The Effects of Tilt on the Images of Black Hole Accretion Flows

We use the GRMHD code Athena++ (White et al. 2016) to evolve two similar accretion flows around a black hole with spin a = 0.9, one aligned with the black hole spin and the other tilted. In both cases we use spherical Kerr–Schild coordinates (t, r, θ, ϕ) with a statically refined grid. The root grid has 56 cells geometrically spaced in radius from $r\approx 0.926\ {r}_{\mathrm{hor}}$ (the horizon is at ${r}_{\mathrm{hor}}\approx 1.44$) to r = 100, 32 cells uniformly spaced in θ from pole to pole, and 44 cells uniformly spaced in ϕ from 0 to 2π. Three nested levels of mesh refinement are added, each doubling resolution in three dimensions. The highest effective resolution of 448 × 256 × 352 (239 cells per decade in radius) is achieved everywhere within 50625 of the midplane.

The aligned simulation is initialized with a hydrostatic torus according to the prescription of Fishbone & Moncrief (1976), with inner edge r = 15, pressure maximum at r = 25, adiabatic index Γ = 4/3, and peak density ρ = 1. A poloidal magnetic field is added using the vector potential

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\phi } & \propto & {\left(\max ({p}_{\mathrm{gas}}-{10}^{-8},0\right)}^{1/2}{r}^{2}\sin \theta \\ & & \times \,\sin \left(\pi L(r;16,34)\right)\sin \left(\pi L(\theta ;70^\circ ,110^\circ )\right),\end{array}\end{eqnarray} \tag{ 1 }$$

with A_r, A_θ = 0. Here $L(x;{x}_{\min },{x}_{\max })$ is the linear ramp function that runs from 0 for $x\leqslant {x}_{\min }$ to 1 for $x\geqslant {x}_{\max }$.

The tilted simulation applies the Fishbone & Moncrief solution to the coordinates $(t,r,\theta ^{\prime} ,\phi ^{\prime} )$, with

$$\begin{eqnarray}&&\theta ^{\prime} ={\cos }^{-1}(\cos i\cos \theta +\sin i\sin \theta \cos \phi ),\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}\begin{array}{rcl}\phi ^{\prime} & = & {\tan }^{-1}(\sin \theta \sin \phi ,\\ & & -\,\sin i\cos \theta +\cos i\sin \theta \cos \phi )\end{array}\end{eqnarray} \tag{ 2b }$$

being the standard angles obtained by tilting spherical coordinates by an inclination i = 24° toward ϕ = 0°. The torus is no longer in exact equilibrium, though the addition of a magnetic field causes this in any event. Here we apply the vector potential

$$\begin{eqnarray}&&{A}_{\phi ^{\prime} }\propto \max (\rho -0.2,0),\end{eqnarray} \tag{ 3 }$$

with ${A}_{r},{A}_{\theta ^{\prime} }=0$.

In both cases we have a magnetic field consisting of a single set of nested loops in the poloidal plane. We normalize the fields such that the density-weighted average of plasma ${\beta }^{-1}\equiv {p}_{\mathrm{mag}}/{p}_{\mathrm{gas}}$ is 0.01.

Both simulations are run to a time of t = 10,000. As expected, the structures of the flows differ in the two cases. Figure 1 shows midplane slices of density at the end of the simulation. In the tilted case, the surface used for the slicing, denoted with two primes, is warped and twisted in order to be orthogonal to the gas angular momentum at each radius. The angle between the surface normal and the black hole spin direction can exceed the initial inclination, reaching 44° at small radii. Using the same density scale, one can see much stronger density contrasts in the tilted case. These result from the standing shocks created in misaligned disks, as explained by Fragile & Blaes (2008).

Zoom In Zoom Out Reset image size

We use the GR ray tracing code grtrans (Dexter & Agol 2009; Dexter 2016) to process snapshots from the simulations into images. The camera is located at coordinates (r₀, θ₀, ϕ₀). We set r₀ = 50; in all cases there is negligible emission and absorption beyond this radius. A grid of 512² pixels is created in the image plane, with a field of view of $24\ {GM}/{c}^{2}$ on each side. Each ray traced back from the image plane through the simulation is sampled at 1600 points.

Here we only consider total intensity (Stokes I) images (only tracking polarization internally during ray tracing) at $230\ \mathrm{GHz}$ as produced by the emission and absorption of thermal synchrotron radiation along each ray. The scale-free MHD snapshots are given physical units by setting the black hole mass M and fluid density scale $[\rho ]$. The latter is adjusted in order to match observed $230\ \mathrm{GHz}$ flux density, fixing the time-averaged mass accretion rate onto the black hole in the process. In all cases the fluid temperature in the simulation is converted to electron temperature according to the same ansatz as in Mościbrodzka et al. (2016), assuming an ion-to-electron temperature ratio of

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{{\rm{i}}}}{{T}_{{\rm{e}}}}=\displaystyle \frac{{R}_{\mathrm{low}}+{R}_{\mathrm{high}}{\beta }^{2}}{1+{\beta }^{2}},\end{eqnarray} \tag{ 4 }$$

with ${R}_{\mathrm{low}}=1$ and ${R}_{\mathrm{high}}=10$. This is the same prescription used in EHT V, where ${R}_{\mathrm{low}}$ is also kept at 1 and ${R}_{\mathrm{high}}$ is varied over a range of plausible values from 1 to 160. We choose to use a single intermediate ${R}_{\mathrm{high}}$ in order to focus on the effects of accretion flow geometry. While changing this parameter can redistribute intensity within the image, possibly changing the overall size of the observed ring, we do not expect variations in ${R}_{\mathrm{high}}$ to qualitatively change aspects of image morphology that arise due to flow geometry. We note that models developed to capture the effects of turbulent heating may not be appropriate for shock-heated regions in the tilted flows, and that better prescriptions may need to be developed for these cases.

When applying the simulations to a face-on model of Sgr A*, we choose $M=4.152\times {10}^{6}\ {M}_{\odot }$ and place the source at a distance $D=8.178\ \mathrm{kpc}$, following the inferred values in The GRAVITY Collaboration (2019). We seek to have the average flux from our snapshots match ${F}_{\nu }=2.4\ \mathrm{Jy}$ (Doeleman et al. 2008), which results in choosing $[\rho ]=1.7\,\times {10}^{-16}\ {\rm{g}}\,{\mathrm{cm}}^{-3}$ in the aligned case and $[\rho ]=4.8\,\times {10}^{-17}\ {\rm{g}}\,{\mathrm{cm}}^{-3}$ in the tilted case. With these scales we can convert the accretion rates in the simulations, averaged from t = 8000 to t = 10,000, to the physical values of $\dot{M}\,=8.4\times {10}^{-9}\ {M}_{\odot }\,{\mathrm{yr}}^{-1}$ in the aligned case and $\dot{M}=5.1\,\times {10}^{-9}\ {M}_{\odot }\,{\mathrm{yr}}^{-1}$ in the tilted case. Defining an Eddington accretion rate of ${\dot{M}}_{\mathrm{Edd}}=10\cdot 4\pi {{GMm}}_{{\rm{p}}}/c{\sigma }_{{\rm{T}}}$, these correspond to $9.1\times {10}^{-8}\ {\dot{M}}_{\mathrm{Edd}}$ and $5.5\times {10}^{-8}\ {\dot{M}}_{\mathrm{Edd}}$. Here we fix the viewing angle θ₀ = 177° and we choose ϕ₀ = 0°, though the symmetry of the system means the latter has little effect. The images are rotated such that the south pole (the one pointed toward the camera) has a small projection in the plane of the image at a position angle of 180° (toward the bottom), with the flow moving clockwise.

For the inclined model of Sgr A*, we keep the same mass, distance, and flux. The viewing angle is θ₀ = 135°, ϕ₀ = 0°. Again the southern (near) pole points to the bottom of the image and the matter is moving clockwise. In this case, the density scale [ρ] is $1.6\times {10}^{-16}\ {\rm{g}}\,{\mathrm{cm}}^{-3}$ in the aligned case and $5.6\,\times {10}^{-17}\ {\rm{g}}\,{\mathrm{cm}}^{-3}$ in the tilted case. The accretion rates are $7.7\,\times {10}^{-9}\ {M}_{\odot }\,{\mathrm{yr}}^{-1}$ ($8.4\times {10}^{-8}\ {\dot{M}}_{\mathrm{Edd}}$) and $6.0\times {10}^{-9}\ {M}_{\odot }\,{\mathrm{yr}}^{-1}$ ($6.5\times {10}^{-9}\ {\dot{M}}_{\mathrm{Edd}}$), respectively.

With M87 we use the inferred mass $M=6.5\times {10}^{9}\ {M}_{\odot }$ and combined distance measurement $D=16.8\ \mathrm{Mpc}$ from EHT VI (the distance is derived from the measurements in Blakeslee et al. 2009; Bird et al. 2010; Cantiello et al. 2018), as well as the flux ${F}_{\nu }=0.98\ \mathrm{Jy}$ (Doeleman et al. 2012), resulting in $[\rho ]$ being $4.8\times {10}^{-18}\ {\rm{g}}\,{\mathrm{cm}}^{-3}$ and $6.9\times {10}^{-19}\ {\rm{g}}\,{\mathrm{cm}}^{-3}$ in the aligned and tilted cases, respectively. This implies physical accretion rates of $\dot{M}=5.8\times {10}^{-4}\ {M}_{\odot }\,{\mathrm{yr}}^{-1}$ ($4.0\,\times {10}^{-6}\ {\dot{M}}_{\mathrm{Edd}}$) and $\dot{M}=1.8\,\times {10}^{-4}\ {M}_{\odot }\,{\mathrm{yr}}^{-1}$ ($1.2\times {10}^{-6}\ {\dot{M}}_{\mathrm{Edd}}$), respectively. In both cases we use a viewing angle ${\theta }_{0}=163^\circ$ (agreeing in magnitude with Mertens et al. 2016), fixing ϕ₀ = 0°. Here we rotate the image such that an approaching jet aligned with the spin axis will have a position angle of 288° (toward the right and slightly up), in agreement with the large-scale jet seen in M87 (Walker et al. 2018). The accretion flow is clockwise.

In the process of ray tracing, we can delineate the boundary between rays that trace back into the black hole and those that do not. We will refer to this boundary as the "geometrical ring," also known as the photon ring.

In several cases we will consider a set of 21 snapshots uniformly sampled in time from t = 8000 to t = 10,000. The separation of Δt = 100 between snapshots is larger than the correlation time at the small radii of interest in these turbulent accretion flows, so these samples can be considered independent.

First we consider the face-on case. Here we use parameters suitable for Sgr A*, where the inclinations of both the spin axis and the gas angular momentum to the line of sight are currently poorly constrained. For example, one might expect the system to prefer alignment with the clockwise disk (that is more edge-on than face-on) of stars within 0.3 Mpc (Paumard et al. 2006; Beloborodov et al. 2006), but at the same time orbital motion in a nearly face-on system is consistent with the near-infrared centroid motion found by the GRAVITY instrument (GRAVITY Collaboration 2018). We choose to first model the system with face-on spin, where the connection between shock structure and image features is clearest.

Figure 2 shows images constructed from high-resolution snapshots at the end of the simulation (t = 10,000). Here we can clearly see a breaking of axisymmetry caused by the standing shocks in the tilted disk, as well as by Doppler shifts in the parts of the disk moving toward or away from the camera. Importantly, the standing shock seen here is in the foreground (the disk is geometrically thick with the inner part not entirely optically thin, and the other shock in the background is obscured and distorted), enabling it to be seen at small projected radii where there is only an uninterrupted, circular shadow in the aligned case.

Zoom In Zoom Out Reset image size

The resolution of Figure 2 is far higher than that of the EHT. We therefore blur the image with a Gaussian kernel with an FWHM of $20\,\mu \mathrm{as}$ to see if there remain any features capable of distinguishing tilted from aligned flows. This kernel is appropriate for modeling EHT data in the image plane (EHT II). In particular, we expect the bright, nonaxisymmetric feature near the center of the lower panel of Figure 2 to affect the shadow shape even if the ring remains relatively circular.

The top panels in Figure 3 show the results of this blurring process on five snapshots from the aligned simulation, equally spaced in time from t = 8000 to t = 10,000 (a span of approximately $11\ \mathrm{hr}$ for Sgr A*). The bottom panels show the corresponding snapshots from the tilted simulation. By eye, adding tilt appears to make the ring brightness slightly less symmetric and to make the shadow less circular.

Zoom In Zoom Out Reset image size

We use the following procedure to quantify the shapes of the rings and shadows in the blurred images. From the image center we resample intensity onto 128 radial rays, each with 128 sample points. A ridgeline is found, consisting of each point that is the local maximum of intensity in its ray. A new center is calculated as the centroid of the pixels contained inside this ridgeline, not weighted by intensity, and the ridgeline is recalculated from this new center, ensuring that even a feature offset from the image center is uniformly sampled in angle. This ridgeline is taken to be the ring. We define a ring "roughness" ${R}_{\mathrm{ring}}$ to be the standard deviation of the set of distances from the new center to the ridgeline, divided by the mean of the set. A perfect circle would have a roughness of 0. Our procedure so far is similar to that described in The Event Horizon Telescope Collaboration (2019c, EHT IV), their Section 9.1, for defining the ring.

Next, we define the "shadow" to be the dimmest quartile of pixels inside the ring. Taking the zeroth and first moments of this set of pixels, again not weighted by intensity, yields the size S and center $({\bar{x}}^{1},{\bar{x}}^{2})$ of the shadow:

$$\begin{eqnarray}&&S=\sum _{\mathrm{shadow}}1,\end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray}&&{\bar{x}}^{i}=\displaystyle \frac{1}{S}\sum _{\mathrm{shadow}}{x}^{i}.\end{eqnarray} \tag{ 5b }$$

A shadow contour can be defined by again sampling along rays emanating from this central point, this time finding the radius where the intensity crosses the aforementioned quartile value. This boundary also has an associated roughness value ${R}_{\mathrm{shadow}}$.

We are particularly interested in the shape of the shadow, so we measure it in another way. Define the second moments

$$\begin{eqnarray}&&{M}_{{ij}}=\displaystyle \frac{4}{S}\sum _{\mathrm{shadow}}({x}^{i}-{\bar{x}}^{i})({x}^{j}-{\bar{x}}^{j}).\end{eqnarray} \tag{ 6 }$$

Arranging the second moments into a matrix, the eigenvalues can be taken to be the squares of the semimajor and semiminor axes of an ellipse, with the corresponding eigenvectors indicating the axes' orientations.⁵

Figure 4 shows the contours for the rings and shadows corresponding to the snapshots shown in Figure 3. The contours match what can be seen by eye in the blurred images. We also plot the ellipses obtained from the second moments, and they are generally good fits to the shadow regions.

Zoom In Zoom Out Reset image size

Using 21 blurred images over the span of time from t = 8000 to t = 10,000 (again, roughly $11\ \mathrm{hr}$ for Sgr A*), we calculate the roughness parameter for the ring to be ${R}_{\mathrm{ring}}=0.027\pm 0.010$ in the aligned case and ${R}_{\mathrm{ring}}\,=0.052\pm 0.019$ in the tilted case. The reported numbers are the mean plus or minus the standard deviation over the set of snapshots. The tilted rings are slightly more uneven, but only by about one standard deviation.

The shadow roughness parameters for the same set of snapshots are ${R}_{\mathrm{shadow}}=0.021\pm 0.011$ and ${R}_{\mathrm{shadow}}=0.136\,\pm 0.055$ for the aligned and tilted cases, respectively. Here there is a significant difference: the shadows in the tilted images are distinctly noncircular. The same result is seen when examining the eccentricities of the best-fit ellipses to the shadows, which are measured to be ${e}_{\mathrm{shadow}}=0.291\pm 0.085$ for the aligned images and ${e}_{\mathrm{shadow}}=0.69\pm 0.11$ for the tilted images. By looking at the shape of the shadow inside the ring, rather than the ring's ridgeline itself, a difference between aligned and tilted accretion flows can be discerned.

The orientations of the major axes of the ellipses fitting the aligned shadows change drastically from one snapshot to the next (a separation of $17\,\mathrm{minutes}$). This is not surprising given the circular nature of these shadows. For the shadows in tilted flows, however, these orientations change slowly in time. The average magnitude by which they change from one snapshot to the next is only 13°.

Given the potential rapid variability of Sgr A* relative to the duration of EHT observations (${GM}/{c}^{3}=20\ {\rm{s}}$), we repeat the above analyses on a time-averaged image. That is, we first apply grtrans to 21 snapshots as before, then average the resulting images in time, then blur the averaged image, and finally measure the ring and shadow properties for the single blurred image. The blurred images are shown in Figure 5, and the corresponding contours are highlighted in Figure 6. The ring roughness is ${R}_{\mathrm{ring}}=0.0147$ in the aligned case and ${R}_{\mathrm{ring}}=0.0304$ in the tilted case. The shadow again shows a greater difference: ${R}_{\mathrm{shadow}}$ is 0.0203 and 0.0998 in the aligned and tilted cases, respectively, and ${e}_{\mathrm{shadow}}$ is 0.209 and 0.641, respectively.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Given the present uncertainty in the orientation of Sgr A*, we consider another viewing angle: 45° off the black hole spin axis. Figure 7 shows high-resolution images created from a single snapshot in both the aligned and tilted cases. Unlike in the face-on case, both images are similarly complex and there are no immediately distinguishable features.

Zoom In Zoom Out Reset image size

As in Section 3.1, we take the blurred, time-averaged image as an appropriate proxy for EHT data. Time averaging the same 21 snapshots spanning t = 8000 to t = 10,000 ($11\,\mathrm{hr}$ given the mass $M=4.152\times {10}^{6}\ {M}_{\odot }$) and applying a $20\ \mu \mathrm{as}$ Gaussian filter, we obtain the images shown in Figure 8. Images from this inclined viewing angle do not have shadows inside rings that are as well-defined as those in the face-on case when blurred to match the EHT resolution. Still, the shadow can be seen to be significantly eccentric in the tilted image, as with the face-on viewing angle discussed in Section 3.1. The images display a distinct crescent morphology. Importantly, the crescent is symmetric in the aligned case, whereas in the tilted case the southern tail extends farther from the brightness peak.

Zoom In Zoom Out Reset image size

Using the blurred images, we can define a ridgeline as before, though in some directions the algorithm will fail to find a well-defined intensity maximum. We can still define parts of a ring, as shown in Figure 9. From these plots, it is apparent that the tilted image has a larger ridgeline. The mean distance from the ring center to the ridgeline is ${r}_{\mathrm{ring}}=19.8\ \mu \mathrm{as}$ in the aligned case and ${r}_{\mathrm{ring}}=26.2\ \mu \mathrm{as}$ in the tilted case.

Zoom In Zoom Out Reset image size

In order to sample intensity along the ridgeline, we extend the procedure: whenever a sample location is needed but no ridgeline can be found, the distance of the ridgeline from the central point is linearly interpolated in angle from the nearest angles in either direction for which a local maximum exists. We can then measure intensity along a closed arc. The results, plotted as a function of position angle, are shown in Figure 10. Again, the skewness in the ridgeline intensity in the tilted case is apparent.

Zoom In Zoom Out Reset image size

The same two simulations can be used to model emission from M87, changing the black hole mass, distance, and average flux parameters when ray tracing. Here we do not have as much freedom to choose the viewing angle inclination θ₀, given the observed large-scale jet oriented 17° off our line of sight. That is, we assume the large-scale jet is aligned with the black hole spin axis, though there are some simulations that suggest this is not the case (Liska et al. 2018). Our choice produces images—both aligned and tilted—broadly consistent with M87 observations, with more emission coming from the southern part of the image. Moreover, changing the viewing angle to align the disk normal with the observed jet would not qualitatively change the fact that tilted flows differ from aligned ones by containing intrinsically nonaxisymmetric structure. For example, a high-resolution image from each simulation is shown in Figure 11, where even the aligned case now has nonaxisymmetric structure. Still, the tilted case displays a qualitative difference by having a prominent spiral shock penetrate the shadow.

Zoom In Zoom Out Reset image size

Figure 12 shows five blurred images taken from each simulation, equally spaced over a time span of 8000 ≤ t ≤ 10,000 (a $2.0\ \mathrm{yr}$ range for M87's mass). As with the inclined viewing angle for Sgr A*, we often find the intensity ridgeline does not completely enclose the shadow; that is, moving outward from the shadow center the intensity sometimes monotonically decreases. Thus the simple procedure we employed for face-on Sgr A* images fails to define a shadow here. We note this may be an artifact of our simple smoothing prescription; there is a ridgeline in the unblurred image, and a smaller smoothing kernel keeps it intact.

Zoom In Zoom Out Reset image size

Figure 13 shows the ridgelines corresponding to the blurred snapshots of Figure 12. The lines break where there is no local maximum in the given direction. Using just the parts of the ridgeline that do not require interpolation, the calculated roughness parameter is ${R}_{\mathrm{ring}}=0.048\pm 0.012$ in the aligned images and ${R}_{\mathrm{ring}}=0.083\pm 0.048$ in the tilted images. The reported numbers are the mean plus or minus the standard deviation over 21 snapshots.

Zoom In Zoom Out Reset image size

Even though the ring and shadow are somewhat less well defined than for the face-on viewing angle, there are two immediately apparent differences between the sets of aligned images and tilted images. First, the latter has ridgelines that are located farther from the ring center in all directions. As in Section 3.2, we quantify this by measuring a mean distance from ring center to ridgeline, averaging over all rays originating from the ring center. We then take the mean and standard deviation over the set of snapshots. In the aligned case, the average ridgeline radius is ${r}_{\mathrm{ring}}=12.47\pm 0.75\ \mu \mathrm{as};$ in the tilted case, it is ${r}_{\mathrm{ring}}=22.0\pm 3.4\ \mu \mathrm{as}$.

The other difference is in the regularity of the blurred aligned images relative to the tilted ones. The latter often have multiple distinct bright locations around the ring. To illustrate this better, we walk along the ridgelines and note the brightness temperature as a function of position angle. Figure 14 shows the resulting ring intensities for 21 snapshots in both cases, with dotted lines denoting where interpolation is used to define a sampling location. All aligned ridgelines have a single peak, reflecting the fact that each image consists of a well-defined crescent. On the other hand, most tilted ridgelines have two or three local maxima, reflecting the clumpy nature of the images.

Zoom In Zoom Out Reset image size