The Galactic Centre in Infra-Red

The European Southern Observatory’s (ESO) GRAVITY instrument is a beam combiner in the infra-red K-band that operates as a part of the Very Large Telescope Interferometer, combining infra-red light received by four different telescopes, out of the eight operated (four 8.2 metre fixed telescopes and four 1.8 metre movable telescopes).

The latest measurements of the stars orbiting the Milky Way’s Galactc Core Super-Massive Black Hole (SMBH), otherwise known as Sagittarius A*, by the GRAVITY instrument have determined its mass and distance to new levels of accuracy:

[2101.12098] Improved GRAVITY astrometric accuracy from modeling of optical aberrations

R_{o} = 8,275 parsecs (+\-) 9.3 parsecs and a mass of (in 10^{6} M_{sol}) 4.297 +\- 0.013.

In round figures, that’s 27,000 light-years and 4.3 million Solar masses. The closest that light can approach a Black Hole and still escape is the Event Horizon, which is the spherical boundary at the distance of the Schwarzschild Radius, which is a radius of 2.95325 kilometres per solar mass. Thus 4.3 million solar masses is wrapped in an Event Horizon 12.7 million kilometres in radius. In aeons to come, when the Milky Way and M31 have collided and their black holes have coalesced, the combined Super Massive Black Hole (SMBH) will mass 100 million solar masses with an event horizon almost 300 million kilometres in radius.

**Into the Maelström**

Mass-energy, so General Relativity tells us, puts dents into Space-time. Most concentrations of mass-energy, like stars, planets and galaxies, form shallow dents. Black Holes – like the future SMBH – go deeper, forming an inescapable waterfall of space-time inwards to their centres, the edge of which is the Event Horizon.

Light follows the curvature of space-time, traveling the shortest pathways (geodesics). At the Event Horizon the available geodesics all point towards the “middle” of the Black Hole. For particles with rest mass, like atoms, dust and space-ships, geodesics can’t be followed, merely approached, so they follow different pathways just as inexorably towards the centre.

Instead of flying radially inwards towards the future SMBH’s Centre, let’s ponder orbiting it. For most orbital distances from any Black Hole any small mass in orbit will experience nothing different to orbiting around any other large mass. Too close and you’ll experience extreme tidal forces if the black hole is small, so to avoid being torn to shreds when approaching really close a really big Black Hole is needed. The future SMBH massing 100 million solar masses with a Schwarzschild radius of 300 million kilometres has very mild Tidal Forces at the Event Horizon, though potentially significant for things as big as stars and planets.

We have multi-year ‘movies’ of stars orbiting around our SMBH, though none as close as we will explore. Close orbits get measured in multiples of M – which is half the Schwarzschild radius. At an radius of \(r = 3M\) space-time is so curved that the geodesics form a circle around the Black Hole. Light can thus orbit indefinitely, building up to potentially extraordinary energy densities if nothing else gets in its way, forming a so-called ** Photon Sphere**. But the centre of the Galaxy is full of dust and gas, so something is always getting in the way. Eventually even photons get so energetic they perturb each other out of the Sphere.

For particles that don’t follow geodesics, merely approximate them, the **Innermost Stable Circular Orbit (ISCO)** is further out, at \(r = 6M\). Objects here travel at half the speed of light. Other shapes of orbits can dip a bit closer in, down to \(r = 4M\). Deeper in and motion near the Black Hole is no longer “orbital”. You must point away from the Black Hole and apply thrust or in-fall is inevitable.

The equation of orbital motion from the ISCO radius (\(r_I\)) all the way to the centre of was only recently worked out in closed form for rotating and non-rotating (stationary) Black Holes. Previously numerical Relativity methods were used, complicating modelling of Accretion Disks around astrophysical Black Holes. The Equation of Motion of a test particle (i.e. very small mass) around a non-rotating Black Hole, which our future SMBH might approximate, is quite simple:

$$x = \frac{r}{r_I} = \frac{r}{6M} = \left(\frac{1}{1+\frac{\phi^2}{12}}\right)$$

\(\Phi\) is the angular distance travelled, with a range from negative infinity to zero, by convention. I’ve plotted \(r\) against \(\Phi\) here:

The Red Circle at \(r = 3M\) is the Photon Sphere and the Yellow Circle at \(r = 2M\) is the Schwarzschild Radius aka Event Horizon. In this case the plot starts at \(r = 5.95 M\) with the test particle circling the Black Hole 6 times before hitting the central point. The proper time experienced by an observer spiralling into the Centre is a bit more complicated. We can parameterise \(x\) as follows to make the mathematics easier:

$$x = \frac{1}{2}(1-cos\psi) = sin^2(\frac{\psi}{2})$$

with \(\psi\) running from an angle \(\pi\) to 0. Then the Proper-Time \(\tau\) of the inspiral trajectory is:

$$\tau = \sqrt{\frac{3r_I^3}{2M}}\left[\frac{1}{2}(3\psi-sin\psi)-2tan(\frac{\psi}{2})\right]$$

The above equation is true for any black-hole, spinning or stationary. For a stationary Black Hole, \(r_I = 6M\), so the equation simplifies to:

$$\tau = 18M\left[\frac{1}{2}(3\psi-sin\psi)-2tan(\frac{\psi}{2})\right]$$

But what is \(M\)? It’s the “geometrised” mass of the Black Hole, which is derived by muliplying the mass by \(\frac{G}{c^2}\). Similarly the proper time is in units of “geometrised” time, so it needs to be divided by the speed of light, \(c\), to convert to seconds.

In the case of the fall from \(r = 5.95M\) to \(r = 0\), thus \(\psi = (5.95/6)*(\pi)\) to \(\psi = 0\), the total time is \(\tau = 1291.14 M\). In the case of our Galaxy’s SMBH a proper time of \(M\) is 493 seconds. So the inspiral time is 176.6 hours and the Event Horizon is reached with 1.32 hours to go.

**Reference:**

Mummery, A. & Balbus, S. “Inspirals from the innermost stable circular orbit of Kerr black holes: Exact solutions and universal radial flow” (2022) https://doi.org/10.48550/arXiv.2209.03579

[accepted to Physical Review Letters: Phys. Rev. Lett. 129, 161101 – Published 12 October 2022]

**Surviving the Plunge?**

Falling into a Black Hole is probably fatal. However, like any fall, it’s not gravity that’ll kill you, but the sudden stop at the end. The final destination is the concentration of mass at the very centre. As the Centre is approached the first derivative of gravitational acceleration with respect to the radial distance vectors – *the tidal forces* – that will be experienced **will** become extreme.

Black holes are the pointy end of a spectrum of astrophysical objects. Stars exist due to their dynamic balance between the outward pressure from their fusion energy production and the inward pressure from their self gravity. When fusion energy production ends, the cores of stars begin collapsing, held above the Abyss of gravitational collapse by successive fusion energy reactions, then **electron degeneracy pressure** from squeezing free electrons too close together (via the Pauli Exclusion Principle), and when that isn’t enough, **neutron degeneracy pressure** and beyond.

Pressure is a measure of the ‘expansive’ energy packed in a volume. Dimensionally we can see that \(F/m^2\) *(Pressure)* = \(E/m^3\) *(Energy per unit volume)*, so that as the mutual gravitational squeeze pushes inwards on a mass of particles which are pushing back against each other thanks to the Pauli Exclusion Principle for fermions (electrons, protons, neutrons etc) that pressure increases and increases, in a feed-back loop. Too much and equilibrium is never achieved. Thanks to Special Relativity we know that energy has mass, so that Pressure adds to the inward squeeze of gravity as particles are squeezed harder together. When the Gravity Squeeze – Push-Back Pressure process self-amplifies and runs away, the mass collapses ‘infinitely’ inwards forming a Singularity. Such a Singularity cuts itself of from the rest of the Universe when it squeezes inwards past the mass’s *Schwarzschild Radius*:

$$r_{EH} = \frac{2GM}{c^2}$$

The resulting Event Horizon defines a Black Hole, by being a **‘surface of no return’** for everything, including light. Nothing escapes from within the Event Horizon. The minimum mass to cause such an inwards collapse and form a Black Hole for a mass of fermions (i.e. the same particles that make Stars, humans and space-ships) in the present day Universe is about 3 solar masses, squished into a volume smaller than 18 kilometres across.

Before we get to that point there are **White Dwarfs** and **Neutron Stars** – objects supported against collapse by Electron Degeneracy Pressure and Neutron Degeneracy Pressure, respectively. White Dwarfs are typically composed of carbon and oxygen – the ashes of helium fusion – and have observed masses anywhere between 0.1 and 1.3 Solar masses. Their radius is proportional to the inverse 1/3 power of their mass:

$$R(M) = \frac{R*}{M^\frac{1}{3}}$$

R* is a reference radius. For a cool white dwarf of 1 solar mass, the radius is about 0.8 Earth’s – 5,600 km. A space vehicle falling from infinity, on a flyby very close to such a star’s surface will rush past the lowest point of its orbit at 6,900 km/s, experiencing over 430,000 gee acceleration. In free-fall however it feels only the first derivative of that acceleration:

$$\dot g(R) = -\frac{2GM}{R^3}$$

Which in this example is 0.15 gee per metre of radial stretching directed outwards and inwards along the direction of the radial distance to the white dwarf and a squeezing force half that directed laterally inwards from the sides. Easily resisted by small structures, like bodies and space-ships.

Neutron stars are smaller again – typically 20 km wide for a 1.3 solar mass neutron star. A near surface flyby isn’t recommended, since the tidal forces are thus almost a million times stronger. Close proximity to a magnetic neutron star is probably lethal anyway due to the intense magnetic fields long before the tidal forces rip you to shreds. Heavier neutron stars get smaller – just like white dwarfs – until they totally collapse as a black hole.

Black holes reverse the trend. The Event Horizon gets bigger linearly with their mass and there’s no upper limit to their mass. Our future Galactic SMBH’s Event Horizon will be 295.325 million kilometres in radius, give or take. Substituting the Schwarzschild Radius equation into the Tidal force equation gives us:

$$\dot g(R) = -\left(\frac{c}{r_{EH}}\right)^2$$

So the tidal force at the Event Horizon is 0.1 microgee per metre. The Moon could almost enter the Event Horizon peacefully…

How far into the SMBH can we, as Observers, then fall? If we can brace ourselves against 1,000 gees per metre of squeezing and stretching, then quite a long way…

$$R = \left(\frac{r_EH.c^2}{\dot g(R)}\right)^{\frac{1}{3}}$$

Which gives a distance of 139,430 kilometres from the centre. In other words 99.953% of the way to the central Singularity.

What wonders might we see? Quantum Gravity is yet to give a clear answer. Traditionally an imploding mass ends in the Singularity, which is a geometrical point. But quantum particles can’t be reduced to a singular point and retain quantum information. A possibility, due to the massively distorted space-time around the collapsing mass, is that ultimately the quantum particles all “bounce” after hitting Planck density and explode back outwards. To external Observers this is seen, in time-dilated fashion, as the slow-leak from the Event Horizon that is Hawking Radiation. Or, if the particles “twist” in a higher dimension, so they bounce as a new Big Bang forming another Universe. This can be seen as an emergence from a **White Hole**, as White Holes must keep expanding else they collapse into another Black Hole.

None of those options are ‘healthy’ to be around as flesh-and-blood Observer, so presently surviving the plunge is in doubt.