Sam Solod's Northwestern CIERA RUE Research Page

About Me

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Hello! I'm Sam Solod, an aspiring astrophysicist studying at DePaul University in Chicago. My journey into the mysteries of the cosmos began in a small town in Vermont, where the dark, starry skies ignited my fascination with the vast unknown.

At DePaul University I study astrophysics and mathematics. My studies allow me to delve into the wonders of the cosmos, from black holes to the fundamental forces that govern our universe. Each day, I am driven by the desire to uncover the secrets of space and contribute to our understanding of the vast, uncharted territories that lie beyond our planet.

I was honored to be accepted into Northwestern University’s Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) Research Experience for Undergraduates (REU) program during the summer of 2024. I had the incredible opportunity to work in Dr. Alexander Tchekhovskoy's group under the guidance of Dr. Deepika Bollimpalli. My research focused on exploring accretion disks around black holes in X-ray binary (BHXRB) systems.

Abstract: Observations of black hole X-ray binaries (BHXRBs) in the ‘hard’ X-ray state reveal aperiodic variability on timescales ranging from a few milliseconds to seconds. Such variability has been modelled as inward-propagating fluctuations of the mass accretion rate. Our work aims to compare 2D and 3D general relativistic magnetohydrodynamical (GRMHD) simulations of magnetized accretion disks to investigate the propagating fluctuations and explore their likely origin as being driven by dynamo processes in magnetized accretion discs.

Introduction

Black hole X-ray binaries (BHXRBs) consist of a star and a stellar-mass black hole orbiting around a common centre of mass. In these binary systems, the black hole pulls plasma off of the companion star and into the orbit of the black hole. When plasma enters the gravitational radius of the black hole, a disk-like structure forms due to the central gravitational force of the black hole (BH), which acts along the line connecting the particle and the BH. The horizontal component of this force is balanced by the centrifugal force from rotation, while the vertical component causes matter to settle toward the midplane until the gas pressure balances the vertical gravity. Plasma in a Keplerian orbit around a BH moves inward when angular momentum is transported outward due to turbulence and viscosity. As the plasma falls inward, its gravitational energy is converted to heat, part of which is radiated away. Consequently, the inner regions of the disk are the hottest, and we observe most of this heat radiation in the X-ray portion of the electromagnetic spectrum.

Many BHXRBs exhibit luminous outbursts on timescales of months to years, during which the luminosity of the system varies by orders of magnitude. When BHXRBs are observed at a relatively low luminosity, the energy spectra show that the radiation of the system peaks in the 'hard' X-ray state (peaking at approximately 100 keV). As the luminosity increases, the energy spectra evolve towards a peak in the 'soft' X-ray state (peaking at approximately 10 keV). This phenomenon, where the spectral states transition simultaneously with changes in luminosity, is called a state transition (see, e.g., the review by Dunn et al., 2007). In this paper, we focus on the variability observed prominently in the hard spectral state.

Observed power spectral densities (PSD) of systems in the hard state exhibit a broadband of power over a wide range of frequencies, known as broadband variability. Observationally, many accreting systems such as young stellar objects, active galactic nuclei, and gamma-ray bursts all display broadband variability in their respective dominant wavelengths that is characterized by a power law behavior of variability power with Fourier frequency (P ∝ ν-1). The widely accepted model for this variability is propagating fluctuations in the mass accretion rate. Although the physical process driving these fluctuations is unclear, the fact that all these systems exhibit the same characteristic feature suggests that the underlying physics of all the systems is likely the same.

The structure of accretion disks was theorized by Shakura & Sunyaev (1973) as geometrically thin, optically thick. However, new models have been developed to account for observed spectral and temporal variability. Thin disks have a relatively cool temperature because the disk can radiate efficiently, making them highly luminous (soft state). This does not describe observations of BHXRBs in the hard state, so a model that does not allow photons to radiate efficiently is necessary to correctly explore broadband variability. A geometrically thick, optically thin disk with advection-dominated accretion flow (ADAF) was theorized to account for low luminosity (Ichimaru, 1977; Narayan & Yi, 1994). Being optically thin, ADAFs are not efficient at radiating heat (Dunn et al., 2007). Much of this heat is advected onto the black hole instead of being radiated away. Our simulations are designed to mimic ADAFs.

Magneto-rotational instability (MRI) is believed to be a leading factor of turbulence influencing accretion and the loss of angular momentum in magnetized accretion disks (Balbus & Hawley, 1991). The turbulence driven by MRI significantly increases the effective viscosity within the accretion disk, therefore facilitating the efficient outward transport of angular momentum. This enhanced viscosity allows for fluctuations in the mass accretion rate (Ṁ) generated by MRI and other dynamic processes to propagate radially inward. The timescale over which matter moves radially due to these viscous forces is referred to as the viscous timescale. According to the propagating fluctuations theory, fluctuations generated on timescales longer than the viscous timescale can propagate inward and imprint onto the fluctuations generated at smaller radii, leading to strong coherence between the accretion rates at different radii. The goal of this study is to compare 2D and 3D general realistic magneto-hydro dynamical (GRMHD) simulations of magnetized ADAFs in the search for the underlying mechanisms that drive propagating fluctuations in BHXRBs. Bollimpalli et al. (2020) found that an ADAF torus disk produces propagating fluctuations when simulated in three dimensions. We explore if propagating fluctuations are evident in two dimensions. Hogg & Reynolds (2018) suggest that the large-scale dynamo that regenerates the poloidal field from the toroidal field could be driving the propagating fluctuations. Therefore, if a two-dimensional simulation showcases propagating fluctuations with an asymmetric Bφ component, we may need to reconsider what is truly producing propagating fluctuations.

Description of Simulations

The non-radiative GRMHD code HARMPI (Gammie et al., 2003; Noble et al., 2006) was used to solve the relativistic GRMHD equations of motion on a spherical polar grid (r, θ, φ) for a stationary black hole spacetime in Kerr-Schild coordinates. Two separate simulations were run with a sampling frequency of Δt = 10 GM/c3 for both. Both simulations are uniform in log(r), θ, and φ. The numerical grid spans from 1.27 rg to 8.41 × 104 rg, such that the boundary does not affect our results. We use outflow boundary conditions in the radial direction, polefix in the θ direction, and for the 3D simulation, we use periodic boundary condition in the φ-direction. Since the focus of this research is on the accretion disk, for the 3D simulation, we use cylindrification to focus the grid towards the mid-plane of the disk.

We use dimensionless units such that G = M = c = 1, where M is the mass of the black hole. This allows the gravitational radius to be rg = GM/c2 = 1. We initiate the simulation with a hydrostatic equilibrium torus given by the Fishbone-Moncrief solution (Fishbone & Moncrief, 1976), around a spinning black hole with a = 0.9. The torus has an inner edge at 6 rg and the pressure maximum is set at 13 rg. A poloidal magnetic field is used to initiate MRI. For the 2D simulation, we use a grid resolution of Nr×Nθ×Nφ = 256×128×1. Ṁ was computed over a time window [0, 100,000] GM/c3. For the 3D run, we use a grid resolution of Nr×Nθ×Nφ = 256×256×128. Ṁ was computed over a time window [0, 20,000] GM/c3.

3D Simulation Density Contour Plot with Magnetic Field Lines:

2D Simulation Density Contour Plot with Magnetic Field Lines:

Analysis & Results

Given that HARMPI is a non-radiative code, we do not have a direct estimation of the luminosity that is required to analyze the simulation. We take the mass accretion rate, Ṁ, as a proxy for luminosity. Ṁ at a certain radius and time is computed by integrating the mass flux over a spherical shell:

Ṁ(r, t) = - ∬ ρ ur √-g dθ dφ,

where ρ is the rest mass density, ur is the radial 4-velocity, and g is the metric determinant. The negative sign is tacked on in order to make Ṁ positive when matter is falling inwards towards the black hole. The first row of Figure 1 shows spacetime plots for each respective simulation. We use a subset of the 2D Ṁ data that matches the time domain of the 3D simulation to yield a better comparison. In both diagrams, there is a clear distinction between inflows and outflows at 10 rg. As the simulation evolves, outer radii begin to accrete in the 3D simulation, whereas the accretion rate in the 2D simulation exhibits small-scale variability that can be attributed to magnetic pressure buildup, which halts accretion. Due to the lack of a dynamo process in the 2D simulation, the large-scale poloidal fields required for MRI decay over time, leading to a decrease in the accretion rate.

We utilize power spectra in the search for variability in both simulations. The power spectra are computed using the Fourier transformation binning methods described by Uttley et al. (2014). At each radius, the time series data (Ṁ) is binned into a certain number of segments with the requirement that the number of time intervals per segment (N) be an integral power of 2. For each segment, a normalized periodogram, P is computed. Ṁ(ν) is the Fourier Transform of Ṁ.

P = (2Δt / ⟨Ṁ⟩2 N) * |Ṁ(ν)|2

The power spectral densities of each simulation at radius 1.677rg are shown in Figure 1. Both simulations exhibit power law behavior with the Fourier frequency power law index between 1-2. Knowing this, we are confident that both simulations exhibit broadband variability. The propagating fluctuation model states that fluctuations driven at different radii should modulate fluctuations at smaller radii as they propagate on the viscous time-scale and any two radii should exhibit high coherence below the viscous frequency and low coherence above the viscous frequency. Coherence measures the statistical similarity of fluctuations between two radii. In our case, we take two Ṁ curves at any particular radius, h(t) and s(t), and compute their discrete Fourier transform denoted by H(ν) and S(ν), respectively. In order to compute the radial coherence (γ2) of Ṁ, we once again use the work from Uttley et al. (2014):

γ2(νj) = |CHS(νj)|2 / PH(νj) PS(νj),

where CHS = H*S is the cross-spectrum averaged over multiple time segments and frequency bins, and PH(νj) and PS(νj) are the power spectra that come from h(t) and s(t). These power spectra are also time-averaged. If these two curves are exactly coherent (γ2 = 1) then they are related through a linear transformation in time. When these curves are unrelated then γ2 = 0 and they are incoherent. The radial coherence plots of each simulation are in the bottom row of Figure 2. The red dotted line represents the viscous frequency. We see extremely strong coherence below this line at all radii for each respective simulation. Both simulations show strong coherence below the viscous frequency, indicating propagating fluctuations in both. According to Hirose et al. (2018), the fluctuations at a given radius are driven at the dynamo frequency. However, our 2D simulation exhibits coherence below the viscous frequency even though it lacks the large-scale dynamo to generate MRI, thus accretion. Therefore, our results suggest that the driving mechanism of these fluctuations is independent of the dynamo process and we plan to investigate this further.

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Figure 1. Power spectral densities of each simulation at a specified radius value. Note the power-law behaviour of variability power.

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Figure 2. Space-time diagram (top row) and radial coherence diagram (bottom row) of the mass accretion rate. The left column represents the 3D simulation results, while the right column shows the 2D simulation results. Strong coherence is observed between the mass accretion rate at any given radius and the innermost radius below the local viscous frequency.

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. AST2149425, a Research Experiences for Undergraduates (REU) grant awarded to CIERA at Northwestern University. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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