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

Michal Karamazov Institute of Theoretical Physics, Faculty of Mathematics and
Physics, Charles University,
V Holešovičkách 2, 18000 Praha 8, Czech Republic  David Heyrovský Institute of
Theoretical Physics, Faculty of Mathematics and Physics, Charles University,
V Holešovičkách 2, 18000 Praha 8, Czech Republic<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the gravitational lensing influence of a massive object in a dark
matter halo, using a simple model of a point mass embedded in a spherical
Navarro–Frenk–White halo. Building on the analysis of critical curves and
caustics presented in the first part of this work, we proceed to explore the
geometry of images formed by the lens. First, we analyze several lensing
quantities including shear, phase, and their weak-lensing approximations,
illustrating the results with image-plane maps. We derive formulae and present
a geometric interpretation for the shear and phase of a combination of two
axially symmetric mass distributions. In the case of our lens model, we
describe the occurrence of zero-shear points and specify the conditions under
which they become umbilic points. Second, we use the eigenvalue decomposition
of the inverse of the lens-equation Jacobian matrix to compute the
magnification and flattening of lensed images. Based on this, we introduce the
convergence–shear diagram, a novel and compact way of visualizing the
properties of images formed by a particular gravitational lens. We inspect
relative deviations of the analyzed lensing quantities in order to evaluate
the perturbing effect of the point mass and the applicability of the weak-
lensing approximation. We explore the dependence of the results on the point-
mass parameters by studying grids of plots for different combinations of its
position and mass. We provide analytical explanations for important patterns
arising in these plots and discuss the implications for the lensing influence
of isolated compact bodies in dark matter halos.

##  1 Introduction

Dark matter halos form the basic building blocks in the bottom-up structure
formation of $\Lambda$CDM cosmology. They constitute the dominant matter
component of the astrophysical objects they are associated with: the largest
halos with galaxy clusters; smaller halos with individual galaxies and dwarf
galaxies. Their properties can be studied by “observations” in the virtual
universes arising from large-scale-structure formation simulations (e.g.,
Zandanel et al., 2018). In our universe, main observational constraints on
galaxy-cluster dark matter halos come from the study of the kinematics of
cluster galaxies (starting from Zwicky, 1933), from measurements of X-ray
emission from intracluster baryonic gas (e.g., Ettori et al., 2013), and from
analyses of weak and strong gravitational lensing of background galaxies
(e.g., Limousin et al., 2007; Okabe et al., 2013). Gravitational lensing
analyses are particularly useful as tools for studying the finer-scale
substructure of cluster halos, such as subhalos of individual cluster
galaxies, local clumps or other inhomogeneities. A detailed analysis of 11
galaxy clusters by Meneghetti et al. (2020) revealed a surprisingly high
efficiency of substructure lensing, more than an order of magnitude higher
than expected from CDM simulations. This result indicates the need for a
better understanding of the lensing effects of individual bodies within the
cluster.

The goal of our work is to study the gravitational lensing influence of a
compact massive body in a dark matter halo. For this purpose, we use a simple
model consisting of a point mass embedded in a spherical Navarro–Frenk–White
(NFW) density profile (Navarro et al., 1996). In the first part of this work
(Karamazov et al., 2021, hereafter Paper I), we studied the critical curves
and caustics of the lens model as a function of the mass and position of the
point mass. We discovered that the model exhibited a rich diversity of
critical-curve topologies and caustic geometries. We mapped the boundaries
separating the corresponding lensing regimes in the point-mass parameter space
and identified the accompanying caustic metamorphoses. Among other findings,
we demonstrated the existence of a critical value of the mass parameter. For
centrally positioned lighter (sub-critical) point masses, the lens has two
radial critical curves. Heavier (super-critical) point masses are strong
enough to fully eliminate the radial critical curves. For critical point
masses the lens has a single radial critical curve with peculiar properties,
which are described in Appendix B of Paper I. We discussed the relevance of
the model to the lensing by galaxies in galaxy-cluster halos as well as other
astrophysical scenarios, such as the lensing influence of a satellite galaxy
or a (super-)massive black hole in a galactic dark matter halo.

In this sequel to Paper I, we explore other lensing properties of the model.
Here we concentrate on the shear and phase and their relation to the geometric
distortions of images formed by the lens. In the weak-lensing regime the
relation is tight, with the shear specifying the semi-axis ratio and the phase
specifying the orientation of the major axis of the image. However, this will
not be the case in the regions with high convergence (near the halo center) or
high shear (near the point mass and near the halo center). In Section 2 we
describe the shear, phase, and image geometry for a NFW-halo-only lens. We
study the images in Section 2.3, starting from the eigenvalue decomposition of
the inverse of the Jacobian matrix and utilizing the convergence–shear
diagram, a new tool described in detail in Appendix A. In Section 2.4 we
introduce the weak shear and weak phase and compare these weak-lensing
estimates with the shear and phase.

In Section 3 we proceed with the analysis of the NFW halo + point-mass lens in
a similar manner. More specifically, in Section 3.1 we derive formulae for the
shear and phase of the combined mass distribution. We describe the emergence
and occurrence of points with zero shear, which may constitute umbilic points
under conditions discussed in Section 3.2. In Section 3.3 we study the
properties of images using the convergence–shear diagram. We present the main
results in Section 3.5 in the form of grids of image-plane maps of different
lens characteristics and convergence–shear diagrams, utilizing the same point-
mass parameter grid as in Paper I. We discuss the results and their broader
relevance in Section 4 and summarize our findings in Section 5.

##  2 Lensing by a NFW Halo

###  2.1 Convergence, Shear, and Phase

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 1996; Paper I),

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

Figure 1: Gravitational lensing of a circular source. Top row: lensing by a
NFW halo with convergence parameter $\kappa_{\text{s}}\approx 0.239035$.
Bottom row: lensing by the same halo with an additional point mass with mass
parameter $\kappa_{\text{P}}\approx 2.714\cdot 10^{-4}$ positioned at
$\bm{x_{\text{P}}}=(0.2,\,0)$. Left column: source-plane plots indicating the
position of the source (black circle) with respect to the lens caustic (red
lines). Right column: image-plane plots indicating the positions of images
(black patches) with respect to the critical curve (solid black lines). The
cyan circle in the bottom right panel marks the Einstein circle of the point
mass; the dashed black lines in the right panels mark the unit-convergence
circle.

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
(Meneghetti et al., 2003; Golse & Kneib, 2002):

$\psi_{\text{\tiny
NFW}}(x)=\begin{cases}\frac{2\,\kappa_{\text{s}}\,r_{\text{s}}^{2}}{D_{\text{l}}^{2}}\left[\ln{x}\ln{\frac{x}{4}}-(\displaystyle\operatorname{arctanh}{\sqrt{1-x^{2}}}\,)^{2}\right]\\\
\hfil 0\\\
\frac{2\,\kappa_{\text{s}}\,r_{\text{s}}^{2}}{D_{\text{l}}^{2}}\left[\ln{x}\ln{\frac{x}{4}}+(\displaystyle\arctan{\sqrt{x^{2}-1}}\,)^{2}\right]\,,\end{cases}$
(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 Meneghetti et al. (2003) and Golse & Kneib (2002) are higher by the
constant $-\psi_{\text{\tiny NFW}}(0)$ and thus they start at zero.

Figure 2: Image-plane color maps of lensing characteristics of the NFW halo
from the top row of Figure 1. For orientation, solid black circles mark the
critical curves in all panels. First panel: shear $\gamma_{\text{\tiny
NFW}}(\bm{x})$ with dotted contours from inside to outside marking
$95\%,90\%,$ and $85\%$ of the central shear $\gamma_{\text{\tiny
NFW}}(0)=\kappa_{\text{s}}\approx 0.239035$. Second panel: weak shear
$\gamma_{\text{w,\tiny NFW}}(\bm{x})$ computed from image flattening, using
the same color bar as in the first panel. Third panel: relative weak-shear
deviation from the shear, $\gamma_{\text{w,\tiny
NFW}}(\bm{x})/\gamma_{\text{\tiny NFW}}(\bm{x})-1$. Fourth panel: weak phase
$\varphi_{\text{w,\tiny{NFW}}}(\bm{x})$ defined by image orientation, with
orange corresponding to images oriented counterclockwise, and blue to images
oriented clockwise from the horizontal. Dot-dashed lines mark the unit-
convergence circle, and locations of horizontal (white background) and
vertical images (high-saturation blue/orange boundary).

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 Wright & Brainerd (2000). In order to understand its behavior
close to the origin, we expand Equation (LABEL:eq:NFW_gamma) 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.

###  2.2 Jacobian

Figure 3: Geometry of an image of a small circular source. Left panel: source
with radius $y_{\text{r}}$; the black and white points lie in the directions
of the eigenvectors of the inverse Jacobian matrix $\mathbb{A}$ parallel and
perpendicular to the phase $\varphi$. Right panels: elliptical image for
$(\kappa,\gamma)=(0.08,0.16)$ and the three other combinations producing an
ellipse of the same shape and size, as marked at the bottom left of each
panel. The sizes of the semi-axes are marked in red; the positions of the
images of the two points marked on the circumference are determined by the
signs of the eigenvalues marked at the bottom right of each panel.

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 (e.g., Schneider et al., 1992)

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

###  2.3 Geometry of Images

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.

Figure 4: CS diagram illustrating the geometry of images formed by the NFW
halo from the top row of Figure 1 with $\kappa_{\text{s}}\approx 0.239035$.
The green line marks all $\left(\kappa_{\text{\tiny
NFW}}(x),\gamma_{\text{\tiny NFW}}(x)\right)$ combinations of the lens, with
radial positions $x$ marked by tick marks, starting from $x=0.002$ near the
right edge and ending with $x=2$ close to the origin of the diagram. The
orange dots mark the $(\kappa,\gamma)$ combinations at the positions of the
three images in the top right panel of Figure 1. For more details on the
diagram and its interpretation see Figure 15.

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
(LABEL:eq:NFW_gamma). 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.

###  2.4 Weak Shear and Phase

Weak-lensing cluster-mass reconstructions are based on statistical analyses of
the images of background galaxies (Kaiser & Squires, 1993). 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 (LABEL:eq:NFW_gamma). 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 3.5.

##  3 Lensing by a NFW Halo + Point Mass

###  3.1 Convergence, Shear, and Phase

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 (for more details, see Paper I, ). 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 Paper I.

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 Paper I. 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 (LABEL:eq:NFW_gamma) 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)

Figure 5: Viewing angle $\omega(\bm{x})$ of the line segment from the halo
center to the point mass, appearing in Equation (31) for computing the shear
$\gamma(\bm{x})$ of the combined lens. Points on the horizontal axis mark the
halo center and the point-mass position $\bm{x_{\text{P}}}$; the dot at the
vertex of the angle marks the position $\bm{x}$. Contours of constant $\omega$
are symmetric pairs of circular arcs connecting the centers of the two lens
components, with the dashed line marking the arc passing through position
$\bm{x}$. The values of $\omega$ range from $\pi$ along the line segment to
$0$ along the rest of the horizontal axis.

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 2.1, 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 3.5.1.

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), (LABEL:eq:NFW_gamma),
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
(3.1) are valid. For example, the case of two point masses was studied by
Schneider & Weiss (1986), and the case of two isothermal spheres was studied
by Shin & Evans (2008).

###  3.2 Jacobian and Umbilic Points

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 Paper I, 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
3.1 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
Paper I, 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 $\kappa_{\text{P}}$.

###  3.3 Geometry of Images

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 2.3 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.

Figure 6: CS diagram illustrating the geometry of images formed by the NFW
halo with a $\kappa_{\text{P}}\approx 2.714\cdot 10^{-4}$ point mass at
$x_{\text{P}}=0.2$, from the bottom row of Figure 1. The purple-shaded area
marks the range of $\left(\kappa(\bm{x}),\gamma(\bm{x})\right)$ combinations
of the lens; the green line marks the $\left(\kappa_{\text{\tiny
NFW}}(x),\gamma_{\text{\tiny NFW}}(x)\right)$ combinations of the halo-only
lens from Figure 4. The green tick marks and labels indicate the radial
distance $x$ along vertical lines in this diagram. The top purple line marks
the maximum shear along a circle with radius $x$ centered on the halo, which
always occurs in the direction of the point mass. The bottom purple line marks
the minimum shear along the circle; for larger radii $x$ (to the left of the
purple tick mark) this occurs in the direction opposite the point mass; for
smaller radii $x$ (to the right of the purple tick mark) this occurs at two
symmetric off-axis points along the circle. The orange dots mark the
$(\kappa,\gamma)$ combinations at the positions of the four full images in the
bottom right panel of Figure 1.

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 3.5.3.

###  3.4 Weak Shear and Phase

Following the example in Section 2.4, 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 (3.1) 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 3.5.4 and Section 3.5.7,
respectively.

###  3.5 Lens Characteristics as a Function of Point-mass Parameters

Table 1: List of Symbols Symbol | Description; First Appearance
---|---
γ | Shear (combined model or general); Equation (18)
γ_NFW | Shear of NFW-halo lens; Equation (LABEL:eq:NFW_gamma)
γ_P | Shear of a point-mass lens; Equation (29)
γ_w | Weak shear (combined model or general); Equation (24)
γ_w,NFW | Weak shear of NFW-halo lens; Section 2.4
δφ_w | Weak-phase deviation due to the point mass; Section 3.5.8
κ | Convergence (combined model or general); Equation (18)
κ_NFW | Convergence of NFW-halo lens; Equation (1)
κ_P, κ_PC | Mass parameter of a point mass and its critical value;
|  Equation (26)
κ_s | NFW halo convergence parameter; Equation (1)
φ | Phase (combined model or general); Equation (18)
φ_NFW | Phase of NFW-halo lens; Equation (16)
φ_w | Weak phase (combined model or general); Equation (41)
φ_w,NFW | Weak phase of NFW-halo lens; Equation (25)
ψ | Lens potential (combined model or general); Equation (6)
ψ_NFW | Lens potential of NFW-halo lens; Equation (8)
ω | Viewing angle of the line segment connecting the halo center and the point-mass position; Equation (32)
Figure 7.A: Image-plane maps of the shear $\gamma(\bm{x})$ of a NFW halo +
point-mass lens, described in Section 3.5.1. Columns correspond to sub-
critical, critical, and super-critical mass parameters $\kappa_{\text{P}}$
marked at the top; rows correspond to point-mass positions $x_{\text{P}}$
marked along the left side. Critical curves are plotted in black, and the
point-mass location is marked by its Einstein ring (cyan). Magenta marks all
positions with $\gamma\geq 1.5$.

In Sections 3.1–3.4 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 Paper I. 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 3.5.1–3.5.8 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 Paper I; they are marked by red crosses in the
parameter-space plot in Figure 6 of Paper I. 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
Paper I.

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.

####  3.5.1 Shear

Figure 7.B: Image-plane maps of the shear $\gamma(\bm{x})$ of a NFW halo +
point-mass lens, for a finer grid of point-mass positions than in Figure 7.A.
Notation same as in Figure 7.A.

Image-plane maps of the shear $\gamma(\bm{x})$ are presented in Figure 7.A.
The shear color scale is the same as in the first two panels of Figure 2,
ranging from white for $\gamma=0$ to magenta for all positions with
$\gamma\geq 1.5$. In the absence of the point mass the shear varies very
slowly in this region, as indicated by the featureless plot in the first panel
of Figure 2.

In the bottom row ($x_{\text{P}}=0$) of Figure 7.A the point mass is located
at the center of the halo and the whole system thus exhibits axial symmetry.
From Equations (31) and (32) it follows that in this case the total shear at
any position is a simple sum of the NFW and point-mass shears. Near the halo
center the NFW shear $\gamma_{\text{\tiny{NFW}}}(x)$, shown in the first panel
of Figure 2, is surpassed by the point-mass shear $\gamma_{\text{P}}(\bm{x})$
which diverges at the origin. A comparison of the bottom row in the three
columns shows that the magenta high-shear region with the strongest point-mass
influence naturally increases with its mass parameter $\kappa_{\text{P}}$.

Next, we focus on the left column illustrating the sub-critical case with
$\kappa_{\text{P}}=10^{-4}$. Already in the second plot from the bottom
($x_{\text{P}}=0.05$), many phenomena described in detail in Section 3.1 can
be clearly seen. A pale circle corresponding to viewing angle $\omega=\pi/2$
connects the point mass and the halo center, marking a region with decreased
shear. The shear drops to zero at two points of this circle located above and
below the horizontal axis of symmetry. These zero-shear points lie inside the
perturbed NFW radial critical curve close to the point mass. Their presence
restricts the bright red high-shear region around the point-mass divergence to
a smaller extent than in the $x_{\text{P}}=0$ case. The value of the shear in
the vicinity of the halo center depends on the direction of approach. Maximum
shear can be seen in the horizontal and minimum shear can be seen in the
vertical direction, tangent to the $\omega=\pi/2$ circle.

Going further to $x_{\text{P}}=0.1$, we see that the paler lower-shear circle
and the directional dependence near the origin become less pronounced, and the
zero-shear points move even closer to the point mass, lying almost vertically
above and below it. They are positioned inside the pair of tiny critical
curves seen in the white low-shear areas. This hints at the fact that umbilics
can only occur at zero-shear points that lie at the intersection of the
$\omega=\pi/2$ circle and the $\kappa=1$ circle, as explained in Section 3.2.

For even higher values of $x_{\text{P}}$ in the sub-critical case, the
directional dependence at the halo center becomes indiscernible and the
pattern close to the point mass becomes more regular. The region of high shear
around the point mass has a horizontally elongated oval shape and the zero-
shear points lie above and below it in the white spots outside the critical
curve. With increasing $x_{\text{P}}$, this shear pattern around the point
mass resembles the total shear of the Chang–Refsdal model, which consists of a
point mass and a constant external shear (Chang & Refsdal, 1984).

The preceding discussion made for the sub-critical case holds also for the
critical case ($\kappa_{\text{P}}=\kappa_{\text{PC}}\approx 2.714\cdot
10^{-4}$) in the central column. In terms of shear, the corresponding plots
portray qualitatively the same sequence of situations as in the left column;
the difference is merely quantitative. More specifically, the pattern around
the point mass is considerably larger and the directional dependence around
the halo center is more pronounced, observable even for higher values of
$x_{\text{P}}$.

These patterns are even larger and more distinct in the right column
illustrating the super-critical case with $\kappa_{\text{P}}=10^{-3}$.
However, there are some important differences. In the second plot from the
bottom, $x_{\text{P}}=0.05$ does not exceed the threshold value of
$\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}\approx 0.0647$. Hence, there are
no zero-shear points. Minimum shear, which is now non-zero in the central
region, can be found at the halo center when approached vertically. For point-
mass positions $x_{\text{P}}\gtrsim 0.0647$ this minimum shear drops to zero
and its position detaches from the origin, moving along the $\omega=\pi/2$
circle. In addition, the third plot from the bottom now depicts the situation
before the detachment of the two small critical curves and, thus, the zero-
shear points still lie inside the perturbed NFW radial critical curve. Note
that in this case the directional dependence of the shear at the halo center
can be seen up to the top row.

The dependence of the shear $\gamma(\bm{x})$ on the point-mass position
$x_{\text{P}}$ can be examined in more detail in Figure 7.B, which includes
plots for a finer grid in terms of $x_{\text{P}}$. Zero-shear points appear at
the halo center at point-mass positions $\\{0.0205,0.0337,0.0647\\}$ in the
sub-critical, critical, and super-critical cases. Other key values of
$x_{\text{P}}$ correspond to changes in the critical-curve topology, as
indicated by the color boundaries in Figure 6 of Paper I.

Notice that the zero-shear points always occur in a positive-Jacobian region,
as indicated by Equation (19): inside the perturbed NFW radial critical curve,
inside the symmetric pair of small critical-curve loops, or outside all
critical-curve loops. Apart from this, there is hardly any correlation between
the shear pattern and the critical-curve geometry.

####  3.5.2 Shear Deviation Due to the Point Mass

In Figure 8.A, we present image-plane maps of the shear deviation
$\gamma/\gamma_{\text{\tiny{NFW}}}-1$ caused by the presence of the point
mass. As indicated by the formula, this quantity represents the relative
difference between the shear $\gamma(\bm{x})$ of the NFW halo + point-mass
lens (shown in Figure 7.A) and the shear $\gamma_{\text{\tiny{NFW}}}(x)$ of
the NFW halo alone (shown in the first panel of Figure 2). As the deviation
falls rather quickly with increasing distance from the point mass, we
introduce a semi-logarithmic color scale to visualize even minor changes in
the deviation. In the positive yellow- and orange-hued regions the shear is
increased, while in the negative blue regions it is decreased by the point
mass. Darkest blue is used for $-1$, the lowest possible deviation. From $-1$
to $-10^{-3}$, blue saturation decreases logarithmically, and then to $0$
linearly, where it reaches white. From $0$ to $10^{-3}$, yellow saturation
increases linearly, and then to $10^{-1}$ logarithmically. The logarithmic
scale then continues to color red at deviation $10$, beyond which the color is
kept constant even though the shear deviation can reach arbitrarily large
values near the point mass.

For better orientation, we also include contours for a few specific values of
the shear deviation. The dot-dashed lines represent the zero-deviation
contour, along which the shears are equal. Paler and darker shades of orange
are used for positive-deviation contours with values $10^{-2}$ and $10^{-1}$,
respectively. Similarly, paler and darker shades of blue indicate negative
deviations $-10^{-2}$ and $-10^{-1}$, respectively.

We first inspect the deviation map for a centrally positioned sub-critical
point mass (bottom left plot). In this case the deviation is equal to
$\gamma_{\text{P}}/\gamma_{\text{\tiny{NFW}}}$, which is positive in the
entire image plane, i.e., the shear is globally increased by the point mass.
The deviation diverges at the halo center, since the point-mass shear
increases to $\infty$ while the halo shear tends to the constant
$\kappa_{\text{s}}$. Further from the halo center the deviation approaches
zero, as the point-mass shear given by Equation (29) falls quickly with
distance. Contours representing deviations $10^{-1}$ and $10^{-2}$ are
slightly larger than the outer radial and tangential critical curves,
respectively.

In the second row ($x_{\text{P}}=0.05$) the point mass is displaced from the
center and a pair of blue regions with negative deviation appears. These
regions reach the halo center from the vertical direction, while the deviation
is positive along the full horizontal axis, as indicated by the orange and
yellow color and by the dot-dashed zero contour pinched at the halo center. In
fact, the deviation along the horizontal axis is always positive for any
$x_{\text{P}}$ and $\kappa_{\text{P}}$, since here
$\gamma/\gamma_{\text{\tiny{NFW}}}-1=\gamma_{\text{P}}/\gamma_{\text{\tiny{NFW}}}$
according to the discussion in the paragraphs following Equation (31). The
pattern near the halo center arises from the directional dependence of the
shear shown in Equation (33). Places with the darkest blue color lie above and
below the point mass, with the lowest shear deviation $-1$ occurring at the
zero-shear points. The region with a deviation larger than $10\%$ in absolute
value is now roughly centered on the point mass, while the region with a
deviation lower than $1\%$ in absolute value lies outside the near-circular
pale orange contour and in narrow bands along the dot-dashed zero-deviation
contour.

At $x_{\text{P}}=0.1$ on the third row we see that the affected area becomes
more asymmetric, with the pale orange contour with $10^{-2}$ deviation now
broken into two lobes extending to the left of the center and to the right of
the point mass. In this case there is a single region with deviation lower
than $1\%$ in absolute value, reaching inside the critical curves, including
the zero-deviation contour, and reaching the halo center along it.

For higher values of $x_{\text{P}}$, the blue regions of negative deviation
expand as the point mass shifts to the right. Their borders indicated by the
zero-deviation contour become more and more circular except for the vicinity
of the point mass, where they enclose the zero-shear points but avoid the
vicinity of the point mass. These dot-dashed contours intersect at the halo
center at a right angle and the deviation remains positive in the spindle-
shaped region along the horizontal axis from the halo center to the point
mass. The orange and blue contours gradually detach from the halo center and
for $x_{\text{P}}\geq 0.25$ they form a four-lobed structure around the point
mass, with positive lobes extending horizontally from the point mass and
negative lobes separated vertically from the point mass. The single region
with deviation lower than $1\%$ in absolute value includes the halo center as
well as a progressively larger area around it, including the entire image
plane except the four lobes around the point mass.

In the critical and super-critical cases in the two right columns, the plots
look similar to those in the sub-critical case, with the colors getting
progressively more saturated indicating higher shears $\gamma_{\text{P}}$ from
heavier point masses. Naturally, the orange and blue contours also expand with
increasing mass. On the other hand, the blue regions of negative deviation
inside the dot-dashed contours do not expand with increasing
$\kappa_{\text{P}}$. On the contrary, they shrink as they recede from a
heavier point mass. Away from the point mass, the dot-dashed zero contours are
almost circular. They intersect at the halo center at a right angle and reach
nearly to the point mass before avoiding it.

Comparing the columns in the different rows, we see that the geometry of the
zero-deviation contour is generic, with increasing $\kappa_{\text{P}}$
affecting only the vicinity of the point mass, and increasing $x_{\text{P}}$
only enlarging the scale. The pattern arises naturally from Equation (31) in
the regime $\gamma_{\text{P}}\ll\gamma_{\text{\tiny{NFW}}}$ valid anywhere
except in the immediate vicinity of the point mass. In this case we expand the
shear and get the simple result
$\gamma/\gamma_{\text{\tiny{NFW}}}-1\simeq(\gamma_{\text{P}}/\gamma_{\text{\tiny{NFW}}})\,\cos{2\,\omega}$.
The ratio in the parentheses is always positive, hence, the zero-deviation
contour is purely given by the condition on the viewing angle requiring
$\cos{2\,\omega}=0$. Figure 5 shows that the corresponding $\omega=\pi/4$ and
$\omega=3\pi/4$ circles describe the dot-dashed contours seen in Figure 8.A
practically exactly, except in the point-mass vicinity where
$\gamma_{\text{P}}\gtrsim\gamma_{\text{\tiny{NFW}}}$.

In the bottom row of the critical and super-critical columns we see the same
pattern of globally positive shear deviation as in the sub-critical column.
What differs is the larger extent of the orange contours. In fact, in the
super-critical case the entire paler-orange contour of deviation $10^{-2}$
lies outside the plotted area. Within the plotted area in all other panels in
the right column, the regions with a deviation lower than $1\%$ in absolute
value are limited to a band along the zero-deviation contour. This band
expands as $x_{\text{P}}$ increases, and eventually connects with the outer
low-deviation region. For $x_{\text{P}}=0.05$ in the super-critical case we
can see very small blue regions of negative deviation without zero-shear
points inside, as these appear at a higher separation, for $x_{\text{P}}\geq
0.0645$. Moreover, here the angle of intersection of the dot-dashed contours
is very different from a right angle. In this case the influence of the point
mass at the halo center is too strong
($\gamma_{\text{P}}/\gamma_{\text{\tiny{NFW}}}\approx 1.7$), so that the
expansion illustrating the generic shape of the zero-deviation contour is not
valid here.

Figure 8.A: Image-plane maps of the relative shear deviation,
$\gamma/\gamma_{\text{\tiny{NFW}}}-1$, caused by the presence of the point
mass, described in Section 3.5.2. The color scale changes from logarithmic to
linear in the interval $[-10^{-3},10^{-3}]$. All positions with deviation
greater than $10$ are marked in red. Contours are plotted for five deviation
values indicated in the color bar. Remaining notation as in Figure 7.A.

A more detailed view of the changing deviation patterns with point-mass
position can be seen in Figure 8.B. Its closer inspection reveals that the
deviation is globally positive not only for centrally-positioned point masses,
but also for plots up to $x_{\text{P}}=0.01$ in the sub-critical case, up to
$0.02$ in the critical case, and up to $0.04$ in the super-critical case.
Imposing the condition $\gamma\leq\gamma_{\text{\tiny{NFW}}}$ on Equation (31)
reveals that negative deviation first appears at the halo center in the
$\omega=\pi/2$ vertical direction once the point-mass shear at the center
decreases to $\gamma_{\text{P}}=2\,\gamma_{\text{\tiny{NFW}}}$. Using Equation
(13) and Equation (29) with $\bm{x}=(0,0)$ then yields the condition for the
existence of negative-deviation regions:
$x_{\text{P}}\geq\sqrt{\kappa_{\text{P}}/(2\,\kappa_{\text{s}})}$. In the sub-
critical case we find $x_{\text{P}}\geq 0.0145$, in the critical
$x_{\text{P}}\geq 0.0238$, and in the super-critical $x_{\text{P}}\geq
0.0457$, in agreement with the deviation maps.

The sizes of the contours can be used to estimate the areas with a strong
effect on the shear due to the presence of the point mass. As an example, for
the three different masses we find that at the moment of separation of the
critical curve surrounding the point mass from the perturbed NFW tangential
critical curve, the darker contours of deviation $\pm\,10^{-1}$ extend roughly
seven Einstein radii from the point mass.

####  3.5.3 Convergence–Shear Diagrams

Figure 8.B: Image-plane maps of the relative shear deviation,
$\gamma/\gamma_{\text{\tiny{NFW}}}-1$, caused by the presence of the point
mass, for a finer grid of point-mass positions than in Figure 8.A. Notation
same as in Figure 8.A.

In Figure 9.A we present a grid of CS diagrams, which provide a description
complementary to the image-plane plots of the shear, its deviation due to the
point mass, and the quantities discussed in the following sections. For a
general understanding of CS diagrams see Appendix A with Figure 15, Section
2.3 with Figure 4, and, in particular, Section 3.3 with Figure 6.

The purple-shaded area marks the full range of $(\kappa,\gamma)$ combinations
of the NFW halo + point-mass lens. Its intersection with the green curve
corresponds to the dot-dashed zero-deviation curve in Figure 8.A. The part of
the area above the green curve then corresponds to the yellow- and orange-hued
positive-deviation regions, and the part below the green curve corresponds to
the blue negative-deviation regions in Figure 8.A.

We start by describing the sub-critical case shown in the left column. For
$x_{\text{P}}=0$ the point mass lies at the center of the halo and the system
therefore has axial symmetry. In addition, the radial dependence of the
convergence is monotonic. This implies that only one value of the shear
$\gamma(x)$ can occur for any value of the convergence $\kappa(x)$. These
combinations $\left(\kappa(x),\gamma(x)\right)$ are plotted here as the bold
purple curve. For positions far from the halo center (at the left side of the
plot), this curve closely follows the unperturbed-halo green curve, which
starts at the origin of the plot. Proceeding to the right (closer to the halo
center), the purple curve reaches the bold black line with slope $-1$
representing the tangential critical curve. Here the magnification is infinite
and the flattening reaches $1$.

Figure 9.A: Convergence–shear (CS) diagrams of a NFW halo + point-mass lens,
described in Section 3.5.3. Combinations
$\left(\kappa(\bm{x}),\gamma(\bm{x})\right)$ occurring in each of the lens
configurations are marked by the purple regions or curves. The green curve
corresponds to the NFW-halo lens from Figure 4. For further details on the
notation see Figure 6; for the interpretation of CS diagrams see Figure 15.

Further to the right, the curve enters an area of negative parity, where the
magnification decreases and the flattening drops to $0$ at the bold vertical
$\kappa=1$ line corresponding to the unit-convergence radius $x_{0}\approx
0.0936$. To the right of this line, images are elongated perpendicularly to
the phase. Roughly here, the purple curve of
$\left(\kappa(x),\gamma(x)\right)$ combinations starts to deviate
significantly from the green curve. It rises rapidly and eventually leaves the
plot, as the shear diverges at the location of the point mass. Close to the
point mass the parity is always negative and both magnification and flattening
approach zero. Before this happens, the purple curve intersects the bold black
line with slope $1$ twice. The first intersection represents the outer radial
critical curve and the second intersection represents the inner radial
critical curve. Between them, images have positive parity.

Increasing the point-mass position from $x_{\text{P}}=0$ to
$x_{\text{P}}=0.05$ brings about several important changes. As the system
loses its axial symmetry, for each value of convergence $\kappa(x)$ there is a
continuous interval of shear values $\gamma(\bm{x})$ in the image plane and
the set of convergence–shear combinations is represented by a two-dimensional
region. At the left side of the plot, far from the halo center, these
combinations remain limited to the close vicinity of the green curve of the
unperturbed halo. At the right side, close to the halo center, the set of
combinations forms a horizontal band, with shear values
$\gamma\in[\gamma_{\text{\tiny{NFW}}}-\gamma_{\text{P}},\gamma_{\text{\tiny{NFW}}}+\gamma_{\text{P}}]$
corresponding to its directional dependence at the halo center, demonstrated
by Equations (31) and (33). In the vicinity of $x=x_{\text{P}}$ we see the
broadest range of shear values. On the one hand, the shear diverges at the
position of the point mass while on the other hand, the shear drops to $0$ at
the zero-shear points occurring here at a radius slightly lower than
$x_{\text{P}}$. Overall, the set of
$\left(\kappa(\bm{x}),\gamma(\bm{x})\right)$ combinations looks similar to the
case illustrated in Figure 6 and described in detail in Section 3.3, with one
important difference. The shear divergence and the zero-shear points both lie
in the area of $\kappa>1$, meaning that nearby images would now be elongated
perpendicularly to the phase.

Figure 9.B: Convergence–shear (CS) diagrams of a NFW halo + point-mass lens,
for a finer grid of point-mass positions than in Figure 9.A. Notation same as
in Figure 9.A. Figure 10.A: Image-plane maps of the weak shear
$\gamma_{\text{w}}(\bm{x})$ of a NFW halo + point-mass lens, described in
Section 3.5.4. These maps also illustrate image flattening with values
$f(\bm{x})=2\,\gamma_{\text{w}}(\bm{x})$. Maximum weak shear
$\gamma_{\text{w}}=0.5$ occurs exclusively along the critical curves.
Remaining notation as in Figure 7.A.

As $x_{\text{P}}$ increases in the following rows, both the shear divergence
and the zero-shear point shift to the left, indicating that the point mass
moves to locations with progressively lower halo convergence. By the third row
from the bottom ($x_{\text{P}}=0.1$), the purple region touches the horizontal
axis to the left of the vertical $\kappa=1$ line, indicating that the lens
underwent an umbilic transition at a slightly lower $x_{\text{P}}$ value. The
surroundings of the shear divergence and the zero-shear point now lie in the
region $\kappa<1$, where images are elongated parallel to the phase. The
purple regions in the diagram also become narrower with increasing
$x_{\text{P}}$. In the case of the divergence this is due to the changing
scale of CS diagrams in terms of image-plane positions, as indicated by the
green ticks. In the case of the horizontal band this is due to the decreasing
value of $\gamma_{\text{P}}(0)$, i.e., the shear due to the point mass at the
halo center.

There are a few differences to notice in the critical case, which is shown in
the central column of Figure 9.A. For a centrally positioned point mass
($x_{\text{P}}=0$), the purple curve of the $\left(\kappa(x),\gamma(x)\right)$
combinations deviates from the green curve of halo combinations similarly as
it does in the sub-critical case. However, instead of intersecting the bold
black line with slope $1$, the curve merely touches it at a single point. This
indicates the disappearance of the inner positive-parity region and the
presence of a degenerate radial critical curve, described in detail in
Appendix B of Paper I. The plot in the third row from the bottom
($x_{\text{P}}=0.1$) depicts a configuration extremely close to the elliptic
umbilic, as the zero-shear points now occur almost precisely at
$(\kappa,\gamma)=(1,0)$. Generally speaking, the purple regions of
convergence–shear combinations are broader than in the sub-critical case,
meaning that at a given distance from the halo center a greater range of shear
values occurs.

For the super-critical mass in the right column, in the axially symmetric
$x_{\text{P}}=0$ case the purple curve of the
$\left(\kappa(x),\gamma(x)\right)$ combinations does not touch the black line
with slope $1$ at all. This means that there are no radial critical curves.
The second plot from the bottom ($x_{\text{P}}=0.05$) is now profoundly
different than in previous cases. Here, the purple area does not touch the
horizontal axis of the plot. As discussed in Section 3.5.1, in this case there
are no zero-shear points and minimum shear can be found at the halo center
when approached vertically. Moreover, unlike in the lower-mass cases, at the
halo center the horizontal band does not spread symmetrically around the green
line. In this case, at the center the shear due to the point mass is higher
than the shear due to the halo, so that
$\gamma\in[\gamma_{\text{P}}-\gamma_{\text{\tiny{NFW}}},\gamma_{\text{P}}+\gamma_{\text{\tiny{NFW}}}]$
as shown by Equation (33) and discussed in the following paragraphs and in
Section 3.3.

Figure 10.B: Image-plane maps of the weak shear $\gamma_{\text{w}}(\bm{x})$ of
a NFW halo + point-mass lens, for a finer grid of point-mass positions than in
Figure 10.A. Notation same as in Figure 10.A.

The variation of the CS diagrams with point-mass position can be inspected in
more detail in Figure 9.B. Note that the shear interval at the halo center is
centered on $\gamma_{\text{P}}$ rather than on $\gamma_{\text{\tiny{NFW}}}$ in
all cases with $x_{\text{P}}<\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}$,
namely: from $x_{\text{P}}=0$ to $x_{\text{P}}=0.02$ in the sub-critical case;
to $x_{\text{P}}=0.03$ in the critical case; to $x_{\text{P}}=0.06$ in the
super-critical case. The $x_{\text{P}}=0.01$ diagrams illustrate the nature of
the transition from the axially symmetric lens configurations at
$x_{\text{P}}=0$, best seen in the right panel. The bold purple curve from
$x_{\text{P}}=0$ gradually expands to a broader band at lower radii for
$x_{\text{P}}=0.01$. The remaining structure of the purple region lies outside
the plotted area in the right panel, but it has a similar nature to the plot
in the left panel (also similar to the right panels for higher $x_{\text{P}}$
values): shear divergence at the lens position $(x_{\text{P}}=0.01)$, and the
shear interval shrinking to a horizontal band centered on $\gamma_{\text{P}}$.

Another feature to notice for the lower $x_{\text{P}}$ values is that the
entire purple region lies above the green curve, which means that for such
configurations the shear is higher than in the absence of the point mass
everywhere in the image plane. As shown in Section 3.5.2, this is the case for
$x_{\text{P}}<\sqrt{\kappa_{\text{P}}/(2\,\kappa_{\text{s}})}$, namely: plots
up to $x_{\text{P}}=0.01$ in the sub-critical case; up to $x_{\text{P}}=0.02$
in the critical case; up to $x_{\text{P}}=0.04$ in the super-critical case.
For higher values of $x_{\text{P}}$ the shear may be lower than in the absence
of the point mass, but only in limited parts of the image plane. For example,
along the horizontal axis of the lens the shear always stays higher than in
the absence of the point mass, as shown in Section 3.5.2. For lower masses at
higher separations $x_{\text{P}}$, the generic geometry of the zero-deviation
contour discussed in Section 3.5.2 shows that the regions of lower shear are
limited to radial distances $x<x_{\text{P}}\,\sqrt{2}$.

####  3.5.4 Weak Shear

The plots in Figure 10.A show image-plane color maps of the weak shear
$\gamma_{\text{w}}(\bm{x})$, which we defined in Section 2.4 as the shear that
would be measured from image deformations using weak-lensing analysis. At the
same time, these maps illustrate image flattening, since
$f(\bm{x})=2\,\gamma_{\text{w}}(\bm{x})$, as shown in Equation (24). This
equality also implies that weak shear values range from $0$ (corresponding to
white color in the maps) to $0.5$ (bright red color), with the maximum value
occurring exclusively along the full length of the critical curve. In spite of
this limited range, for purposes of comparison we retain the same color scale
as in the first two panels of Figure 2 and in Figure 7.A.

We begin our description in the left column of Figure 10.A with a sub-critical
point mass, starting from the bottom row corresponding to its central position
in the NFW halo. Directly at the center the weak shear is equal to zero, since
the image cannot be flattened in any direction due to the axial symmetry of
the lens configuration. Going further from the halo center, the weak shear
reaches $0.5$ at the small inner radial critical curve, then it decreases
slightly before returning to $0.5$ at the outer radial critical curve. The
weak shear then drops to zero at the unit-convergence circle, along which
images are undistorted, increasing again to $0.5$ at the tangential critical
curve, beyond which it drops asymptotically to zero.

In the second row ($x_{\text{P}}=0.05$), the outer red ring of high weak shear
along the perturbed NFW tangential critical curve is preserved. The same holds
for the white $\gamma_{\text{w}}=0$ ring along the unit-convergence circle,
which is in fact preserved exactly in all the configurations for all point
masses. However, several changes can be seen closer to the halo center, where
there are now four points with zero weak shear. These points coincide with the
halo center, the point-mass position, and the pair of zero-shear points. This
can be understood by inspecting Equation (22) and taking its limit at these
points. At the halo center, the difference of signs before $\gamma$ is
suppressed by diverging $\kappa$, which also suppresses the directional
dependence of the shear close to the center. Both fractions in Equation (22)
tend to $1$, which results in zero flattening. At the position of the point
mass, both numerators and denominators are dominated by the diverging shear
and the fractions thus again approach unity. At the zero-shear points, the
fractions for $\gamma=0$ are directly equal to $1$. It is worth noting here
that zero shear $\gamma$ implies zero weak shear $\gamma_{\text{w}}$, but not
vice versa. All four points are connected by a paler low-weak-shear region
corresponding to the $\omega=\pi/2$ circle, which is interrupted between the
zero-shear points and the point-mass position by the perturbed NFW radial
critical curve, along which the weak shear reaches $0.5$.

At $x_{\text{P}}=0.1$ the region of low weak shear surrounding the point mass
is superimposed over the white ring of the unit-convergence circle, while the
pair of $\gamma_{\text{w}}=0$ zero-shear points is now trapped inside tiny
critical curves, here very close to the elliptic umbilic transition. For
higher values of $x_{\text{P}}$, a lobe forms on the perturbed NFW tangential
critical curve with the point mass inside and the zero-shear points outside.
By $x_{\text{P}}=0.25$ the curve splits and the point mass is surrounded by a
small oval critical-curve loop elongated toward the halo center. Three points
with zero weak shear remain associated with this loop: the point-mass position
inside, and the two zero-shear points directly above and below the loop.
Similar but larger weak-shear patterns near the tangential-critical-curve
lobes and the detached ovals can be seen in the two right columns showing the
critical and super-critical cases, in the rows with $x_{\text{P}}\geq 0.15$.

In the critical case with a centrally located point mass (bottom plot in
central column), there is only one $\gamma_{\text{w}}=0.5$ circle along the
single radial critical curve between the white center and the white $\kappa=1$
circle. The third row ($x_{\text{P}}=0.1$) illustrates the peculiar situation
in the presence of elliptic umbilic points (technically, these occur at
$x_{\text{P}}\approx 0.0996$, but the plot is visually identical). These are
zero-shear points lying directly on the unit-convergence circle. At these
points the weak shear (and flattening) is undefined, as can be seen by
substituting $\gamma=0$ and $\kappa=1$ in Equation (22). In the plot we can
see them as point-like interruptions of the white unit-convergence circle.

In the super-critical case with $x_{\text{P}}=0$ (bottom right plot) there is
no radial critical curve. Hence, the weak shear increases only slightly
between the center and the $\kappa=1$ circle without reaching $0.5$. In the
$x_{\text{P}}=0.05$ plot the pair of vertically offset zero-weak-shear points
is missing, since the condition
$x_{\text{P}}>\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}$ for the existence of
zero-shear points is not fulfilled here. Note in the same plot that the weak-
shear pattern near the halo center resembles the central directionally
dependent shear pattern. However, while the shear is undefined at the center,
the weak shear reaches $0$ from any direction, with only the rate of
convergence depending on the direction. The rate is slowest in the horizontal
direction with
$\gamma_{\text{w}}\sim(\gamma_{\text{P}}+\gamma_{\text{\tiny{NFW}}})/(\kappa-1)$,
and fastest in the vertical direction with
$\gamma_{\text{w}}\sim|\gamma_{\text{P}}-\gamma_{\text{\tiny{NFW}}}|/(\kappa-1)$
to first order in $1/(\kappa-1)$. Clearly, the directionality will be most
pronounced when $\gamma_{\text{P}}=\gamma_{\text{\tiny{NFW}}}$ at the halo
center, i.e., at the appearance of the zero-shear points. The directionality
will be least pronounced when either of the component shears
$\gamma_{\text{\tiny{NFW}}},\gamma_{\text{P}}$ dominates over the other at the
halo center.

The emergence and evolution of the features and structures discussed above can
be studied in more detail in Figure 10.B. For example, the formation of the
tiny critical-curve loops around the zero-(weak)-shear points is well visible
in the sub-critical case at $x_{\text{P}}=0.08$ or in the critical case at
$x_{\text{P}}=0.09$. The location of the white zero-(weak)-shear points inside
these loops is best visible for $x_{\text{P}}=0.12$ for all three masses, as
well as for $x_{\text{P}}=0.13$ in the sub-critical case.

####  3.5.5 Weak-shear Deviation Due to the Point Mass

The plots in Figure 11.A depict the weak-shear deviation caused by the
presence of the point mass, given by
$\gamma_{\text{w}}/\gamma_{\text{w,\tiny{NFW}}}-1$. As indicated by the
formula, it is defined as the relative difference between the weak shear
$\gamma_{\text{w}}(\bm{x})$ of a NFW halo with a point mass (see Figure 10.A)
and the weak shear $\gamma_{\text{w,\tiny{NFW}}}(\bm{x})$ of a NFW halo alone
(see the second panel of Figure 2). This means that the plots are the weak-
shear equivalents of the plots from Figure 8.A described in Section 3.5.2.
Therefore, we use the same color scale and the same set of contours as in
Figure 8.A. Note that due to Equation (24), the plots in Figure 11.A also
exactly portray the relative image-flattening deviation, including the color
bar and the values of the contours. The yellow- and orange-hued regions thus
correspond to higher flattening and weak shear, while the blue regions
correspond to lower flattening and weak shear than in the absence of the point
mass.

The striking patterns of the colored areas in Figure 11.A look remarkably
complex at first. However, especially in the top rows, away from the halo
center and from the vicinity of the point mass, the plots are very similar to
those in Figure 8.A. This should be expected, since in these regions with low
values of $\gamma$ and $\kappa$ the weak shear is a good approximation of the
shear. Closest to the point mass, the plots differ from those in Figure 8.A
fundamentally. In Figure 8.A, the relative shear deviation diverges to
$\infty$ at the position of the point mass. In Figure 11.A, even inside the
positive weak-shear deviation contours in the top rows, there is a blue
negative area in which the deviation falls to the minimum possible value of
$-1$ at the point-mass position. This is a consequence of the weak shear
$\gamma_{\text{w}}$ converging to $0$, while the shear $\gamma$ diverges
there. Interestingly, this result holds even for $x_{\text{P}}=0$ at the halo
center, where even $\gamma_{\text{w,\tiny{NFW}}}$ converges to $0$.
Nevertheless, in the immediate vicinity of the center the weak-shear ratio
$\gamma_{\text{w}}/\gamma_{\text{w,\tiny{NFW}}}\simeq
4\,(x\,\ln{x})^{2}\,\kappa_{\text{s}}/\kappa_{\text{P}}\to 0$, so that even in
this case the weak-shear deviation at the point-mass position is $-1$.

In order to decipher the alternating positive and negative regions in Figure
11.A, we concentrate on their boundaries which are indicated by the dot-dashed
zero-weak-shear-deviation contour. Especially in the top rows, for higher
$x_{\text{P}}$, we can see that parts of the contours look identical to the
zero-shear-deviation contours from Figure 8.A. However, here there is an
additional strong effect closely associated with the critical curve, unlike in
the case of the shear-deviation patterns which show very little influence of
the critical curve. More specifically, this additional effect reflects the
relative deformations of the critical curve caused by the point mass.

Figure 11.A: Image-plane maps of the relative weak-shear deviation,
$\gamma_{\text{w}}/\gamma_{\text{w,\tiny{NFW}}}-1$, caused by the presence of
the point mass, described in Section 3.5.5. The maps also exactly portray the
relative deviation in image flattening due to the point mass. Color scale,
contours, and remaining notation as in Figure 8.A.

The zero-weak-shear-deviation contour can be defined using the flattening from
Equation (22) by setting
$f(\kappa,\gamma)=f(\kappa,\gamma_{\text{\tiny{NFW}}})$. Since the convergence
is the same with or without the point mass (except at the point-mass
position), we immediately see that the zero-shear-deviation contour

$\gamma(\bm{x})=\gamma_{\text{\tiny{NFW}}}(x)$ (42)

automatically also forms a component of the zero-weak-shear-deviation contour.
All dot-dashed contours from Figure 8.A thus appear also in Figure 11.A. The
remaining components can be obtained by solving the flattening equality, which
yields the additional non-trivial solution

$\gamma(\bm{x})\,\gamma_{\text{\tiny{NFW}}}(x)=\left[\,1-\kappa(x)\,\right]^{2}\,.$
(43)

This equation describes all components of the zero-weak-shear-deviation
contour that do not appear in Figure 8.A.

In particular, these include components closely associated with the critical
curves, which is best illustrated when the NFW critical curves are only weakly
perturbed by the point mass. Along the NFW tangential critical curve we recall
that $\gamma_{\text{\tiny{NFW}}}(x)=1-\kappa(x)$, as discussed in Section 2.2.
Similarly, along the perturbed NFW tangential critical curve which lies
outside the unit-convergence circle, the shear satisfies
$\gamma(\bm{x})=1-\kappa(x)$, as discussed in Section 3.2. It is
straightforward to show that the product of the two shears is lower than
$\left[\,1-\kappa(x)\,\right]^{2}$ along a section of one of these critical
curves and at the same time it is higher than
$\left[\,1-\kappa(x)\,\right]^{2}$ along the corresponding section of the
other critical curve. Due to continuity, there is a contour between the
critical curves of the two models along which Equation (43) is satisfied. When
the point-mass perturbation is weak, such as in the top left plot in Figure
11.A, the corresponding component of the zero-weak-shear-deviation contour is
indistinguishable from the perturbed NFW tangential critical curve. The
contour can be distinguished for example in the top right plot near the
horizontal axis in the direction of the point mass.

A similar argument can be made for the NFW radial critical curve, along which
$\gamma_{\text{\tiny{NFW}}}(x)=\kappa(x)-1$, as discussed in Section 2.2, and
for the perturbed NFW radial critical curve inside the unit-convergence circle
with $\gamma(\bm{x})=\kappa(x)-1$, as discussed in Section 3.2. Even in this
case there is a contour between the two critical curves along which Equation
(43) is satisfied. The contour can be distinguished from the perturbed NFW
radial critical curve in Figure 11.A for example in the super-critical
$x_{\text{P}}=0.15$ plot near the horizontal axis in the direction of the
point mass. Note that Equation (43) also accounts for other components of the
zero-weak-shear-deviation contour, such as the loop around the point mass in
the top row of Figure 11.A.

At the mutual intersections of the components of the zero-weak-shear-deviation
contour, Equation (42) and Equation (43) have to be satisfied simultaneously.
This implies that these points also represent the intersections of the
critical curves of the NFW halo with and without the point mass. The combined
geometry of the components with their mutual intersections explains the
partitioning of the image plane into the color patterns seen in Figure 11.A.

Figure 11.B: Image-plane maps of the relative weak-shear deviation,
$\gamma_{\text{w}}/\gamma_{\text{w,\tiny{NFW}}}-1$, caused by the presence of
the point mass, for a finer grid of point-mass positions than in Figure 11.A.
Notation same as in Figure 11.A.  Figure 12.A: Image-plane maps of the
relative weak-shear deviation from the shear, $\gamma_{\text{w}}/\gamma-1$, of
a NFW halo + point-mass lens, described in Section 3.5.6. The maps illustrate
the relative error of the weak-lensing shear estimate. All positions with
deviation greater than $1.5$ are marked in red. Remaining notation as in
Figure 7.A.

Starting with a centrally positioned sub-critical point mass (bottom left
plot), the deviation shows a very small negative region around the point mass
inside the inner radial critical curve, followed by a positive annulus
reaching just beyond the outer radial critical curve, a negative annulus
almost to the tangential critical curve, and a positive outer region. All
three boundaries separating these regions are described by Equation (43). At
$x_{\text{P}}=0.05$, the negative region around the point mass is connected
with the larger negative annulus. In addition, the zero-shear-deviation
boundary given by Equation (42) can be seen, introducing negative areas above
and below the halo center, with positive crescents where it reaches beyond the
zero-weak-shear-deviation contour associated with the perturbed NFW radial
critical curve. At $x_{\text{P}}=0.1$, strong positive deviation can be seen
close to the two tiny critical-curve loops, along which
$\gamma_{\text{w}}=0.5$ while $\gamma_{\text{w,\tiny{NFW}}}\approx 0$.
However, note that at the zero-shear points inside the loops the weak-shear
deviation equals $-1$, with the indiscernible negative regions around them
bordered by contour loops obeying Equation (43). At $x_{\text{P}}=0.15$, the
region inside the zero-shear-deviation contour flips color again as it crosses
the zero-weak-shear-deviation contour associated with the perturbed NFW
tangential critical curve. At $x_{\text{P}}=0.25$, the point-mass critical-
curve loop is detached from the NFW-halo critical curve, showing the
characteristic four-lobed contour pattern seen in Figure 8.A, here with the
central negative region described above.

A similar sequence can be seen in the two right columns, in which the
structures are larger and the deviations more prominent. In particular, the
zero-weak-shear-deviation contours near the critical curves are better visible
in some of the plots here. Note also the change with increasing mass for a
central position of the point mass in the bottom row. In the critical case
there is only a single degenerate radial critical curve, with the accompanying
zero-weak-shear-deviation contour well separated from it. The positive-
deviation annulus is narrower as the central negative region is larger, and
for a higher mass it disappears entirely. This can be seen in the super-
critical case, where there is only a single large negative-deviation region
reaching almost to the tangential critical curve.

Figure 12.B: Image-plane maps of the relative weak-shear deviation from the
shear, $\gamma_{\text{w}}/\gamma-1$, of a NFW halo + point-mass lens, for a
finer grid of point-mass positions than in Figure 12.A. Notation same as in
Figure 12.A.

The transitions of the patterns with increasing point-mass position
$x_{\text{P}}$ can be studied in more detail in Figure 11.B. For example, note
the appearance of a positive-deviation region in the super-critical case at
$x_{\text{P}}=0.01$, as soon as the point mass is displaced from the halo
center. For larger separations the boundary of this region forms the zero-
weak-shear-deviation contour associated with the perturbed NFW radial critical
curve. The small negative regions inside the tiny critical-curve loops can be
seen here at $x_{\text{P}}=0.13$ in the sub-critical case, or at
$x_{\text{P}}=0.12$ in all cases. In the critical and super-critical cases at
$x_{\text{P}}=0.13$, we see that these regions persist even after the loops
merge with the outer critical curve.

Studying the colored contours, we see that the regions with the strongest
negative weak-shear deviation occur close to the point mass and near the zero-
shear points (except when they lie between the perturbed NFW radial and
tangential critical curves). The strongest positive deviation occurs close to
the point mass along the horizontal axis outside its Einstein radius, and just
outside the tiny critical-curve loops enclosing the zero-shear points. A
closer inspection of the contours reveals that some of them display sharp
bends at specific positions. These may appear at three types of locations.
First, at the unit-convergence circle, as can be seen for example in the
critical case at $x_{\text{P}}=0.15$ on the pale blue, pale orange, and dark
orange contours. These kinks are caused by the switching of the minimum
fractions in Equation (22) at $\kappa=1$, which causes a discontinuity in the
derivatives.

Second, at the critical curves, as can be seen for example in the super-
critical case at $x_{\text{P}}=0.25$ on the pale orange and pale blue contours
at the left side of the plot, or in the sub-critical case at
$x_{\text{P}}=0.15$ on the pale orange and pale blue contours above and below
the halo center. These kinks are caused by crossing the zero points of the
absolute values in Equation (22), which also causes a discontinuity in the
derivatives. Third, at the NFW-halo critical curves, as can be seen for
example in the super-critical case at $x_{\text{P}}=0.15$ on the dark blue
contours around the zero-shear points above and below the point mass, or in
the critical case at $x_{\text{P}}=0.20$ on the dark blue contour extending
from the point mass toward the halo center. These kinks are similar to the
previous ones, being caused by crossing the zero points of the absolute values
in Equation (22) when evaluating the NFW halo weak shear.

Regarding the extent of the contours, they are generally slightly smaller than
those in Figure 8.A, indicating that the weak-shear deviation falls more
quickly with the distance from the point mass than the shear deviation. As a
rough estimate, at the moment of separation of the critical curve surrounding
the point mass from the perturbed NFW tangential critical curve, the
$\gamma_{\text{w}}/\gamma_{\text{w,\tiny{NFW}}}-1=\pm\,10^{-1}$ contours
extend roughly six Einstein radii from the point mass.

####  3.5.6 Weak-shear Deviation from the Shear

The relative deviation $\gamma_{\text{w}}/\gamma-1$ of the weak shear (shown
in Figure 10.A) from the shear (shown in Figure 7.A) is plotted in the image-
plane color maps in Figure 12.A. We use the same color scale as in the third
panel of Figure 2, which shows the same quantity plotted for the NFW halo
alone. By its definition, this deviation shows the relative error of shear
estimation using the weak-lensing approximation.

In the case of the sub-critical mass (left column), the plots are very similar
to the plot in the third panel of Figure 2, with the point mass affecting only
its nearby surroundings. Note that the weak shear is always zero at the
position of the point mass, so that the deviation reaches its minimum value of
$-1$ there, leading to a dark blue spot similar to the one associated with the
point mass in Figure 11.A. In the bottom row, the central negative spot is
thus more prominent than in the absence of the point mass and it includes even
the inner radial critical curve. At $x_{\text{P}}=0.05$, there is a pair of
red spots with a strong positive deviation close to the zero-shear points
above and below the point mass. At these points, both $\gamma_{\text{w}}$ and
$\gamma$ drop to $0$, but their ratio $\gamma_{\text{w}}/\gamma$ converges to
$1/|\kappa-1|$. The deviation is thus positive as long as $\kappa<2$,
corresponding to zero-shear points located more than circa $0.011$ from the
center of our fiducial halo.

As the point mass crosses the unit-convergence circle near $x_{\text{P}}=0.1$,
the red spots shrink but their peak deviation increases. When the zero-shear
points approach the perturbed NFW tangential critical curve at
$x_{\text{P}}=0.15$, the spots expand to their largest as they merge with the
positive-deviation band along the critical curve. When the point-mass
critical-curve loop is detached from the perturbed tangential critical curve
of the NFW halo as seen in the two top rows, the positive-deviation spots
remain associated with the zero-shear points above and below the point mass,
even though their peak deviation declines with increasing point-mass position
$x_{\text{P}}$. The pattern around the halo center at $x_{\text{P}}=0.3$ is
virtually identical to the unperturbed-halo pattern in the third panel of
Figure 2.

The plots for a critical mass in the central column follow a similar sequence,
with larger affected regions around the point mass. When it is positioned at
the halo center (bottom row), we see that the single radial critical curve
shows practically zero deviation, and between it and the blue unit-convergence
circle the positive deviation reaches lower values. At $x_{\text{P}}=0.1$, the
red areas of high deviation extend from the perturbed NFW radial critical
curve past the zero-shear points and beyond the unit-convergence circle. The
pinched pattern at its intersection with the circle is indicative of the
elliptic umbilic at which the weak shear is undefined, as discussed in Section
3.5.4.

The regions strongly influenced by the super-critical mass in the right column
are much larger, with nearly half of the pattern around the NFW halo critical
curve affected for $x_{\text{P}}=0.1,0.15,$ and $0.2$. For $x_{\text{P}}=0$
there is no region of positive deviation between the halo center and the unit-
convergence circle. For $x_{\text{P}}=0.05$ a positive region is present but
there are no red spots, as there are no zero-shear points at this
configuration. At $x_{\text{P}}=0.1$ the red spots are very large and
prominent. At $x_{\text{P}}=0.15$, in addition to the zero-shear-point red
spots there are two adjacent smaller red spots along the perturbed NFW radial
critical curve.

The changing patterns can be studied with a finer step in point-mass positions
in Figure 12.B. For example, the pattern around the tiny critical-curve loops
can be seen at $x_{\text{P}}=0.12$ and in a few neighboring panels. We also
point out the $x_{\text{P}}=0.03$ panel in the sub-critical case and the
$x_{\text{P}}=0.04$ panel in the critical case. In these panels there are no
red spots close to the zero-shear points, which lie still too close to the
halo center. However, in the super-critical case at $x_{\text{P}}=0.06$ we see
red (or rather orange) spots at the tips of the perturbed NFW radial critical
curve even though the zero-shear points are not present yet. One panel higher,
at $x_{\text{P}}=0.07$, these red spots are more prominent while the zero-
shear points are located near their edge closer to the halo center. Clearly,
the red spots develop at the perturbed NFW radial critical curve even for
lower point-mass positions at which there are no zero-shear points. With a
slight increase in $x_{\text{P}}$, the points reach the red spots, which then
remain associated with them at more distant point-mass positions.

Overall, the pattern of the blue areas shows that the weak shear
underestimates the shear near the halo center, along the unit-convergence
circle, and roughly within an Einstein radius of the point mass (extending
further when it overlaps with the unit-convergence circle). Practically
everywhere else the weak shear overestimates the shear, most prominently in
the red and orange areas: close to the zero-shear points, along the perturbed
NFW tangential critical curve, and along the perturbed NFW radial critical
curve (except when a super-critical mass is positioned close to the halo
center). Further from the halo center, e.g., close to the right edge of the
plots in Figure 12.A for our fiducial halo, the positive deviation is very
low, so that the weak shear may serve as a good approximation of the shear.
However, the agreement fails in the vicinity of the point mass. This occurs
primarily inside its Einstein radius, but also further to the tangentially
offset zero-shear points. Note that for
$\kappa_{\text{\tiny{NFW}}}(x_{\text{P}})\ll 1$ the deviation at the zero-
shear points reaches a value
$\gamma_{\text{w}}/\gamma-1\approx\kappa_{\text{\tiny{NFW}}}(x_{\text{P}})$,
so that their influence decreases as the halo convergence declines for higher
$x_{\text{P}}$.

####  3.5.7 Weak Phase

In Figure 13.A we present image-plane plots of the weak phase
$\varphi_{\text{w}}$ of the NFW halo + point-mass lens, given by Equation
(41). We defined the weak phase in Section 2.4 as the phase that would be
measured from the orientation of image deformations using weak-lensing
analysis. For an elliptical image of a small circular source the weak phase is
simply the angle between its major axis and the horizontal axis in the plots.
Thus, $\varphi_{\text{w}}=0$ corresponds to horizontally and
$|\varphi_{\text{w}}|=\pi/2$ to vertically elongated images. The weak phase of
the NFW-halo lens is described in Section 2.4 and shown in the fourth panel of
Figure 2. Without the point mass, outside the unit-convergence circle the weak
phase is equal to the phase and images are oriented tangentially. Inside the
unit-convergence circle the weak phase is perpendicular to the phase and
images are oriented radially.

In Figure 13.A we use the same weak-shear color scale as in the fourth panel
of Figure 2, with white corresponding to horizontal images, orange to images
oriented counterclockwise from the horizontal, and blue to images oriented
clockwise from the horizontal. The saturation of both colors increases with
the angle from the horizontal, from zero saturation for angle $0$ to maximum
saturation for angle $\pi/2$. The dot-dashed contour marks several special
orientations: $\varphi_{\text{w}}=0$, $|\varphi_{\text{w}}|=\pi/2$, and
undefined $\varphi_{\text{w}}$. The weak phase is undefined at all points with
zero weak shear, as described in Section 3.5.4. These include the halo center,
the point-mass position, the zero-shear points, and the unit-convergence
circle. All of these points thus play an important role in the patterns seen
in the color maps in Figure 13.A.

Thanks to the axial symmetry of the lens configurations with $x_{\text{P}}=0$
in the bottom row of Figure 13.A, the color maps are identical to that of the
unperturbed halo, as seen in the fourth panel of Figure 2. This means that
outside the $\kappa=1$ circle images are oriented tangentially, while inside
the circle the weak phase is flipped by $\pi/2$ and images are oriented
radially. Exactly at the halo center the weak phase is undefined. Note the
colors alternating around the center with orange in the first quadrant and the
saturation varying from zero along the horizontal to maximum along the
vertical axis. Such a pattern corresponds to radial orientation of images
around the center.

Figure 13.A: Image-plane maps of the weak phase $\varphi_{\text{w}}(\bm{x})$
of a NFW halo + point-mass lens, described in Section 3.5.7. Orange and blue
correspond to images oriented counterclockwise and clockwise, respectively,
from the horizontal. Dot-dashed curves indicate exactly horizontal, exactly
vertical, or undefined image orientation. Remaining notation as in Figure 7.A.

The variation of the weak-phase maps with point-mass position $x_{\text{P}}$
is best seen in the super-critical case (right column). In the top row with
$x_{\text{P}}=0.3$ the point mass is well separated from the halo center. The
left part of the plot closer to the halo center is similar to the unperturbed
pattern seen also in the bottom row. Note here the slight shift toward the
point mass of the dot-dashed contour extending vertically outward from the
unit-convergence circle, which corresponds to horizontal images. The main new
feature is the oval region with inverted colors bordered by a loop of the dot-
dashed contour, which passes through the position of the point mass at its
right side and extends in the direction of the halo center on its left side.
The pattern seen around the point mass also has colors alternating around the
center, however, here with blue in the first quadrant. In addition, the
saturation varies from zero along the vertical to maximum along the horizontal
axis. This pattern corresponds to tangential orientation of images around the
point mass.

The left part of the dot-dashed border of the oval separates high-saturation
regions and, thus, corresponds to vertically oriented images. The right part
of the border lies in the white band separating low-saturation regions and,
thus, corresponds to horizontally oriented images. The two parts of the
boundary meet at the zero-shear points that lie above and below the point
mass. In this panel, horizontal images occur along the horizontal axis inside
the unit-convergence circle, along the vertical dot-dashed contour outside the
unit-convergence circle, and along the right boundary of the oval region
connecting the zero-shear points vertically through the point-mass position.
Vertical images occur along the horizontal axis outside the unit-convergence
circle, along the vertical dot-dashed contour inside the unit-convergence
circle, and along the left boundary of the oval region connecting the zero-
shear points and passing vertically through a point between the halo center
and the point mass.

Going down in the right column, the inverted-color oval shrinks slightly as it
moves with the point mass closer to the halo center. At $x_{\text{P}}=0.15$,
the oval partly overlaps the unit-convergence circle. The very pale color in
the region of their overlap indicates near-horizontal orientation of images
there. At the boundary of the oval in this plot, images are oriented
horizontally along its left part inside the unit-convergence circle and along
its right part from the zero-shear points to the point-mass position. Along
the segments extending from the zero-shear points to the left till the unit-
convergence circle, images are oriented vertically. Image orientations along
the other parts of the dot-dashed contour remain unchanged.

Figure 13.B: Image-plane maps of the weak phase $\varphi_{\text{w}}(\bm{x})$
of a NFW halo + point-mass lens, for a finer grid of point-mass positions than
in Figure 13.A. Notation same as in Figure 13.A. Figure 14.A: Image-plane maps
of the weak-phase deviation $\delta\varphi_{\text{w}}(\bm{x})$ caused by the
presence of the point mass, described in Section 3.5.8. The maps also exactly
portray the change in image orientation due to the point mass. Red and blue
indicate counterclockwise and clockwise deviation, respectively. The color
scale changes from logarithmic to linear in the interval
$[-\pi/2000,\pi/2000]$. Four colored contours correspond to
$\delta\varphi_{\text{w}}$ values indicated in the color bar; the dot-dashed
contour corresponds to $\delta\varphi_{\text{w}}=0$ and
$\delta\varphi_{\text{w}}=\pm\pi/2$. Remaining notation as in Figure 7.A.

At $x_{\text{P}}=0.1$, the zero-shear points now lie inside the unit-
convergence circle, so that horizontally oriented images now occur along the
left boundary of the oval up to the zero-shear points and along the part of
the right boundary outside the unit-convergence circle passing through the
point-mass position. Vertically oriented images occur along the segments
extending from the zero-shear points to the right till the unit-convergence
circle. The pattern inside the overlap of the oval and the unit-convergence
circle reflects two general properties of its boundaries. Crossing the
boundary of the oval inverts only the color at $\varphi_{\text{w}}=0$ or
$|\varphi_{\text{w}}|=\pi/2$, which corresponds to a continuous change in
orientation. Crossing the unit-convergence circle inverts the color as well as
the saturation, which corresponds to a $\pi/2$ flip in orientation. Note here
also the more prominent bulging of the vertical dot-dashed contour passing
through the halo center as the point mass lies closer.

Between $x_{\text{P}}=0.1$ and $x_{\text{P}}=0.05$, the boundary of the
inverted-color oval undergoes reconnection with the bulging vertical dot-
dashed contour. At $x_{\text{P}}=0.05$ this contour passes through the point-
mass position, while a small inverted-color oval lies to the left of it, with
its left boundary passing through the halo center. Note that at
$x_{\text{P}}=0.05$ there are no zero-shear points, so that here the entire
boundary of the small oval corresponds to horizontal images. What is more
striking in this plot is the pattern around the point mass, which now lies
inside the unit-convergence circle. Orange in the first quadrant and zero
saturation along the horizontal axis indicate that images are now oriented
radially around the point mass. As the point mass moves closer to the halo
center, at a certain value $0\leq x_{\text{P}}<0.05$ the small oval disappears
and the dot-dashed vertical curve passes through the origin for
$x_{\text{P}}=0$.

In the critical and sub-critical cases (the central and left columns,
respectively) the same sequence of changes occurs as in the super-critical
case, although for slightly different values of $x_{\text{P}}$. Note that the
inverted-color oval increases in size with increasing mass parameter
$\kappa_{\text{P}}$. In the top rows, the point at which the left side of its
boundary crosses the horizontal axis marks a boundary of influence of the halo
and the point mass (at least as far as image orientation is concerned). Both
boundaries passing through this point correspond to vertically oriented
images. However, images located off the horizontal axis to the left of this
point are tilted tangentially to the halo, while those to the right of this
point are tilted tangentially to the point mass.

Details of the changing patterns and contours can be studied in more detail in
Figure 13.B. For example, it can be seen that the reconnection of the vertical
and oval dot-dashed contours occurs at the zero-shear points. At higher values
of $x_{\text{P}}$ the points lie on the larger oval passing through the point-
mass position, while at lower values they lie on the smaller oval passing
through the halo center. For progressively lower point-mass distances, the
zero-shear points move along the smaller oval to the center, where they
disappear at $x_{\text{P}}=\sqrt{\kappa_{\text{P}}/\kappa_{\text{s}}}$.

Figure 14.B: Image-plane maps of the weak-phase deviation
$\delta\varphi_{\text{w}}(\bm{x})$ caused by the presence of the point mass,
for a finer grid of point-mass positions than in Figure 14.A. Notation same as
in Figure 14.A.

The plot for the critical case at $x_{\text{P}}=0.1$ illustrates the situation
at the elliptic umbilics, which occur for a slightly lower point-mass
distance. In this case, the full length of the oval contour corresponds to
horizontal image orientations (except the point-mass position and the zero-
shear points with undefined orientation). Similarly, the $x_{\text{P}}=0.09$
plots for the critical and super-critical cases indicate the interesting
pattern near the point mass when it lies on the unit-convergence circle (in
both cases for a slightly higher point-mass distance). To the right of the
point mass images are oriented vertically (tangentially), to the left
horizontally (radially), and above and below the orientation is undefined
(i.e., the images are circular).

The weak-phase plots presented in Figure 13.A can also be used to visualize
the phase, which we do not present in a separate plot. The reason is indicated
by Equation (41), which shows that outside the unit-convergence circle the
phase is equal to the weak phase, while inside it differs by $\pi/2$. The
phase plots would thus differ from the weak-phase plots in Figure 13.A by
having the colors and saturations flipped inside the unit-convergence circle
around the halo center. The circle would disappear in such plots, and the
color and saturation outside would extend continuously inside all the way to
the halo center. The dot-dashed contour would then consist only of the
horizontal axis, the perturbed vertical line through the halo center, and the
large oval associated with the point mass reconnecting to the small oval
associated with the halo center.