Gravitational Lensing By a Massive Object in a Dark Matter Halo.
II. Shear, Phase, and Image Geometry

The surface density of a halo with a spherical NFW profile expressed in units of the critical surface density $\Sigma_{\text{cr}}$ yields the dimensionless convergence profile (Bartelmann 1; 8),

$$\kappa_{\text{\tiny NFW}}(x)=2\,\kappa_{\text{s}}\;\frac{1-\mathcal{F}(x)}{x^{2}-1}\,,$$

(1)

where $x$ is the plane-of-the-sky distance from the halo center expressed in units of the halo scale radius $r_{\text{s}}$, and $\kappa_{\text{s}}$ is the halo convergence parameter. The function

$$\mathcal{F}(x)=\begin{cases}\frac{\displaystyle\arctanh{\sqrt{1-x^{2}}}}{\displaystyle\sqrt{1-x^{2}}}&\text{for $x<1$}\,,\\ \hfil 1&\text{for $x=1$}\,,\\ \frac{\displaystyle\arctan{\sqrt{x^{2}-1}}}{\displaystyle\sqrt{x^{2}-1}}&\text{for $x>1$}\,,\end{cases}$$

(2)

has a similar radial behavior to the convergence $\kappa_{\text{\tiny NFW}}(x)$: both decrease monotonically from $\infty$ at the halo center to $0$ for $x\gg 1$ (see 8 for details). The radius $x_{0}$ at which the convergence is equal to $1$ can be computed numerically from

$$\frac{1-x_{0}^{2}}{\mathcal{F}(x_{0})-1}=2\,\kappa_{\text{s}}\,.$$

(3)

This unit-convergence radius and the circle that it defines play a key role when studying the geometry of images formed by a lens. For the NFW halo $x_{0}$ increases monotonically with the convergence parameter $\kappa_{\text{s}}$, as illustrated in 8. We note here that for $\kappa_{\text{s}}=3/2$ the unit-convergence radius is equal to the scale radius, $x_{0}=1$. For lower values of $\kappa_{\text{s}}$ the unit-convergence circle lies inside the scale-radius circle; for higher values outside.

A light ray passing through the halo at a position $\bm{x}$ in the plane of the sky is deflected by an angle

$$\bm{\alpha}(\bm{x})=\frac{4\,\kappa_{\text{s}}\,r_{\text{s}}\,D_{\text{s}}}{D_{\text{l}}\,D_{\text{ls}}}\,\left[\ln{\frac{x}{2}}+\mathcal{F}(x)\right]\,\frac{\bm{x}}{x^{2}}\,,$$

(4)

where $D_{\text{l}}$, $D_{\text{s}}$, and $D_{\text{ls}}$ are the angular-diameter distances from the observer to the lens, from the observer to the source, and from the lens to the source, respectively. Expressed in units of the angular scale radius, the position of a source $\bm{y}$ and the position of its image $\bm{x}$ formed by the gravitational field of the NFW halo are connected by the lens equation

$$\bm{y}=\bm{x}-4\,\kappa_{\text{s}}\,\left[\ln{\frac{x}{2}}+\mathcal{F}(x)\right]\,\frac{\bm{x}}{x^{2}}\,.$$

(5)

For illustration, in the top row of Figure 1 we show the lensing of a circular source by a NFW halo with convergence parameter $\kappa_{\text{s}}\approx 0.239035$, the fiducial value used in 8. As seen in the top left panel, in this example the black circular source centered at $\bm{y_{\text{c}}}=(0.015,\,-0.005)$ with radius $y_{\text{r}}=0.005$ lies inside the radial caustic without overlapping the central point-like tangential caustic. Solving the lens Equation (5) numerically for a point $\bm{y}$ on the circumference of the source yields three points $\bm{x}$ on the boundaries of the three black images shown in the top right panel. One image lies outside the tangential critical curve, a second image lies between the tangential and radial critical curves, and the smallest third image lies inside the radial critical curve. The dashed circle with radius $x_{0}\approx 0.0936$ is the unit-convergence circle of this halo. In this case, the first two images are elongated in the tangential (azimuthal) direction, while the third image is elongated in the radial direction.

For a source lying inside the radial caustic and overlapping the central tangential caustic, the first two images would merge along the tangential critical curve, forming an Einstein ring. For a smaller source positioned close to the inner side of the radial caustic, the second image would lie inside the unit-convergence circle and be elongated in the radial direction. For a source positioned on the radial caustic, the second and third images would merge at the radial critical curve. For a source lying outside the radial caustic, these two images would vanish, leaving only the first image.

The deformations and orientations of the images are best studied by computing the lens shear and its phase, quantities that may be introduced by means of the lens potential. The deflection angle can be written in terms of the gradient of the lens potential $\psi(x)$, which in this case is circularly symmetric,

$$\bm{\alpha}(\bm{x})=\frac{D_{\text{s}}}{D_{\text{ls}}}\,\nabla_{\bm{\theta}}\,\psi(x)=\frac{D_{\text{s}}\,D_{\text{l}}}{D_{\text{ls}}\,r_{\text{s}}}\,\frac{\bm{x}}{x}\,\frac{{\rm d}\psi}{{\rm d}x}\,,$$

(6)

where we converted the angular position in radians $\bm{\theta}$ to the angular position in scale-radius units, $\bm{x}=\bm{\theta}\,D_{\text{l}}/r_{\text{s}}$. By substituting for the deflection angle from Equation (4) we can express the lens-potential derivative

$$\frac{{\rm d}\psi_{\text{\tiny NFW}}}{{\rm d}x}=\frac{4\,\kappa_{\text{s}}\,r_{\text{s}}^{2}}{D_{\text{l}}^{2}}\,\frac{1}{x}\,\left[\ln{\frac{x}{2}}+\mathcal{F}(x)\right]\,.$$

(7)

Integration yields the following expressions for the NFW halo lens potential [10, 6]:

$$\psi_{\text{\tiny NFW}}(x)=\begin{cases}\frac{2\,\kappa_{\text{s}}\,r_{\text{s}}^{2}}{D_{\text{l}}^{2}}\left[\ln{x}\ln{\frac{x}{4}}-(\displaystyle\arctanh{\sqrt{1-x^{2}}}\,)^{2}\right]\\ \hfil 0\\ \frac{2\,\kappa_{\text{s}}\,r_{\text{s}}^{2}}{D_{\text{l}}^{2}}\left[\ln{x}\ln{\frac{x}{4}}+(\displaystyle\arctan{\sqrt{x^{2}-1}}\,)^{2}\right]\,,\end{cases}$$

(8)

where the expression in the first row holds for $x<1$, in the second row for $x=1$, and in the third row for $x>1$. For $x\ll 1$, the potential close to the halo center

$$\psi_{\text{\tiny NFW}}(x)=-\,\frac{2\,\kappa_{\text{s}}\,r_{\text{s}}^{2}}{D_{\text{l}}^{2}}\left(\ln^{2}{2}+\frac{x^{2}}{2}\,\ln{\frac{x}{2}}\right)+\mathcal{O}(x^{4}\,\ln{x})\,,$$

(9)

starting from a finite negative value and increasing monotonically outward, crossing zero at the scale radius. Note that the expressions for the potential in [10] and [6] are higher by the constant $-\psi_{\text{\tiny NFW}}(0)$ and thus they start at zero.

For a circularly symmetric lens potential the lens shear $\gamma$ can be computed as

$$\gamma=\frac{D_{\text{l}}^{2}}{2\,r_{\text{s}}^{2}}\left|\frac{{\rm d}^{2}\psi}{{\rm d}x^{2}}-\frac{1}{x}\frac{{\rm d}\psi}{{\rm d}x}\right|\,.$$

(10)

For the NFW profile the second derivative of the lens potential,

$$\frac{{\rm d}^{2}\psi_{\text{\tiny NFW}}}{{\rm d}x^{2}}=\frac{4\,\kappa_{\text{s}}\,r_{\text{s}}^{2}}{D_{\text{l}}^{2}}\\ \times\left[\frac{1}{x^{2}-1}+\frac{2x^{2}-1}{x^{2}(1-x^{2})}\,\mathcal{F}(x)-\frac{1}{x^{2}}\ln{\frac{x}{2}}\right]\,,$$

(11)

can be used together with the first derivative from Equation (7) in Equation (10) to yield the shear of the NFW halo,

$$\gamma_{\text{\tiny NFW}}(x)=2\,\kappa_{\text{s}}\left[\frac{2}{x^{2}}\ln{\frac{x}{2}}+\frac{1}{1-x^{2}}+\frac{2-3x^{2}}{x^{2}(1-x^{2})}\,\mathcal{F}(x)\right]\,,\\$$

(12)

as shown by [19]. In order to understand its behavior close to the origin, we expand Equation (12) for $x\ll 1$ and obtain

$$\gamma_{\text{\tiny NFW}}(x)=\kappa_{\text{s}}\,\left[\,1+\frac{3}{2}\,x^{2}\,\ln{\frac{x}{2}}+\frac{13}{8}\,x^{2}\,\right]+\mathcal{O}(x^{4}\,\ln{x})\,.$$

(13)

By setting $x=0$ we see that the NFW shear at the center is equal to the convergence parameter of the halo, $\gamma_{\text{\tiny NFW}}(0)=\kappa_{\text{s}}$. From this value the shear decreases outward monotonically. An expansion close to the scale radius shows that for $x\to 1$

$$\gamma_{\text{\tiny NFW}}(x)=2\,\kappa_{\text{s}}\,\left[\,\frac{5}{3}-2\,\ln{2}-4\,\left(\frac{11}{15}-\ln{2}\right)\,(x-1)\,\right]\\[4.30554pt] +\mathcal{O}((x-1)^{2})\,,$$

(14)

which yields $\gamma_{\text{\tiny NFW}}(1)\approx 0.561\,\kappa_{\text{s}}$. The NFW shear decreases for $x\gg 1$ to zero,

$$\gamma_{\text{\tiny NFW}}(x)=2\,\kappa_{\text{s}}\,\left[\,\frac{2}{x^{2}}\,\ln{\frac{x}{2}}-x^{-2}+\frac{3\pi}{2}\,x^{-3}-4\,x^{-4}\,\right]\\ +\mathcal{O}(x^{-5})\,.$$

(15)

The first panel in Figure 2 shows a contour plot of the shear $\gamma_{\text{\tiny NFW}}(\bm{x})$ in the central part of a NFW halo. Going outward from the center, the dotted contours correspond to 95%, 90%, and 85% of the central shear $\gamma_{\text{\tiny NFW}}(0)=\kappa_{\text{s}}$. Clearly, the shear changes very slowly on this scale, as indicated also by the practically homogeneous color, with the color bar set for comparison with further figures. The solid black circles mark the radial (smaller) and tangential (larger) critical curve for the fiducial halo convergence parameter $\kappa_{\text{s}}\approx 0.239035$.

The NFW shear can be written in terms of its two components, defined as

$$(\gamma_{\text{\tiny NFW}1},\gamma_{\text{\tiny NFW}2})=\gamma_{\text{\tiny NFW}}\,(\cos{2\varphi_{\text{\tiny NFW}}},\sin{2\varphi_{\text{\tiny NFW}}})\,,$$

(16)

where the trigonometric functions of the phase $\varphi_{\text{\tiny NFW}}$ can be computed for a point $\bm{x}=(x_{1},x_{2})=x(\cos{\phi},\sin{\phi})$ in the image plane as

$$(\cos{2\varphi_{\text{\tiny NFW}}},\sin{2\varphi_{\text{\tiny NFW}}})=x^{-2}(x_{2}^{2}-x_{1}^{2},-2x_{1}x_{2})\\ =-(\cos{2\phi},\sin{2\phi})\,.$$

(17)

The negative sign in front of the last parentheses indicates that the phase $\varphi_{\text{\tiny NFW}}=\phi+\pi/2+k\pi$, i.e., its orientation is always perpendicular to the position vector of the point. Note that this also means that the phase and the shear components are undefined at $x=0$, since the phase depends on the direction of approach to the center.

The Jacobian matrix of a general lens equation expressed in terms of the convergence $\kappa(\bm{x})$, shear $\gamma(\bm{x})$, and phase $\varphi(\bm{x})$ has the form [16, e.g.,]

$$J(\bm{x})=\frac{\partial\,\bm{y}}{\partial\,\bm{x}}=\begin{pmatrix}1-\kappa-\gamma\cos{2\varphi}&-\gamma\sin{2\varphi}\\ -\gamma\sin{2\varphi}&1-\kappa+\gamma\cos{2\varphi}\end{pmatrix}\,.$$

(18)

Its determinant, the Jacobian, can be computed from the convergence and the shear:

$$\mathrm{det}\,J(\bm{x})=\left[\,1-\kappa(\bm{x})-\gamma(\bm{x})\,\right]\,\left[\,1-\kappa(\bm{x})+\gamma(\bm{x})\,\right]\,.$$

(19)

The critical curves, explored in detail in 8, can be obtained by setting $\mathrm{det}\,J(\bm{x})=0$. We note here merely that for an axially symmetric lens such as the studied NFW halo, the first term in Equation (19) yields the tangential critical curve and the second term yields the radial critical curve. Hence, the shear is related to the convergence by $\gamma(x_{\text{T}})=1-\kappa(x_{\text{T}})$ at the tangential critical curve, and by $\gamma(x_{\text{R}})=\kappa(x_{\text{R}})-1$ at the radial critical curve, with the two critical curves separated by the unit-convergence circle.

To study the geometry of the images, we invert the Jacobian matrix to obtain the mapping from the source plane to the image plane. We can write the inverse matrix in terms of its two eigenvalues,

$$\lambda_{\parallel}(\bm{x})=(1-\kappa-\gamma)^{-1},\qquad\lambda_{\perp}(\bm{x})=(1-\kappa+\gamma)^{-1}\,,$$

(20)

as

$$\mathbb{A}(\bm{x})=\lambda_{\parallel}\begin{pmatrix}\cos^{2}{\varphi}&\cos{\varphi}\sin{\varphi}\\ \cos{\varphi}\sin{\varphi}&\sin^{2}{\varphi}\end{pmatrix}\\ +\lambda_{\perp}\begin{pmatrix}\sin^{2}{\varphi}&-\cos{\varphi}\sin{\varphi}\\ -\cos{\varphi}\sin{\varphi}&\cos^{2}{\varphi}\end{pmatrix}\,,$$

(21)

where $\kappa$, $\gamma$, and $\varphi$ are functions of the image-plane position $\bm{x}$. The matrix accompanying $\lambda_{\parallel}$ is a projection matrix onto the eigenvector $(\cos{\varphi},\sin{\varphi})$; the matrix accompanying $\lambda_{\perp}$ is a projection matrix onto the eigenvector $(-\sin{\varphi},\cos{\varphi})$. Equation (21) shows that an image at position $\bm{x}$ is scaled by a factor $\lambda_{\parallel}$ in the direction parallel to the phase $\varphi$, and by a factor $\lambda_{\perp}$ in the direction perpendicular to the phase, $\varphi+\pi/2$.

A small circular source with radius $y_{\text{r}}$ centered at $\bm{y}_{\text{c}}$ is thus portrayed by the lens as a set of $n$ small elliptical images with semiaxes $y_{\text{r}}/|1-\kappa(\bm{x}^{[i]}_{\text{c}})-\gamma(\bm{x}^{[i]}_{\text{c}})|$ and $y_{\text{r}}/|1-\kappa(\bm{x}^{[i]}_{\text{c}})+\gamma(\bm{x}^{[i]}_{\text{c}})|$. Their positions $\bm{x}^{[i]}_{\text{c}}(\bm{y}_{\text{c}})$, $i=1\ldots n$, can be found by solving the lens equation, i.e., Equation (5) for the NFW halo. We illustrate the geometry of one such image in the second panel of Figure 3 for convergence $\kappa(\bm{x}^{[i]}_{\text{c}})=\kappa=0.08$ and shear $\gamma(\bm{x}^{[i]}_{\text{c}})=\gamma=0.16$. In the three right panels we include all other $(\kappa,\gamma)$ combinations that lead to the same combination of semiaxes $|\lambda_{\parallel}|\,y_{\text{r}}$ and $|\lambda_{\perp}|\,y_{\text{r}}$, i.e., they generate an elliptical image of the same shape and size. The images differ in their orientation and parity. For the combinations in the two right panels with $\kappa>1$ the major axis is oriented perpendicular to the phase $\varphi$ rather than parallel to it. The images in the third and fourth panels have negative parity, as indicated by the positions of the images of the black and white points on the circumference of the source. The signs of the eigenvalues are marked in each panel, with negative values indicating mirroring along the corresponding eigenvector.

For a general source, the distortion of its image can be quantified by the dimensionless flattening,

$$f(\kappa,\gamma)=1-\min\left({\left|\frac{1-\kappa-\gamma}{1-\kappa+\gamma}\right|,\left|\frac{1-\kappa+\gamma}{1-\kappa-\gamma}\right|}\right)\,,$$

(22)

defined here using the ratio of the smaller to larger eigenvalues, with their definitions taken from Equation (20). For an elliptical image of a circular source, $f$ is equal to its ellipticity. For the sample images in Figure 3, $f\approx 0.30$. While the absolute value of the ratio of the eigenvalues determines the distortion of the image, their product determines its magnification and parity. Hence, it is sufficient to know the convergence $\kappa$ and shear $\gamma$ at the position of an image of a small source in order to fully determine its distortion, magnification, parity, and orientation with respect to the phase.

For a given gravitational lens, this information can be condensed into the convergence–shear (hereafter CS) diagram, introduced in Appendix A. The geometry of images formed by a NFW halo lens as a function of their radial position $x$ can be easily identified from its CS diagram, shown in Figure 4 for the fiducial convergence parameter $\kappa_{\text{s}}\approx 0.239035$. The green curve connecting the $\left(\kappa_{\text{\tiny NFW}}(x),\gamma_{\text{\tiny NFW}}(x)\right)$ points is obtained by using the NFW halo convergence from Equation (1) and the NFW halo shear from Equation (12). The center of the halo corresponds to position $(\kappa,\gamma)=(\infty,\kappa_{\text{s}})$ in the diagram, while for $x\to\infty$ the green curve reaches the origin, $(\kappa,\gamma)\to(0,0)$. The tick marks along the curve correspond to radii (from the right side of the plot) $x\in\{0.002,0.003,\ldots,0.01,0.02,\ldots,0.1,0.2,\ldots,1,2\}$.

In order to interpret the image geometries in a NFW halo from Figure 4, we follow the green curve, starting from the origin for very distant images and progressing toward the halo center. We first recall the discussion following Equation (17), which implies that for the NFW profile the direction of the phase corresponds in polar coordinates to the tangential, and the direction perpendicular to the phase to the radial direction, respectively. Initially, the image has positive parity with magnification increasing from $1$ and the flattening increasing from $0$. The image is expanded tangentially ($|\lambda_{\parallel}|>1$) but contracted radially ($|\lambda_{\perp}|<1$), as indicated by the position above the dashed line.

Between $x=2$ and $x=1$ the green line crosses the diagonal and the image becomes expanded also radially ($|\lambda_{\perp}|>1$). For lower $x$, the green curve approaches the solid black tangential-critical-curve line, along which the magnification and $|\lambda_{\parallel}|$ become infinite, the flattening increases to $1$, and $|\lambda_{\perp}|$ remains finite. After crossing the tangential critical curve at $x=x_{\text{T}}\approx 0.155$, the image parity changes to negative (due to the sign of $\lambda_{\parallel}$ changing to negative) and the magnification and flattening decrease. The maximum distortion stays oriented tangentially until reaching the unit-convergence circle, $x=x_{0}\approx 0.0936$, corresponding to the vertical solid black line at $\kappa=1$. To the right of this line the maximum distortion is oriented radially. At the unit-convergence radius the negative-parity image is close to its lowest magnification (though it stays higher than $16$ in this case, as indicated by the hyperbolic contours), and with zero flattening its shape is undistorted.

Proceeding further toward the halo center, the magnification and flattening increase again, with the radial expansion $|\lambda_{\perp}|$ growing rapidly while the tangential expansion $|\lambda_{\parallel}|$ decreases. At the solid black radial-critical-curve line, which corresponds to $x=x_{\text{R}}\approx 0.056$, the magnification and $|\lambda_{\perp}|$ become infinite, the flattening increases to $1$, and $|\lambda_{\parallel}|$ remains finite. For lower radii, image parity changes back to positive (due to the sign of $\lambda_{\perp}$ changing to negative), and the magnification and flattening decrease. The expansion in both perpendicular directions decreases, until the intersection with the dashed black line at $x\approx 0.019$. Images lying closer to the halo center are tangentially contracted rather than expanded ($|\lambda_{\parallel}|<1$). After crossing the solid black unit-magnification hyperbola at $x\approx 0.0105$, all images are demagnified. The last important intersection occurs at $x\approx 0.0068$; images to the right of the last dashed black line are contracted even radially ($|\lambda_{\perp}|<1$), and their magnification and flattening decrease to $0$ at the halo center.

The three orange points marked along the green curve in Figure 4 correspond to the radial positions of the centers of the three images in the top right panel of Figure 1. The first point at $x\approx 0.188$ corresponds to the image outside the tangential critical curve at the right side of the panel. The second point at $x\approx 0.115$ corresponds to the image between the tangential critical curve and the unit-convergence circle at the left side of the panel. The third point at $x\approx 0.014$ corresponds to the smallest image inside the radial critical curve. The magnification, parity, flattening, orientation, and the two scaling factors of the corresponding images can be determined from the positions of these points in the diagram in Figure 4. Note that the values obtained from the diagram are technically valid at the positions of the source-center images and thus correspond to the local “point-source” values. Taking into account the radial extent of each image, their position (and the relevant range of their properties) should be marked by line segments along the green line in Figure 4 rather than by points.

Weak-lensing cluster-mass reconstructions are based on statistical analyses of the images of background galaxies [7]. The convergence map is computed from maps of the shear components. These are constructed from the shear and the phase, which are in turn determined from the shapes and orientations of the images. In the weak-lensing limit, the geometry of the elliptical image of a small circular source corresponds to the second panel of Figure 3. Its axis ratio yields the shear and the orientation angle of its major axis is equal to the phase. The semi-minor to semi-major axis ratio $b/a$ is obtained by expanding the ratio $\lambda_{\perp}/\lambda_{\parallel}$ of the eigenvalues from Equation (20) to first order in $\kappa$ and $\gamma$. The ellipticity, which is equal to the flattening of the image defined in Equation (22), reduces in this limit to

$$f=1-b/a\simeq 2\,\gamma\,,$$

(23)

i.e., double the value of the local shear. For illustration, for the image in the second panel of Figure 3 Equation (23) yields an approximate flattening $2\,\gamma=0.32$, which is an $8\%$ overestimate of the actual value $f\approx 0.30$ from Equation (22).

Based on the weak-lensing regime, we introduce the weak shear and weak phase, which are computed from the images using the weak-lensing relations from the previous paragraph. We define the weak shear as

$$\gamma_{\text{w}}(\bm{x})=\frac{1}{2}\,f\left(\kappa(\bm{x}),\gamma(\bm{x})\right)\,,$$

(24)

where the flattening $f$ is computed from Equation (22) using lens-specific convergence and shear functions. Note that $\gamma_{\text{w}}$ by its definition attains only values from the interval $\left[0,\,0.5\right]$. In the weak-lensing regime $\gamma_{\text{w}}\approx\gamma$, but as $\gamma$ and $\kappa$ increase, the weak shear computed from the image flattening deviates from the shear.

For the NFW halo we compute the weak shear $\gamma_{\text{w,\tiny NFW}}(\bm{x})$ from Equation (24) using its convergence from Equation (1) and shear from Equation (12). The second panel in Figure 2 shows a plot of $\gamma_{\text{w,\tiny NFW}}(\bm{x})$ in the central part of a NFW halo, using the same color scale as for the shear in the first panel of the figure. Due to its relation to the flattening in Equation (24), the plot can be interpreted following the $f$ values along the green curve in the diagram in Figure 4. At the halo center the flattening and thus also the weak shear are equal to zero. Going outward from the center, the weak shear increases to its maximum value of $0.5$ at the radial critical curve, marked by the inner black circle. From there it drops to $0$ at the unit-convergence radius and increases back to $0.5$ at the tangential critical curve, marked by the outer black circle. Beyond the tangential critical curve the weak shear drops asymptotically to $0$. Comparison with the left panel shows the substantial difference between the shears in the central region of the NFW halo, in terms of amplitude as well as radial pattern. Here the variations in image distortion are primarily driven by the convergence rather than by the shear.

In the third panel of Figure 2 we illustrate the difference between the first two panels by plotting the relative deviation of the weak shear from the shear, $\gamma_{\text{w,\tiny NFW}}/\gamma_{\text{\tiny NFW}}-1$. The blue regions in which the weak shear underestimates the shear are limited to the vicinity of the origin and the vicinity of the unit-convergence circle. In both cases the weak shear drops to zero and the relative deviation thus reaches $-1$, its minimum possible value. Everywhere else the weak shear overestimates the shear, with the positive deviation peaking at the critical curves and dropping to $0$ asymptotically. Note that the maxima at the critical curves may be negative for NFW halos with a sufficiently high convergence parameter $\kappa_{\text{s}}$, for which $\gamma_{\text{\tiny NFW}}(x_{\text{R}})$ or even $\gamma_{\text{\tiny NFW}}(x_{\text{T}})$ exceeds the weak-shear value at critical curves (i.e., $0.5$).

In addition to the weak shear, we define the image-based weak phase $\varphi_{\text{w}}$ as the angle between the major axis of the image of a small circular source and the horizontal ($x_{1}$) axis of the image plane. Taking into account Figure 3 and the discussion preceding Equation (22), in the case of the NFW halo it is related to the phase $\varphi_{\text{\tiny NFW}}$ as follows:

$$\varphi_{\text{w,\tiny{NFW}}}(\bm{x})=\Bigg\{\begin{array}[]{ll}\,\varphi_{\text{\tiny NFW}}(\bm{x})&\,\text{for $\kappa_{\text{\tiny NFW}}(\bm{x})<1$}\,,\\ \,\varphi_{\text{\tiny NFW}}(\bm{x})+\pi/2&\,\text{for $\kappa_{\text{\tiny NFW}}(\bm{x})>1$}\,.\end{array}$$

(25)

We use values from the interval $\left[-\pi/2,\,\pi/2\right]$ for both $\varphi_{\text{\tiny NFW}}$ and $\varphi_{\text{w,\tiny{NFW}}}$.

The fourth panel of Figure 2 shows a color map of the weak phase $\varphi_{\text{w,\tiny{NFW}}}$ of the NFW halo. The white regions with $\varphi_{\text{w,\tiny{NFW}}}=0$ correspond to horizontally elongated images, in the orange regions with $\varphi_{\text{w,\tiny{NFW}}}>0$ the images are oriented counterclockwise and in the blue regions with $\varphi_{\text{w,\tiny{NFW}}}<0$ the images are oriented clockwise from the horizontal. The bright orange/blue boundaries correspond to images elongated exactly vertically, with $|\varphi_{\text{w,\tiny{NFW}}}|=\pi/2$. The weak phase flips by $\pi/2$ along the unit-convergence circle, separating the inner radially oriented from the outer tangentially oriented images. The weak phase is undefined along this circle as well as at the origin, which corresponds to zero flattening. The dot-dashed lines added for orientation mark the positions of all images with exactly horizontal, exactly vertical, or undefined orientation.

Since the weak phase differs from the phase only by the $\pi/2$ flip inside the unit-convergence circle, a similar color map of the phase $\varphi_{\text{\tiny NFW}}(\bm{x})$ would differ merely by having inverted color and saturation inside the circle. In other words, the color and saturation outside the circle in the fourth panel of Figure 2 would be radially extended to the halo center. Hence, the first and third quadrants would be entirely blue and the second and fourth quadrants would be entirely orange.

The plots of the different quantities in Figure 2 are presented as reference plots to aid the interpretation of the results for the NFW halo + point-mass lens model presented in Section III.5.

Adding a compact massive object modeled by a point mass positioned at $\bm{x_{\text{P}}}$ changes the convergence to

$$\kappa(\bm{x})=2\,\kappa_{\text{s}}\;\frac{1-\mathcal{F}(x)}{x^{2}-1}+\pi\,\kappa_{\text{P}}\,\delta(\bm{x}-\bm{x_{\text{P}}})\,,$$

(26)

where the mass parameter $\kappa_{\text{P}}$ corresponds to the ratio of the solid angles subtended by the point-mass Einstein circle and by the halo scale-radius circle [8, for more details, see]. Hence, $\sqrt{\kappa_{\text{P}}}$ is the point-mass Einstein radius in units of the halo scale radius. The convergence in Equation (26) is identical to the NFW halo convergence from Equation (1), except exactly at the position of the added point mass. The lens equation can be written as

$$\bm{y}=\bm{x}-4\,\kappa_{\text{s}}\,\left[\ln{\frac{x}{2}}+\mathcal{F}(x)\right]\,\frac{\bm{x}}{x^{2}}-\kappa_{\text{P}}\,\frac{\bm{x}-\bm{x_{\text{P}}}}{|\bm{x}-\bm{x_{\text{P}}}|^{2}}\,,$$

(27)

in the form used in 8.

For illustration, in the bottom row of Figure 1 we show the lensing of the same circular source as in the top row, by the same NFW halo with an additional point mass with mass parameter $\kappa_{\text{P}}\approx 2.714\cdot 10^{-4}$ positioned at $\bm{x_{\text{P}}}=(0.2,\,0)$. This parameter combination is selected from the parameter-space grid used in 8. The position of the point mass is indicated by its Einstein circle (cyan) in the bottom right panel. As seen in the bottom left panel, the black circular source lies inside the weakly perturbed radial caustic, with its upper part lying also inside the strongly perturbed tangential caustic. For a source not lying on the caustic, lens Equation (27) yields $2,4,$ or $6$ images. For a source lying on the caustic, several images appear combined into a lower number of macro-images. In the example shown in the bottom right panel there are five macro-images. Four of them are images of the full source; the fifth macro-image to the top left of the point mass consists of two additional images of the upper part of the source joined along the critical curve. Comparing the images with those in the top right panel, we see that the left and central images are affected only weakly by the point mass. The right image is affected more strongly, plus there are two new images closer to the point mass. For these images in particular, their distortion cannot be simply classified as tangential or radial.

In order to compute the shear we start from the lens potential, which has an additional term due to the point mass,

$$\psi(\bm{x})=\psi_{\text{\tiny NFW}}(x)+\frac{r_{\text{s}}^{2}}{D_{\text{l}}^{2}}\,\kappa_{\text{P}}\,\ln{|\bm{x}-\bm{x_{\text{P}}}|}\,,$$

(28)

where $\psi_{\text{\tiny NFW}}(x)$ is given by Equation (8).

The point-mass shear has the simple form

$$\gamma_{\text{P}}(\bm{x})=\frac{\kappa_{\text{P}}}{|\bm{x}-\bm{x_{\text{P}}}|^{2}}\,,$$

(29)

divergent at the point-mass position and dropping rapidly outward. Since the NFW halo shear peaks at its central value $\kappa_{\text{s}}$, the added point mass will dominate the lens shear in its vicinity, wherever it may be positioned. The shear can be generally computed from the second derivatives of the lens potential, namely

$$\gamma=\frac{D_{\text{l}}^{2}}{r_{\text{s}}^{2}}\,\sqrt{\frac{1}{4}\left(\,\psi,_{\scriptscriptstyle 11}-\psi,_{\scriptscriptstyle 22}\right)^{2}+\left(\psi,_{\scriptscriptstyle 12}\right)^{2}}\,,$$

(30)

where the commas denote partial derivatives with respect to image-plane coordinates $(x_{1},x_{2})$. For our combined lens we compute the derivatives of the lens potential from Equation (28) and get

$$\gamma(\bm{x})=\sqrt{\left[\gamma_{\text{\tiny{NFW}}}(x)-\gamma_{\text{P}}(\bm{x})\right]^{2}+4\,\gamma_{\text{\tiny{NFW}}}(x)\,\gamma_{\text{P}}(\bm{x})\,\cos^{2}{\omega(\bm{x})}}\,,$$

(31)

where $\gamma_{\text{\tiny{NFW}}}(x)$ and $\gamma_{\text{P}}(\bm{x})$ are given by Equation (12) and Equation (29), respectively, and

$$\cos{\omega(\bm{x})}=\frac{{\bm{x}}\cdot(\bm{x}-\bm{x_{\text{P}}})}{x\,|\bm{x}-\bm{x_{\text{P}}}|}$$

(32)

is the dot product of the unit vectors pointing to $\bm{x}$ from the halo center and from the point-mass position. In terms of image-plane geometry, $\omega$ is the viewing angle from point $\bm{x}$ of the line segment connecting the halo center and the point-mass position. As shown in Figure 5, curves of constant $\omega$ are circular arcs connecting symmetrically the halo center and the point-mass position. Note that $\omega$ is also equal to the angle between the tangent to the arc at either of its end points and the outward horizontal direction, as follows from the tangent–chord theorem (alternate segment theorem).

Along the line segment connecting the center and the point mass the viewing angle reaches its maximum, $\omega=\pi$, while along the rest of the horizontal axis it reaches its minimum, $\omega=0$. In both cases, $\cos^{2}{\omega}=1$ and the total shear from Equation (31) is $\gamma(\bm{x})=\gamma_{\text{\tiny{NFW}}}(x)+\gamma_{\text{P}}(\bm{x})$. In this case both shears act in the same orientation, so that their combination is maximal. Along the circle bisected by the line segment, we find $\omega=\pi/2$ according to Thales’s theorem and the total shear is $\gamma(\bm{x})=|\gamma_{\text{\tiny{NFW}}}(x)-\gamma_{\text{P}}(\bm{x})|$. In this case the two shears act in perpendicular directions, so that their combination is minimal. For the pair of small arcs in Figure 5 with $\omega=3\pi/4$ and for the pair of large arcs with $\omega=\pi/4$ we get $\cos^{2}{\omega}=1/2$ and the total shear is $\gamma(\bm{x})=\sqrt{\gamma_{\text{\tiny{NFW}}}^{2}(x)+\gamma_{\text{P}}^{2}(\bm{x})}$.

As discussed in Section II.1, for the NFW halo the central shear is defined, $\gamma_{\text{\tiny{NFW}}}(0)=\kappa_{\text{s}}$, while the phase and shear components are undefined. The same holds for the central properties of the point-mass lens. However, for the combined lens even the shear at the halo center is undefined. For $x\to 0$ Equation (31) yields

$$\gamma(\bm{x})\to\sqrt{\left[\kappa_{\text{s}}-\kappa_{\text{P}}\,x_{\text{P}}^{-2}\right]^{2}+4\,\kappa_{\text{s}}\kappa_{\text{P}}\,x_{\text{P}}^{-2}\,\cos^{2}{\omega}}\,,$$

(33)

a value that depends on the direction of approach to the center, due to the directional dependence of $\omega$. As seen from Figure 5 and as explained in the discussion above, approaching the center along the horizontal axis leads to the highest value (the sum of the two shears) while an approach along the vertical axis leads to the lowest value (the absolute value of the difference of the two shears). The situation at the position of the point mass is similar, though here the angular differences are suppressed by the divergence of $\gamma_{\text{P}}$.

The range of values of the shear occurring in the studied central region of the image plane is larger than for the halo or the point mass separately. Its upper limit is $\infty$, due to the divergence of $\gamma_{\text{P}}$ at the point-mass position $\bm{x_{\text{P}}}$. Its lower limit may reach $0$. As seen from the form of Equation (31), this may occur only along the $\omega=\pi/2$ circle at points where $\gamma_{\text{\tiny{NFW}}}(x)=\gamma_{\text{P}}(\bm{x})$. Following the circle from the halo center to the point mass, $\gamma_{\text{\tiny{NFW}}}$ decreases while $\gamma_{\text{P}}$ increases. Zero-shear points thus exist only if $\gamma_{\text{\tiny{NFW}}}$ is equal to or larger than $\gamma_{\text{P}}$ at the halo center. Hence, for $x_{\text{P}}<\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}$ there are no zero-shear points. In this range, for point masses closest to the halo center, the minimum shear $\gamma=\kappa_{\text{P}}\,x_{\text{P}}^{-2}-\kappa_{\text{s}}$ occurs at the halo center when approached along the vertical axis.

Zero-shear points exist for all larger point-mass distances from the halo center. For $x_{\text{P}}=\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}$ there is one zero-shear point located directly at the halo center. For any $x_{\text{P}}>\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}$ there are two zero-shear points lying symmetrically above and below the horizontal axis on the $\omega=\pi/2$ circle. With increasing distance of the point mass from the halo center the zero-shear points shift along the circle toward the position of the point mass, so that for larger distances they lie nearly vertically above and below the point mass. Their separation from the point mass, which is approximately $\sqrt{\kappa_{\text{P}}/\gamma_{\text{\tiny{NFW}}}(x_{\text{P}})}$ in this regime, increases with distance as the halo shear decreases.

Image-plane maps of the shear $\gamma(\bm{x})$ for different masses and positions of the point mass are presented and discussed in Section III.5.1.

The shear components are defined by

$$(\gamma_{1},\gamma_{2})=\gamma\,(\cos{2\varphi},\sin{2\varphi})\,,$$

(34)

where the trigonometric functions of the phase $\varphi$ can be computed for a point $\bm{x}=(x_{1},x_{2})$ in the image plane as

	$\displaystyle\cos{2\varphi}$	$\displaystyle=$	$\displaystyle\frac{1}{\gamma(\bm{x})}\left[\frac{x_{2}^{2}-x_{1}^{2}}{x^{2}}\,\gamma_{\text{\tiny{NFW}}}(x)\right.$
			$\displaystyle\left.\hskip 28.45274pt+\frac{(x_{2}-x_{\text{P}2})^{2}-(x_{1}-x_{\text{P}1})^{2}}{|\bm{x}-\bm{x_{\text{P}}}|^{2}}\,\gamma_{\text{P}}(\bm{x})\right]$
	$\displaystyle\sin{2\varphi}$	$\displaystyle=$	$\displaystyle\frac{-2}{\gamma(\bm{x})}\left[\frac{x_{1}x_{2}}{x^{2}}\,\gamma_{\text{\tiny{NFW}}}(x)\right.$
			$\displaystyle\left.\hskip 28.45274pt+\frac{(x_{1}-x_{\text{P}1})(x_{2}-x_{\text{P}2})}{|\bm{x}-\bm{x_{\text{P}}}|^{2}}\,\gamma_{\text{P}}(\bm{x})\right]\,,$

where the shears $\gamma(\bm{x}),\gamma_{\text{\tiny{NFW}}}(x),$ and $\gamma_{\text{P}}(\bm{x})$ are given by Equations (31), (12), and (29), respectively. Note that in this case the phase and the shear components are undefined at the halo center and at the point-mass position, since the phase as well as the shear $\gamma(\bm{x})$ depend on the direction of approach to these points.

We would like to point out that Equation (31) is a special case of the more general formula

$$\gamma=\sqrt{(\gamma_{\text{A}}-\gamma_{\text{B}})^{2}+4\,\gamma_{\text{A}}\gamma_{\text{B}}\cos^{2}{(\varphi_{\text{A}}-\varphi_{\text{B}})}}$$

(36)

for the shear of a combination of two mass distributions with shears $\gamma_{\text{A}},\gamma_{\text{B}}$ and phases $\varphi_{\text{A}},\varphi_{\text{B}}$. For two circularly symmetric mass distributions with the same sign of the expression $\psi^{\prime\prime}-x^{-1}\psi^{\prime}$ that appears in Equation (10), the absolute value of the phase difference in Equation (36) is equal to the viewing angle $\omega$.

The expression $\psi^{\prime\prime}-x^{-1}\psi^{\prime}$ is globally negative for a range of mass distributions, such as for the NFW profile, for a point mass, for a singular or a non-singular (cored) isothermal sphere. For a combination of two such distributions, the formula for the shear in Equation (31), the following discussion, and the formulae for the phase in Equation (III.1) are valid. For example, the case of two point masses was studied by [15], and the case of two isothermal spheres was studied by [17].

The Jacobian of the lens equation can be computed from Equation (19) using the convergence from Equation (26) and the shear from Equation (31). Its explicit form is presented in 8, together with a detailed analysis of the critical curves which are obtained by setting the Jacobian equal to zero. The parts of the critical curve lying outside the unit-convergence circle ($x>x_{0}$) satisfy the equation

$$\gamma(\bm{x})=1-\kappa(\bm{x})\,,$$

(37)

which yields the tangential critical curve in absence of the point mass. The parts lying inside the unit-convergence circle ($x<x_{0}$) satisfy the equation

$$\gamma(\bm{x})=\kappa(\bm{x})-1\,,$$

(38)

which yields the radial critical curve in absence of the point mass.

Equation (19) also indicates that for the Jacobian to be equal to zero at a point lying directly on the unit-convergence circle, the shear must be zero at such a point. From the properties of zero-shear points discussed in Section III.1 it follows that such critical-curve points must lie at the intersections of the unit-convergence circle and the $\omega=\pi/2$ circle extending from the halo center to the point-mass position.

For $x_{\text{P}}<x_{0}$ these circles have no intersection and, thus, there are no critical-curve points along the unit-convergence circle. For $x_{\text{P}}=x_{0}$ these circles have an intersection exactly at the position of the point mass. However, at this point the shear is not zero, so that even in this case there is no critical-curve point along the unit-convergence circle. For any $x_{\text{P}}>x_{0}$ these circles have two intersections. In this case the requirement of zero shear leads to the condition

$$x_{\text{P}}=x_{0}\,\sqrt{1+\kappa_{\text{P}}\,/\left[4\kappa_{\text{s}}\left(1+\ln{\frac{x_{0}}{2}}\right)+2-3x_{0}^{2}\right]}\,.$$

(39)

We conclude that for any value of the mass parameter $\kappa_{\text{P}}$ Equation (39) yields a single corresponding distance of the point mass from the halo center, for which the critical curve has points lying on the unit-convergence circle.

If we place the point mass along the horizontal axis in the image plane at $\bm{x_{\text{P}}}=(x_{\text{P}},0)$, the positions of these critical-curve points are

$$\bm{x}=\left(\frac{x_{0}}{x_{\text{P}}},\pm\sqrt{1-\frac{x_{0}^{2}}{x_{\text{P}}^{2}}}\right)\,x_{0}\,,$$

(40)

with the value of $x_{\text{P}}$ given by Equation (39). These points with $\kappa=1$ and $\gamma=0$ have special significance. As discussed in Appendix A, such critical-curve points correspond to umbilics. Equation (39) thus presents a condition for the existence of umbilics in the studied lens system. In the $(\kappa_{\text{P}},x_{\text{P}})$ parameter-space plots in Figure 6 of 8, Equation (39) describes the green and violet umbilic boundary, starting at $x_{\text{P}}=x_{0}$ at the $\kappa_{\text{P}}=0$ vertical axis and increasing monotonically for higher $\kappa_{\text{P}}$. In the image plane, the umbilic points lie along the unit-convergence circle. For $\kappa_{\text{P}}\ll 1$ they are located close to the horizontal axis in the direction of the point mass. Their displacement from the axis increases with increasing $\kappa_{\text{P}}$.

The geometry of the images can be studied using the eigenvalue decomposition of the inverse of the Jacobian matrix given by Equation (21) and the convergence–shear (CS) diagram introduced in Appendix A. For a point mass placed at the center of the halo, the convergence and shear are purely radial functions. Hence, the range of their possible combinations is limited to the $\left(\kappa(x),\gamma(x)\right)$ curve in the CS diagram. In this case the analysis of possible image geometries can directly follow the example presented in Section II.3 for images formed by the NFW halo.

Even when the point mass is positioned away from the halo center, the convergence given by Equation (26) preserves its circular symmetry (with the exception of the single point at the position of the point mass). This means that any convergence value $\kappa$ can be one-to-one translated to the corresponding radial distance from the halo center $x$. However, the shear given by Equation (31) loses circular symmetry. In the CS diagram this results in the range of possible $\left(\kappa(\bm{x}),\gamma(\bm{x})\right)$ combinations covering a two-dimensional region. In terms of image distortions and orientations, the lack of symmetry means that instead of the terms “tangential” and “radial” we revert to the more general “in the direction of the phase” and “perpendicular to the phase”, respectively.

For illustration, in Figure 6 we present the CS diagram for a NFW halo with a $\kappa_{\text{P}}\approx 2.714\cdot 10^{-4}$ point mass located at $x_{\text{P}}=0.2$. The purple-shaded region bounded by the bold purple lines shows the range of $(\kappa,\gamma)$ combinations occurring in the image plane. Added for orientation is the green curve from Figure 4 showing the $(\kappa,\gamma)$ combinations of the NFW halo without the point mass. As in Figure 4, the tick marks along the curve mark radial distances from the halo center. At any value of $x$ along this axis, the vertical extent between the bold purple lines indicates the range of shear values $\gamma(\bm{x})$ occurring along the circle $|\bm{x}|=x$. The maximum shear always occurs in the direction of the point mass, i.e., for $\bm{x}=(x,0)$. At large distances $x$, the minimum shear occurs in the direction opposite to the point mass, i.e., for $\bm{x}=(-x,0)$. At lower distances, to the right of the purple vertical tick mark (in the case of Figure 6 near $x=0.4$), minimum shear occurs at two points offset symmetrically from the axis connecting the halo center and the point mass. Close to the halo center, at the right edge of the diagram, minimum shear occurs at points offset nearly perpendicularly from the halo center, i.e., for $\bm{x}\approx(0,\pm\,x)$. Equation (33) shows that at the center of the halo the maximum shear is $\kappa_{\text{s}}+\kappa_{\text{P}}\,x_{\text{P}}^{-2}$ and the minimum shear is $|\kappa_{\text{s}}-\kappa_{\text{P}}\,x_{\text{P}}^{-2}|$.

In the example shown in Figure 6, the shear range at large radii does not visibly deviate from the green NFW halo shear. At radii lower than the purple tick mark near $x=0.4$, the maximum shear starts to deviate substantially from the green curve. The minimum shear starts to deviate visibly between $x=0.3$ and $x=0.2$. Along the circle with the radius of the point-mass distance, $x=x_{\text{P}}=0.2$, the maximum shear diverges at the position of the point mass. At a slightly lower radius, the minimum shear reaches $0$ at the positions of the off-axis zero-shear points. For lower radii, the shear interval shrinks back toward the NFW shear at the green curve. However, instead of reaching the central NFW shear $\gamma_{\text{\tiny{NFW}}}(0)=\kappa_{s}\approx 0.2390$, the limiting shear at $x=0$ varies within the interval $[0.2322,0.2458]$, as discussed in the previous paragraph.

The four orange points marked in the purple region in Figure 6 correspond to the positions of the four images of the source center in the bottom right panel of Figure 1. Note that the fifth macro-image lying on the critical curve in Figure 1 does not include an image of the source center. The point appearing at $(\kappa,\gamma)\approx(0.637,1.255)$ corresponds to the image just to the right of the point mass in Figure 1. The three remaining points in Figure 6 correspond to perturbed versions of the three images appearing in the absence of the point mass in the top right panel of Figure 1. The point at $(\kappa,\gamma)\approx(0.694,0.194)$ corresponds to the lower right image outside the critical curve in the bottom right panel of Figure 1. The point at $(\kappa,\gamma)\approx(0.897,0.233)$ corresponds to the image just outside the unit-convergence circle at the left side of the panel in Figure 1. The fourth point at $(\kappa,\gamma)\approx(1.962,0.244)$ corresponds to the small image close to the halo center in Figure 1.

The properties of the images can be determined from the positions of the points in the diagram; the changes in the properties of the latter three due to the presence of the point mass can be studied by comparing the diagrams in Figures 6 and 4. Note that, in this case, taking into account the full extent of each image would require marking them in the CS diagram by exact patches covering the corresponding range of $(\kappa,\gamma)$ combinations instead of by the points used in Figure 6. This would permit including even partial images that do not contain an image of the source center, such as the fifth macro-image in the bottom right panel of Figure 1.

CS diagrams for different masses and positions of the point mass are presented and discussed in Section III.5.3.

Following the example in Section II.4, we use the geometry of image distortions to introduce the weak-lensing shear and phase estimates for the NFW halo + point-mass lens. We compute the weak shear $\gamma_{\text{w}}$ from Equation (24), substituting the convergence from Equation (26) for $\kappa(\bm{x})$, and the shear from Equation (31) for $\gamma(\bm{x})$.

The angle between the major axis of the image of a small circular source and the $x_{1}$ axis of the image plane is equal to the phase outside the unit-convergence circle; it is perpendicular to the phase inside the unit-convergence circle. Hence, the weak phase $\varphi_{\text{w}}$ is related to the phase $\varphi$ as follows:

$$\varphi_{\text{w}}(\bm{x})=\Bigg\{\begin{array}[]{ll}\,\varphi(\bm{x})&\,\text{for $\kappa(\bm{x})<1$}\,,\\ \,\varphi(\bm{x})+\pi/2&\,\text{for $\kappa(\bm{x})>1$}\,,\end{array}$$

(41)

where $\varphi(\bm{x})$ is given by Equation (III.1) and $\kappa(\bm{x})$ by Equation (26). We use values from the interval $\left[-\pi/2,\,\pi/2\right]$ for both $\varphi$ and $\varphi_{\text{w}}$.

Image-plane maps of the weak shear $\gamma_{\text{w}}(\bm{x})$ and weak phase $\varphi_{\text{w}}(\bm{x})$ for different masses and positions of the point mass are presented and discussed in Section III.5.4 and Section III.5.7, respectively.

In Sections III.1–III.4 we defined the lensing quantities of interest and described their general properties. In this section we present plots illustrating these lens characteristics for different point masses embedded in a NFW halo with a fiducial convergence parameter $\kappa_{\text{s}}\approx 0.239035$. With the exception of Figure 9.A and Figure 9.B, all of the plots are presented as color maps in the image plane. For better orientation in these maps, we plot the critical curves (solid black) and mark the point-mass position by its Einstein ring (cyan).

In each of the following figures, the three columns of the plot grid correspond to the same three values of the mass parameter $\kappa_{\text{P}}$ of the point mass used in 8. These differ in the number of radial critical curves they generate for $x_{\text{P}}=0$: sub-critical $\kappa_{\text{P}}=10^{-4}$ with two radial critical curves; critical $\kappa_{\text{P}}=\kappa_{\text{PC}}\approx 2.714\cdot 10^{-4}$ with one radial critical curve; super-critical $\kappa_{\text{P}}=10^{-3}$ with no radial critical curve.

For each of the characteristics discussed in Sections III.5.1–III.5.8 we present two plot grids. The rows in the first grid correspond to seven values of the point-mass position $x_{\text{P}}$ increasing in steps of $0.05$ from $0$ to $0.3$. These parameter combinations correspond to the critical-curve and caustic gallery in Figure 5 of 8; they are marked by red crosses in the parameter-space plot in Figure 6 of 8. The rows in the second grid correspond to nineteen values of $x_{\text{P}}$ increasing in steps of $0.01$ from $0$ to $0.15$, then in steps of $0.05$ to $0.3$ in the top row. These parameter combinations are marked by red and black crosses in Figure 6 of 8.

For better orientation in the notation of the different shears, convergences, phases, and other lensing quantities, we list selected symbols together with their first appearance in the text in Table 1.