B.3.2. Circular Lenses
While one of the most important lessons about modeling gravitational lenses in the real world is that you can never (EVER!) ^{2} safely neglect the angular structure of the gravitational potential, it is still worth starting with circular lens models. They provide a basic introduction to many of the elements which are essential to realistic models without the need for numerical calculation. In a circular lens, the effective lens potential (Part 1) is a function only of the distance from the lens center = ||. Rays are radially deflected by the angle
(B.3) |
where we recall from Part 1 that () = () / _{c} is the surface density in units of the critical surface density, D_{ds} and D_{s} are the lens-source and observer-source comoving distances and = D_{d}^{ang} is the proper distance from the lens. The bend angle is simply twice the Schwarzschild radius of the enclosed mass, 4GM( < ) / c^{2}, divided by the impact parameter and scaled by the distance ratio D_{ds} / D_{s}.
The lens equation (see Part 1) becomes
(B.4) |
where
(B.5) |
is the average surface density interior to in units of the critical density. Note that there must be a region with <> > 1 to have solutions on both sides of the lens center. Because of the circular symmetry, all images will lie on a line passing through the source and the lens center.
The inverse magnification tensor (or Hessian, see Part 1) also has a simple form, with
(B.6) |
where = (cos, sin). The convergence (surface density) is
(B.7) |
and the shear is
(B.8) |
The eigenvectors of M^{-1} point in the radial and tangential directions, with a radial eigenvalue of _{+} = 1 - + = 1 - d / d and a tangential eigenvalue of _ = 1 - - = 1 - / = 1 - <>. If either one of these eigenvalues is zero, the magnification diverges and we are on either the radial or tangential critical curve. If we can resolve the images, we will see the images radially magnified near the radial critical curve and tangentially magnified near the tangential critical curve. For example, all the quasar host galaxies seen in Figs. B.3 and B.4 lie close to the tangential critical line and are stretched tangentially to form partial or complete Einstein rings. The signs of the eigenvalues _{±} give the parities of the images and the type of time delay extremum associated with the images. If both eigenvalues are positive, the image is a minimum. If both are negative, the image is a maximum. If one is positive and the other negative, the image is a saddle point. The inverse of the total magnification µ^{-1} = | M^{-1}| is the product of the eigenvectors, so it is positive for minima and maxima and negative for saddle points. The signs of the eigenvalues are referred to as the partial parities of the images, while the sign of the total magnification is referred to as the total parity.
It is useful to use simple examples to illustrate the behavior of circular lenses for different density profiles. In most previous lensing reviews, the examples are based on lenses with finite core radii. However, most currently popular models of galaxies and clusters have central density cusps rather than core radii, so we will depart from historical practice and focus on the power-law lens (e.g. Evans & Wilkinson [1998]). Suppose, in three dimensions, that the lens has a density distribution r^{-n}. Such a lens will produce deflections of
(B.9) |
as shown in Fig. B.9, with convergence and shear profiles
(B.10) |
The power law lenses cover most of the simple, physically interesting models. The point mass lens is the limit n 3, with deflection = b^{2} / , convergence = 0 (with a central singularity) and shear = b^{2} / r^{2}. The singular isothermal sphere (SIS) is the case with n = 2. It has a constant deflection = b, and equal convergence and shear = = b / 2. A uniform critical sheet is the limit n 1 with = , = 1 and = 0. Models with n 3/2 have the cusp exponent of the Moore ([1998]) halo model. The popular 1 / r NFW (Navarro, Frenk & White [1996], see Section B.4.1) density cusps are not quite the same as the n 1 case because the projected surface density of a 1 / r cusp has ln rather than a constant. Nonetheless, the behavior of the power law models as n 1 will be very similar to the NFW model if the lens is dominated by the central cusp. The central regions of galaxies probably act like cusps with 1 n 2.
The tangential magnification eigenvalue of these models is
(B.11) |
which is always equal to zero at = b _{E}. This circle defines the tangential critical curve or Einstein (ring) radius of the lens. We normalized the models in this fashion because the Einstein radius is usually the best-determined parameter of any lens model, in the sense that all successful models will find nearly the same Einstein radius (e.g. Kochanek [1991a], Wambsganss & Paczynski [1994]). The source position corresponding to the tangential critical curve is the origin ( = 0), and the reason the magnification diverges is that a point source at the origin is converted into a ring on the tangential critical curve leading to a divergent ratio between the "areas" of the source and the image. The other important point to notice is that the mean surface density inside the tangential critical radius is <> 1 independent of the model. This is true of any circular lens. With the addition of angular structure it is not strictly true, but it is a very good approximation unless the mass distribution is very flattened. The definition of b in terms of the properties of the lens galaxy will depend on the particular profile. For example, in a point mass lens (n 3), b^{2} = (4GM / c^{2} D_{d}^{ang})(D_{ds} / D_{s}) where M is the mass, while in an SIS lens (n = 2), b = 4(_{v} / c)^{2} D_{ds} / D_{s} where _{v} is the (1D) velocity dispersion of the lens. For the other profiles, b can be defined in terms of some velocity dispersion or mass estimate for the lens, as we will discuss later in Section B.4.9 and Section B.6. The radial magnification eigenvalue of these models is
(B.12) |
which can be zero only if n < 2. If n < 2 the deflection goes to zero at the origin and the lens has a radial critical curve at = b(2 - n)^{1/(n-1)} < b interior to the tangential critical curve. Models with n 2 have constant (n = 2) or rising deflection profiles as we approach the lens center and have negative derivatives d / d at all radii.
A nice property of circular lenses is that they allow simple graphical solutions of the lens equation for arbitrary deflection profiles. There are two parts to the graphical solution - the first is to determine the radial positions _{i} of the images given a source position , and the second is to determine the magnification by comparing the area of the images to the area of the source. Recall first, that by symmetry, all the images must lie on a line passing through the source and the lens. Let now be a signed radius that is positive along this line on one side of the lens and negative on the other. The lens equation (Eqn.B.4) along the line is simply
(B.13) |
where we have rearranged the terms to put the deflection on one side and the image and source positions on the other. One side of the equation is the bend angle (Fig. B.9), while the other side of the equation, - , is simply a line of unit slope passing through the source position . The solutions to the lens equation for any source position are the radii _{i} where the line crosses the curve.
For understanding any observed lens, it is always useful to first sketch where the critical lines must lie. Recall from the discussion of caustics in Part 1, that images are always created and destroyed on critical lines as the source crosses a caustic, so the critical lines and caustics define the general structure of the lens. All our power law models have a tangential critical line at = b, which is the solution (b) = b and corresponds to the source position = 0. The origin, as the projection of the critical curve onto the source plane, is the tangential caustic (strictly speaking a degenerate pseudo-caustic) corresponding to the critical line. A point source at the origin is transformed into an Einstein ring of radius _{E} = b.
The second step of the graphical construction is to determine the angular structure of the image. For simplicity, suppose the source is an arc with radial width and angular width . By symmetry, the angle subtended by an image relative to the lens center must be the same as that subtended by the source. For an image at _{i} and a source at , the tangential extent of the image is |_{i}| while that of the source is . The tangential magnification of the image is simply |_{i}| / = (1 - |(_{i}) / _{i}|)^{-1} after making use of the lens equation (Eqn. B.13), and this is identical to the tangential magnification eigenvalue (Eqn. B.11). The thickness of the arc requires finding the image radii for the inner and outer edges of the source, _{i}() and _{i}( + ). The ratio of the thickness of the two arcs is the radial magnification,
(B.14) |
and this is simply the inverse of the radial eigenvalue of the magnification matrix (Eqn. B.12) where we have taken the derivative of the lens equation (Eqn. B.13) with respect to the source position to obtain the final result. Thus, the tangential magnification simply reflects the fact that the angle subtended by the source is the angle subtended by the image, while the radial magnification depends on the slope of the deflection profile with declining deflection profiles (d / d < 0) demagnifying the source and rising profiles magnifying the source.
In Fig. B.10 we illustrate this for the point mass lens (n 3). From the shape of the deflection profile, it is immediately obvious that there will be only two images, one on each side of the lens. If we assume > 0, the first image is a minimum located at
(B.15) |
with _{1} > _{E} and positive magnification
(B.16) |
while the second image is a saddle point located at
(B.17) |
with - _{E} < _{2} < 0 and negative magnification
(B.18) |
As the source approaches the tangential caustic ( 0) the magnifications of both images diverge as ^{-1} and the image radii converge to _{E}. As the source moves to infinity, the magnification of the first image approaches unity and its position approaches that of the source, while the second image is demagnified by the factor (1/2)(b / ) and converges to the position of the lens. The image separation
(B.19) |
is always larger than the diameter of the Einstein ring and the total magnification
(B.20) |
is the characteristic light curve expected for isolated Galactic microlensing events (see Part 4). The point mass lens has one peculiarity that makes it different from extended density distributions like galaxies in that it has two images independent of the impact parameter of the source and no radial caustic. This is a characteristic of any density distribution with a divergent central deflection (n > 2).
Figure B.11. Graphical solutions for the SIS (n = 2) lens when < b and there are two images. |
Figure B.12. Graphical solutions for the SIS (n = 2) lens when > b and there is only one image. |
The SIS (n = 2) model is the "standard" lens model for galaxies. Figs. B.11 and B.12 show the geometric constructions for the images of an SIS lens. If 0 < < b, then the SIS lens also produces two images (Fig. B.11). The first image is a minimum located at
(B.21) |
and the second image is a saddle point located at
(B.22) |
The image separation |_{1} - _{2}| = 2b is constant, and the total magnification |µ_{1}| + |µ_{2}| = 2b / is a simple power law. The magnification produced by an SIS lens is purely tangential since the radial magnification is unity. If, however, > b, then there is only one image, corresponding to the minimum located on the same side of the lens as the source (see Fig. B.12). This boundary on the source plane at = b between having two images at smaller radii and only one image at larger radii is a radial (pseudo)-caustic that can be thought of as being associated with a radial critical curve at the origin. It is a pseudo-caustic because there are neither images nor a divergent magnification associated with it.
Figure 13. Graphical solutions for the Moore profile cusp (n = 3/2) lens when > b/4 and there is only one image. |
Historically the next step is to introduce a core radius to have a model with a true radial critical line and caustic (see Part 1, Blandford & Kochanek [1987b], Kochanek & Blandford [1987], Kovner [1987a], Hinshaw & Krauss [1987], Krauss & White [1992], Wallington & Narayan [1993], Kochanek [1996a]). Instead we will consider the still softer power law model with n = 3/2, which would correspond to the central exponent of the "Moore" profile proposed for CDM halos (Moore et al. [1998]). As fig. B.13 shows, there is only one solution for || > b / 4, a minimum located at
(B.23) |
and with _{1} > b assuming is positive. The magnification expressions are too complex to be of much use, but the magnification µ_{1} diverges at = b when the source is on the tangential pseudo-caustic at = 0. As Fig. B.14 shows, we find two additional images once || < b / 4. The first additional image is a saddle point located at
(B.24) |
with - b < _{2} < - b / 4, which has a negative magnification that diverges at both _{2} = - b (the tangential critical curve) and _{2} = - b / 4. This latter radius defines the radial critical curve where the magnification diverges because the radial magnification eigenvalue 1 - + = 1 - d / d = 0 at radius = b / 4. The third image is a maximum located at
(B.25) |
with - b / 4 < _{2} < 0 and a positive magnification that diverges on the radial critical curve. As we move the source outward from the center we would see images 2 and 3 approach each other, merging on the radial critical line where they would have divergent magnifications, and then vanishing to leave only image 1. We would see the same pattern if instead of softening the exponent we had followed the traditional path and added a core radius to the SIS model. With a finite core radius the central deflection profile would pass through zero, and this would introduce a radial critical curve and a third image which would be a maximum of the time delay surface.
In Fig. B.14 we also illustrate the geometric meaning of the partial parities (the signs of the magnification eigenvalues). A source structure (the L) defines the reference shape. Image 1 is a minimum with positive partial parities (+ +) defined by the signs of the tangential and radial eigenvalues. The orientation of image 1 is the same as the source. Image 2 is a saddle point with mixed partial parities (- +) because the tangential eigenvalue is negative while the radial eigenvalue is positive. This means that the image is inverted in the tangential direction relative to the source. Image 3 is a maximum with negative partial parties (- -), so the image is inverted in both the radial and tangential directions relative to the source. The total parity, the product of the partial parities, is positive for maxima and minima so the orientation of the image can be produced by rotating the source. The total parity of the saddle point image is negative, so its orientation cannot be produced by a rotation of the source.