe affected by this procedure.

A different approach to obtain the light distribution ofthe bar was pursued by Schmidt (1996). Using an HST I-band image taken with the Planetary Camera prior to thefirst servicing mission (Westphal 1992), the galaxy wasdecomposed into bulge, disc and bar with analytical profiles


© 1998 RAS, MNRAS 295, 488–496

Figure 4. Model parameters along the valley of best fits. Plottedagainst v are the total mass and the masses of bulge and bar insidea circle of 0.9 arcsec in 1010 hÐ1

75 M>, as well as the semiminor baraxis b in arcsec and the total magnification m. Different line styleshave been used for different values of l as indicated in the b–vpanel.


© 1998 RAS, MNRAS 295, 488–496

that have been convolved with the point-spread function ofthe telescope. Exponential profiles (Andredakis & Sanders1994) were fitted to bulge and disc. After these componentswere removed from the image, the bar and the spiral armsremained. The bar light distribution could be fitted with a

Table 1. Model parameters for Q2237+0305 for three values of the bar expo-nent l and the bulge exponent v. The columns for l\0.5, 1.0 and 2.0 areindicated. Each table entry contains the value for v\1.05 and the differences tothe corresponding values for v\1.15 (upper index) and 0.95 (lower index).Only one value is given for a line if the differences between the bar models arebelow the rounding precision. x2 is divided by 3, the number of degrees offreedom. E0, b and the source position (b1, b2) are given in arcsec. m total is thetotal magnification, mij are the relative magnification ratios between the images.Dtij are the relative time delays between the images in hÐ1

75 hours. Mbulge(s0.9pp)and Mbar(s0.9pp) are the masses of bulge and bar inside a circle of 0.9 arcsec,and Mbar,total is the total mass of the bar. Masses are given in 1010 hÐ1

75 M>. Thesemimajor bar axis was assumed as a\9 arcsec. The VLA flux ratios are the3.6-cm radio flux ratios measured by Falco et al. (1996).

Table 2. Local lensing parameters atthe positions of the quasar images. Thevalues have been determined with al\1 bar, but they are identical forl\0.5 and 2 within ¹0.01. Each tableentry contains the value for v\1.05 andthe differences to the correspondingvalues for v\1.15 (upper index) and0.95 (lower index).

Figure 5. Contour plot of confidence regions in the parameterspace of bar normalization kc and semiminor axis b (in arcsec) fora bulge exponent v\1.05, ellipticity e\0.31 and bar exponentl\2. The indicated contours contain 68.3, 95.4 and 99.7 per cent ofnormally distributed models around the minimum of x2.

l\2 Ferrers profile with a\9.5¹1.0 arcsec andb\1.0¹0.3 arcsec. This light model can be combined withthe mass model for l\2 from Table 1 due to the similarvalue for b; the I-band mass-to-light ratios of these modelcomponents inside a circle of 0.9 arcsec are given by M/LI24.8 h75 for bulge plus disc and M/LI25.0 h75 for the bar.The bar mass detected with gravitational lensing and the barlight in the I band in this model both constitute a 5 per centfraction of the total mass respectively light inside 0.9 arc-sec.

This result is, however, dependent on the bulge model.For a de Vaucouleurs bulge (de Vaucouleurs 1948), al\0.5 Ferrers profile fitted the bar light much better with asimilar value for a, but a much larger value b\3.1¹0.9arcsec which cannot be combined with the lensing modelfrom Table 1, since b is very different. The unrefurbishedHST point-spread function inhibited a clear distinctionbetween the quality of fit of these different bulge models, sothat the question about the minor bar axis is not decidedyet.

A value of bE1 arcsec could, in connection with thevalues for b in Table 1, be taken as evidence in favour ofsteeper bar models, lE2. There is observational evidencefrom the light distributions of real bars for more boxy massdistributions (Freeman 1996), so that it has to be ascer-tained that this result is not a relic of the approximation ofthe bar with an elliptical shape. At the positions of thequasar images, however, an ellipse is a good approximationof a box due to the large semimajor axis of the bar, a29arcsec. A lensing model with a steeper Ferrers profile withl\10 leads to a larger bar with b\2.0. This indicates that ameasurement of the extent of the bar could in principleconstrain the mass distribution of the bar. If one wants touse this method, high-resolution images with small seeinghave to be used, since an accuracy of the bar width on theorder of tenths of an arcsecond is needed.

4 DISCUS SION

In this paper we have presented a barred galaxy model forthe gravitational lens 2237+0305. We used a power-lawelliptical mass distribution for the bulge and chose themodel for which the observed ellipticity is predicted. Itturned out that this model has an exponent close to unity,which is compatible with other determinations of lens massprofiles through gravitational lensing (Kochanek 1995; Gro-gin & Narayan 1996). The bar represents a small perturba-tion of the deflection field of the bulge of the galaxy,amounting to (7.5¹1.5)Å108 hÐ1

75 M> or about 5 per cent ofthe bulge mass in the critical region inside the quasarimages.

The relative magnifications our model predicts for thequasar images can be compared with the 3.6-cm radio fluxratios published by Falco et al. (1996). Their results are alsogiven in Table 1. Falco et al. argue that it is unlikely that theradio flux densities are variable from microlensing due tothe larger size of the radio-emitting region as compared tothe optical continuum-emitting region, although they can-not completely rule out microlensing as an important effectin the radio. If microlensing is not important, the modelmagnification ratios should be identical to these measuredflux ratios. It can be seen that only the ratio mDA between

images D and A is not compatible with their results,although no effort was made to fit the flux ratios.

In order to find out more about the discrepancy of theratio mDA between observation and model, one has to makethe relatively large error bars from Falco et al. smallerthrough longer radio observation of the object. Unfortuna-tely, Q2237+0305 has a radio flux density of only 21 mJy(Falco et al. 1996), so that radio observations of this objectare very time-consuming; Falco et al. observed for 11 h, ofwhich only 5 h could be used eventually due to weatherconditions.

To get additional, independent arbiters for the model, itwould be very helpful to measure the time delays in thissystem. Since the time delays are of the order of severalhours, this has to be done in a wavelength domain where thenecessary intraday variability is likely to occur for a radio-quiet quasar, for example in the X-ray regime as proposedby Wambsganss & Paczynski (1994). Also, monitoring in theradio would show if the quasar image flux densities vary atthese frequencies. Unless we learn more about the radioflux densities and the time delays, it is not possible to decidewhether or not it is microlensing that causes the low magni-fication of image D. In the optical, image D has always beenthe faintest quasar image. In fact, in the first resolved imageof the quasar, image D was not visible at all (Tyson &Gorenstein 1985). There is also spectroscopic evidencefrom optical data that image D is undergoing demagnifica-tion (Lewis et al. 1996).

If image D is in fact microlensed in the radio, the conse-quences are interesting. The scale size of the radio regioncould be less than the characteristic scale of the causticnetwork. Alternatively, the radio source could have anasymmetric structure like a jet that would have differingmicrolensing properties for different paths of the micro-lenses across the source.

Yet another way to change the radio flux density of imageD significantly would be a globular cluster or black hole(Lacey & Ostriker 1985) with a mass of about 106 M> in thehalo of the lensing galaxy that is situated close to image D.An object of this mass would magnify or demagnify theradio image of the quasar, depending on its location withrespect to the direction of the local shear. This effect, theperturbation of lens models by 106 M> objects, has recentlybeen treated by Mao & Schneider (1997). With this, we canestimate that a surface mass density of globular clusters orblack holes of approximately 0.04 Scrit (Scrit is defined inSection 2.1) or 470 h75 M> pcÐ2 is needed to observe ademagnification of image D by 40 per cent or more with aprobability of 20 per cent. Higher surface mass densitieswould make it more likely. This is much more than theglobular cluster surface mass density of about 1 M> pcÐ2

seen in our Galaxy within 5 kpc of the Galactic Centre (Mao& Schneider 1997). It thus seems unlikely that the demagni-fication is due to a globular cluster.

The question of the existence of such a massive objectnear image D could be solved with a method that was pro-posed by Wambsganss & Paczynski (1992). They showedthat these objects would bend or even create holes in theradio maps of milliarcsecond jets of gravitationally lensedquasars. If we could observe extended structure of theimages of the radio-weak Q2237+0305, features due toglobular cluster or black hole lensing could be easily identi-


© 1998 RAS, MNRAS 295, 488–496

fied, because they would not be seen in the other gravita-tionally lensed images of the quasar.

It has recently been suggested (Keeton, Kochanek &Seljak 1997; Witt & Mao 1997) that a strong lensing pertur-bation is necessary for a number of lenses in addition to anelliptical mass distribution in order to model the observedimage geometry. In our model for Q2237+0305, the addi-tional perturbation from the bar is small, but the system is,nevertheless, unique in that we can see the perturbingagent. For the lens systems mentioned in these papers, thebetter fit for the models was obtained with an additionalexternal shear. We saw, when we compared our results withWambsganss & Paczynski (1994), that the predictions formagnifications or time delays from shear models differ byup to a factor of 2 from the predictions from elliptical massdeflectors. If one wants to get lensing models with reliablepredictions for magnifications or time delays, the necessaryperturbations will ultimately have to be generated in themodels by mass components like dark matter haloes (Kee-ton et al. 1997) or bars.

ACKNOWLEDGMENTS

We thank Hans-Jorg Witt, Joachim Wambsganß and theanonymous referee for their comments on the manuscript.RS gratefully acknowledges support by a Melbourne Uni-versity Postgraduate Scholarship, an Australian OverseasPostgraduate Research Scholarship and by the Studienstif-tung des deutschen Volkes. This research was supported inpart by the Deutsche Forschungsgemeinschaft (DFG)under Gz. WA 1047/2-1.


© 1998 RAS, MNRAS 295, 488–496

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APPENDI X A : POW ER-L AW ELLIPTICA L M AS S DISTR IBUTIONS

The lensing properties of the elliptical power-law mass dis-tributions from equation (2) with v\1 have been describedby Kassiola & Kovner (1993) and Kormann, Schneider &Bartelmann (1994). Kormann et al. determined these bysolving the Poisson equation for the deflection potential c(see Schneider et al. 1992 for the definition of the deflectionpotential) in polar coordinates y\Zy2

1+y22 and h,

1

yqqy 2y

qcqy3+ 1

y 2

q2c

qh 2\2k (y, h). (A1)

Their approach can be generalized for arbitrary realnumbers v with 0Rvs2. Define n\2Ðv, sn\sin np/2,cn\cos np/2 and

D (h)\X cos2 h

(1+e)2+

sin2 h

(1Ðe)2. (A2)

Using the integrals

N1\hh

0

cos nhpDv(hp)

dhp,

N2\hp/2

h

sin nhpDv(hp)

dhp,

N3\N1+hp/2

h

cos nhpDv(hp)

dhp, (A3)

the deflection potential for 0RhRp/2 is given by

c (y, h)\E0y n&1n N1 sin nh+

1

n 2N2+cn

sn

N33 cos nh'.(A4)

The potential for the other quadrants can be taken from thisresult because of the elliptical symmetry of the mass distri-bution. The deflection angles a\Hc for this quadrant canbe calculated from equation (A4). The Cartesian compo-nents are

a1(y, h)\E0y nÐ1 &N1(sin nh cos hÐcos nh sin h)

+2N2+cn

sn

N33 (cos nh cos h+sin nh sin h)',a2(y, h)\E0y nÐ1 &N1(sin nh sin h+cos nh cos h)

+2N2+cn

sn

N33 (cos nh sin hÐsin nh cos h)'.(A5)

These expressions can be evaluated numerically. Forv\n\1 the integrals are analytically solvable, and the for-mulae become identical to the results of Kassiola & Kovner(1993) and Kormann et al. (1994). After this work wascompleted, we discovered that Grogin & Narayan (1996)also used power-law elliptical mass profiles in their model ofthe lens of the double quasar 0957+561. In their paper theypresent the deflection angles of power-law elliptical massdistributions in a complex-valued lensing formalism interms of the complex hypergeometric function.

APPENDI X B: FERR ERS PROFILES

The deflection potential c0 and the deflection angles a0 ofan elliptical slice of constant surface density, a Ferrers pro-file from equation (4) with l\0, are given by

c0(y1, y2)\Ð1

2 abkc(Q00Ðy 2

1 Q10Ðy 22 Q01), (B1)

and the Cartesian components

a01(y1, y2)\abkcy1Q10,

a02(y1, y2)\abkcy2Q01, (B2)

where

Q00\2 ln (Za 2+r+Zb2+r)Ð1

Q01\2

a 2Ðb2 2Xa 2+r

b2+rÐ13

Q10\2/D (r)ÐQ01. (B3)

r is the positive solution of

y 21

a 2+r+

y 22

b2+r\1 (B4)

outside the bar, and r\0 inside the bar. D(r) is given by

D (r)\Z(a 2+r) (b2+r). (B5)

a0 has first been derived by Schramm (1990) by using theknown force field for a homogeneous ellipsoid and lettingthe largest axis go to infinity in order to get the correspond-ing two-dimensional equations. To derive the potential c0

in terms of real-numbered coordinates y1, y2, we appliedSchramm’s method to the results by Pfenniger (1984) forthe potential and force fields of three-dimensional Ferrersellipsoids. This enables us to calculate deflection potentialand deflection angles for a whole family of Ferrers profiles.The deflection potential for Ferrers surface-density profileswith integer exponents l is given by

cl(y1, y2)\Ðabkc

2 (l+1) hl

r

du

D (u)

Å21Ðy 2

1

a 2+uÐ

y 22

b2+u3l+1

. (B6)

The corresponding three-dimensional expression was firstderived by Ferrers (1877). For integers j, k, j80 or k80,this integral can be split up into a sum containing thecoefficients

Qjk\hl

r

du

D (u)

1

(a 2+u) j(b2+u)k. (B7)

For Q00 use equation (B3); a remaining contribution fromthe infinite axis had to be subtracted here. Pfenniger (1984)solved his corresponding integrals using recurrence rela-tions. Translated into two dimensions, the Qjk obey therelation

Qjk\(QjÐ1, kÐQj, kÐ1)/(a 2Ðb2), (B8)

as well as for na0

Qn0\1

2nÐ1 & 2

D (r) (a 2+r)nÐ1ÐQnÐ1,1', (B9)

Q0n\1

2nÐ1 & 2

D (r) (b2+r)nÐ1ÐQ1, nÐ1'. (B10)

With these equations, the deflection potential cl as well asthe deflection angles al\Hcl can be calculated from equa-tion (B6). The coefficients Qij can be treated as constant forthe derivation with respect to y1 and y2 since the definitionof r in equation (B4) implies that qcl /qr\0.


© 1998 RAS, MNRAS 295, 488–496

Weighing a galaxy bar in the lens Q2237+=0305

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