Scaling Relations Between Supermassive Black Hole Mass and Properties of Their Host Stellar Components - astro-wiki/astrowiki GitHub Wiki

In the local universe, profound correlations $-$ Scaling Relations $-$ have been found between the mass of supermassive black holes (SMBH) and velocity dispersion as well as stellar mass of the bulge component inside the galaxies they reside (e.g., Magorrian et al. 1998; Ferrarese & Merrit 2000; Gebhardt et al. 2000; Kormendy & Ho 2013), often expressed in the following form (Kormendy & Ho 2013):

$$ \begin{align} {\rm log},(\frac{M_{\rm BH}}{10^9, M_{\odot}})=-(0.509 \pm 0.049)+(4.384 \pm 0.287),{\rm log},(\frac{\sigma}{200,{\rm km},{\rm s}^{-1}}),,{\rm intrinsic},{\rm scatter}=0.29,{\rm dex} \end{align} $$

$$ \begin{align} {\rm log},(\frac{M_{\rm BH}}{10^9, M_{\odot}})=-(0.310 \pm 3.901)+(1.170 \pm 0.080),{\rm log},(\frac{M_{\rm bulge}}{10^{11},M_{\odot}}),,{\rm intrinsic},{\rm scatter}=0.28,{\rm dex} \end{align} $$

Such correlation shows the smallest scatter for ellipticals or classical bulges, both of which are dispersion dominated systems that fall onto the fundamental plane. Extending similar measurement to the entire galaxy that includes potential disks or pseudobulges would significantly increase the intrinsic scatter, resulting in the following relations (Greene et al. 2020, all galaxy types, no upper-limits included):

$$ \begin{align} {\rm log},(\frac{M_{\rm BH}}{10^9, M_{\odot}})=-(0.699 \pm 0.055)+(4.340 \pm 0.240),{\rm log},(\frac{\sigma}{200,{\rm km},{\rm s}^{-1}}),,{\rm intrinsic},{\rm scatter}=0.53,{\rm dex} \end{align} $$

$$ \begin{align} {\rm log},(\frac{M_{\rm BH}}{10^9, M_{\odot}})=-(0.713 \pm 0.113)+(1.390 \pm 0.130),{\rm log},(\frac{M_{*}}{10^{11},M_{\odot}}),,{\rm intrinsic},{\rm scatter}=0.79,{\rm dex} \end{align} $$

SMBH, with $M_{\rm BH}\gtrsim 10^6,M_{\odot}$, should dominate the gravitation field within $R_{\rm e}\lesssim \frac{GM_{\rm BH}}{\sigma}$, in which $\sigma$ represents central velocity dispersion of the bulges inside their host galaxies. This radius usually varies from $1\sim 100, {\rm pc}$, depending on different stellar mass. On the other hand, the value of $\sigma$ in equation (1), by convention, is usually taken as the averaged stellar velocity dispersion within the effective radius of the bulge, tracing gravitation field at spatial scale $\gtrsim 1,{\rm kpc}$. The tight correlation in equation (1) and (2), with only an intrinsic scatter of $0.3\pm 0.1,{\rm dex}$, should therefore indicate a physical mechanism besides gravity that is capable of linking the two components whose sphere of influences differ by a factor of $\sim 1000$ in size. The black hole-host scaling relations, since their discoveries, have led to the rapid expansion of coevolution scenario, in which mutual influence between galaxies and the SMBH at galactic nuclei constrain their growth in lockstep across cosmic time. Instead of merely being a high-energy laboratory for testing exotic physics, SMBH are gradually being merged into the entire picture of galaxy ecosystem.

Black hole mass

Dynamical Measurement of the Black Hole Mass

The motion of matter at a certain radius is influenced by the enclosed mass in the following way described by collisionless Boltzmann equation:

$$ \begin{align} -\frac{d}{dr}(\rho \sigma_r^2)+\frac{2 \rho}{r}(\sigma^2_t-\sigma^2_r)+\frac{\rho v_{rot}^2}{r}=\frac{GM(r)\rho}{r^2} \end{align} $$

in which $\sigma_r$ and $\sigma_t$ are the tangential and radial velocity dispersion, $v_{rot}$ representing the rotation velocity. Therefore, detecting SMBH requires dynamical measurement with spatial resolution comparable to the size of black hole sphere of influence, which can be obtained by conducting long-slit spectroscopy at different radius near the galactic nuclei, or through IFU observation.

The very first detection was from Tonry 1987, who placed the slit at different location of M32 to map the radial distribution of rotation velocity and velocity dispersion. Their results suggested an dark massive object located within the central $\sim 1,{\rm kpc}$, with $\sim 10^6,M_{\odot}$ and extremely large mass-to-light ratio. More solid evidence came later with the aid of HST, which can resolve the central one parsec (van der Marel et al. 1997, 1998). Adopting the formalism $\psi_{\rm dark}=GM_{\rm dark}(r^2+\epsilon^2)^{-1/2}$ to describe the gravitational potential of the central dark object, van der Marel et al. 1998 found that the velocity dispersion radial distribution can be best fitted with $M_{\rm dark}=4\times 10^6,M_{\odot}$ and $\epsilon=0$, indicating the potential of a point-mass (figures below).

Black hole in M32

A major uncertainty in black hole dynamical detection and measurement lies in the two terms regarding velocity dispersion in equation (5), $\sigma_r$ and $\sigma_t$. As spectroscopic observations can only measure velocity information along the line of sight, velocity anisotropy needs to be estimated based on various assumptions. Such uncertainty is non-trivial since the second term in equation (5) solely depends on the difference between the tangential and radial value of $\sigma$. This can be partly alleviated by the orbital superposition technique (Schwarzschild 1979, 1993), in which an orbit template library is constructed for the gravitational potential of the central point mass $M_{\rm BH}$ (to be determined) and stellar density (derived from the observed surface brightness distribution and stellar mass-to-light ratio which is estimated from color). An optimal templates combination is then calculated to best-match observed distribution of line-of-sight velocity and velocity dispersion. Such method is usually applied to model IFU datacubes.

Measurement of the Host Galaxy Properties

Stellar Velocity Dispersion

Low temperature gas in stellar atmosphere absorbs the stellar continuum, producing absorption lines that can trace stellar dynamics. A major problem is the influence from stellar internal kinematics. Since being produced in the stellar atmosphere, different absorption lines are subjected to pressure and/or rotation broadening by various degree.

Ca triplets in nuclei and outskirt

The most ideal tracer would be the Ca II IR triplet lines at rest-frame 8498, 8542 and 8662 ${\rm \AA}$, as they are prominent in a wide variety of stellar types (as shown in the figures below, Ca triplets profiles vary little from bluest galactic nuclei to relatively red outskirt) while being intrinsically narrow compared to galactic velocity dispersions, with intrinsic width of $\sigma \sim 20-30,{\rm km}, {\rm s}^{-1}$ ranging from K giant stars to supergiant stars (e.g., Dressler 1984; Barth et al. 2002; Filippenko & Ho 2003; Greene et al. 2006). Other choices includes Mg I $b$ triplet at 5167, 5173 and 5184 ${\rm \AA}$, which contain both a few strong lines and a forest of weak features that are particularly useful in measuring dispersions less than 150 km ${\rm s}^{-1}$, as well as Ca II H and K lines ($\sim 4000,{\rm \AA}$), which are extremely strong but intrinsically very broad, making them suitable only for measurements of large velocity dispersions ($\gtrsim 300,{\rm km},{\rm s}^{-1}$).

In technical terms, line-of-sight velocity dispersion is estimated by convolving the stellar template spectrum with a Gaussian kernel representing velocity dispersion broadening, and then being matched to the observed spectrum for optimization:

$$ \begin{align} M(x)={[T(x),\otimes G(x)]+C(x)}P(x) \end{align} $$

In the above equation, $M$ is the model spectrum, $T$ is the stellar template spectrum, $G$ is the Gaussian broadening function, $C$ is a featureless continuum and $P$ is a polynomial responsible for possible reddening between template and data spectrum.

Host Galaxy Stellar Mass Measurement

Galaxies in the local universe are generally well-resolved. Stellar mass and luminosity can be derived either from stellar population modeling using multi-band photomety or from color-related mass-to-light ratio. One of the dilemma is the bulge luminosity and stellar mass measurement for spiral galaxies, in which accurate bulge-disk image decomposition is required.

Sun Wen will be invited to finish the rest of the content in this subsection

Scaling Relations in Distant Universe

The scaling relations serve as a paradigm for studying the coevolution between SMBH and their host galaxies. One way to understand the early formation of black holes and galaxies is to study their redshift evolution relative to the local scaling relations. Currently, results for high redshift scaling relations are still ambiguous due to several major difficulties.

To start with, the sphere of inflence for the central SMBH can hardly be resolved once going beyond the local universe. Identification of their presence as well as measuring $M_{\rm BH}$ therefore relies on type 1 Active Galactic Nuclei (AGN), which shows prominent broad permitted emission lines with full width at half maximum (FWHM) $\gtrsim 1000,{\rm km},{\rm s}^{-1}$ (e.g., Weedman 1977; Hao et al. 2005) that are thought to originate from gas moving at high speed near the black hole (broad-line region, BLR). With multi-epoch time-domain observations, reverbration mapping technique can model the structure of the broad-line region as well as its distance to the SMBH. Together with the emission line width which tells us the speed, black hole mass can be estimated by:

$$ \begin{align} M_{\rm BH}=f\frac{R_{\rm BLR}(\Delta V)^2}{G} \end{align} $$

where $f$ is the virial factor representing the structure and velocity distribution of the BLR clouds, $R_{\rm BLR}$ is the distance between the BLR and SMBH and $\Delta V$ is the speed traced by emission line FWHM. The virial factor $f$, which is usually treated as a constant (e.g., Netzer 1990), is also shown to vary from object to object (e.g., Xiao et al. 2018; Yu et al. 2019). Reverbration mapping requires long-term observations that can be expensive and time-consuming. A more advanced techniques named near-infrared interferometry (GRAVITY Collaboration 2024) can spatially resolve the broad-line region through differential phase signal, but is currently limited to only a few specified targets. Alternatively, an empirical relation between BLR size and AGN continuum luminosity (usually measured at 5100 ${\rm \AA}$), namely the R-L relation (Kaspi et al. 2000; Bentz et al. 2013), is adopted in many works to estimtate black hole mass using single-epoch data. Overall, measuring black hole mass in high redshift AGNs is still hindered by limited observational resources as well as various model assumptions.

Regarding host galaxy properties, a critical aspect is to disentangle the emission from the host and central nuclei. For type 1 AGNs, their relatively unobscured nature makes the nuclei extremely bright, sometimes overshining the entire host galaxy. AGN-host decomposition are performed either using integrated spectrum (e.g., Shen et al. 2015; Ren et al. 2024) and SED from multi-band photometry (e.g., Merloni et al. 2010), which fits the continuum and/or absorption line features using different sets of AGN and stellar population synthesis model (see more detailed information), or by decomposing multi-band images using parametric surface brightness profiles (e.g., Ding et al. 2020, 2023; Kocevski et al. 2023; Chen et al. 2024) (see more detailed information).

High-z Scaling Relation

Currently, scaling relations beyond the local universe are still ambiguous. In the pre-JWST era, some works report consistent scaling relations as the local universe for X-ray selected type 1 AGNs at $0<z<2$ (e.g., Jahnke et al. 2009; Shen et al. 2015; Sun et al. 2015;), while there have also been mounting evidence that high-redshift quasars appear to have systematically overmassive black holes (e.g., Walter et al. 2004; Peng et al. 2006; Merloni et al. 2010; Kormendy & Ho 2013). The advent of JWST allows the detection of rest-frame optical emission for AGNs at $z>4$, enabling the detection of resolved host galaxy stellar population. The majority of works find overmassive black holes a factor of $10\sim 100$ compared with local relations (shown in the above figure) (e.g., Harikane et al. 2023; Maiolino et al. 2023; Pacucci et al. 2023; Chen et al. 2024; Furtak et al. 2023), where the most extreme case has been reported in Goulding et al. 2023, claiming that it is possible for the X-ray luminous AGN at $z=10.1$, UHZ-1, to have $M_{\rm BH}/M_\sim 1$. On the other hand, some study indicate consistent $M_{\rm BH}-M_$ and $M_{\rm BH}-\sigma_*$ relations as the local universe in low-luminosity quasars (Ding et al. 2023), with modeled star-formation history even pointing towards undermassive black holes in the past 0.3 Gyr (Onoue et al. 2024).