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Detecting Black Hole Occultations by Stars with Space Interferometric Telescopes

We show that the occultation of Sagittarius A* by stars can be detected with space-based or space-ground very-long-baseline-interferometers (SVLBIs), with an expected event rate that is high due to relativistic precession. We compute the tell-tale signal of an occultation event, and describe methods to flag non-occultation events that can masquerade as the signal.


INTRODUCTION
The Event Horizon Telescope (EHT) has demonstrated the utility of ground based very-long-baselineinterferometers (VLBIs) in studying supermassive black hole very close to their event horizons (Event Horizon Telescope Collaboration et al. 2019a,b,c,d,e,f). The first EHT observations of the supermassive black hole at the center of the M87 galaxy (henceforth M87) were conducted at 1.3mm, with the longest baseline providing a resolution of ∼ 25µas (Event Horizon Telescope Collaboration et al. 2019b). While this is sufficient to resolve the two largest black holes in the sky, M87 and Sagittarius A* (Sgr A*), at scales comparable to their event horizons, the first EHT observations cannot observe phenomena that generate small scale spatial variations on much smaller scales.
There are two main avenues for improving the resolution of a VLBI observatory: by utilizing higher observing frequencies or through the addition of stations that provide longer baselines. On the second front, it is possible to add observing stations in space to the current EHT array, thus upgrading it by the addition of baselines that can in principle be much longer than the Earth's diameter. Another possibility is to perform observations with only stations in space, by combining multiple orbiting satellites. Besides their significantly higher resolution, space-based VLBIs (SVLBIs) have the advantage of not being affected by atmospheric effects.
The first SVLBI observations were conducted by combining elements of the Tracking and Data Relay Satellite System (TDRSS) with ground-based observatories on Earth (Levy et al. 1986(Levy et al. , 1989Linfield et al. 1989Linfield et al. , 1990. This inaugural SVLBI array included a 4.9m radio antenna on a geostationary orbit as its space-based antenna at 2.3 and 15 GHz. The first dedicated SVLBI station was the Highly Advanced Laboratory for Communications and Astronomy (HALCA) satellite as part of the VLBI Space Observatory Programme (VSOP) (Hirabayashi et al. 1998(Hirabayashi et al. , 2000. The HALCA satellite carried an 8m antenna in an elliptical orbit with an apogee at 28, 000km from the Earth's center and further demonstrated the practical possibility of groundspace SVLBIs. The most recent SVLBI array to be deployed is the RadioAstron project 1 , which included a 10m observatory onboard the satellite Spektr-R (Kardashev et al. 2013). The RadioAstron project possessed a maximum baseline of ∼ 3.3 × 10 5 km, and a resolution that is in principle higher than that of the first EHT observations. However, RadioAstron utilizes an observing frequency that is too low to pierce through the plasma surrounding M87 or Sgr A*.
Theoretical studies utilizing synthetic observations of general relativistic simulations show that few satellites on medium Earth orbits can provide resolutions enough to probe small scale spatial structure of the emission structure around Sgr A* as long as the obits of the satellites can be reconstructed accurately (Roelofs et al. 2019). Further, general relativistic simulations also show that the emission from the vicinity of a black hole include a thin 'photon ring' component consisting of subrings indexed by the number of half-orbits around the black hole that the photons underwent (Johnson et al. 2020). Detecting the interferometric signatures of individual subrings requires the resolution provided by SVL-BIs.
In this paper, we provide another motivation for an SVLBI observatory. Occultation events, where a dim object covers a portion of the supermassive black hole emission will produce a small scale variation on the black hole emission profile that could be observed by SVLBIs.
In Section 2, we discuss the population and sizes of stars close to Sgr A* and their role as possible occulters. In Section 3, we show that general relativistic precession can greatly increase the transit probabilities of objects in orbit around Sgr A*. In Section 4 we describe our model for the signal of a stellar occultation event. In Section 5, we discuss how such a signal is seen by an SVLBI and provide methods to reject false positives using only SVLBI observables. Finally, in Section 6, we provide our concluding remarks.

Population of stellar occulters
The most probable occulters are stars orbiting close to Sgr A*. In particular, the existence of the S stars, massive B-stars orbiting around Sgr A*, suggests the the possible existence of a cluster of stars orbiting very close to Sgr A* (Genzel et al. 2010;Sabha et al. 2012). While the immense tidal force of Sgr A* prohibits standard star formation mechanisms from forming these stars in situ (Morris 1993), this star cluster can be the formed in a gas disk from a disrupted molecular cloud (Genzel et al. 2010;Alexander 2017;Tsuboi et al. 2018), as well as through processes such as dynamical interactions of two stellar disks (Löckmann et al. 2008) or multiple 3body exchanges between Sgr A* and an in-falling binary (Gould & Quillen 2003), perhaps through the aid of a massive perturber (Perets et al. 2007). In this section, we will argue that the existence of this star cluster implies that might lead to an appreciable number of occulters near the tidal disruption radius at ∼ 10R S , where R S ≈ 10 12 cm is the Schwarzschild radius of Sgr A*.
If we assume the star cluster to be described by the Kroupa mass function (Kroupa 2001), where m is the stellar mass and NdM is the number of stars in the mass range m to m + dm, then we can relate the number of stars enclosed within radius r from Sgr A* of an arbitrary mass M to the number of stars with mass 10M per logarithmic mass bin, where N enc (m, < r) indicates the number of stars of mass M enclosed within a radius r from Sgr A* per logarithmic bin. The mass function of the star cluster could be more top-heavy, although simulations of stelar dynamics show that a Kroupa mass function is still consistent for a star cluster near Sgr A* (Löckmann et al. 2009). Assuming that the present moment is not a special time in the star formation history at the Galactic Center (i.e., the Galactic Center is not currently experiencing a starburst of massive stars), we can relate N (M, < r) to N obs (10M , < r), the observed number of enclosed stars in a logarithmic mass bin centered at 10M , where T (m) is the main sequence lifetime of a star of mass m, which can be fitted by the formula (Buzzoni 2002), This extra T (M )/T (10M ) factor takes into account the fact that high mass stars do not live as long as low mass stars, and evolve into compact remnants that are unobservable.
The fact that we have observed ∼ 1 star of mass ∼ 10M within 10 −2 pc from Sgr A* (i.e., the apocenter of star S2) then implies that The density of stars as a function of radius at the Galactic Center can be fitted by the Nuker model (Lauer et al. 1995;Merritt 2010;Gallego-Cano et al. 2018;Schödel et al. 2018;Baumgardt et al. 2018), which consists of an inner power-law with index γ, an outer power law with index β, and a smooth transition region around the break radius, r b with α parameterizing the sharpness of the transition. Extrapolating to our regime of interests, the Nuker model reduces to its inner power law form, where γ = 1.75 represents the Bahcall-Wolf cusp (Bahcall & Wolf 1976). While γ has been measured for stars located further from Sgr A* Merritt 2010), no measurement of γ has been made for r 100R S . Using equation (5) for our normalization, Figure 1 plots the enclosed number of 1M stars per logarithmic bin, N enc (M , < r) as a function of distance from Sgr A* for a variety of γ's. We find that a star cluster at the center of the Milky Way Galaxy results up to tens of Solar mass stars orbiting within 100R S . Such a star cluster might also host an even larger number of dwarf mass stars, which could dominate the occultation event rate.
We note that the normalization equation (5) overshoots the recent GRAVITY limit by a factor of ∼ 2, and thus is most likely an overestimate (GRAVITY Collaboration et al. 2020). This might be the result of deviations from our assumptions of the mass function or star formation history close to Sgr A*. However, even with a normalization that is smaller by an order of magnitude, we still predict an appreciable number of stars of Solar mass and lower orbiting very close to Sgr A*. The event rate of Sgr A* occultations depends on this normalization, and thus in principle can be used to calibrate the currently unknown mass function and star formation history close to Sgr A*.

Sizes of stellar occulters
Occultations by larger stars provide a stronger signal, but owing to the mass function and the short mainsequence lifetimes of massive stars they represent much rarer events. On the other hand, while less massive stars produce smaller occultation signal, they might be numerous enough to dominate the number of detected occultations. For example, stars like Proxima Centauri, with a mass of ∼ 0.12M and a radius of ∼ (1/6)R (Kervella et al. 2016(Kervella et al. , 2017a, will generate occultation signal that is weaker than occultations by Sun-like stars, but are ∼ 20 times more numerous. Using the observational fit that the mass-radius relation of a star is approximately linear (Rauch 1999), the angular radius of an occulter at the Galactic Center distance of ∼ 8kpc (Gravity Collaboration et al. 2019) is, . The star orbits Sgr A* with an initial semi-major axis of 20RS and initial inclination of 52 degrees off the equatorial plane. The black hole is located at (0, 0, 0). Axes labels are in Schwarzschild radius. The star's orbital node precesses around the spin axis, and causes the orbit to cover a large fraction of 4π solid angle after a precession period.
Unlike the case with exoplanet occultation of stars, where the occultation signal is mainly observed as a flux reduction (transit depth) that scales with the area of the occulter, the SVLBI signal that we are considering include effects that scale with the diameter of the occulter. This is because the interferometry signal of a baseline, by the projection-slice theorem, is sensitive to the one dimensional size of the occulter along the baseline axis. Further, the metallicities of stars in the Galactic Center can be much greater than Solar values (Najarro et al. 2009;Do et al. 2018). As the metallicity of a star determines the opacity of its atmosphere, there is a positive correlation between the sizes of stars and their metallicities (Houdebine et al. 2016;Kesseli et al. 2019). This means that occulters close to Sgr A* potentially possess significantly greater radii than that predicted by equation (8), and thus have correspondingly larger occultation signals.

ENHANCEMENT OF TRANSIT PROBABILITIES BY GENERAL RELATIVISTIC PRECESSION
The probability for an object A to be seen in transit around another object B is, where Ω O is the solid angle, as seen from B, covered by the track of object A as it orbits around B. For exoplanets, the probability of a planet orbiting a star of radius R * at orbital radius a p to be seen in transit is therefore given by ∼ R * /a p . Using this formula for the probability of a Sun-like star to be seen in transit across the supermassive black hole Sgr A* gives a sizeable transit probability of ∼ 50% if the star is orbiting just beyond its tidal disruption radius at ∼ 10R S . However, due to general relativistic precessions, an object in close orbit around a supermassive black hole possesses an even greater transit probability. Obits inclined with respect to the black hole equatorial plane will have their orbital plane precess due to frame dragging and quadrupole precessions. These nodal precessions cause the star orbital path to fill a significant portion of the solid angle around the black hole. For a wide range of orbital parameters, Ω O in equation (9)  If a star is located far from the black hole, however, the precession timescales are long, and thus will not increase the transit probability appreciably over a typical observational campaign. Therefore, unlike the exoplanet case, whether a star is seen in transit across Sgr A* or not is less a function of its inclination and position of orbital nodes, but rather whether the star orbits the black hole close enough to undergo significant precession. The frame dragging precession has a period of (Merritt 2013;Psaltis et al. 2013), where P is the Keplerian period, a is the black hole's spin, e the orbital eccentricity, r the orbital semi-major axis, and M BH the mass of the black hole. Figure 3 plots T J as a function of orbital semi-major axis around a ∼ 4 × 10 6 M black hole with a = 0.5. The frame dragging precession timescale for stars orbiting such a black hole at semi-major axis r ∼ 50R S is ∼ 1 year for non-eccentric orbits. If the orbit is highly eccentric (e ≥ 0.9), this point is reached at r ∼ 100R S . A secondary nodal precession is supplied by the orbital interaction with the spacetime's quadrupole mo-  ment. The period for the quadrupole precession is given by (Merritt 2013;Psaltis et al. 2013), where |q| is the spacetime's quadrupole moment. For a Kerr black hole, q = −a 2 . Figure 3 plots T Q as a function of orbital semi-major axis around a ∼ 4 × 10 6 M black hole with a = 0.5. In our regime of interest, the quadrupole precession is weaker than the frame dragging precession, and is only important for stars orbiting with semi-major axis r 50R S for eccentric orbits and r 20R S for non-eccentric orbits.

OCCULTATION MODEL
We employ for simplicity a crescent model for the black hole emission, obtained by subtracting a disc of radius R n from within a larger disc of radius R p in the image plane. The complex visibility of this model is given by (Kamruddin & Dexter 2013), where I 0 is the surface brightness, a 1 and b 1 are the horizontal and vertical offsets, respectively, of the inner disc from the center of the larger disc, J 1 (x) the Bessel function of the first kind, and When the the center of the smaller disc coincides with that of the larger disc, a 1 = b 2 = 0, we will refer this model as the ring model. Due to the small angular size of a typical occulter, its effects will only be seen in long baselines with lengths > 50Gλ. The validity of using a crescent instead of a full GRMHD simulation in modeling our emission source in this regime is supported by the fact that while accretion astrophysics results in complex structures appearing in the visibility of short baselines, visibilities of long baselines where the occultation signal is most prominent are dominated by the clean signature of an emission ring (Johnson et al. 2020). The occulter is modeled as a non-emitting disc of radius R o offset from the center of the larger emitting disc by a 2 in the horizontal direction and b 2 in the vertical direction in the image plane. This disc subtracts the flux from the emission model, so that the total visibility is given by where where X and Y are coordinates on the image plane, H(x) the Heaviside function, and F U the Fourier transform where the domain is restricted to be within the emitting ring, U .

VLBI SIGNAL
To gain an understanding of how the occultation signal, equation (14), is manifested in observations of an array with a limited coverage of uv-space, consider the case of a single SVLBI baseline. The orientation of this baseline with respect to Sgr A* defines an observational axis on the image plane. When an occulter crosses the emission region, the VLBI signal along that axis will be modified by the presence of the occulter by the projection-slice theorem, where F is the Fourier transform, P the projection operator that acts on the image plane by an integral with x and y being the direction parallel and perpendicular, respectively, to the observational axis of the baseline, and S is the slice operator, that acts on uv-space as where u x and u y are the directions corresponding to x and y, respectively, in uv-space. The right-hand-side of equation (16) is the signal measured by the SVLBI baseline, while the left-hand-side is simply an integral on the image plane. By the projection-slice theorem during an occultation event, the signal detected by a single VLBI baseline will be a the projection of the 'hole' created on the emission region due to the presence of an occulter to its observational axis. As a concrete example, Figure 4, shows the orbital motion of an occulter of radius ∼ 2R as it moves across an observational axis (aligned to Y = 0 on the image plane) during a baseline crossing event. That the occulter crosses an observational axis is in actuality not important, as by the projection-slice theorem, a baseline on this observational axis would still be able to detect this occulter even if the occulter is offset from said axis. In Figure 4, before the circular occulter crosses the baseline, it is located above and to the left of the observational axis, while after the crossing it is located below and to the right of said axis. As the projection-slice theorem states that the baseline is blind to positional information along the vertical axis, the baseline just detects an occulter moving from the left to the right in the coordinates of Figure 4. Figure 5 shows a snapshot of the normalized visibility amplitude, |V | norm , in the VLBI uv-space along such an axis when an occulter is in the middle of the axis. The amplitudes are normalized so that the first peak of the un-occulted model is unity. The presence of the occulter is imprinted on the uv-space as both a change in the visibility amplitudes, as well as a shift in the locations of the troughs and peaks of the signal.
The situation for multiple SVLBI baselines is similar. If the baselines all lie across one observational axis, then multiple points in the x-axis of Figure 5 can be sampled. This improves the detectability of an occultation event.
If instead the baselines lie on multiple observational axes on the image plane, by the projection-slice theorem, the signal on each axes will be sliced onto each observational axes' corresponding uv axes.   (12) with a1 = b1 = 0, R1 = 22.1µas, R2 = 19.9µas, and the occulter is a star of radius ∼ 2R orbiting at 100RS. Due to the perojection-slice theorem, the signal as seen by a baseline along this observational axis depends only on the projection of the occulter along the observational axis (see text for details).

Event timescales
A baseline crossing event occurs over a timescale, where θ LOS is the angle between the orbital velocity and the line-of-sight, while v o is the velocity of the occulter given by, where r i is the instantaneous orbital distance from Sgr A*, and r is again the orbital semi-major axis. The approximate sign in equation (20) refers to the fact that we are neglecting higher order relativistic effects. For a circular orbit, v 0 ≈ GM/r, and This timescale can be significantly different for eccentric orbits. For example, an orbit at apnegricon with r i = (1 + e)r, r = 100R S , and e = 0.7 possesses a baseline crossing time of Further, due to the geometric factor sin θ LOS , this timescale can be very long for eccentric orbits with a geometry where the orbital velocity is mostly along the line-of-sight. Related to the baseline crossing time is the transit time, defined as the time it takes the occulter to travel across a single section of the emitting region.
For an occulter crossing normal to an emitting ring, this is the time it takes to cross the width of the ring. The transit time sets the timescale for the duration over which an occultation event can be detected for. If L limb is the characteristic size of a limb of the emitting region, the transit time is given by, if the limb is larger than the occulter. If the occulter is larger than the limb (2R o > L limb ), then the transit time is equal to the baseline crossing time, While L limb takes a large range of values that can depend on the geometry of the crossing or, in the case of a crescent emission region, which side of the crescent the occulter passes through, its minimum value is of order minutes. While we do not know the future capabilities of SVLBIs, we note that the EHT scans are minutes long (Event Horizon Telescope Collaboration et al. 2019b). Finally, the timescale for the occulter to cross the entire emission region (i.e., the entire ring or crescent), is  Figure 5. A snapshot of the normalized visibility amplitude in Fourier space for an occulter crossing the axis, as in the middle panel of Fig. 4 (dashed), when there is an occulter located along the ux = 0 line of the axis (dotted), and without an occulter (dashed) for a Sun-like star of radius R = 7 × 10 10 cm (top) and for a slightly larger occulter of radius ∼ 2R (bottom), both orbiting at 100RS. The left panel shows the signal from 50 − 80 Gλ, while the right panel shows the same signal from 70 − 80 Gλ. The emission model is given by equation (12) with a1 = b1 = 0, R1 = 22.1µas, R2 = 19.9µas, and I0 normalized so that the first peak at u = v = 0 of the un-occulted model is unity. The occulter is placed at a2 = 21µas and b2 = 0 for the dashed lines, and at a2 = 0 and b2 = 21µas for the dotted lines.
though this duration can be significantly shorter if the occulter crosses near grazing incidence of the emitting region. If an occulter is detected to be crossing the observational axis of a baseline, it will cross other observational axes within time T e , which for circular orbits, For eccentric orbits, as we are now integrating over an appreciable fraction of the orbit, we adopt the average orbital speed, where T K is the Keplerian period and E(m) is the complete elliptical integral of the second kind. On average, a higher eccentricity will increase the time taken for the occulter to cross the emission region. As before, there are special orbits where the geometry is just right for the sin θ LOS term to cause an extremely long T e . However, these orbits require specific geometries and are rare. Neglecting these rare orbits as well as unbound orbits (where T e → ∞), the maximum timescale for the occulter to cross the emission region is, Beyond this timescale, one would have to wait for at least an orbital time before the next occultation event by the same occulter.

Methods for identifying false positives
As discussed in the previous subsections, multiple baselines along the same observational axis provide more sampling points of the signal during an occultation event, while multiple baselines on different observational axes provide the opportunity for multiple baseline transit events to be observed by different baselines, separated by time at most T e,max . However, another important reason to require multiple SVLBI baselines is to break the degeneracy between an occultation event and inherent time-variations in the emission profile.
Along one observational axis, the qualitative signal due to an occulter can be mimicked by, for example, a transient thinning of the emission profile. We plot one such a false positive in Figure 6. As the orbital motion of the occulter produces a specific time variability in the signal, breaking the degeneracy between an occultation event and a false signal would be possible with one baseline if one has enough time resolution to resolve the transit. However, it could be challenging to use this method to characterize the shortest transit events, as those lasts for T t = T e ∼ minutes.
If multiple baselines along the same observation axis are available, but not enough time resolution is achieved to resolve the transit, a mimicking event can be distinguished from an occulter through model fitting. Only very specific changes to the emission profile, ones that create circular holes in the emission profile, can mimic the full uv-signature of an occulter. Signals generated by thinning the emission ring along the observational axis, for example, can produce the same salient features as an occulter crossing the axis: a reduction in the overall amplitude and pushing the peaks and troughs of the signal to higher baselines (c.f. Figure 6). However, such an event cannot produce the same ratio of the offsets of the locations of the peaks/troughs to the reduction in amplitude as a genuine occultation event.
If multiple baselines along different observational axes are available, one can again rule out false signals even if not enough time resolution is achieved in units of T t . As implied by the projection-slice theorem, the signal of an occulter appears on all baselines through the Fourier transform of their projection on each baselines' observational axis. Suppose a candidate occultation event is detected at a particular baseline. If the occultation event is genuine, every other axis must, at the same instance, also have their signal modified by the amount demanded by the projection-slice theorem. An event created by generic transient variations in the emission region (e.g., changes in the thickness in a part of the emission region) will not create such a signal across multiple observational axes. As in the previous method, only very specific events, such as inherent variations that create circular holes in the emission profile, can pass this test.
Unlike an event resulting from changes in the emission profile, an occulter has to move along a straight line in the image plane. As such, when an occulter candidate event is detected, its time variation must behave as demanded by a linear motion. This fact allows us to further reject false positives using baselines on multiple observational axes, even if we do not have enough time resolution on the scale of a transit time, T t . If a single limb crossing is detected, within time T e,max , there will potentially be a secondary limb crossing event at a different location on the image plane, due to the occulter moving across the emission region. The detection of a candidate secondary crossing event within time T e,max increases the likelihood that both events are genuine. Conversely, a detection of a secondary event long after T e,max means that the two events are likely not occultation events. The tradeoff of this method is that it will mischaracterize genuine occultation events that cross the emission profile tangentially along chords far from the center of the emission region, as those occulters will only have a single limb crossing event. This method will also mischaracterize occulters along a hyperbolic orbit, or those on the rare orbits with geometry that allow them to take more than T e,max to cross the emission profile.
Suppose now that we have both multiple baselines on multiple observational axes and enough time resolution to resolve a transit time. Because occulters have to move along a straight line in the image plane, in a genuine occultation event, every single baselines must detect signal that are geometrically consistent with the motion of the occulter. For example, if the occulter's orbit is moving parallel to an observational axis, baselines on that axis must all detect the same motion moving in the same direction, and baselines on axes rotated with respect to the first must detect the same motion projected on their axes. If a secondary limb crossing event is also detected, the location of the secondary limb crossing can also be checked for consistency with the previously deduced motion of the occulter.
Finally, an occultation event will repeat over an orbital period until general relativistic precession brings the orbit out of transit. As such, the likelihood of an occultation event is improved through the detection of correlated occultation signals with a periodicity comparable to an orbital period (∼ 4 days for an orbit with a semi-major axis of 100R S ). As false positives might also exhibit variations with a timescale comparable to the orbital period, this method is unreliable if used alone, and is best used as a secondary method to strengthen the confirmation of an occultation event that passed the other checks.

CONCLUSIONS
We have shown that the stellar cluster around Sgr A* potentially results in an appreciable population of stars down to the tidal disruption radius at ∼ 10R S . These stars are close enough to the supermassive black hole to facilitate occultation events, and modify the emission signal of Sgr A* as seen from a SVLBI. We have shown how the transit probabilities of stars orbiting Sgr A* are greatly enhanced by general relativistic precession. Even for modest Sgr A* spins of a ∼ 0.5, the precession timescales are short for stars orbiting with semi-major axis r 100R S , especially on eccentric orbits. Further, we computed how an occultation affect the SVLBI signal during a transit event. We found that, due to the size of a typical stellar occulter, the occultation signal is prominent at long baselines 50Gλ. The most generic signature of an occultation signal is the modulations of the amplitude, as well as the locations of the peaks and troughs of the signal. Having baselines that are oriented along different axes in the image plane is the best way to distinguish genuine occultation events from transient variations in the emission profile of the region around the black hole.