Black Hole Shadow Observations with Space-Ground Interferometers

Nowadays the observation of the black hole (BH) shadows is one of the hottest problems in the astrophysics (see review [1]-[3]). The pioneer observations of the BH shadow in the innermost area of M87 by the Event Horizon Telescope (EHT)¹¹1http://eventhorizontelescope.org have just been made ([4]-[9]). The resulting images clearly demonstrate the exceptional capabilities of modern instruments. At the same time, progress in the study of black holes and their nearby surroundings requires the continuation of research, i. e. the observation of BHs in other radio sources and at other (higher) frequencies.

At present, when the BH interferometry is developing very rapidly, the terms used to designate the BH images are: “shadow”, “silhouette”, and “photon ring” (the last term was first used in [10]). Sometimes their meaning is shared, but often mixed. So, in [11] the term “silhouette” is used, which is understood as the image of the BH event horizon, in [12] the term “shadow” is used, which refers to the image of the so-called photon sphere ($r=3GM$ for a non-spinning BH). In English-language literature, terms “shadow”and “silhouette”are often confused. The authors follow this practice and name the shadow of BH its image for the specific model, without any detailing what the border of the shadow is. The image of the black hole depends both on its mass and rotation, and on the properties of the source that illuminates the black hole (disk, jet, bright spot, etc.). Some aspects of the building of the BH shadow in the General Relativity (GR) and extended theories of gravity have been discussed in [13] (see also references therein), as well as in [14]-[18].

However, regardless of the definition, the BH shadows have tiny angular sizes even for the closest BHs. This means that an VLBI technique should play a key role in shadow observations. Important parameters are also the magnitude of the baseline projection and the frequency at which the receiving equipment operates. The ground-based interferometers are limited by the Earth diameter and their baselines cannot exceed 12800 km. However, the great advantage of this construction is that we can relatively easy control it, repair, develop and collect the big data arrays. The ground-based VLBI array is realized in EHT array, which has been carried out the first observations of the BH shadow in the center of M87 at 230 GHz.

An angular resolution of interferometer can be improved by increasing the baseline or by drifting to higher frequency. So, it would be a more productive idea to build the array of space-based radiotelescopes, which could form an interferometer with huge baselines ([19]-[21]). A more detailed consideration of possible satellite orbits can be found in [22]-[24]. The angular resolution of that interferometer might thousand times exceed the one for the ground-based devices. But, being realized, the mission appears to be very expensive and its technical support can hardly be realized as easy as it is executed for the ground-based antennas. In addition to the technical and financial problems there are a lot of difficulties in the filling of $(u,v)$ plane. In ground-based observations it is possible to achieve a relatively homogeneous coverage of the $(u,v)$ plane because the overall configuration of these tracks is almost the same every 24 hours, whereas in space-ground observations, especially with large projections of the base, the coverage is expected to be rather heterogeneous. It leads to the additional difficulties in imaging procedure.

Nevertheless, the idea of space-ground interferometer is very tempting and now it is under heat consideration and discussion (see [22]-[24]). It can be realized in future space missions. One of them is the Millimetron space mission [25] with cooled 10-meter mirror operating in millimeter-  and submillimeter bands, which is planned to be launched in late 2020s. The angular resolution of this instrument in the interferometric mode is assumed to be so high that we can clearly observe, in principle, the shadows of the BHs in many galaxies. However, there are reasons that worsen this ideal picture. For example, there are limitations on the sensitivity of the instrument, the radiation scattering by plasma inhomogeneities may occur, etc. In addition, all these effects depend drastically on the frequency. In the paper, we focus on the fundamental possibility of observing the shadow of a BH and neglect the nuances associated with the characteristics of specific observational instruments, as well as specific astronomical objects. The preliminary catalog of supermassive BHs can be found in [26], where the observational possibilities of Millimetron mission were taken into account. The catalog is based on the extended catalog [27].

The interpretation of the interferometric observations of BH shadows requires the simulation of the expected image. This problem, in turn, requires a lot of effort to develop the radiation source models. As it is shown in many papers ([2], [28]-[36]) the BH image depends significantly on the BH surrounding, i. e. on the structure of the accretion disk, the dependence of its temperature on the radial coordinate, on the existence of relativistic plasma jet, etc. It is also necessary to take into account the radiation of the corona, the geometry of the magnetic field, the presence of synchrotron radiation, and more. Except that, the image depends also on the interstellar scattering processes ([37],[38]). In our simulation we do not take into account all these effects and use the simplest model of the shadow image, which depends only on the mass and spin of a BH and the source of photons - the bright plane behind the BH. Nevertheless, it allows us to reveal the characteristic features of the results of ground-based observations and the observations in space-ground interferometer. We do not consider an image of the real source, SgrA^∗ (or M87^∗, M31^∗), but a model of a BH shadow with the same angular size.

The recent observation of the BH shadow was provided by the ground-based interferometer. This means that we need to study the prospects for future research of BHs. The most evident way to do it is to join the experiences of EHT and RadioAstron mission [39] in future space experiments. In the paper we compare the images of the BH shadow which can be obtained by the interferometers with different configurations.

Main goal of the paper is to discuss the preferred satellite orbit, which should allow to obtain the shadow of a BH with high resolution quickly. Such a high-quality image is able to deliver the important information about the physical processes in the very vicinity of supermassive BHs (structure of the inner disk and base of a plasma jet), inhabiting the innermost parts of massive galaxies. This also can be applied to test the General Relativity in strong gravitational fields. We do not discuss the technical problems of interferometry and leave their solution to VLBI specialists.

We consider the spinning BH and its shadow, or silhouette, with a simple geometry of photons source. We assume that a BH is described by the Kerr metric and its spin is close to maximal, $a=0.9981$ (dimensionless parameter describing the ratio of the angular momentum of the BH to its mass). The spin axis is perpendicular to the view line of the distant observer (see fig. 1). Behind the BH and far away there is a bright flat screen, which emits the quanta uniformly to a hemisphere (in solid angle $2\pi$). If the screen plane is perpendicular to the view line of the distant observer then the BH silhouette looks like the one shown in fig. 2²²2If the BH was irradiated from a solid angle $4\pi$, the shadow image would look different: a wide dark ring between narrow photonic arcs and a contrasting edge (the red ring in the color system fig. 2) would be light.. The similar image might appear if a BH of the stellar mass orbits a red giant star and passes in front of the star.

To build the photon trajectories we solved the equations of motion under the General Relativity assumptions for each quantum. The system of six ordinary differential equations can be found in [40, 41]. Ordinary differential equation solvers are included in many packages and freely distributed in Internet ([42, 43]³³3https://www.mcs.anl.gov/petsc/). The simulated image counts the trajectories of approximately 10 million quanta.

This image has got a number of characteristic details: it is asymmetric, its brightness is inhomogeneous and, finally, it includes the thin annuli inside, formed by the quanta, which came to the observer after a few revolutions around the BH. The left hand side of the annuli is actually presents, but cannot be adequately displayed because the width of all the annuli is much smaller than the pixel diameter. On the right hand side the annuli can be seen separately and their total width is only 17-18 times less than the shadow diameter. Thus we consider the image of a BH shadow, which has enough small size details to elaborate and discuss the data processing technique. Certainly, such small and dim details will be highly likely blurred by the scattering processes, but this fact will be considered later (see Conclusion).

We presume that the spin axis of the BH in the center of the Milky Way is perpendicular to the galactic plane and coincides with the axis of the Milky Way rotation. It means, that we know its orientation on the celestial sphere. As to M87 and M31, their spin axes and the spin axes of their BHs keep some uncertainty yet, so we accepted that their spins are oriented along the declination axis. Finally, we assume that our sources are time-independent.

To reconstruct the images we used the well-established clean procedure ([44, 45]). This algorithm is widely used in astrophysics and allows to extract the image from the Fourier coefficients on a finite number of $(u,v)$ plane points (which implies a smoothing procedure). Mathematically, we deal with an incorrect problem because a coverage of $(u,v)$ plane is incomplete. As it has been shown, for example, in [7], the BH shadow images have some deviations between different image recovery methods and their different implementations. Nevertheless, as it has been demonstrated there, the morphology of images remains unchanged. In the paper, we also focus on the morphology of the image and do not consider the features that may be associated with the use of a specific procedure of image recovery or with the source model. The clean method does give a general idea of the shadow image.

We found that the interferometer, which includes both the ground-based telescopes and a low-orbit satellite has the advantage in comparison with other reviewed cases. The key role here plays an ability to fill the $(u,v)$ plane with high density during the relatively short time (a day or week). And a low-orbit interferometer is able to successfully solve this problem because the good coverage of the $(u,v)$ plane takes here less than a week. Moreover, if the orbit lays at about 300-400 km above the Earth the revolution lasts 1.5 hour and after 5-7 cycles the coverage of the $(u,v)$ plane becomes dense enough. At that case the observation procedure may last only about 10 hours and, in principle, we can get the BH image with satisfactory quality and resolution twice a day. A satellite in the orbit with $R\sim 2R_{\oplus}$ has about three times longer orbital period, so a similar coverage of the $(u,v)$ plane will be achieved in about a day (see fig. 4). The simultaneous use of two or three satellites can further reduce this time.

The ground-based arrays have an enough $(u,v)$ coverage, but the most objects cannot be observed all day long because they are below the horizon during some period. At those periods the coverage of $(u,v)$ plane is, certainly, stopped and we have the traces which look like a half of ellipse (see fig. 3). The great advantage of ground-based instruments is, obviously, their low cost.

The longer is a baseline of the interferometer the more difficult is the correlation process. In particular, this is due to the difficulty in synchronizing of ground and on-board timers. It means that the space-ground interferometer should have much more sophisticated equipment than a ground instrument.

A case of a high-orbit satellite differs from others. Indeed, its angular resolution is incredibly high. Theoretically, it is so high that we would be able to see the annuli inside the shadow area in fig. 2 separately, i. e. the resolution could be even greater than it is reproduced in the Figure. However, we face here a problem with a poor $(u,v)$ plane coverage. The fact is that the information obtained in the interferometric observation is not the image of the object itself. It is a Fourier Transform of the true image at limited number points of the $(u,v)$ plane. The reliable image reconstruction is possible only with a dense enough coverage of $(u,v)$ plane. One turn around the Earth of the satellite in Lagrangian point $L_{2}$ takes a full year. And even an increase in the duration of observations will not have a radical effect on the coverage of the $(u,v)$ plane. A spacecraft moving in the orbit near point $L_{2}$ every year almost repeats its track on the $(u,v)$ plane. The displacement of the spacecraft up and down from the ecliptic plane only increases slightly the coverage of the $(u,v)$ plane.

Another interesting idea is to place the radiotelescope at the Lunar pole [21]. The orbital period of the natural Earth satellite lasts about a month, so one third or a half of a year (a few turns around) is enough to cover the $(u,v)$ plane. In any case, this is much less than that is required for the satellite located at the Lagrange point $L_{2}$. However, for the implementation of such a colossal project there are still many technical problems to be solved.

At the present only two successful space VLBI missions have been implemented: the VLBI Space Observatory Programme (VSOP) [46] and RadioAstron [39]. Both were equipped by the receivers in centimeter wave bands, so it means that we have not enough experience yet.

The long-awaited detections of the BH shadow by EHT is the first step to test the General Relativity in strong gravitational fields. However, one has found that its quantitative features are not sufficient to distinguish between BHs using different theories of gravity. It is highlighting the fact that the great caution is needed when interpreting the BH images as the tests of General Relativity [47]. Since the BH shadow can be measured more precisely by the space-ground interferometer than by the ground-based one, the space-ground VLBI mission does allow to carry out the stronger tests of the General Relativity and the accretion models.

This activity has been partly supported by Russian Foundation for Basic Research, grant 19-02-00199 and the RAS project KP 19-270 “Questions of the origin and evolution of the Universe using methods of ground observations and space research”.

The authors are grateful to Dr. A. Andreanov for his kind help in preparation of the BH shadow images and numerous useful discussion. We are grateful to Dr. V.N. Strokov Dr. I.N. Pashchenko for their help with the manuscript preparation. We also grateful to Dr. V.I. Kostenko and Dr. S.F. Likhachev for useful discussions. One author (S.R.) is very grateful to Dr. O.N. Sumenkova, Dr. R.E. Beresneva and Dr. O.A. Kosareva for the possibility of fruitful working on this problem.