© The Authors 2025

1. Introduction

The Event Horizon Telescope (EHT) collaboration recently reported 1.3 mm very long baseline interferometry (VLBI) observations of the center of both the M87* galaxy and the object Sgr A* in the core of our own galaxy. The EHT collaboration reported the observation of an annular feature centered on M87* with a typical radius of ∼21 μas. The brightness along the ring is asymmetric mostly because of the Doppler boosting from the matter orbiting around the jet axis, which is inclined by 17 degrees relatively to the line of sight (Event Horizon Telescope Collaboration 2019a,b,c,d,e,f, 2021a,b). The ring diameter of the central source of our Milky Way, Sgr A*, located at a distance of 8 kpc, is reported as ∼50 μas, with an image that is dominated by a bright, thick ring with a diameter of 51.8±2.3 μas. This ring has a modest azimuthal brightness asymmetry and a comparatively dim interior. Preferably, the dominant model of the accretion disk that reproduces these observations would be a magnetically arrested disk with an inclination of i≤30°. (Event Horizon Telescope Collaboration 2022b,a,c,d,e).

The shadow and surrounding photon ring involve a special curve known as the apparent boundary (Bardeen 1973), and it is well known as the critical curve on the image plane. A light ray with an impact parameter equal to that of the critical curve asymptotically approaches a bound photon orbit when traced backwards from its observation by a distant observer. The existence of such a critical curve produces an observable photon ring: a bright and narrow ring-shaped feature near the critical curve due to the emission of radiation by the optically thin matter from the region of the central object, whose photons travel close to the unstable photon orbit before reaching the observer (Jaroszynski & Kurpiewski 1997; Falcke et al. 1999; Johannsen & Psaltis 2010).

Additionally, the light rays associated with the photon ring typically orbit the central object multiple times before reaching the observer. This can lead to a significant increase in brightness that may continue to grow without limits in the absence of absorption, as the optical path lengths near the unstable photon orbit become increasingly long; nevertheless, the specific form of the visible photon ring is not influenced by the astronomical source profile (Gralla et al. 2020). Therefore, it can be used as a probe to test general relativity and models for compact objects. In the absence of rotation, one obtains a single unstable circular photon orbit corresponding to an extremum of the effective potential, which is called a photon sphere. When spherical symmetry is lost, the set of unstable photon orbits is deformed and thickened, giving rise to a photon shell (Misner et al. 1973).

Separating the contributions of the background geometry and the influence of the surrounding flow in shadow images has become a critical challenge in the field. In the case of a spherical accretion configuration, the size of the observed shadow for a Schwarzschild black hole is closely influenced only by the space-time geometry, without being affected by the emission profile (Narayan et al. 2019). However, considering an optically and geometrically thin disk around a compact object, later studies indicated that the black-hole image is composed of a dark region and a bright region consisting of several components: direct emission, lensing rings, and photon rings, and the size of the shadow depends on the location of the inner edge of the accretion disk and the intensity of the observed luminosity (Gralla et al. 2019). Those important features may be regarded as a way to distinguish between black holes and black-hole mimickers in the Universe (Mazur & Mottola 2023; Visser & Wiltshire 2004; Casadio et al. 2024). In fact, the brightness and width of the lensing ring depend on the background geometry and the emission features of the accretion disk surrounding the compact object, and the observed image is dominated by the direct component (Rosa & Rubiera-Garcia 2022). Nonetheless, this is only true for axial observations, as the direct and lensed contributions lose their circular structure for inclined observations (Rosa et al. 2023, 2024; Rosa 2023).

If the space time under consideration features an innermost stable circular orbit (ISCO), i.e., a radius below which circular orbits become unstable, this radius accurately predicts where the inner edge of the accretion disk lies (Afshordi & Paczyński 2003). However, in models of optically thin and geometrically thick accretion disks, for which the inner edge of the disk can penetrate inside the unstable photon orbit, the minimum size of the shadow is not necessarily limited to the critical curve and can be strongly reduced (Zhang et al. 2024; Vincent et al. 2022).

One of the main advantages of shadow observations is that they provide a framework with which to probe deviations with respect to the Kerr or Schwarzschild solutions. However, the limited precision of the first set of observations does not allow for a precise identification of the underlying geometry (Wielgus 2021; Daas et al. 2023; Lin et al. 2022). Therefore, it is difficult to underestimate theoretical efforts required to calculate the forms and properties of shadows produced by black holes or other alternative compact objects in astrophysical environments. Besides these two well-studied Schwarzschild and Kerr hypotheses, at the theoretical level, several alternative models for compact objects have been proposed, featuring a wide range of properties that could potentially fill the gap in the interpretation and understanding of the observational data (Cardoso & Pani 2019).

In this work, we were interested in studying the shadow properties in the background of a distorted and deformed compact object due to quadrupole moments. This solution is a generalization of the q-metric previously analyzed in other publications (e.g., Erez & Rosen 1959; Faraji 2022). We showed that moderate changes in the quadrupole moment could lead to significant alterations in the properties of the shadow.

On one hand, we observed that the radius of the photon ring, the innermost stable circular orbit (ISCO), and the optical appearance can be different depending on the parameters of the geometry and intensity profile. On the other hand, we can deduce information about the source from its observed intensity profile. The latter is of crucial importance when testing the nature of compact objects and, therefore, testing the strong-field regime in the general theory of relativity.

Another important issue in astrophysics is the observation of abnormally high velocities in the outer region of galaxies due to the surrounding invisible dark matter (Brand & Blitz 1993; Sofue & Rubin 2001). The gravitational tidal forces induced by dark matter influence the propagation of light. Therefore, the presence of dark matter can be analyzed through the properties of the shadow via the assumption of different equations of state, (e.g., Hou et al. 2018a, b; Haroon et al. 2019). However, in these studies the results are model-dependent. To explore the effect of dark matter, we considered its behavior as an effective mass, which leads to modifications in the background geometry, (e.g., Leung et al. 1997; Konoplya & Zhidenko 2011; Konoplya et al. 2019; Konoplya 2019).

The paper is organized as follows: in Section 2, we introduced the generalized q-metric; in Section 3, we reviewed the geodesic motion of time-like and null particles in this background, we introduced the accretion-disk models, and we produced the observed shadow images and intensity profiles; in Section 4, we introduced the dark-matter model as a modification of the radial mass function and obtain the corresponding observed shadow images and intensity profiles; and in Section 5 we summarized our results and trace prospects for future work. Throughout the paper, we used a system of geometric units in which G = 1= c unless stated otherwise.

2. Generalized q-metric space time

One of the generalizations of the Schwarzschild space time recurring in the Weyl family of solutions (Weyl 1917) is the q-metric. This asymptotically flat solution of the Einstein field equations represents the exterior gravitational field of an isolated static and axisymmetric mass distribution containing a quadrupole moment, and it can be used to investigate the exterior fields of slightly deformed astrophysical objects in the strong-field regime (Quevedo 2011). This metric was further generalized by relaxing the assumption of an isolated compact object, or equivalently asymptotic flatness, by considering the presence of an exterior distribution of mass in its vicinity (Faraji 2022). Since this space time is not asymptotically flat by construction, the solution presented here is a vacuum solution of the Einstein field equations in the region near the central deformed object. The extension of this vacuum region depends on the chosen values of the parameters; i.e., smaller parameter values lead to a more extended vacuum region. Beyond this vacuum region, the space time could be extended by introducing a proper stress-energy tensor, which can be interpreted as a general external distribution of matter. Such a generalization resembles a magnetic surrounding (Ernst 1976), and it is also characterized by multipole moments. However, due to the subdominant effects of the higher multipole moments, it is appropriate to consider only the contributions to the metric up to the quadrupole moment for analyses in an astrophysical context. Therefore, in this space time, there are two independent quadrupole moments: one responsible for the deformation of the central object, α, and the other associated with the relaxation of asymptotic flatness, linked to the concept of an external matter distribution, β. Here, we adopted a set of coordinates analogous to the usual spherical coordinates $\left(t,r,\theta ,\varphi \right)$ in spherically symmetric space times¹. the line element describing this generalized q-metric is written as

$$\begin{array}{cc}\hfill {\mathit{ds}}^2& ={\left(1-\frac{2M}{r}\right)}^{1+\alpha }{e}^{2\hat{\psi }}{\mathit{dt}}^2-{\left(1-\frac{2M}{r}\right)}^{-\alpha }{e}^{-2\hat{\psi }}\hfill \\ \hfill & [{\left(1+\frac{M^2{sin}^2\theta }{r^2-2\mathit{Mr}}\right)}^{-\alpha (2+\alpha )}{e}^{2\hat{\gamma }}\left(\frac{{\mathit{dr}}^2}{1-\frac{2M}{r}}+r^{2}d{\theta }^2\right)\hfill \\ \hfill & +r^2{sin}^2\theta d{\varphi }^2],\hspace{0.5em}\hfill \end{array}$$(1)

where M is a parameter that plays the role of the mass of the central object, and the metric functions $\hat{\psi }$ and $\hat{\gamma }$ control the distortion of the space time, also up to quadrupole, and read as follows:

$$\hat{\psi }=\frac{\beta }{2}\left[{\left(\frac{r}{M}-1\right)}^2\left(3{cos}^2\theta -1\right)+{sin}^2\theta \right],$$(2)

$$\begin{array}{cc}\hfill \hat{\gamma }& =\beta {sin}^2\theta [-2(1+\alpha )\left(\frac{r}{M}-1\right)\hfill \\ \hfill & +\frac{\beta r}{4M}\left(\frac{r}{M}-2\right)\left((-9{cos}^2\theta +1){\left(\frac{r}{M}-1\right)}^2-{sin}^2\theta \right)]\hfill \end{array}$$

where α and β are quadrupole parameters. The coordinates range in the intervals t∈(−∞,+∞), r∈(0,+∞), θ∈[0,π], and ϕ∈[0,2π). Under the assumptions outlined above, this metric contains three free parameters: the deformation parameter, α; the distortion parameter, β; and the total mass, M, which can be absorbed through a redefinition of the radial coordinate. In the equatorial plane, described by θ=π/2, the metric functions reduce to

$$\hat{\psi }=\frac{\beta }{2}\left[-{\left(\frac{r}{M}-1\right)}^2+1\right],$$(3)

$$\hat{\gamma }=\beta \left[-2(1+\alpha )\left(\frac{r}{M}-1\right)+\frac{\beta r}{4M}\left(\frac{r}{M}-2\right)\left({\left(\frac{r}{M}-1\right)}^2-1\right)\right].$$

In fact, the presence of a quadrupole can have a notable impact on the geometric properties of space time when compared to the Schwarzschild solution (see, e.g., Faraji & Trova 2021, 2022). Mathematically, the deformation parameter is bounded by α∈(−1,∞). However, in an appropriate astrophysical context, other studies (see, e.g., Faraji & Trova 2021, 2022) have demonstrated that the most relevant domain for α is almost (−1,1). It is important to note that geodesics are influenced by parameters α and β simultaneously. In fact, calculating the radius of the innermost stable circular orbit (ISCO) and of the unstable photon orbit, which is done by setting the second and first derivatives of the respective effective potentials to zero, leads to a relationship between α and β. This happens because, for a given value of α the range of the parameter β – specifically in the equatorial plane – depends on α and arises directly from solving the mentioned equations. In addition, the range of the parameter β turns out to be significantly smaller than the latter. Additionally, in this work, we were interested in configurations that are approximately spherically symmetric, and, consequently, the quadrupole parameters should be relatively small in magnitude. To guarantee that this approximation of spherical symmetry is valid, one needs to keep the absolute values of the parameters α and β below the values 0.01 and 10⁻⁶, respectively, which guarantees that the deviations from spherical symmetry of the g_tt, g_rr, and g_ϕϕ components of the space-time metric are always below 1% in the region where photons propagate (see Appendix A). This solution is the exact superposition of an oblate or prolate compact object, surrounded by an external distribution of matter. Recently, the impact of superposed fields in the motion of free test particles in this space time was studied (Faraji 2022), and it was shown how the positions of important astrophysical orbits, such as ISCO for particles, and photon circular orbits, depend on the parameters of the metric. In Figure 1, we plotted the second derivative of the particle potential $V_{\mathrm{part}}=-g_{\mathit{tt}}\left(\frac{L^2}{r^2}+1\right)$, where L is the angular momentum of the particle and the first derivative of the photon effective potential $V_{\mathrm{phot}}=-\frac{g_{\mathit{tt}}}{r^2}$. The results indicate that an increase in α shifts the inflection points of the particle's effective potential and the extrema of the photon effective potential to larger radial coordinates. In contrast, an increase in β causes the behavior of these two features to shift in opposite directions due to the effect of external distribution. In this case, the unstable photon orbit, being closer to the object than ISCO, feels the gravitational pull of the central object more strongly than the influence of the external fields. For instance, if β>0, the external quadrupole tends to push the unstable photon orbit inward, increasing the curvature near the central object. The ISCO, located farther out, feels a weaker pull from the central object but a stronger influence from the external quadrupole, which causes it to shift outward.

Fig. 1. Second radial derivative of particle effective potential, $V_{\mathrm{part}}^{″}$ (top row), and first radial derivative of the photon potential, $V_{\mathrm{phot}}^{\prime }$, both as a function of the radius, r, with β = 0 and varying α (left panels) and with α = 0 and varying β (right panels), at the equatorial plane $\theta =\frac{\pi }{2}$. The radius of the ISCO is defined as the radius at which $V_{\mathrm{part}}^{″}=0$, and the radius of the photon orbit is defined as the radius at which $V_{\mathrm{phot}}^{\prime }=0$. We observed that a change in α causes the radii of the ISCO and the photon orbit to change in the same direction, whereas a change in β causes these radii to change in opposite directions.

Moreover, from an astrophysical perspective, the negative values of the quadrupole parameters are rather counterintuitive, since they describe prolate compact objects, while positive values correspond to oblate compact objects that are widely observed in the Universe.

3. Shadows and light rings

The black-hole shadow for a far-away observer is defined as the set of directions in the sky, propagated backwards in time, that cross the event horizon. Its boundary consists of rays that asymptotically approach the surfaces of the underlying space time on which photons undergo unstable closed orbits, i.e., the photon shell (PS) or critical curves, with the same specific energy and angular momentum that are conserved quantities. In fact, the shape and size of the shadow depend on the geometry of the background space time, which requires the analysis of the light-ray or null geodesics of a test particle.

3.1. Null geodesics and impact parameter

The Lagrangian $\mathcal{L}=\frac{1}{2}{g}_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }$ describing the motion of a particle, in a space time described by a diagonal metric in spherical coordinates and restricted to the equatorial plane, reads as

$$\mathcal{L}=\frac{1}{2}\left[g_{\mathit{tt}}{\dot{t}}^2-g_{\mathit{rr}}{\dot{r}}^2-g_{\varphi \varphi }{\dot{\varphi }}^2\right],$$(4)

where ${\dot{x}}^{\mu }\equiv \frac{\mathrm{d}{x}^{\mu }}{\mathrm{d}\tau }$ and τ is proper time. In addition, there exist two conserved quantities associated with time translation and angular momentum, which are defined as

$$E=\frac{\partial \mathcal{L}}{\partial \dot{t}}=g_{\mathit{tt}}\dot{t},L=\frac{\partial \mathcal{L}}{\partial \dot{\varphi }}=-g_{\varphi \varphi }\dot{\varphi }.$$(5)

When solving Eq. (5) with respect to $\dot{t}$ and $\dot{\varphi }$ and inserting the result into Eq. (4) under the restriction $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=0$ for null geodesics, one obtains

$${\dot{r}}^2=\frac{1}{g_{\mathit{rr}}}\left[\frac{E^2}{g_{\mathit{tt}}}-\frac{L^2}{g_{\varphi \varphi }}\right].$$(6)

For a given photon propagating in the background space time, if $\dot{r}$ vanishes at a radius larger than the unstable photon orbit, this implies that the photon reaches a radius of closest approach and subsequently propagates back to infinity. If this situation is never verified, the photon is captured by the central object. By introducing the impact parameter $b=\frac{L}{E}$ and rewriting Equation (6), one can obtain the so-called critical impact parameter b_crit, which discriminates between captured and uncaptured photons. This can be done by solving the equations $\dot{r}=0$ and $\ddot{r}=0$ for circular orbits, which leads to

$$g_{\varphi \varphi }-g_{\mathit{tt}}{b}_{\mathrm{crit}}^2=0,$$(7)

$$g_{\mathit{tt}}^{\prime }{g}_{\varphi \varphi }^2-b_{\mathrm{crit}}^2{g}_{\varphi \varphi }^{\prime }{g}_{\mathit{tt}}^2=0;$$(8)

here, a prime denotes a derivative with respect to r, and the result does not depend on g_rr. For example, in the Schwarzschild case, $b_{\mathrm{crit}}=3\sqrt{3}$. In our setup, as the radius of the PS depends on the quadrupole parameters, different choices imply different values of b_crit. If the impact parameter b<b_crit for a given photon, the photon is captured by the compact object. Instead, for b=b_crit the photon revolves infinitely around the central compact object. Finally, for b>b_crit the photon is scattered back and may reach the observer.

In addition, Eq. (6) can be conveniently rewritten in the following form from which one can obtain the trajectories of the light rays:

$$\frac{\mathrm{d}\varphi }{\mathrm{d}r}=\pm \frac{b}{g_{\varphi \varphi }}\frac{\sqrt{-g_{\mathit{tt}}{g}_{\mathit{rr}}}}{\sqrt{1+\frac{b^2{g}_{\mathit{tt}}}{g_{\varphi \varphi }}}}.$$(9)

The signs (±) represent ingoing (−) and outgoing (+) null geodesics. In what follows, we performed a numerical integration of Eq. (9) using a Mathematica-based ray-tracing code that has been used in several publications to simulate the observational properties of compact objects (Rosa & Rubiera-Garcia 2022; Rosa et al. 2023, 2024; Rosa 2023; Olmo et al. 2023, 2022; Guerrero et al. 2022a,b,2021; Asuküla et al. 2024; Macedo et al. 2024). Note that a given ingoing geodesic transitions into an outgoing geodesic at the radius of the closest approach, i.e., when dϕ/dr diverges, at which point one should invert the sign of Eq. (9) to proceed with the integration. Contrary to what happens in spherically symmetric space times, the right hand side of Eq. (9) depends explicitly on complicated functions of θ, which is a major obstacle in the numerical resolution of the equation. To simplify this procedure and improve the computational time, we employed the spherically symmetric approximation; i.e., we considered small values of the quadrupole parameters and restricted the analysis to the equatorial plane: $\theta =\frac{\pi }{2}$. The errors incurred by taking this approximation are always smaller than 1% for the situations analyzed in this work (see Appendix A for more details on the validity of this approximation). Concerning the study of circular geodesics, i.e., $\dot{r}=0$ and $\ddot{r}=0$, the radius of these circular geodesics is related to α and β through the following relation (Faraji 2022)

$$\alpha =\beta \frac{r}{M}\left(\frac{r}{M}-1\right)\left(\frac{r}{M}-2\right)+\frac{1}{2}\left(\frac{r}{M}-3\right),$$(10)

where in the limit β = 0 one recovers the radius of the PS for the q-metric r_LR=M(3+2α), and in the limit α=β = 0 one recovers the Schwarzschild radius of the PS r_LR = 3M. In addition, one can have one bound unstable photon orbit in the equatorial plane for any given valid $\beta <-\frac{1}{2}$ with

$$\beta =-\frac{1}{2\left[3{\left(\frac{r}{M}-1\right)}^2-1\right]}.$$(11)

This is not the case in the Schwarzschild space time, nor in the q-metric. In fact, this arises due to the existence of a quadrupole related to the external source and indicating that, for certain negative values of β, an unstable bound photon orbit exists in the equatorial plane. It is necessary to exclude these orbits in order to conduct a study on shadows.

3.2. Accretion-flow models and intensity profiles

The observation of the shadow requires a large enough angular resolution; however, there is a backdrop of sufficiently bright light sources against which the shadow can be seen as a dark spot. For example, the emission of the surrounding matter can partially obscure any shadow. In this section, we considered optically and geometrically thin accretion disks in the equatorial plane surrounding the compact object in order to investigate the observed specific intensity. Furthermore, since the optically thin disk is transparent to its own radiation, photons turning more than one half-orbit around the compact object produce a thin photon ring in superposition with direct emission. Consequently, the observed photon ring is broken into an infinite sequence of self-similar photon rings forming several loops and stacking on the direct emission.

We assumed that the accretion disk is located in the static rest frame and emits isotropically. More importantly, the whole setup should be close enough to the horizon since photons emitted from this vicinity with a small enough impact parameter are absorbed by the central compact object and cast a shadow. In fact, the inner part of the accretion-disk flow has the greatest density; therefore, most of the radiation comes from this region with the highest temperature. If the disk has no magnetic field with sub-Eddington luminosities, then, with high accuracy, the place of the innermost stable circular orbit (ISCO) is considered as the inner edge of the thin disk (Afshordi & Paczyński 2003) where most of the luminosity comes from. Therefore, it can be used to infer possible luminosity and intensity profiles for different accretion-disk models. However, in what follows, we considered emission from an optically thin and geometrically thin region near the compact object, with three different stationary accretion-disk models. In addition, the observer stands on the vertical axis, θ = 0. Under those considerations, the intensity models are as follows:

1. The ISCO model: In this model, the luminosity of the accretion disk increases monotonically from r=∞ to r_ISCO, where it achieves a maximum, and then abruptly decays in the region, r<r_ISCO.

2. The LR model: Even though the circular orbits for r<r_ISCO are unstable, one can still argue that unstable orbits exist in the region, r_LR<r<r_ISCO. Thus, in this model, it is supposed that the luminosity of the accretion disk increases monotonically from r=∞ to r_LR, where it peaks, and then abruptly decays in the region, r<r_LR.

3. The EH model: In this model, the emissivity extends to the event horizon and is not truncated at any larger radius. This model is motivated by the expectation that the region, r<r_LR, is populated by matter falling into the central compact object all the way down to the event horizon (EH), which for the Schwarzschild space time (α=β = 0) stands at r_EH = 2M. We thus considered an additional intensity model for which the luminosity increases monotonically from r=∞ and peaks at r_EH.

Note that for α≠0 the space time does not have an event horizon, but a naked singularity at r = 2M. However, for the sake of notation simplicity, we adhered to the symbol r_EH = 2M and pictorial representation for these solutions. To model the three intensity profiles described above, we referred to an emission profile published in Gralla et al. 2020, and shown to be in close agreement with observational predictions and general relativistic magneto-hydrodynamics (GRMHD) simulations. The intensity profile, $I\left(r\right),$ of this model is derived from Johnson's SU distribution – which is a free function with a smooth profile, mostly concentrated within a few Schwarzschild radii of the central object – and increases as we approach the compact object. This emission profile is given, by

$$I\left(r,\gamma ,\mu ,\sigma \right)=\frac{e^{-\frac{1}{2}{\left[\gamma +\text{arcsinh}\left(\frac{r-\mu }{\sigma }\right)\right]}^2}}{\sqrt{{\left(r-\mu \right)}^2+{\sigma }^2}},$$(12)

where γ, μ, and σ are free parameters that shape the behavior of the function $I\left(r\right)$. The parameter γ controls the rate of increase of the luminosity from infinity down to the peak; the parameter μ induces a translation of the profile and controls the radius at which the luminosity peaks; and the parameter σ controls the dilation of the profile. The values of the parameters γ, μ, and σ for each of the three disk models mentioned above – i.e., the ISCO, LR, and EH disk models – are specified in Table 2, r_LR and r_ISCO are provided in Table 1, and the respective intensity profiles are plotted in Figure 2. In this figure, different values of α and β are not plotted separately as they correspond solely to a translation of the ISCO and LR models in the horizontal axis. However, we adapted these models in our setup considering the fact that the place of ISCO and the light ray changes with α and β (Faraji 2022).

Fig. 2. Luminosity profiles for three models of accretion disks considered: ISCO model (solid red), LR model (dashed blue), and EH model (dotted purple) for α = 0 and β = 0.

Note that the intensity profiles plotted in Figure 2 correspond to the intensity profiles in the reference frame of the accretion disk. The frequencies in the reference frame of the observer and the emitter are denoted by ν_o and ν_e, respectively. These two frequencies are related via the gravitational redshift as ${\nu }_0=\sqrt{g_{\mathit{tt}}}{\nu }_{\mathrm{e}}$; then, the associated specific intensity $i\left(\nu \right)$ scales as

$$i\left({\nu }_{\mathrm{o}}\right)={\left({\nu }_0/{\nu }_{\mathrm{e}}\right)}^{3}i\left({\nu }_{\mathrm{e}}\right)=g_{\mathit{tt}}^{\frac{3}{2}}i\left({\nu }_{\mathrm{e}}\right).$$(13)

For each time that a light ray intersects the equatorial plane, it gathers additional brightness from the disk emission. Due to the extra brightness contributions, the observed intensity is described as the sum of intensities from each intersection. Consequently, the total intensity, I, in the reference frame of the observer can be written as

$$I_{\mathrm{o}}(r)={\displaystyle \sum _n}{g}_{\mathit{tt}}^{\frac{3}{2}}\left(r\right)I_{\mathrm{e}}\left(r\right){|}_{r=r_n(b)},$$(14)

where $n=\frac{\theta }{2\pi }$ and r_n(b) is the demagnification factor, $\frac{\mathrm{d}r}{\mathrm{d}b}$, which represents the intersection radius with the equatorial plane for the nth time on its backwards journey, where b is the impact parameter. The continuous version of Eq. (14) is also taken into consideration in our Mathematica-based code to produce the intensity profiles in the reference frame of the observer.

3.3. Shadows and observed intensity profiles

To analyze the effects of the parameters α and β in the observational properties of the background space time given in Eq. (1) in the regime where the spherically symmetric approximation is valid, we ran the ray-tracing code for several combinations of setups. In particular, we considered α={−0.01;0;0.01} and β={−10⁻⁵;0;10⁻⁵} for the three accretion-disk luminosity profiles given in Table 2. The accretion disk is placed at the equatorial plane, i.e., at θ=π/2, while the observer is placed at the vertical axis, θ = 0, and at a radius of r = 600M. Note that this approximation is only valid locally due to the alterations in the metric components caused by the parameter β. These alterations cause the space time to lose its asymptotic flatness, and thus the distance from the central object to the observer cannot be made arbitrarily large. The value r = 600M was chosen in such a way that the effects of losing the asymptotic flatness are still negligible, while maintaining a distance large enough for the light rays to be approximately parallel when they reach the observer. In the equatorial-plane validity region for β<0 is wider than for β>0, and an increase in β leads to a decrease in the size of this region. We also chose relatively small values of β with the goal of having a wider validity region allowing for the comparison of the results among different cases. Our results indicate that the effects of the parameter β are negligible on the observed shadow image and luminosity profiles. Thus, in this section, we restricted our analysis to the effects of the parameter α. For a complete set of results including the effects of the parameter β, we refer the reader to Appendix B. The shadow images for the three disk models are presented in Figure 3, whereas the observed intensity profiles for the three disk models are presented in Figure 4. As for photon rings, their size and location can be quite different depending on the features of the parameters. In addition, the corresponding luminosity is also strongly related to the background geometry and the properties of the disk.

Fig. 3. Shadow images for ISCO disk model (top row), LR disk model (middle row), and EH disk model (bottom row), for α=−0.01 (left column), α = 0 (middle column), and α=+0.01 (right column).

Fig. 4. Observed intensity I(r) as function of radius r for a varying α with constant β = 0, and for the ISCO model (left panel), LR model (middle panel), and EH model (right panel).

The first important feature to note is that the observed intensity profiles feature multiple peaks of intensity, in contrast to the emitted intensity profiles, which feature a single peak of intensity (see Figure 2). These multiple peaks of intensity at the observer are caused by photons that reach the observer after circling around the central object with a different number of half-orbits. In this peak structure, we can identify three main components:

1. The direct component: corresponding to the wider and dominant component, it is composed of the photons emitted directly from the accretion disk at θ=π/2; i.e., the equatorial plane to the observer at θ = 0. These photons have travelled a total of a quarter orbit around the central object, Δθ=θ_e−θ_o=π/2, and are defined as the primary image.

2. The lensed component: corresponding to a narrower component placed either inside (for the ISCO model) or outside (for the LR and EH models) the inner edge of the direct component, is composed of the photons that have been emitted by the accretion disk in the direction opposite the observer, and then have been lensed around the central object once before reaching the observer. These photons have travelled an extra half-orbit more than the photons of the direct component, Δθ = 3π/2, and are defined as the secondary image.

3. The photon ring component: corresponding to the narrowest of the three components, it is composed of the photons that have been emitted by the accretion disk and have orbited at least once around the central object, close to the LR, before reaching the observer². This component presents an infinite substructure of peaks corresponding to the photons that have orbited a total of Δθ = 5π/2+nπ, for n≥0, an integer, which leads to additional brightness once again. This infinite substructure of peaks converges to critical curve, b=b_c, which becomes exponentially closer to each other with an increase in the number of orbits.

For the ISCO model, these three components are all distinguishable in the observed emission profiles (see, e.g., the top row in Figure 3 and the left panel in Figure 4). However, for the LR model, the direct and lensed components appear superimposed, and for the EH model, the photon ring component is also superimposed with the direct emission.

In all three models considered for the accretion disk, one verifies that the parameter α affects the results more prominently than the parameter β. Indeed, since a variation in α induces a non-negligible translation of the radii of the ISCO and the LR of the background space time, one observes that the size of the shadow is strongly affected by this parameter, with the shadow radius increasing for α>0 and decreasing for α<0. Furthermore, one also verifies that the maximum amplitude of the observed intensity profiles for the direct emission is anticorrelated to the value of α. This behavior is expected as, for negative values of both quadrupole parameters, the radius of the ISCO is smaller than for the positive values (Faraji 2022); thus, the radiation is being emitted closer to the central object. Indeed, for all of the accretion-disk models, one verifies that the rate of change of the intensity profile as a function of the radial coordinate is more abrupt for larger values of α, which is also expected. As a consequence, the lensed and photon ring components of the intensity profiles for negative values of alpha are wider, and thus more visible in the shadow images. On the other hand, the parameter β presents a minor effect on the observational properties of this setting, inducing smaller changes in the observed intensity profiles in comparison with the effect of the parameter α (see Appendix B). Our results show that the observed intensity profiles are slightly closer to the central object for negative values of β, while positive values of beta tend to bring them farther away. The effects are particularly clear for the ISCO model, for which the direct and lensed components of the observed intensity profile are separated, but it is also visible (although less contrasting) in the LR and EH models, for which the direct and lensed components of the observed intensity profile are superimposed.

Figure 5 shows how the null geodesic congruences are affected by parameters α and β. Although all calculations and plots extend up to an observer distance of 600M, in this figure we only displayed results out to 100M for convenience. The left panel shows the effect of α for a wide range of impact parameters. Most trajectories cross the equatorial plane once, but photons that propagate close to the unstable photon orbit on the image plane wrap around the central object and cross the plane twice, three times, or more, with photons appearing exactly on the critical curve describing trajectories that asymptotically approach the unstable bound orbits composing the photon shell. The size of the shadow is defined by the range of impact parameters corresponding to the trajectories that do not cross the equatorial plane before intersecting the black disk. Thus, one can see that the size of the shadow produced by an oblate compact object is larger than the shadow produced by the Schwarzschild black hole, and the reverse is true for a prolate compact object. In the right panel, the impact of the parameter β is shown. It is evident that the effect of the parameter β in this setting and in the equatorial plane is much smaller in comparison with the effects of the parameter α as anticipated by the analysis of the intensity profiles and shadow images.

Fig. 5. Photon trajectories that reach a distant observer in far right z = 100M. The black disk at x = 1 represents the event horizon for α = 0 or the naked singularity for α≠0. Photon trajectories are colored according to the different parameters of the metric. Left panel: β = 0 and prolate object α=−0.01 (blue lines), Schwarzschild solution α = 0 (black lines), and oblate object α = 0.01 (red lines). Right panel: α = 0 and β=−10−6 (blue lines), Schwarzschild solution β = 0 (black lines), and β = 10−6 (red lines). Note that each set of three geodesics is emitted with the same impact parameter at z = 100M, but, due to the fact that the space time is not asymptotically flat, they reach the vicinity of the central object visible in the plot with slight deviations from the asymptotically flat α = 0 or β = 0 cases.

4. Dark-matter model

The standard model of cosmology characterizes the Universe as geometrically flat, with dark energy, dark matter, and ordinary matter comprising approximately 68%, 27%, and 5% of its total mass-energy density, respectively. Dark matter most likely consists of unknown elementary particles beyond the standard model of particle physics that originated in the early Universe, resulting in negligible effects of its velocity dispersion on structure formation (referred to as cold dark matter). Additionally, unlike baryonic matter, dark matter is not expected to form accretion disks due to its low interaction cross-section. Furthermore, due to the absence of the coupling with the electromagnetic field, light rays may travel through dark-matter distributions virtually unaffected by any nongravitational means³. To date, there has been no direct detection of dark matter. The evidence supporting its existence is primarily based on cumulative measurements, such as galactic rotation curves (Rubin et al. 1980) and the cosmic microwave background (Planck Collaboration XVI 2014). In this study, we focused only on its fundamental property; more specifically, its manifestation through gravitational effects (Konoplya 2019) via the definition of an additional effective mass function as

$$m(r)=\{\begin{array}{cc}M\hfill & r<r_{\mathrm{s}}\hfill \\ M+\mathcal{S}(r)\mathrm{\Delta }M\hfill & r_{\mathrm{s}}\le r\le r_{\mathrm{s}}+\mathrm{\Delta }r\hfill \\ M+\mathrm{\Delta }M\hfill & r_{\mathrm{s}}+\mathrm{\Delta }r<r\hfill \end{array},$$(15)

where r_s = 2M is the Schwarzschild radius, ΔM is the mass of the dark-matter halo, Δr is the thickness of the dark-matter halo, and $\mathcal{S}$ is a radial function defined in such a way as to preserve the continuity of the mass function and its first derivative with respect to r. This function takes the form

$$\mathcal{S}(r)=\left(3-2\frac{r-r_s}{\mathrm{\Delta }r}\right){\left(\frac{r-r_s}{\mathrm{\Delta }r}\right)}^2.$$(16)

In this definition, ΔM>0 corresponds to the positive dark-matter mass-energy density and ΔM<0 to the negative one. In this work, we solely considered the positive case and replaced M by m(r) in the line element in Eq. (1).

Figure 6 shows the behavior of the g_tt and g_rr components for the dark-matter model for different combinations of ΔM and Δr. To see the effect of the dark-matter distribution on the spherically symmetric black-hole case, we considered the combination α=β = 0. Compared to the vanishing dark-matter case, we see that the metric coefficients have two additional local extrema: a local minimum close to the inner edge of the disk corresponding to the inner edge of the dark matter halo; and a local maximum at the outer edge of the halo. Furthermore, the magnitude of these extrema is anticorrelated to the thickness of the halo, with large values of Δr inducing only slight modifications to the space-time geometry close to the central object. Furthermore, previous publications have shown that, under certain conditions, the space time presents an additional pair of unstable photon orbits with opposite stability characters (Konoplya 2019). Note that if r_o≤r_LR<r_s, where r_o is the radius at which the observer stands, the observer can only see the purely compact object effect on the shadow without the contribution of dark matter. Also, if r_s+Δr<r_LR<r_o, the dark-matter halo is constrained in the vicinity of the central object, but it is assumed not to fall into the event horizon, which is a rather nonphysical assumption. We thus chose to discard the two cases above and restrict our analysis to the combination r_s<r_LR<r_s+Δr.

Fig. 6. Metric components gtt (first row) and grr (second row) for the dark-matter model with Δr = 50M with varying ΔM (left plot) and for ΔM = 12M with varying Δr (right plot); α=β = 0.

The methods outlined in the previous section for the ray-tracing and production of the observed intensity profiles and shadow images were also applied to the model introduced in this section. However, since the radius of the ISCO is highly sensitive to the mass distribution of the halo and it is a discontinuous function of ΔM for configurations in which the mass of the halo is much larger than the mass of the central object, we decided to restrict our analysis to the LR and EH models for the accretion disk. The observed intensity profiles, I(r), for the particular case of α=β = 0 for the LR and EH models are given in Figure 7. We chose α=β = 0 to analyze the effect of the dark-matter model on the intensity profiles separately. Our results indicate that both the LR and the EH models are only slightly sensitive to both ΔM and Δr, with variations in the smaller values of Δr and larger values of ΔM being the most prominent. Figure 8 shows the shadow images for the LR and EH disk models for the same particular case, α=β = 0. This figure compares the shadow images in the absence of dark matter with two mass distributions of the same length scale with different masses. One can observe a slight increase in the size of the shadow for both disk models with an increase in the mass of the halo.

Fig. 7. Intensity I(r) as a function of radius, r, for the LR disk model (top row) and for the EH disk model (bottom row), with α=β = 0 for Δr = 50M, with ΔM={7M,14M,21M} (left column), and ΔM = 12M with ΔM={30M,60M,90M} (right column).

Fig. 8. Shadow images for LR disk model (top row) and EH model (bottom row) with α=β = 0. The first column depicts the absence of dark matter, while the second and third columns show Δr = 50M and variations in ΔM = 7M,21M, respectively.

Turning now to the analysis of how the parameters α and β influence the results, Figure 9 shows the observed intensity profiles for varying α with β = 0 for a given set of parameters of the dark-matter model. Similarly to what was observed in the absence of the DM halo in Figures 3 and 4, variations in α induce radial translations of the observed intensity profiles. This is due to the fact that the accretion disk is closer to the compact object for smaller values of α. Furthermore, upon comparing the LR panel in Figures 4 and 9 for the cases in which β = 0, it is evident that for any constant value of α, the radial position of the peak shifts to larger values. We found the same pattern in the EH model panels of Figures 9 and 4. Furthermore, in accordance with the results of the previous section, we observed that variations in the chosen values for the parameter β induce barely noticeable changes in the observed intensity profiles. To clarify this point, in Figure 10 we showed the observed intensity profiles for α = 0 for both the DM model, and in the absence of DM with varying β. These results indicate that the presence of DM does not alter the sub-dominant character of the parameter β in the observational properties of these models. In Figure 11, we depicted the shadow images for varying α and vanishing β for EH and LR models. The results are comparable with those of Figure 9.

Fig. 9. Intensity I(r) as function of radius, r, for LR disk model (left plot) and for EH disk model (right plot), with ΔM = 20M, Δr = 50M for α={−0.01,0,0.01}, and β = 0.

Fig. 10. Comparing intensity I(r) for ER disk model with α = 0 and varying β, both in the presence and absence of dark matter.

Fig. 11. Shadow images for LR disk model (top row) and EH disk model (bottom row) with ΔM = 20M and Δr = 50M, for α = 0.01 (left column), α = 0 (middle column), and α=−0.01 (right column).

Summarizing, our results indicate that considering the DM model that we chose in this work, the presence of DM halos affects the observed intensity profiles and shadows of the models considered only slightly; thus, it is a subdominant effect in comparison with the effects of the dominant deformation parameter, α. Nevertheless, for a given model with a set value of the deformation parameter, the effects of the DM halo increase with the mass and the proximity of the halo from the central compact object.

5. Summary and conclusion

In this work, we analyzed the shadow images and photon ring structure of a novel class of compact objects that are analytically tractable, with the goal of extending the previous research conducted in different space times. This space time is a solution to the Einstein field equation and an extension of the familiar q-metric, which is static and axisymmetric and includes a distortion parameter, β, and a deformation parameter, α, as briefly outlined in Section 2. This analysis was done both in the absence and the presence of a dark-matter halo, the latter being represented by a modification in the radial mass function with two free parameters controlling the mass, ΔM, and the typical length-scale Δr of the halo.

In our analysis, we assumed the gas medium to be optically thin. The distinctive aspects of the observed images are primarily influenced by the location and characteristics of the emitting material close to the central object. In general, it can be inferred that the inclusion of a quadrupole in the metric significantly impacts the size and intensity of shadow images. Specifically, the oblateness of the central compact object, represented by the α parameter, plays a more pronounced role in influencing the characteristics of the shadow. This is due to the fact that modifications in α induce large variations in the radius of the ISCO and the unstable photon orbit, which control the interior edge of the accretion-disk models. On the other hand, we showed that the distortion parameter β plays a negligible role in affecting the observational properties of these space times. These results indicate that the analysis of the sizes of shadows observed experimentally could be used to constrain the oblateness of the central compact object, but in the absence of oblateness, one can distinguish the effect of distortion due to the external matter distribution.

Furthermore, we analyzed the effects of the presence of a dark-matter halo surrounding the central object and the possibility of identifying this presence from the physical properties of emission from an optically thin disk region. Our results indicate that dark matter has the ability to either enhance or diminish the size of the observed shadow, although these effects are sub-dominant in comparison with the effects of oblateness, considering the chosen parameter's range. Note that this outcome is contingent upon the specific dark-matter model being considered. In this work, we considered a rather simple model for the dark-matter halo; however, one can consider more complicated models (e.g., Hernquist 1990; Kiselev 2003; Li & Yang 2012; Xu et al. 2020) for further investigation.

The work considered in this manuscript provides the first step in the analysis of more realistic and astrophysically motivated models. Indeed, several assumptions made in this work could be dropped. Larger variations of the deformation and distortion parameters, for which the spherically symmetric approximation ceases to be valid, could be considered. Furthermore, in a realistic astrophysical scenario, we expect compact objects to be rotating, in contrast with the static configurations considered here. Finally, our analysis assumed the observer to be face-on with respect to the equatorial plane. While this assumption is acceptable if one wants to perform a comparison with the experimental data from M87, a generalization for inclined observations could be beneficial to allow for a comparison with the Sgr A* data. We leave these extensions for future publications.

Acknowledgments

Sh.F. is funded by the University of Waterloo and in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities at Perimeter Institute. Also, thanks the excellence cluster QuantumFrontiers funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2123 QuantumFrontiers. J.L.R. acknowledges the European Regional Development Fund and the programme Mobilitas Pluss for financial support through Project No. MOBJD647, project No. 2021/43/P/ST2/02141 co-funded by the Polish National Science Centre and the European Union Framework Programme for Research and Innovation Horizon 2020 under the Marie Sklodowska-Curie grant agreement No. 94533, Fundação para a Ciência e Tecnologia through project number PTDC/FIS-AST/7002/2020, and Ministerio de Ciencia, Innovación y Universidades (Spain), through grant No. PID2022-138607NB-I00.

References

Appendix A: Validity of the spherical symmetry approximation

In this section, we proved that the approximation taken in this work, i.e., that one can integrate the geodesic equation and obtain the trajectory of the light rays by taking the spacetime to be approximately spherically symmetric. For this purpose, and since the geodesic equation for the photons depends on the g_rr, g_tt, and g_ϕϕ components of the metric, we calculated the relative errors between these metric components of the spherically symmetric Schwarzschild solution (defined by the parameters α = 0 and β = 0) and those of the solutions with non-zero α and β. We defined the relative error as:

$${\nu }_i=\left|\frac{g_{\mathit{ii}}^S-g_{\mathit{ii}}}{g_{\mathit{ii}}^S}\right|,$$(A.1)

where ν is the relative error, the subscript i can denote either r, t, or ϕ, and $g_{\mu \nu }^S$ is the Schwazrschild spacetime metric. In the region of interest where photons are propagating between the observer and the central object and not falling into the event horizon, i.e., for a range $r\in \left[3M,100M\right]$ and $\theta =\left[0,\frac{\pi }{2}\right]$, the relative errors are plotted in Figure A.1. These results show that, in the regions where the photons propagate, the relative errors are consistently smaller than 1%, thus validating the approximation used.

Fig. A.1. Relative errors νr (top row), νt (middle row), and νϕ (bottom row) as defined in Eq. (A.1) for α=∓0.01 and β=±10−6 (left column), α = 0 and β=±10−6, and α=±0.01 and β=±10−6. The shades represent, from darker to lighter, the regions where the errors are below 0.02%, 0.05%, 0.1%, 0.2%, 0.5%, and 1%, with the largest errors being of the order of 2%.

Appendix B: Complete shadow images and intensity plots

For completeness, in this appendix we showed the complete set of shadow images and intensity plots for all of the parameter combinations, i.e., α={−0.01,0,0.01} and β={−10⁻⁵,0,10⁻⁵}, and for all of the accretion disk models, i.e., the ISCO, the LR, and the EH model in Figures B.1 -B.6. These results serve to illustrate the negligible effect of the parameter β to the observable properties of the system.

Fig. B.1. Shadow images for the ISCO disk model with different combinations of α and β, namely α=−0.01 (top row), α = 0 (middle row), and α=+0.01 (bottom row); and β=−10−5 (left column), β = 0 (middle column), and β = 10−5 (right) column).

Fig. B.2. Intensity I(r) as a function of the radius r for the ISCO disk model for a varying α with constant β (top row) and for a varying β with constant α (bottom row)

Fig. B.3. Shadow images for the LR disk model with different combinations of α and β, namely α=−0.01 (top row), α = 0 (middle row), and α=+0.01 (bottom row); and β=−10−5 (left column), β = 0 (middle column), and β = 10−5 (right) column).

Fig. B.4. Intensity I(r) as a function of the radius r for the LR disk model for a varying α with constant β (top row) and for a varying β with constant α (bottom row)

Fig. B.5. Shadow images for the EH disk model with different combinations of α and β, namely α=−0.01 (top row), α = 0 (middle row), and α=+0.01 (bottom row); and β=−10−5 (left column), β = 0 (middle column), and β = 10−5 (right) column).

Fig. B.6. Intensity I(r) as a function of the radius r for the EH disk model for a varying α with constant β (top row) and for a varying β with constant α (bottom row).

All Tables

All Figures

Fig. 1. Second radial derivative of particle effective potential, $V_{\mathrm{part}}^{″}$ (top row), and first radial derivative of the photon potential, $V_{\mathrm{phot}}^{\prime }$, both as a function of the radius, r, with β = 0 and varying α (left panels) and with α = 0 and varying β (right panels), at the equatorial plane $\theta =\frac{\pi }{2}$. The radius of the ISCO is defined as the radius at which $V_{\mathrm{part}}^{″}=0$, and the radius of the photon orbit is defined as the radius at which $V_{\mathrm{phot}}^{\prime }=0$. We observed that a change in α causes the radii of the ISCO and the photon orbit to change in the same direction, whereas a change in β causes these radii to change in opposite directions.

In the text

	Fig. 2. Luminosity profiles for three models of accretion disks considered: ISCO model (solid red), LR model (dashed blue), and EH model (dotted purple) for α = 0 and β = 0.
In the text

	Fig. 3. Shadow images for ISCO disk model (top row), LR disk model (middle row), and EH disk model (bottom row), for α=−0.01 (left column), α = 0 (middle column), and α=+0.01 (right column).
In the text

	Fig. 4. Observed intensity I(r) as function of radius r for a varying α with constant β = 0, and for the ISCO model (left panel), LR model (middle panel), and EH model (right panel).
In the text

Fig. 5. Photon trajectories that reach a distant observer in far right z = 100M. The black disk at x = 1 represents the event horizon for α = 0 or the naked singularity for α≠0. Photon trajectories are colored according to the different parameters of the metric. Left panel: β = 0 and prolate object α=−0.01 (blue lines), Schwarzschild solution α = 0 (black lines), and oblate object α = 0.01 (red lines). Right panel: α = 0 and β=−10−6 (blue lines), Schwarzschild solution β = 0 (black lines), and β = 10−6 (red lines). Note that each set of three geodesics is emitted with the same impact parameter at z = 100M, but, due to the fact that the space time is not asymptotically flat, they reach the vicinity of the central object visible in the plot with slight deviations from the asymptotically flat α = 0 or β = 0 cases.

In the text

	Fig. 6. Metric components gtt (first row) and grr (second row) for the dark-matter model with Δr = 50M with varying ΔM (left plot) and for ΔM = 12M with varying Δr (right plot); α=β = 0.
In the text

	Fig. 7. Intensity I(r) as a function of radius, r, for the LR disk model (top row) and for the EH disk model (bottom row), with α=β = 0 for Δr = 50M, with ΔM={7M,14M,21M} (left column), and ΔM = 12M with ΔM={30M,60M,90M} (right column).
In the text

	Fig. 8. Shadow images for LR disk model (top row) and EH model (bottom row) with α=β = 0. The first column depicts the absence of dark matter, while the second and third columns show Δr = 50M and variations in ΔM = 7M,21M, respectively.
In the text

	Fig. 9. Intensity I(r) as function of radius, r, for LR disk model (left plot) and for EH disk model (right plot), with ΔM = 20M, Δr = 50M for α={−0.01,0,0.01}, and β = 0.
In the text

	Fig. 10. Comparing intensity I(r) for ER disk model with α = 0 and varying β, both in the presence and absence of dark matter.
In the text

	Fig. 11. Shadow images for LR disk model (top row) and EH disk model (bottom row) with ΔM = 20M and Δr = 50M, for α = 0.01 (left column), α = 0 (middle column), and α=−0.01 (right column).
In the text

	Fig. A.1. Relative errors νr (top row), νt (middle row), and νϕ (bottom row) as defined in Eq. (A.1) for α=∓0.01 and β=±10−6 (left column), α = 0 and β=±10−6, and α=±0.01 and β=±10−6. The shades represent, from darker to lighter, the regions where the errors are below 0.02%, 0.05%, 0.1%, 0.2%, 0.5%, and 1%, with the largest errors being of the order of 2%.
In the text

	Fig. B.1. Shadow images for the ISCO disk model with different combinations of α and β, namely α=−0.01 (top row), α = 0 (middle row), and α=+0.01 (bottom row); and β=−10−5 (left column), β = 0 (middle column), and β = 10−5 (right) column).
In the text

	Fig. B.2. Intensity I(r) as a function of the radius r for the ISCO disk model for a varying α with constant β (top row) and for a varying β with constant α (bottom row)
In the text

	Fig. B.3. Shadow images for the LR disk model with different combinations of α and β, namely α=−0.01 (top row), α = 0 (middle row), and α=+0.01 (bottom row); and β=−10−5 (left column), β = 0 (middle column), and β = 10−5 (right) column).
In the text

	Fig. B.4. Intensity I(r) as a function of the radius r for the LR disk model for a varying α with constant β (top row) and for a varying β with constant α (bottom row)
In the text

	Fig. B.5. Shadow images for the EH disk model with different combinations of α and β, namely α=−0.01 (top row), α = 0 (middle row), and α=+0.01 (bottom row); and β=−10−5 (left column), β = 0 (middle column), and β = 10−5 (right) column).
In the text

	Fig. B.6. Intensity I(r) as a function of the radius r for the EH disk model for a varying α with constant β (top row) and for a varying β with constant α (bottom row).
In the text