Detecting Black Hole Occultations by Stars with Space Interferometric Telescopes

Pierre Christian; Abraham Loeb

doi:10.3847/1538-4357/ab9cbc

1. Introduction

A novel avenue for studying astrophysical black holes on event horizon scales is through high-resolution electromagnetic observations (Bardeen 1973; Luminet 1979; Falcke et al. 2000; Takahashi 2004; Johannsen & Psaltis 2010). The feasibility of this technique using ground-based very-long-baseline interferometers (VLBIs) has been demonstrated by the Event Horizon Telescope (EHT; Event Horizon Telescope Collaboration et al. 2019a, 2019b, 2019c, 2019d, 2019e, 2019f). The first EHT observations of the supermassive black hole at the center of the M87 galaxy (henceforth M87) were conducted at 1.3 mm, with the longest baseline providing a resolution of ∼25 μas (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.

Prior to the 1.3 mm observations, the intrinsic size of the Sgr A* radio source was measured to be ∼2 astronomical units (au) at 7 mm (Bower et al. 2004) and ∼1 au at 3 mm (Shen et al. 2005). While these sizes are dominated by scattering, the wavelength dependence of the intrinsic size implies that measurements at shorter wavelengths have the potential to resolve the source at scales that are comparable to the physical size of the central object. Indeed, a measurement at 1.3 mm showed structure at scales comparable to the event horizon of a ∼4 × 10⁶M_⊙ black hole (Doeleman et al. 2008). This measurement was followed by VLBI detections of time variability (Fish et al. 2011), resolved magnetic field structure (Johnson et al. 2015), and asymmetry (Fish et al. 2016) at 1.3 mm. Another observation at the same wavelength using a larger array with longer baselines also confirmed the detection of a highly compact source (Lu et al. 2018).

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. Utilizing their much longer baselines, space-based or space–ground very-long-baseline interferometers (SVLBIs) can operate at longer wavelengths while still improving their resolutions compared to Earth-based observatories. Besides their significantly higher resolution, 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 with ground-based observatories on Earth (Levy et al. 1986, 1989; Linfield et al. 1989, 1990). This inaugural SVLBI array included a 4.9 m 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 (Hirabayashi et al. 1998, 2000). The HALCA satellite carried an 8 m antenna in an elliptical orbit with an apogee at 28,000 km from the Earth's center and further demonstrated the practical possibility of ground–space SVLBIs. The most recent SVLBI array to be deployed was the RadioAstron project,³ which included a 10 m observatory on board the satellite Spektr-R (Kardashev et al. 2013). The RadioAstron project possessed a maximum baseline of ∼3.3 × 10⁵ 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 orbits of the satellites can be reconstructed accurately (Roelofs et al. 2019). Furthermore, emission from the vicinity of a black hole include a thin "photon ring" component consisting of subrings indexed by the number of times that the photons underwent around the black hole (Gralla et al. 2019). While numerical calculations show that this component produces a strong interferometric signature, the current EHT does not possess the necessary resolution to resolve it, and detection of the individual subrings requires the resolution provided by SVLBIs (Johnson et al. 2020).

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 roles 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.

2. Analysis of Possible Occulters

2.1. Population of Stellar Occulters

The most probable occulters are stars orbiting close to Sgr A*. In particular, the existence of the S stars, both detected (Schödel et al. 2002; Ghez et al. 2003, 2005; Eisenhauer et al. 2005; Gillessen et al. 2009, 2017; Habibi et al. 2017), as well as expected but hitherto undetected (Waisberg et al. 2018), suggests 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 through processes such as dynamical interactions of two stellar disks (Löckmann et al. 2008) or multiple three-body 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 an appreciable number of occulters near the tidal disruption radius at ∼10R_S, where R_S ≈ 10¹² cm is the Schwarzschild radius of Sgr A*.

If we assume the star cluster to be described by the Kroupa mass function (Kroupa 2001),

$\begin{eqnarray}&&{\rm{N}}^{\prime} \equiv \displaystyle \frac{{dN}}{d\mathrm{log}m}\propto {m}^{-1.3},\end{eqnarray} \tag{ 1 }$

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 10 M_⊙ per logarithmic mass bin,

$\begin{eqnarray}&&{{\rm{N}}}_{{\rm{enc}}}^{{\prime} }(M,\lt r)={{\rm{N}}}_{{\rm{enc}}}^{{\prime} }(10{M}_{\odot },\lt r)\,{\left(\displaystyle \frac{m}{10{M}_{\odot }}\right)}^{-1.3},\end{eqnarray} \tag{ 2 }$

where ${{\rm{N}}}_{{\rm{enc}}}^{{\prime} }(m,\lt r)$ 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 stellar 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 ${\rm{N}}^{\prime} (M,\lt r)$ to ${{\rm{N}}}_{{\rm{obs}}}^{{\prime} }(10{M}_{\odot },\lt r)$ , the observed number of enclosed stars in a logarithmic mass bin centered at 10M_⊙,

$\begin{eqnarray}&&{{\rm{N}}}_{{\rm{enc}}}^{{\prime} }(M,\lt r)\sim {{\rm{N}}}_{{\rm{obs}}}^{{\prime} }(10{M}_{\odot },\lt r)\,\displaystyle \frac{T(M)}{T(10{M}_{\odot })}\,{\left(\displaystyle \frac{m}{10{M}_{\odot }}\right)}^{-1.3},\end{eqnarray} \tag{ 3 }$

where T(m) is the main-sequence lifetime of a star of mass m. While red giants also contribute to the number of S-stars, the contribution of the giant phase lifetime to T(m) is small compared to that of the main-sequence lifetime, and thus they are ignored in this approximation. T(m) can be fitted by the formula (Buzzoni 2002),

$\begin{eqnarray}&&{\mathrm{log}}_{10}[T(m)/\mathrm{yr}]=0.825{\mathrm{log}}_{10}^{2}\,\left(\displaystyle \frac{m}{120{M}_{\odot }}\right)+6.43.\end{eqnarray} \tag{ 4 }$

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 the star S2, an S star of mass ∼10M_⊙ with an apocenter of 10⁻² pc from Sgr A*, has been detected (Ghez et al. 1998; Boehle et al. 2016; GRAVITY Collaboration et al. 2020) implies that

$\begin{eqnarray}&&{{\rm{N}}}_{{\rm{enc}}}^{{\prime} }({M}_{\odot },\lt {10}^{-2}\,{\rm{pc}})\sim 20\displaystyle \frac{T({M}_{\odot })}{T(10{M}_{\odot })}\sim 8\times {10}^{3}.\end{eqnarray} \tag{ 5 }$

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; Fritz et al. 2016; Baumgardt et al. 2018; Gallego-Cano et al. 2018; Schödel et al. 2018),

$\begin{eqnarray}&&\rho (r)\propto {\left(\displaystyle \frac{r}{{r}_{b}}\right)}^{-\gamma }\,{\left[1+{\left(\displaystyle \frac{r}{{r}_{b}}\right)}^{\alpha }\right]}^{\tfrac{\gamma -\beta }{\alpha }},\end{eqnarray} \tag{ 6 }$

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,

$\begin{eqnarray}&&\rho (r)\propto {r}^{-\gamma },\end{eqnarray} \tag{ 7 }$

where γ = 1.75 represents the Bahcall–Wolf cusp (Bahcall & Wolf 1976). While γ has been measured for stars located further from Sgr A* (Gallego-Cano et al. 2018; 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 1 M_⊙ stars per logarithmic bin, ${{\rm{N}}}_{{\rm{enc}}}^{{\prime} }({M}_{\odot },\lt r)$ 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 the Milky Way 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, so Figure 1 should be thought of as an order of magnitude estimate.

**Figure 1.** Enclosed number of stars within a logarithmic mass bin centered at 1M_⊙ as a function of distance from Sgr A* with γ values 1.5 (solid), 1.75 (dotted), and 2 (dashed).
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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*.

2.2. Sizes of Stellar Occulters

Occultations by larger stars provide a stronger signal, but owing to the mass function and the short main-sequence 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, 2017a, 2017b), 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 ∼8 kpc (Gravity Collaboration et al. 2019) is,

$\begin{eqnarray}&&{\theta }_{o}^{* }(M)\approx 0.58\times \left(\displaystyle \frac{M}{{M}_{\odot }}\right)\,\mu \mathrm{as}.\end{eqnarray} \tag{ 8 }$

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.

Furthermore, 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.

3. Enhancement of Transit Probabilities by General Relativistic Precession

The probability for an object A to be seen in transit around another object B is,

$\begin{eqnarray}&&{P}_{{\rm{transit}}}=\displaystyle \frac{{{\rm{\Omega }}}_{O}}{4\pi },\end{eqnarray} \tag{ 9 }$

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) becomes comparable to 4π. An illustrative case is shown in Figure 2, where we followed a numerical integration of the geodesic equation in the Kerr metric representing a star in an inclined orbit around Sgr A*. As shown in Figure 2, a timelike geodesic precesses and covers a large fraction of the entire 4π solid angle.

**Figure 2.** Geodesic integration of a star's orbit around Sgr A* with spin parameter a = 0.5 and the black hole spin pointing in the vertical direction (∼20 orbits are shown over a time of ∼5 days, a fraction of T_J). The star orbits Sgr A* with an initial semimajor axis of 20R_S 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.
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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),

$\begin{eqnarray}&&{T}_{J}=\displaystyle \frac{P}{4a}\,{\left[\displaystyle \frac{{c}^{2}r(1-{e}^{2})}{{{GM}}_{{\rm{BH}}}}\right]}^{3/2},\end{eqnarray} \tag{ 10 }$

where P is the Keplerian period, a is the black hole's spin, e the orbital eccentricity, r the orbital semimajor axis, and M_BH the mass of the black hole. Figure 3 plots T_J as a function of orbital semimajor axis around a ∼4 × 10⁶M_⊙ black hole with a = 0.5. The frame dragging precession timescale for stars orbiting such a black hole at semimajor axis r ∼ 50R_S is ∼1 year for noneccentric orbits. If the orbit is highly eccentric (e ≥ 0.9), this point is reached at r ∼ 100R_S.

**Figure 3.** Precession timescales for the frame dragging (top) and quadrupolar (bottom) precessions as a function of orbital semimajor axis around a black hole of spin a = 0.5 with the mass of Sgr A* (R_S ≈ 10¹² cm). The different lines indicate orbits with eccentricities 0.9 (solid), 0.5 (dotted), 0.3 (dashed), and 0 (dotted–dashed).
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A secondary nodal precession is supplied by the orbital interaction with the spacetime's quadrupole moment. The period for the quadrupole precession is given by (Merritt 2013; Psaltis et al. 2013),

$\begin{eqnarray}&&{T}_{Q}=\displaystyle \frac{P}{3| q| }\,{\left[\displaystyle \frac{{c}^{2}r(1-{e}^{2})}{{{GM}}_{{\rm{BH}}}}\right]}^{2},\end{eqnarray} \tag{ 11 }$

where $| q|$ is the spacetime's quadrupole moment. For a Kerr black hole, q = −a². Figure 3 plots T_Q as a function of orbital semimajor axis around a ∼4 × 10⁶M_⊙ 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 semimajor axis r ≲ 50R_S for eccentric orbits and r ≲ 20R_S for noneccentric orbits.

In addition to the relativistic precessions, other stars can also perturb the orbits of a star orbiting near Sgr A*. This effect would further increase the amount of solid angle covered by the star's orbit beyond its Keplerian estimate (Merritt 2010).

4. Occultation Model

We employ for simplicity a crescent model for the black hole emission, obtained by subtracting a disk of radius R_n from within a larger disk of radius R_p in the image plane. The complex visibility of this model is given by (Kamruddin & Dexter 2013),

$\begin{eqnarray}\begin{array}{rcl}{V}_{e}(u,v) & = & 2\pi {I}_{0}\,\left[\displaystyle \frac{{R}_{p}{J}_{1}(2\pi {{kR}}_{p})}{2\pi k}\right.\\ & & \left.-\,{e}^{-2\pi ({a}_{1}u+{b}_{1}v)}\displaystyle \frac{{R}_{n}{J}_{1}(2\pi {{kR}}_{n})}{2\pi k}\right],\end{array}\end{eqnarray} \tag{ 12 }$

where I₀ is the surface brightness, a₁ and b₁ are the horizontal and vertical offsets, respectively, of the inner disk from the center of the larger disk, J₁(x) the Bessel function of the first kind, and

$\begin{eqnarray}&&k=\sqrt{{u}^{2}+{v}^{2}}.\end{eqnarray} \tag{ 13 }$

When the center of the smaller disk coincides with that of the larger disk, a₁ = b₂ = 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 >50 Gλ. 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 nonemitting disk of radius R_o offset from the center of the larger emitting disk by a₂ in the horizontal direction and b₂ in the vertical direction in the image plane. This disk subtracts the flux from the emission model, so that the total visibility is given by

$\begin{eqnarray}&&V(u,v)={V}_{e}(u,v)-{V}_{o}(u,v),\end{eqnarray} \tag{ 14 }$

where

$\begin{eqnarray}&&{V}_{o}(u,v)={F}_{U}\left[H\left({R}_{o}-\sqrt{{\left(X-{a}_{2}\right)}^{2},{\left(Y-{b}_{2}\right)}^{2}}\right)\right],\end{eqnarray} \tag{ 15 }$

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.

5. 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,

$\begin{eqnarray}&&F\circ P=F\circ S,\end{eqnarray} \tag{ 16 }$

where F is the Fourier transform, P the projection operator that acts on the image plane by an integral

$\begin{eqnarray}&&P[f(x,y)]={\int }_{-\infty }^{\infty }f(x,y){dy},\end{eqnarray} \tag{ 17 }$

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

$\begin{eqnarray}&&S[f({u}_{x},{v}_{y})]=f({u}_{x},0),\end{eqnarray} \tag{ 18 }$

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, $| V{| }_{{\rm{norm}}}$ , 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 unocculted 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.

5.1. Event Timescales

A baseline crossing event occurs over a timescale,

$\begin{eqnarray}&&{T}_{c}=\displaystyle \frac{2{R}_{o}}{{v}_{o}\sin {\theta }_{{\rm{LOS}}}},\end{eqnarray} \tag{ 19 }$

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,

$\begin{eqnarray}&&{v}_{o}\approx \sqrt{{GM}\left(\displaystyle \frac{2}{{r}_{i}}-\displaystyle \frac{1}{r}\right)},\end{eqnarray} \tag{ 20 }$

where r_i is the instantaneous orbital distance from Sgr A*, and r is again the orbital semimajor axis. The approximate sign in Equation (20) refers to the fact that we are neglecting higher order relativistic effects.

For a circular orbit, ${v}_{0}\approx \sqrt{{GM}/r}$ , and

$\begin{eqnarray}&&{T}_{c}\approx 1\,{\rm{minutes}}\times \,\displaystyle \frac{{R}_{o}}{{R}_{\odot }}\times {\left(\displaystyle \frac{100{R}_{S}}{r}\right)}^{1/2}\times {\sin }^{-1}{\theta }_{{\rm{LOS}}}.\end{eqnarray} \tag{ 21 }$

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

$\begin{eqnarray}&&{T}_{c}\approx 2.5\,{\rm{minutes}}\times \,\displaystyle \frac{{R}_{o}}{{R}_{\odot }}\times {\sin }^{-1}{\theta }_{{\rm{LOS}}}.\end{eqnarray} \tag{ 22 }$

Furthermore, due to the geometric factor sinθ_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,

$\begin{eqnarray}&&{T}_{t}=\displaystyle \frac{{L}_{{\rm{limb}}}}{{v}_{o}\sin {\theta }_{{\rm{LOS}}}},\end{eqnarray} \tag{ 23 }$

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,

$\begin{eqnarray}&&{T}_{t}={T}_{c}.\end{eqnarray} \tag{ 24 }$

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

$\begin{eqnarray}&&{T}_{e}\approx \displaystyle \frac{\sqrt{27}{R}_{S}}{{v}_{o}\sin {\theta }_{{\rm{LOS}}}},\end{eqnarray} \tag{ 25 }$

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,

$\begin{eqnarray}&&{T}_{e}\approx 0.8\,{\rm{hr}}\times {\left(\displaystyle \frac{100{R}_{S}}{r}\right)}^{1/2}.\end{eqnarray} \tag{ 26 }$

For eccentric orbits, as we are now integrating over an appreciable fraction of the orbit, we adopt the average orbital speed,

$\begin{eqnarray}\begin{array}{rcl}{\bar{v}}_{o} & = & \displaystyle \frac{4{aE}(e)}{{T}_{{\rm{K}}}}\\ & = & \sqrt{\displaystyle \frac{{GM}}{r}}\left(1-\displaystyle \frac{1}{4}{e}^{2}-\displaystyle \frac{3}{64}{e}^{2}-\displaystyle \frac{5}{256}{e}^{6}-\ldots \right),\end{array}\end{eqnarray} \tag{ 27 }$

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 $\sin {\theta }_{{\rm{LOS}}}$ 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 ${T}_{e}\to \infty$ ), the maximum timescale for the occulter to cross the emission region is,

$\begin{eqnarray}&&{T}_{e,{\rm{\max }}}\approx 1.3\,{\rm{hr}}\times {\left(\displaystyle \frac{100{R}_{S}}{r}\right)}^{1/2}.\end{eqnarray} \tag{ 28 }$

Beyond this timescale, one would have to wait for at least an orbital time before the next occultation event by the same occulter.

5.2. 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, such as those that last for T_t = T_e ∼ minutes.

**Figure 6.** Snapshot of the normalized visibility amplitude in Fourier space for an occulter crossing the observational axis (dashed), without an occulter (solid), and a false positive (dotted) produced by reducing the thickness of the emission ring. For an instance of time, a false positive produces the same qualitative signal as an occulter, i.e., changing the amplitude and positions of the troughs and peaks of the visibility amplitude.
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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 (see 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 semimajor 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.

5.3. A Case Study with Three Baselines

Next we explore the effect of having a sparse baseline coverage on the ability of an SVLBI experiment to distinguish a genuine occultation event from a false signal through a case study of an SVLBI with three baselines. Consider an occultation event where the occulter is a Sun-like star detected by four stations configured in a manner plotted in the top panel of Figure 7. Similar calculations for a different choice of station configuration can be made analogously; this configuration can be considered with no loss of generality, as long as the stations are located far enough from each other for the occultation signature to be detectable. Furthermore, we will limit our analysis to only three baselines, marked red, green, and blue in Figure 7. While the other baselines (e.g., the dotted line in Figure 7) can be used to improve the detection, we limit ourselves to only a subset of the baselines to demonstrate that a genuine occultation event can be distinguished even at times that, due to the orbital motion of the satellites, some of the baselines are too short to be able to detect an occultation signal.

With such a sparse array, we need to leverage the time variability of the signal to screen for genuine occultation events. Even in the chance that random fluctuations in the accretion flow can mimic the signal on all three baselines at a particular time, it is exceedingly unlikely for them to mimic the time variability of a genuine occultation event. This can be understood through the projection-slice theorem: when the occulter moves in image space (e.g., along the negative Y-axis in Figure 7), the portion of the image that has their intensity reduced due to the presence of the occulter moves accordingly on the image plane, and thus its projection on the baseline axes also moves. When Fourier transformed, this generates a predictable signal on each baseline. Assuming the simple model that the occulter moves at a constant velocity along a straight line across the image plane, this signal can then be searched for using standard model fitting algorithms.

In the bottom figures of Figure 7, we plot the amplitude detected on the three baselines (all of them at 75 Gλ) as well as R, the ratio of the signal detected by the red and blue baselines. Because the occulter motion is perpendicular to the green baseline, the ratios between the signal detected by the green and the other baselines are trivial. While the ratios at a particular time can be mimicked by random fluctuations in the accretion flow, the time evolution shown in these plots are characteristic to a circular dark spot in the image plane moving with a constant velocity.

There is the possibility for a cold spot orbiting along the accretion flow that can mimic the time variability of an occultation signal for short times. However, a cold spot is carried along the accretion flow, in contrast to an occulter that moves along its own orbit. Thus, a cold spot will not move along a straight line in the image plane except for a limited amount of time. If the directionality of the accretion flow is known (e.g., via Doppler beaming), a cold spot will also have a motion that is on average along said direction, in contrast to an occulter that can move unconstrained on the image plane. Furthermore, as detailed in the previous subsection, an occulter can produce two occultation events within a short timescale if its motion strikes a chord across the emission region. This behavior is not present for cold spots.

6. 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 an 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 semimajor axis r ≲ 100R_S, especially on eccentric orbits.

Furthermore, we computed how an occultation affects 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 ≳50 Gλ. 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.

The authors thank Dimitrios Psaltis and the anonymous referee for useful comments. A.L. was supported in part by the black hole Initiative at Harvard University, which is funded by JTF and GBMF grants.

Detecting Black Hole Occultations by Stars with Space Interferometric Telescopes

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Abstract

1. Introduction