Difficulties with Recovering the Masses of Supermassive Black Holes from Stellar Kinematical Data

By M. Valluri, D. Merritt, Eric Emsellem. 2002

152 citations recorded by the source corpus.

We investigate the ability of three-integral, axisymmetric, orbit-based modeling algorithms to recover the parameters defining the gravitational potential (mass-to-light ratio ϒ and black hole mass M•) in spheroidal stellar systems using stellar kinematical data. We show that the potential estimation problem is generically underdetermined when applied to long-slit kinematical data of the kind used for most black hole mass determinations to date. A range of parameters (ϒ, M•) can provide equally good fits to the data, making it impossible to assign best-fit values. The indeterminacy arises from the large variety of orbital solutions that are consistent with a given mass model. We demonstrate the indeterminacy using a variety of data sets derived from realistic models, as well as published observations of the galaxy M32. The indeterminacy becomes apparent only when a sufficiently large number of distinct orbits are supplied to the modeling algorithm; if too few orbits are used, spurious minima appear in the χ2(ϒ,M•) contours, and these minima do not necessarily coincide with the parameters defining the gravitational potential. We show that the range of degeneracy in M• depends on the degree to which the data resolve the radius of influence rh of the black hole. For FWHM/2rh ≳ 0.5, where FWHM refers to the instrumental resolution, we find that only very weak constraints can be placed on M•. In the case of M32, our reanalysis demonstrates that when a large orbit library is used, data published prior to 2000 (FWHM/2rh ≈ 0.25) are equally consistent with black hole masses in the range 1.5 × 106 M☉ < M• < 5 × 106 M☉, with no preferred value in that range. Exactly the same data can reproduce previous published results with smaller orbit libraries. While the Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) data for this galaxy (FWHM/2rh ≈ 0.06) may overcome the degeneracy in M•, HST data for most galaxies do not resolve the black hole's sphere of influence and in these galaxies the degree of degeneracy allowed by the data may be greater than previously believed. We investigate the effect of regularization, or smoothness constraints, on the degree of degeneracy of the solutions. Enforcing smoothness reduces the range of acceptable models, but we find no indication that the true potential can be recovered simply by enforcing smoothing. For a given smoothing level, all solutions in the minimum χ2 valley exhibit similar levels of noise; as the smoothing is increased, there is a systematic shift in the midpoint of the χ2 valley, until at a high level of smoothing the solution is biased with respect to the true solution. These experiments suggest both that the indeterminacy is real (i.e., that it is not an artifact associated with nonsmooth solutions) and that there is no known empirical way to choose the smoothing parameter to ensure that the correct solution is selected.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/e8ad37e0a4b567bd8f7169a75458241eecdb8385

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In Astronomy and Astrophysics

1. M. Cappellari
2. P. Hopkins
3. L. Hernquist
4. M. Sarzi
5. T. Naab
6. Luis C. Ho
7. Eric Emsellem
8. J. Kormendy

Citations in the corpus

• Structure and Kinematics of Early-Type Galaxies from Integral Field Spectroscopy (2016, methodology): “…because the DF is a function of the three isolating integrals of motion (Jeans 1915) and one cannot expect to uniquely constrain both the 3-dim DF and the 3-dim mass distribution using only a 3-dim observable, as provided by the LOSVD at every spatial location (e.g. Valluri et al. 2004, section 3).”
• Galaxy Masses: A Review (2013, methodology): “However it is unlikely to robustly constrain additional quantities from the same 63 3D data (Valluri et al. 2004), namely the 3D total mass distribution and the angles at which it is observed.”
• NEW LIMITS ON AN INTERMEDIATE-MASS BLACK HOLE IN OMEGA CENTAURI. II. DYNAMICAL MODELS (2009, background): “…is fairly indirect, and to exploit it one must construct complicated numerical models that retrieve the full phase-space distribution function of the stars (van der Marel et al. 1998; Cretton et al. 1999; Gebhardt et al. 2000; Valluri et al. 2004; Thomas et al. 2004; van den Bosch et al. 2008).”
• THE BLACK HOLE MASS IN M87 FROM GEMINI/NIFS ADAPTIVE OPTICS OBSERVATIONS (2011, methodology): “These models are based on Schwarzschild’s (1979) method, and similar models are presented in Richstone & Tremaine (1984), Rix et al. (1997), Cretton et al. (1999), and Valluri et al. (2004).”
• The SAURON project – X. The orbital anisotropy of elliptical and lenticular galaxies: revisiting the (V/σ, ɛ) diagram with integral‐field stellar kinematics (2007, background): “Similar axisymmetric implementations were developed by other groups (Gebhardt et al. 2003; Valluri et al. 2004; Thomas et al. 2004).”
• The SAURON project - IV. The mass-to-light ratio, the virial mass estimator and the Fundamental Plane of elliptical and lenticular galaxies (2005, methodology): “A similar approach was adopted by us and by other groups to measure the black hole (BH) masses in galaxy nuclei or to analyse the stellar orbital distribution (e.g. Cretton et al. 1999; Cappellari et al. 2002; Verolme et al. 2002; Gebhardt et al. 2003; Valluri et al. 2004; Thomas et al. 2004).”
• Triaxial orbit based galaxy models with an application to the (apparent) decoupled core galaxy NGC 4365 (2007, background): “…and that proper motion kinematic observations can be used (van de Ven et al. 2006), provided that the models have sufficient internal freedom, e.g. the total number of orbits is large enough (Cretton & Emsellem 2004; Thomas et al. 2004; Richstone et al. 2004; Valluri et al. 2004; Magorrian 2006).”
• Supermassive Black Holes in Galactic Nuclei: Past, Present and Future Research (2004, background): “…which have addressed the issue (Ferrarese & Merritt 2000; Merritt & Ferrarese 2001b&c; Graham et al. 2001; Ferrarese 2002a; Marconi & Hunt 2003; Valluri et al. 2004) have concluded that resolving the sphere of influence is an important (although not sufficient) factor: not resolving rh can…”
• The ATLAS3D project XV: benchmark for early-type galaxies scaling relations from 260 dynamical models: mass-to-light ratio, dark matter, fundamental plane and mass plane (2012, background): “It is unlikely for the data to contain enough information to constrain additional parameters, like the dark matter halo shape and the viewing angles (e.g. Valluri et al. 2004).”
• Mass Deficits, Stalling Radii, and the Merger Histories of Elliptical Galaxies (2006, background): “Of course, the uncertainties associated with the Mdef and M values in Figure 7may be large, the former due to uncertainM/L corrections, the latter due to various difficulties associated with SBHmass estimation (e.g., Maciejewski & Binney 2001; Valluri et al. 2004).”
• THE BLACK HOLE MASS, STELLAR MASS-TO-LIGHT RATIO, AND DARK HALO IN M87 (2009, methodology): “Similar orbit-based models are presented, for example, in Rix et al. (1997), van der Marel et al. (1998), Cretton et al. 1999, Valluri et al. (2004).”