The Optical Appearance of Compact Stars: Shadows and Luminous Rings

Abstract

In 2019, the direct imaging of M87* [1] by the Event Horizon Telescope opened the door to understand the nature of the central object and the underlying theory of gravity, since it explore the regions where the gravitational field is extremely strong. In the images obtained by the collaboration, we can observe two distinct regions: a dark circular center called shadow and an enveloping luminous ring produced by the hot accretion disk surrounding the astrophysical object. In this chapter, we want to explain the basic tools to analyse the optical appearance of a compact object. Starting by the light bending near a massive body and the expected detected images when a star illuminates a black hole. Finally, we assume different models of accretions disks as well as the technique to obtain the similar images as the one obtained by the Einstein telescope.

1 Introduction

The optical appearance of a black hole or any other ultra-compact object (UCO) is the image one would expect to obtain after processing the data measured by several telescopes around the world. This technique for acquiring such an image is called imaging and can be done using different methodology [2, 6, 24, 29]. Indeed this is currently done by the Event Horizon Telescope (EHT) collaboration, whose first result of M87$^*$ on April 10, 2019 [1, 2] was a game changer in this field. This collaboration uses radio telescopes spread in few groups around the globe to collect the data. Since every measurement needs to be done by a pair of telescopes, which is linked to the distance between them, there are many gaps on the resulting image as a consequence of the small number of telescopes. The missing data is filled by an algorithm that generates possible ‘realistic’ solutions. The possible resulting images that can fit reasonable the small collected data are huge. Nevertheless, by increasing the number of telescopes, the resolution is going to be better since there are going to have more image ‘covered with data’ and not by the algorithm.

In the corresponding images obtained by the EHT collaboration, which are similar to Fig. 1, we can identify two distinct regions: a nearly circular dark center called shadow and a wide surrounding luminous ring produced by the extremely hot accretion disk surrounding the astrophysical object [13].

Fig. 1

Representation of a shadow and a wide luminous ring using the AI Dall-e

These types of observations are extraordinarily important, since they explore the most extreme conditions of matter and spacetime. In fact, they detect photons that have traveled very close to the horizon of a black hole, where the gravitational field is extremely strong. It is therefore quite possible that these measurements will highlight any small discrepancies between theories, since gravitational effects are more prominent there. Not only black hole images, but also other observations of strong fields, such as gravitational wave signals, are essential for understanding gravity. For example, both allow testing the Kerr hypothesis, according to which the only physically acceptable solution for a rotating, uncharged black hole in General Relativity (GR) is the Kerr geometry. Therefore, if any signal is detected coming from a compact rotating and uncharged object inexplicable by Kerr phenomenology, this would be the signature of the New Physics. This New Physics may come from considering exotic matter (violating energy conditions) or theories beyond GR.

Now that we know why it is relevant to study the optical appearance of UCOs, let us explain the phenomenon. First, black holes have a so-called event horizon, for which if something enters in the region below it, they will never be able to escape, even photons. Therefore, these objects cannot emit light like other astrophysical objects such as stars or any kind of dwarfs. The only way we can directly observe such an object is because it is illuminated by one of the following luminous sources,

• A distant point-like source, for example, a star that is loosely attached to the compact object.

• An accretion disk emitting around it. Such a possibility could occur in a tighter binary system or in a dense center of a galaxy.

Although both are likely to reveal the presence of a black hole, the expected brightness of the ring provided by both sources will not be the same, since the number of photons deflected by the gravitational field differs due to the proximity of the light sources and the emitting surface. The first case is simpler to analyze, since we expect the light rays to concentrate on the critical curves. However, the brightness of the corresponding luminous rings may be fainter than those coming from a surrounding accretion disk. This later scenario is actually the case we have already observed, but it involves much more arduous machinery to model and study the corresponding luminous rings and shadow.

Fig. 2

3D plot of the light bending phenomenon in a spacetime created by a static, chargless (Schwarzschild) black hole in GR

We shall also understand the trajectories followed by the observed photons. These trajectories deviate as they approach a massive body (see Fig. 2 for descriptive image), giving rise to curved trajectories called geodesics. This phenomenon was predicted by GR, and was used as a test of this theory. Sir Arthur Eddington first observed this effect during a solar eclipse in 1919.¹ He measured the displacement between the apparent and real positions of some stars as a consequence of the light deflection produced by the Sun’s gravitational field. This feature is also the main responsible for the optical appearance, since the photons emitted or traveling near the object are deflected as a consequence of the strong gravitational field. Therefore, it is necessary to analyze the geodesics to know what the image of the object will look like.

The theories of gravity predict the angular size and the shape of the shadow, which depends on the geometry of the spacetime. For example, in GR, the shadows are almost circular, but its size and shape depend on the mass and not so strongly on the spin of the Kerr black hole. However, these and other features in the image are sensible to astrophysical properties of the plasma near the black hole as it is going to be shown in Sect. 4.2.

2 Geodesics in GR

Geodesics are those curved trajectories described by the functions $x^\mu =x^\mu (\lambda )$, with $\lambda$ being the affine parameter, whose tangent vector, $t^\mu = dx^\mu /d\lambda$, is invariant under parallel transport (autoparallels) defined by the connection of a metric. Thus, they are described by the following equations

$$\begin{aligned} t^\mu \nabla _\mu t^\nu =\ddot{x}^\alpha +\Gamma ^\mu _{\alpha \beta }\dot{x}^\alpha \dot{x}^\beta =0 \ , \end{aligned}$$

(1)

where $\Gamma ^\mu _{\alpha \beta }$ are the Christoffel symbols of the metric, $g_{\mu \nu }$, and dots denote derivatives with respect to an affine parameter, $\lambda$. As one can already imagine, solving the above equation to find the geodesics can be quite demanding as one has to calculate all the components of the connection. Thankfully, such an equation can also be obtained through the variational procedure by considering the Lagrangian,

$$\begin{aligned} \mathcal {L}=\dfrac{1}{2} g_{\mu \nu }\dot{x}^{\mu }\dot{x}^{\nu } \ , \end{aligned}$$

(2)

where $g_{\mu \nu }$ is a general metric solution of the field equations. Substituting the above Lagrangian into the Euler-Lagrange equations, one can explicitly check that it leads to Eq. (1). For any spherically symmetric spacetime with line element defined as

$$\begin{aligned} ds^2=-A(r) dt^2+B(r) dr^2+ C(r)(d\theta ^2 + \sin ^2\theta d\varphi ^2) \ , \end{aligned}$$

(3)

one can assume that the motion takes place in the plane $\theta =\pi /2$ without loss of generality, because of the symmetry of the geometry we can always redefine the coordinates so that the geodesics happen in such a plane. Then, the geodesic equations become

$$\begin{aligned} A(r) \, \dot{t} = & {} E \ , \end{aligned}$$

(4)

$$\begin{aligned} 2 B(r) \ddot{r}+B'(r) \dot{r}^2+A'(r)\dot{t}^2-C'(r) \dot{\phi }^2 = & {} 0 \ , \end{aligned}$$

(5)

$$\begin{aligned} C(r) \, \dot{\phi } = & {} L \ . \end{aligned}$$

(6)

Since the Lagrangian does not depend on the coordinates t and $\phi$, the first and third equations are constants of motion, also known as Killing symmetries, where E is the total energy and L the angular momentum of a particle per unit of mass. Indeed, if we compute the Hamiltonian, $H=p\dot{q}-\mathcal {L}$, where $\dot{q}$ are the derivatives of the coordinates with respect to the affine parameter,

$$\begin{aligned} 2 H= -\dfrac{E^2}{A(r)}+ \dot{r}^2\, B(r) + \dfrac{L^2}{C(r)} \ . \end{aligned}$$

(7)

We can also check that it is a constant of motion as it does not explicitly depend on the affine parameter. Actually, we can redefine $2H=k$, where $k=-1, 0$ for timelike and null observers, respectively. For timelike observers we mean those that travel slower than light, linked with massive particles. On the contrary, null observers are those that travel at the speed of light or that are massless. With the redefinition of the Hamiltonian, we can rewrite the above equation as

$$\begin{aligned} AB \dot{r}^2 = E^2 -V(r) \ , \end{aligned}$$

(8)

where we have introduced the effective potential

$$\begin{aligned} V(r) = A(r) \left( -k + \dfrac{L^2}{C(r)}\right) \ . \end{aligned}$$

(9)

Remember that geodesics are related with the motion of particles on a curved spacetime. In the following subsections, we are going to turn our attention to the main important aspects of null and timelike geodesics for the study of the optical appearance.

2.1 Null Geodesics and Gravitational Lensing

As we mentioned in the introduction, particles’ trajectories are bent as a consequence of an ample curvature of spacetime generated by a massive body. This is similar to what happens to the light when passes through an optical lens, however in this case the lens is a gravitational source and therefore such a process is called gravitational lensing. Here, we want to explain the mathematical framework that describes such an effect by using the geodesic equation for null (massless) observers.

Consider a light ray starting from spatial infinity and approaching to a gravitational lens. As the photon gets sufficiently close to the gravitational source, due to the spacetime geometry, it begins to deviate from their initial direction until they get to the closest radius and subsequently turn back to spatial infinity again.

To comprehend this effect, we should rewrite Eq. (8) in terms of the impact parameter² defined as $b=L/E$,

$$\begin{aligned} \dfrac{A(r)B(r)}{L^2}\left( \dfrac{dr}{d\lambda }\right) ^2= \dfrac{1}{b^2}-V_{eff}(r) \ge 0 \ , \end{aligned}$$

(10)

where now the effective potential is

$$\begin{aligned} V_{eff}(r)=\dfrac{A(r)}{C(r)} \ . \end{aligned}$$

(11)

Equation (10) describes a one-dimensional trajectory of a photon with impact parameter b governed by a potential $V_{eff}$. This equation gives us an idea of how close to the object we are and helps us to classify the trajectories depending on the number of turns around the center. In order to understand this issue check Fig. 3, where we have depicted a potential for a static, chargeless black hole (blue curve) and a trajectory of a photon (black line) with an impact parameter $b=6.5$. Note that a light ray only propagates in those regions fulfilling $1/b^2 \ge V_{eff}$. Additionally, there is a particular radius where both functions intersect. This is the turning point or the radius of closest approach, $r_0$, as $dr/d\lambda =0$ there. After reaching this point, the photon is going to go from reducing the distance with the black hole to grow it again.

Fig. 3

Schwarzschild effective potential (for a unitary mass, $M=1$), a photon trajectory with impact parameter $b=6.5$ and $b_c$ depicted in blue black and red, respectively. With this plot, it is clear why Eq. (10) hints on how close a photon can be from the object’s center and which the radius of closest approach (given by the intersection between the blue and black lines) depending on the impact parameter

Going back to the mathematics, one can obtain the impact parameter value for a particular $r_0$ as

$$\begin{aligned} b=V_{eff}^{-1/2} (r_0)= \sqrt{\dfrac{C_0}{A_0}}\ , \end{aligned}$$

(12)

where the subscript $_0$ means evaluated at $r_0$. If the effective potential has a maximum, there is a radius of closest approach corresponding to the unstable photon orbit or photon sphere radius, $r_{ps}$ and the impact parameter leading to such curve is

$$\begin{aligned} b_c= \sqrt{\dfrac{C_{ps}}{A_{ps}}}\ , \end{aligned}$$

(13)

called critical impact parameter. Since this corresponds to a maximum of the potential, it effectively splits the space of light rays issued from the observer’s screen into two classes: those with $b>b_c$ are deflected at $r_0$ back towards asymptotic infinity, while those with $b<b_c$ will inspiral down towards the center of the object (thus meeting the event horizon in a black hole case). Since this orbit is unstable, any small perturbation will make the photon to eventually fall into the black hole horizon or escape to the asymptotic infinity. Thus, a photon with an impact parameter arbitrarily close to $b \gtrsim b_c$ will turn a large number of times around the compact object. To calculate $r_{ps}$, we should find the maximum of the potential defined in Eq. (11)

$$\begin{aligned} V'_{eff}(r_{ps})= -\dfrac{A(r)}{C(r)} D(r)\biggr |_{r=r_{ps}} = 0 \quad \text {with} \quad D(r)= \left( \dfrac{C'(r)}{C(r)}-\dfrac{A'(r)}{A(r)}\right) \ , \end{aligned}$$

(14)

where primes denote derivative respect to radial coordinate. Thus, a photon sphere exists (critical curve) if $D(r) = 0$. Since the maxima of the effective potential are the responsible of the photon spheres, they are a useful tool to study the black hole shadow. However, when we analyse the non-spherical static case, the concept of the photon sphere is generalized [8, 30].

If one wants to analyze the optical appearance of a compact object illuminated by the light rays passing close by, one has to suitably rewrite the geodesic Eq. (10) in order to be able to calculate the deflection angle. Thus, the equation must be expressed in terms of the variation of the azimuth angle $\phi$ with respect to the radial coordinate. Using Eq. (6), we obtain

$$\begin{aligned} \dfrac{A(r)B(r)}{C(r)^2} \left( \dfrac{dr}{d\phi }\right) ^2= \dfrac{1}{b^2}-V_{eff}(r) \ . \end{aligned}$$

(15)

The above equation is the one we are going to use when we want to get the optical appearance of a UCO.

2.2 Timelike Geodesics

Remember that timelike geodesics describe the motion of a massive particles. This will play a role when we want to model the accretion disk composed by plasma, an ionized gas formed by ions and free electrons, i.e. massive particles. Recall once again the geodesic Eq. (8), which in this case reduces to

$$\begin{aligned} AB \left( \dfrac{dr}{d\lambda }\right) ^2=E^2-A\left( 1+\dfrac{L^2}{C}\right) \ , \end{aligned}$$

(16)

where the last term of the equation is the effective potential for timelike observers. In this case, the potential usually has a minimum, that is

$$\begin{aligned} \dfrac{dV}{dr}= -\dfrac{A}{C} D(r)-A' =0 \ . \end{aligned}$$

(17)

Such a position is known as the Inermost Stable Circular Orbit (ISCO). Conversely to the photon sphere, the fact that this radius corresponds to a minimum of the potential instead of a maximum translates to a stable orbit for timelike observers. As a consequence, one would expect that the inner edge of the orbit is placed here, since it is stable. However, for supermassive black holes this is not the case, and typically the inner edge is going to be even closer to the event horizon.

Let us now apply this knowledge to the optical appearance.

3 Illumination from a Loosely Bounded Star

Remember that the shadow of an UCO can be observed by two different means; the first one is by a distant orbiting star while the second is by an accretion disk. We begin by the former case, which is simpler and sets the ground of the main phenomenology happening in this framework. In this situation, we assume the star to be a punctual isotropic light emitter.

Figure 4 represent the system star-object with the main photon trajectories coming out of the star that are able to reach us. Another assumption made here is that the UCO corresponds to a Schwarzschild black hole with unitary mass ($M=1$) placed at the origin of coordinates. The star is located at $r=-5$ in two different configurations with respect to the central object and we have also considered our observatory far away on the right hand side of the plot. The light rays depicted in both figures are divided into three different colors: blue, red and green. The first one is the typical example of gravitational lensing since its trajectory is slightly deviated, while the second and third colors can barely be distinguishable as the initial and final parts of the trajectories overlap. The red one only does one turn around the massive body, whereas the green does two. Thus, we can note that, as the radius of the turning point reduces, the deflection angle increases until reaching the critical distance in which light is not able to escape from the object. At such a distance, the trajectory yields a circular orbit around the center depicted in the figures as the yellow dashed circle. This orbit is the so-called photon sphere which is circular because its radius corresponds to the maximum of the potential and, consequently, it will not change if there is no perturbation. If the radius keep reducing, then the light ray will be dragged into the center of the object.³

Fig. 4

Representation of the Schwarzschild black hole-star system and photon trajectories in two different configurations: in a the star his ‘hidden’ behind the black hole while in b the star is above. We assume that we are at asymptotic infinity to the right of each plot. The black disk symbolizes the interior of the event horizon, the dashed yellow circle is the photon sphere, while the purple dashed line is the value of the critical impact parameter for Schwarzschild. The red, green and blue curves correspond to the trajectories of the photons that completely turn around the center one, two and zero times, respectively

After these loops, photons depart from the object recovering its almost straight paths when they are far away, since the gravitational field of the black hole decreases with distance. Thus, when they reach us, their trajectories are practically parallel to each other (see the above figures). Furthermore, we can see that the more turns the light does around the black hole, the closer they get when they leave. Indeed, the distance between them gets exponentially small on each turn, so after some loops, they are going to essentially lead to the critical impact parameter. Let us pay attention to Fig. 4b. This image corresponds to a case where the system is not exactly aligned, i.e. the star, black hole and us do not form a straight line. Normally when the system is slightly unaligned, what we would see is a ‘cross’: four identical images of the star. This can be seen in the Universe, although the gravitational lens is a galaxy instead of a black hole (look for Einstein’s cross, for example). For the extreme case of Fig. 4b, what we would expect to observe is only two identical reproductions of the source at opposite sites, corresponding to the light rays reaching us from above the abscissa axis and from below.

Fig. 5

Representation of the shadow and the luminous ring surrounding it for the configuration seen in Fig. 4a

On the contrary, for the case depicted in Fig. 4a, where the star is completely aligned with the object and us, the image is going to be completely different. In this case, if we perform a revolution around the horizontal axis, we will get the 3D version of the trajectories. Once we have this picture in mind, let’s go one step further. Imagine yourself located at the end of the horizontal axis; what you would see is shown in Fig. 5, where we can see a luminous ring surrounding a black circular area. This thick yellowish ring represents the regions where a large number of photons accumulate and is the one that an observer is expected to see. The radius of these luminous rings is given by the critical impact parameter since the observed photons will reach us with a certain impact parameter and, therefore, the radius of the luminous ring is given by this parameter. This means that the radius of the shadow, the central black region, in this case is given by the critical impact parameter and not by the event horizon. However, if the star would have been placed inside the photon sphere, then a smaller shadow would appear, since photons emitted in that region can still arrive to us if they are emitter as ‘direct’ as possible, that is without turning around the object. Therefore, even though the photon sphere and the event horizon given by the geometry of the spacetime play a main role on determining the size of the shadow, the position of the photon’s emission is also a meaningful factor on the optical appearance.

4 Illumination from an Accretion Disk

Let us now move to the second case of illumination: the one produced by an accretion disk, an extremely hot disk formed by plasma. The first question that may arise is: how is it formed? At the beginning, the black hole is surrounded by a gas that orbits around the object far from the event horizon. Contrary to the common idea we have of a black hole, they can have other bodies rotating at a sufficient distance in the same way as the planets of our Solar System do around the Sun, instead of being a huge astrophysical object that swallows everything that comes near it. Therefore, the surrounding gas would be orbiting around it without falling into it. The reason why we all have the image of a black hole attracting and eating everything is because there is something else besides gravity that makes the gas approach the black hole: friction.

Friction heats up the surrounding gas, which means that the gravitational energy of the system has to be transformed to thermal, causing the gas to fall into the black hole. As a consequence, the gas becomes a hot disk around the black hole. The hotter the accreting material becomes, the more energetic light it emits, as happens, for example, with stars or incandescent light bulbs.

In this scenario, we have to face a more challenging situation. On the one hand, the accretion disk can certainly be well inside the photon sphere and its shape allows to emit all along its surface instead of a point-like seen before, requiring a precise knowledge of the trajectories of light in spacetime. On the other hand, the received luminosity depends strongly on the emission profile of the disk. This demands a realistic model of the disk to establish the luminosity profile, which is obtained by simulations combining GR and Magnetohydrodynamics. In this section, we intend to explain in detail both needed ingredients, the geodesic structure analysis as well as the modelling of the disk itself, and the assumptions we use to simplify the study of the problem.

4.1 Ray Tracing

The first step is to get under control the trajectories followed by photons passing near the compact object. Before getting into the mathematics, let’s start by explaining the fundamental idea of the ray tracing. Its main objective is to classify light rays according to the number of turns they make around the object. This information is necessary because, after all, when an accretion disk is added, the light rays will intersect with it at most twice per turn around the object. Depending on the characteristics of the accretion disk, these could mean additional luminous ‘enhancements’.

Moreover, as we mentioned before, the optical appearance of a compact object is closely related to the impact parameter. This is so because we are so distant of the gravitational source that the light rays come almost parallel among themselves and with a different impact parameters, as we saw in Fig. 4. Using this fact, we assume that each pixel of the image corresponds to a wave detector that received a light with an impact parameter b. The responsible geodesic obtained from Eq. (15) is traced backwards towards the black hole, ending either close to the horizon or when the geodesic escapes again to a large distance from the black hole. With this, we know the total number of revolutions around the compact object. This procedure is known as ray tracing.

To define how we count the number of turns, we take into account the configuration of the object-observer system introduced in the previous section. Recall that we place the observer at asymptotic infinity on the right-hand side, so a photon that was not deflected at all by the compact object would have turned $n=1/2$ times, i.e., it would have gone directly from the left-hand side to the right-hand side of the plot. In fact, the total number of orbits made by a single light ray is the (normalized) change of the azimuth angle, $n(b) \equiv \tfrac{\phi }{2}$. Consequently, the number of intersections with the equatorial plane of a given line is [2n]. Finally, recall that as we approach the critical impact parameter, $b \gtrsim b_c$, a light ray will have a longer trajectory around the neighborhood of a black hole until it is formally there forever at the critical value (or until a perturbation causes it to fall into or out of the object). The number of orbits will obviously depend on how close the impact parameter is to the critical one.

Under the above conditions, typical relevant contributions to the total luminosity on the observer’s screen will be given by three types of trajectories indexed by an integer m which counts the number of intersections of a particular light ray with the vertical axis, i.e.

$$\begin{aligned} \frac{m}{4}-\frac{1}{4} \le n < \frac{m}{2}+\frac{1}{4} \ , \end{aligned}$$

(18)

except for the first case, $m=1$, for which the lower limit corresponds to $n=1/2$ and remember that n is the normalized change of the azimuth angle. We will use this number to classify the different types of emission⁴:

• Direct ($m=1$): represents trajectories that intersect just $m=1$ times the vertical axis ($1/2 \le n < 3/4$), meaning that the light rays emitted by the disk go directly to the observer. This is the dominant contribution to the optical appearance of the object, in terms of luminosity and width of the associated radiation ring. However, it essentially reproduces the characteristics of the accretion disk rather than those of the background geometry and its critical curve.

• Lensed ($m=2$): corresponds to the light rays crossing the equatorial plane for a second time, and it is defined by $3/4<n \le 5/4$, being the subdominant contribution to the luminosity.

• Photon ring ($m=3$): composed by light rays intersecting the equatorial plane at least three times, and is defined by $n>5/4$ .

• Higher order $(m>3)$: typically contribute negligibly to the total luminosity (see [15] for a general discussion), as a consequence of the reduction of their impact parameter range. For this reason, they are usually integrated into the photon ring emission. These modes are much more sensitive to the characteristics of the background geometry than the rest of the emission.

The fact that higher order emissions can be neglected was already depicted in Fig. 4, where the blue curve corresponds to the direct emission, the red to the photon ring (has three cuts with the vertical axis) and the green to a high order emission of $m=5$, where these two later contributions are indistinguishable when they reach the observer.

Note that a light ray with lensed emission contains the direct emission and the photon ring contains both. As mentioned before, the contribution of higher order emissions is negligible. Indeed, their contribution to the total luminosity can be dismissed. Indeed, its contribution exponentially decrease as they approach the critical curve [3] such that beyond $m=3$ all additional emissions are typically accumulated in the $m=3$ mode and thus giving the position of the critical curve [11, 15]. Although we have said that higher order emission is usually omitted, the shape of the effective potential plays an important role in the contribution of these lower order trajectories, for example the geometry studied in [16] is richer compared to the Schwarzschild case, allowing higher-order emission to contribute significantly to the total luminosity.

Last but not least, it should be noted that for impact parameters $b<b_c$ the light ray will also perform a series of half turns. These light beams will be emitted near the central region of the object within the photon sphere. However, when calculating their inner ray-traced trajectories we will see two different cases: if the object has an event horizon like a black hole, the last trajectory we can detect is emitted very close to the event horizon, while for those that they do not have, such trajectories continue their path to the center of the solution. For the case of a black hole, there are light rays that do not intersect the equatorial plane because they cross the event horizon before without encountering the accretion disk on their trajectory. These trajectories form the inner shadow [7], $b=b_{is}$, and defines the brightness depression of a black hole independently of the emission properties of the geometrically thin accretion disk, since we will never detect those photons. However, such an inner shadow may be missing for a compact object without a horizon.

4.1.1 Schwarzschild Black Hole

Let us start by considering a static, spherically symmetric solution: the Schwarzschild black hole. Even though we know black holes are rotating objects, this assumption turns out to be a good approximation since the size and shape of the shadow, as seen by an asymptotic observer, depend very weekly on the spin of the black hole in combination with the inclination with respect to the line of sight, with deviations from circularity lying within $\sim 7\%$ for ultra-fast spinning black holes [31].

As we have already seen, for this case, one needs to calculate first the location of the horizon, since depending on the position of the horizon, the impact parameters that belong to the inner shadow are going to be different,

$$\begin{aligned} r_h= 2M \ , \end{aligned}$$

(19)

with M being the mass of the black hole. Apart from the position of horizon, there is another main radial distance which plays a role when analyzing the optical appearance of a compact object: the critical curve (for which we shall also reserve the word ‘photon sphere’), which for Schwarzschild is

$$\begin{aligned} r_{ps}=3M \ . \end{aligned}$$

(20)

The next step in our analysis is to integrate the geodesic equation for a bunch of light rays spanning the whole region of impact parameter values. The corresponding trajectories can be therefore classified according to the number of (half-)orbits around the solution as follows (Table 1):

Table 1 Impact parameter range for direct, lensed and photon ring emissions for Schwarzschild black hole

We have ordered them from the outermost to the innermost emission. To illustrate this general discussion, the trajectories of a bunch of photons are depicted in Fig. 6 for $b \in (0,10)$. We point out that the observer’s screen is located at the far right side of this plot in all these cases. In these figures one can see the direct (green), lensed (orange) and photon ring (red) trajectories outside the photon sphere (dashed yellow). In addition, we have plotted the photon ring (blue), lensed (purple), and direct (cyan) emission originated from inside the photon sphere, $b<b_c$, while the black trajectories correspond to the inner shadow, those light rays that do not cross at any time the vertical axis.

Fig. 6

Ray tracing of the Schwarzshild black hole of unitary mass (M=1). The observer’s screen is located in the far right side of this plot and the type of emission is defined with respect to the number of intersections with the equatorial plane (vertical line): for $b>b_c$ we have direct (green), lensed (orange) and photon ring (red) emissions reaching to a minimum distance from the photon sphere (dashed yellow circumference) before running away, while for $b<b_c$ we also have direct (cyan), lensed (purple) and photon ring (blue) emissions. The latter three trajectories intersect the BH horizon (black central circle) after crossing the photon sphere. The bunch of black curves do not intersect the equatorial plane and therefore no emission can come out on them no matter the accretion disk model, therefore corresponding to the inner shadow of the solutions

At this point, we already know which is the impact parameter range for each type of photon trajectory. Despite with Fig. 6 we can visually reason why we only classify the null geodesics into these three groups, as we can barely see the photon ring emissions (red and blue), there is a better way to understand it graphically. This is Fig. 7 which depict the transfer functions, $r_m$, in terms of the impact parameter. The transfer functions, $r_m$, account for the location of the m-th intersection between the light ray and the vertical axis (i.e., the future disk). Therefore, the information one can subtract about this plot is how demagnified the light ring will be by the slope of the transfer function; the steeper it is, the lesser the contribution. This is so because the ring’s width will be continuously shrinking since its thickness depends on the impact parameter range. Bearing this in mind, the direct emission is the largest contribution to the total luminosity by far, and the lensed and photon ring are highly diminished as we expected from the previous section.

Fig. 7

The first three transfer functions for the direct (blue), lensed (orange) and photon ring (green) emissions. $b_c$ denotes the location of the corresponding photon sphere. The slope of each curve is interpreted as the demagnification factor of the corresponding emission

This plot of the transfer is also very useful to understand how many rings and how they are going to be distributed depending on the inner edge of the accretion disk. Imagine that the inner edge is placed at $r=6M$, this means that the accretion disk emits from $r\ge 6M$. If we draw an horizontal line in the plot at such a radius,⁵ we are only going to receive those light rays above the horizontal line. Thus, for such a case, we will be able to distinguish three rings, one for each type of emission. The ‘direct’ ring would go from more or less $b=7$ to infinity, whereas the lensed would go from $b\sim [5.5,6.15]$ and the photon ring near the critical value, so the shadow is going to extend up to there. But if we move the inner edge down to $r=2M$, we can see how now all the curves intersect and, therefore, all the rings are going to be overlapped at some points. For this case, the shadow is going to be the smallest, the one we have defined as inner shadow. Now that we completely know the trajectories of the photons and what we can expect of the luminous rings and shadows, we are able to move to the modelling of the accretion disk.

4.2 Accretion Disk Model

Treating the disk can become a real ordeal very easily, as its modeling requires the use of General Relativistic Magneto-Hydrodynamic (GRMHD) simulations, requiring the models fully account for relativistic effects, matter dynamics and photon propagation. Nonetheless, let us begin with a toy model proposed in [15] that will already show the effects of the gravitational lensing and redshift of the emitted photons. Therefore, let us itemize the several considerations we have assumed for this first example:

• Placed on the equatorial plane of the object which is perpendicular to us: the image seen from the observer will be face-on.

• Optically thin: the disk does not re-absorb the photons. On each intersection with the equatorial plane the light ray will ‘pick up’ additional brightness, in the sense that when the photon crosses, a new photon can be emitted. This strongly depends on the particular assumed emission intensity profile of the disk.

• Geometrically thin: the width of the disk is negligible as compared to the radial extension of the disk, which means that most of the matter lies close to the radial plane.⁶ This property produces an infinite sequence of concentric rings from photons that have completed n half-orbits in their approach to the critical curve.

• The specific luminosity only depends on the radial coordinate, $I^{em}=I(r)$.

• Isotropic emission in the rest frame of matter: the intensity does not depend on the frequency, $\nu$, in the static frame.

• Monochromatic emission: the emissivity, $j_\nu$, depends on the frequency as $j_\nu \sim \nu ^2$.

• The intensity profile is higher close to the black hole, where the deflection of the light is strongest and the emitting plasma velocity is close to those of light.

Note that the last assumptions assume a static frame, this can be done because the disk is assumed to be face-on, so the effects between the dynamics of the disk are degenerated by the choice of the radial profile. Additionally, significant progress can also be made by using analytical models of static accretion disks with a localized emission starting from a finite-size region of the disk. Here we have assumed the disk thin, but if it is spherically symmetric, the luminous rings would converge to the critical curve itself and delimit the outer edge of the shadow [25], instead of the infinite sequence of concentric rings proper of an infinitely thin disk (or even thicker [35]).

Once we have defined the properties of emission and absorption to the disk, we can produce a model total intensity image by solving the unpractised radiative transport equation, governed by the Boltzmann equation for photons. The relativistic Boltzmann equation, which is written in terms of invariant quantities or frame-independent, reduces to

$$\begin{aligned} \frac{d}{d \lambda }\left( \frac{dI_{\nu }}{d\nu ^3}\right) =\left( \frac{j_\nu }{\nu ^2}\right) -(\nu \alpha _\nu )\left( \frac{I_\nu }{\nu ^3}\right) \ . \end{aligned}$$

(21)

where $I_{\nu }$ is the intensity for a given frequency $\nu$, $j_{\nu }$ is the emissivity, $\alpha _\nu$ the absorptivity, and quantities inside parenthesis are frame-independent. The resolution of such an equation demands precise knowledge of the fluid forming the disk (i.e. number density, angular momentum, emissivity and absorptivity).

For the purpose of simulating different stages in the temporal evolution of such an accretion disk, we are modelling such a profile by truncating the inner edge of the disk, $r_{ie}$, at different radius. Also, we assume that there the intensity actually takes its maximum value, and smoothly falls off outwards until asymptotic infinity (so that the outer edge of the disk is assumed to be infinitely far away) with a given radial decay. To simplify the analysis of this aspect, typically in the literature different decay profiles for the emission are taken ad hoc depending on how close to the innermost region of the geometry the inner edge of the disk is. Indeed, there are different possibilities to write the expressions that describe the profiles, but here we are interested in their shape:

Fig. 8

Plots of the three different normalized intensity profiles emitted by the accretion disk

• Model I: The emission starts at the ISCO for timelike observers, while vanishing in the region internal to it and falling off asymptotically to zero beyond of it, see Fig. 8

• Model II: The emission has a sharp peak at the critical curve also known as photon ring, (20), having a qualitatively similar central and asymptotic behaviour as Model I, see Fig. 8

• Model III: The emission starts right off the event horizon (in the black hole case⁷) or the innermost region for horizonless UCOs. This profile decays more smoothly to zero than in the previous two cases, see Fig. 8

Once the emission profile of the accretion disk is set, we can turn our attention to the observed intensity. If we assume that the photons emitted reach us without interacting with anything, then the observed intensity is the emitted but altered due to both gravitational redshift and the optical properties of the accretion disk. The former phenomena happens when a light ray is emitted close to a gravitational object and escapes from it, the emitted frequency is going to be affected; in particular, if the frequency of the photon in the rest frame of the plasma in the disk is given by $\nu _e$ with associated intensity $I_{\nu _e}$, then the photon frequency measured by the distant observer will be $\nu _o$ with intensity $I^{ob}$. To relate both intensities we use the assumption of a geometrically (infinitesimally) thin accretion disk, for which Eq. (21) implies that $I_{\nu }/\nu ^3$ is conserved along a photon’s trajectory. Since $I_{\nu }/\nu ^3$ is conserved along a photon’s trajectory, radiation emitted from a radius r and received at any frequency $\nu '$ has specific intensity [15]

$$\begin{aligned} I^{obs} = g^4 I_{\nu }(r) \ , \end{aligned}$$

(22)

where g is the square root of the time metric component; for spherically symmetric spacetimes this would be $g=A(r)^{1/2}$ with A defined in Eq. (3). Thus, in the spherically symmetric geometry considered in this work $I_{\nu '}^{ob}=A^{3/2}(r)I(r)$. On the contrary, the implications of an optically thin disk are less known. The raw idea is that each additional intersection of the trajectories with the accretion disk will contribute to pick up additional luminosities according to the emission profile of the disk. Therefore, the total observed intensity will be

$$\begin{aligned} I^{obs}(b)=\sum _m A^2I_{\vert _{r=r_m(b)}} \ , \end{aligned}$$

(23)

where remember that the transfer function, $r_m(b)$, contains the information about the radius of the disk where a given light ray with impact parameter b will have its m th-intersection with the disk (in the coordinate r).

4.3 The Optical Appearance of an Schwarszchild Black Hole

The first model of the accretion disk is extended up to the ISCO for timelike observers. Its emitted intensity profile is plotted in Fig. 8, and the observed intensity together with the optical appearance with its intensity legend is depicted in left panel of Fig. 9. The fact that the emission starts at the ISCO allows to clearly identify the impact parameter regions on the observed intensity corresponding to (from larger to smaller b’s) a small reproduction of the emission profile and two spikes representing the direct, lensing and photon ring emissions, respectively. This is translated into a clean view of the three kinds of light rings in the optical appearances image (after zooming in a little bit). The direct emission is largely dominating the total luminosity with a broad ring very bright at the inner edge and smoothly fading out for larger impact parameters. This ring encloses a thinner and dimmer ring (the lensed emission) and inside this latter an even thinner photon ring which is barely visible at naked eye.

Fig. 9

The observed luminosity (top) and the optical appearance (bottom) for the Schwarzschild BH with an accretion disk based on the Model I (left), Model II (middle) and Model III (right panel) depicted in Fig. 8, viewed from a face-on orientation

In Model II, depicted in the middle panel of Fig. 9, the direct, lensed, and photon ring types are overlapped in the observed intensity as a consequence of the inner edge location of the accretion disk being on the critical curve itself, which enables the direct emission via the gravitational redshift correction to pierce well inside the critical impact factor region and become the dominant contribution there, while for larger impact parameter values the combined lensed and photon ring emissions occurring roughly at the same location produce a large but narrow spike in the observed emission. Indeed, if we zoom in, we see a split between the photon ring (being fainter and closer to the direct peak) and the lensed spikes. After this luminosity boosts, the direct emission dominates again in a fainter way. The net result is that in the optical appearance the lensing and photon rings are superimposed with the direct emission. The lensing ring contribution can be appreciated in this figure, though the one of the photon ring is highly diluted and barely visible.Lastly, Model III is depicted in right panel of Fig. 9. Since the inner edge of the disk extends all the way down to the event horizon this translates into a much wider region of luminosity in the observed emission, thanks to the stretching of the direct emission to a larger distance. As a consequence, the photon ring and lensed emissions appear now as two separated but superimposed spikes with the direct emission. Another meaningful feature is the enlargement of the range of lensed emission. At the same time, this discussion is reflected in the optical appearance which shows a much wider region of luminosity with the contributions of the direct, lensed and photon ring emission. However, the second type of light rays encloses a wide ring right on the middle of it, whereas another (dimmer) one right on the inner boundary comes from the photon ring emission.

With this we have the basic tools to analyze and understand the techniques used to obtain the shadows and photon rings of a static and spherically symmetric UCO, for example [4, 16,17,18, 26, 27, 33, 34].

5 Discussion

The simplifications assumed here make the employed methods more accessible, but in order to compare their results with real data, the models must be improved. The possibilities for doing so are numerous. On the one hand, in terms of geometry, we can extend the above framework to rotating case [11, 12, 36]. In the cases of considering non-Schawzschild or Kerr geometries, one has to be careful when studies these kind of objects, since if they present an anti-photon sphere, it can produce a non-perturbative instability [5, 10, 19, 20, 23]. On the other hand, one can also improve the modeling of the accretion disk. For example, the disk can be considered to be tilted (e.g. the disk of M87* is likely to have an inclination of 17$^{\circ }$), as well as a larger profile [35]. For a more realistic model used by the EHT collaboration, see [14], for example.