Lecture 13 - Weighing the Invisible

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Speaker: Prof. Martin Bureau - Department of Physics (Astrophysics) and Wadham College

The Significance of Black Holes in Astrophysics

Hierarchical Structure Formation in the Universe

The formation of galaxies involves the following processes:

The universe expands. Overdense regions collapse, leading to the growth and merging of structures.
Dark matter dominates the gravitational dynamics. Gas accretes and accumulates, leading to the formation and evolution of stars and galaxies.
Runaway star formation occurs, yet large blue galaxies (and green ones) are not observed.
Black holes regulate gas accretion and, through feedback mechanisms, self-limit star formation.
Black holes and galaxies co-evolve, preventing the formation of large blue galaxies.

The $M_{\text{BH}}-\sigma_{\text{star}}$ Relation: Current Status

Supermassive black hole (SMBH) masses:

These masses underpin a vast theoretical framework and all indirect measurement methods. However, only a limited number of measurements (approximately 100, all local) exist.
A limited number of methods are available, each subject to unique selection effects and systematic biases.
The sample is highly biased.

Elliptical galaxies: Stellar kinematics are used to estimate SMBH masses.

Spiral galaxies: Ionised gas kinematics are employed for mass estimation.

Seyfert II galaxies: Maser kinematics provide mass measurements.

There is a pressing need for a universal method applicable to all galaxy types, conceptually straightforward (free from systematics), and easily scalable for rapid application.

Light Properties and Telescope Techniques

Light, as an electromagnetic wave, exhibits the following properties:

Intensity: Imaging
Wavelength: Spectroscopy
Polarisation: Polarimetry

Spectroscopic Techniques

Spectral lines (emission or absorption) provide information on:

Chemical composition
Physical state
Kinematics (via the Doppler effect): $\frac{\Delta v}{c} = \frac{\Delta \lambda}{\lambda}$ (non-relativistic approximation)

Spatial Resolution

Limitations in optical resolution include:

Diffraction limit: $\theta \approx \frac{\lambda}{D}$, where $\theta$ is the angular resolution, $\lambda$ is the wavelength, and $D$ is the telescope diameter.

Resolution is proportional to the telescope diameter and inversely proportional to the wavelength.

prop. telescope diametre
inv. prop. wavelength
Natural seeing conditions typically result in an angular resolution of $\theta \approx 1''$ in the optical regime.

Techniques to improve resolution include:

Speckle imaging
Adaptive optics (using laser guide stars or natural stars)
Interferometry

Synthesis Imaging Techniques

Key principles of synthesis imaging include:

A baseline is formed between two antennas.
The number of baselines is given by $= \frac{N(N - 1)}{2}$, where $N$ is the number of antennas.
The u-v plane represents the spatial frequency domain.
The image is obtained as the Fourier transform of the (sparsely populated) u-v plane.
The synthesized beam and resulting image are produced.

Spatial frequency filtering involves:

The maximum baseline determines the highest resolution (smallest detectable structures).
The minimum baseline determines the lowest resolution (largest detectable structures).

The Black Hole at the Centre of the Milky Way

Measuring the Mass of the Central Black Hole

The measurement is based on the following principles:

$F = ma$
$\frac{GMm}{R^{2}} = \frac{mv^{2}}{R}$
$M \propto v^{2}R$
$v \propto \frac{M^{\frac{1}{2}}}{R^{\frac{1}{2}}}$

Stellar Orbits in the Milky Way

The analysis is based on the following principles:

$F = ma = m \frac{\text{d}v}{\text{d}t} = m \frac{\text{d}^{2}x}{\text{d}t^{2}}$

However, the underlying cause of the stars' orbital motion remains a key question.

Observing Black Holes with ALMA

The Atacama Large Millimeter/submillimeter Array (ALMA)

A collaboration between Europe, North America, and East Asia, hosted in Chile.
A $1.5 billion investment.
Now fully operational.
Comprising 50 antennas of 12 metres and 12 antennas of 7 metres.
Multiple large arrays for enhanced resolution.
Capable of observing across a wide range of frequencies.
Located at an exceptional site with minimal atmospheric interference.

Measuring Dark Mass with ALMA