by Paul Gilster | Feb 14, 2023 | Exoplanetary Science |
I want to mention the recent confirmation of K2-415b because this world falls into an interesting category: Planets with major implications for studying their atmospheres. Orbiting an M5V M-dwarf every 4.018 days at a distance of 0.027 AU, this is not a planet with any likelihood for life. Far from it, given an equilibrium temperature expected to be in the range of 400 K (the equivalent figure for Earth is 255 K). And although it’s roughly Earth-sized, K2-415b turns out to be at least three times more massive.
What this planet has going for it, though, is that it transits a low mass star, and at 70 light years, it’s close. Consider: If we want to take advantage of transmission spectroscopy to study light being filtered through the planetary atmosphere during ingress and egress from the transit, nearby M-dwarf systems make ideal targets. Their habitable zones are close in, so we get frequent transits around small stars. But the number of Earth-sized transiting worlds around nearby examples of such stars is small, totalling 14 in eight 8 different systems (limiting the range to 30 pc, or roughly 100 light years). That includes the high-priority seven around TRAPPIST-1.
Image: This is Figure 3 from the paper. Caption: Sensitivity plot (5 ? contrast curve) for K2-415 in the K0 band based on the combined IRCS image. The inset shows the zoomed image of the target with a FoV of 4” × 4”. Credit: Hirano et al.
As we move into ever deepening searches for atmospheric biomarkers, worlds like these are going to be in the vanguard, fodder for the emerging class of 30-meter telescopes and, of course, space-based observatories like the James Webb Space Telescope. We’re going to get to know this small group of worlds well as this decade continues, which will also greatly assist us in understanding how M-dwarf planets evolve. When it comes to atmospheres around small red stars, there are no guarantees.
After all, these stars especially in their youth are known to be intense emitters of extreme ultraviolet and X-ray radiation, with striking levels of flare activity. Can an atmosphere survive this bombardment, or an intense stellar wind? Will these planets evolve secondary atmospheres through surface outgassing from volcanoes and interactions with magma? Doubtless we’ll find examples of various scenarios, which should deepen our knowledge of how the atmospheres of habitable worlds emerge.
It will also be interesting to see if the apparent and recently noted (in TESS data) lack of detected planets around the lowest mass stars will be supported by later datasets. K2-415b forces that question, in that it orbits one of the lowest mass stars hosting an Earth-sized planet. Counting planets of any size, only ten transiting planet-hosting stars are cooler than K2-415. One of these is TRAPPIST-1.
K2, the extended Kepler mission, observed K2-415b, an unusually close system at 70 light years given that of the original Kepler target stars, fewer than one percent were closer than 600 light years (we have much to thank K2 for as it moved out of the original Kepler field, a classic case of making a virtue out of necessity). The follow-up campaign described in today’s paper validates the planet. And the authors note that future radial velocity studies will be on the lookout for additional planets in this system.
This too could get interesting. Lead author Teruyuki Hirano (Graduate University for Advanced Studies, Japan) and colleagues explain::
K2-415b is located slightly inner of the classical habitable zone (Kopparapu et al. 2016) based on the insolation flux onto the planet, but an outer planet, if any, in the system with a slightly longer period (e.g., 10 ? 15 days) could sit inside the habitable zone. Little is known on the properties of multi-planet systems around the lowest mass stars (< 0.3 M), but assuming that their properties are similar to those in the “Kepler-multi” systems, the planets could have a typical spacing of ? 20 mutual Hill radii (Weiss et al. 2018). Recently, Hoshino & Kokubo (2023) also showed that the typical orbital spacing of planets formed by giant impacts is ? 20 mutual Hill radii independently of stellar masses through N-body simulations. Thus, it is quite possible that a secondary planet having ? 1 M? has an orbital period of 10 ? 15 days.
The paper is Hirano et al., “An Earth-sized Planet around an M5 Dwarf Star at 22 pc,” accepted for publication at the Astrophysical Journal and available as a preprint.
by Paul Gilster | Feb 8, 2023 | Missions |
Over the past several years we’ve looked at two missions that are being designed to go beyond the heliosphere, much farther than the two Voyagers that are our only operational spacecraft in what we can call the Local Interstellar Medium. Actually, we can be more precise. That part of the Local Interstellar Medium where the Voyagers operate is referred to as the Very Local Interstellar Medium, the region where the LISM is directly affected by the presence of the heliosphere. The Interstellar Probe design from Johns Hopkins Applied Physics Laboratory and the Jet Propulsion Laboratory’s Solar Gravity Lens (SGL) mission would pass through both regions as they conduct their science operations.
Both probes have ultimate targets beyond the VLISM, with Interstellar Probe capable of looking back at the heliosphere as a whole and reaching distances are far as 1000 AU still operational and returning data to Earth. The SGL mission begins its primary science mission at the Sun’s gravitational lens distance on the order of 550 AU, using the powerful effects of gravity’s curvature of spacetime to build what the most recent paper on the mission calls “a ‘telescope’ of truly gigantic proportions, with a diameter of that of the sun.” The vast amplification of light would allow a planet on the other side of the Sun to be imaged at stunning levels of detail.
Image: This is Figure 1 from the just released paper on the SGL mission. Caption: A visualization of the key primary optical axes (POA) and the projected image plane of the exoplanet. The imaging spacecraft is the tiny element in front of the exoplanet image plane. Credit: Helvajian et al.
Let’s poke around a bit in “Mission Architecture to Reach and Operate at the Focal Region of the Solar Gravitational Lens,” just out in the Journal of Spacecraft and Rockets, which sets out the basics of how such a mission could be flown. Remember that this work has proceeded through the NASA Innovative Advanced Concepts (NIAC) office, with Phase I, II and now III studies resulting in the refinement of a design that can satisfy the requirements of the heliophysics decadal survey. JHU/APL’s Interstellar Probe takes aim at the same decadal, with both missions designed to return data relevant to our own star and, in SGL’s case, a more distant one.
Given that it has taken Voyager 1 well over 40 years to reach 159 AU, getting a payload to the gravitational lens region for operations there and beyond as the craft departs the Sun is a challenge. But the rewards would be great if it can be made to happen. The JPL work and a great deal of theoretical study prior to it have revealed that an optical telescope of no more than meter-class equipped with an internal coronagraph for blocking the Sun’s light would see light from the target exoplanet appearing in the form of an ‘Einstein ring’ surrounding the solar disk. High-resolution imagery of an exoplanet can be extracted from this data. We can also trade spatial for spectral resolution. From the paper:
The direct high-resolution images of an exoplanet obtained with the SGL could lead to insight on the on-going biological processes on the target exoplanet and find signs of habitability. By combining spatially resolved imaging with spectrally resolved spectroscopy, scientific questions such as the presence of atmospheric gases and its circulation could be addressed. With sufficient SNR and visible to mid-infrared (IR) sensing [26], the inspection of weak biosignatures in the form of secondary metabolic molecules like dimethyl-sulfide, isoprene, and solid-state transitions could also be probed in the atmosphere. Finally, the addition of polarimetry to the spatially and spectrally resolved signals could provide further insight such as atmospheric aerosols, dust, and, on the ground, properties of the regolith (i.e., minerals) and bacteria and fauna (i.e., homochirality)…
I won’t labor the issue, as we’ve discussed gravity lens imaging on many an occasion in these pages, but I did want to make the point about spectroscopy as a way of underlining the huge reward obtainable from a mission that can collect data at these distances. The paper is rich in detailing the progress of our thinking on this, but I turn to the mission architecture for today, offering as it does a remarkable new way to conceive of deep space missions both in terms of configuration and propulsion. For we’re dealing here with spacecraft that are modular, reconfigurable and highly adaptable using clusters of spacecraft that practice self-assembly during cruise.
The SGL mission is based on a constellation of identical craft, the primary components being what the authors call ‘proto-mission capable’ (pMC) spacecraft, with final ‘mission capable’ (MC) craft being built as the mission proceeds. Smaller pMC nanosats, in other words, dock during cruise to build an MC; five or perhaps six of the latter are assumed in the mission description in this paper to allow full capability during the observational period within the focal region of the gravity lens. The pMC craft use solar sails for a close pass by the Sun, all of them launched into a parking orbit before deployment toward the Sun. The sailcraft fly in formation following perihelion, dispose of their thermal shielding, then their sails, and begin assembly into MC spacecraft.
How to separate a final, fully functional MC craft into the constituent units from which it will be assembled in flight is no small issue, and bear in mind the need for extreme adaptability, especially as the craft reach the gravitational lensing region. Near-autonomous operations are demanded. The SGL study used simulations based on current engineering methodology (CEM) tools, modifying them as needed. The need for in-flight assembly stood out from the alternative. From the paper;
Two types of distributed functionality were explored: a fractionated spacecraft system that operates as an “organism” of free-flying units that distribute function (i.e., virtual vehicle) or a configuration that requires reassembly of the apportioned masses. Given that the science phase is the strong driver for power and propellant mass, the trade study also explored both a 7.5-year (to ?800 AU) and 12.5-year (to ?900 AU) science phase using a 20 AU/year exit velocity as the baseline. The distributed functionality approach that produced the lowest functional mass unit is a cluster of free-flying nanosatellites (i.e., pMC) each propelled by a solar sail but then assembled to form an MC spacecraft.
Out of all this what emerges is a pMC design with the capability of a 6U CubeSat nanosatellite, self-contained and three-axis stabilized, each of these units to carry a critical part of the larger MC spacecraft. Power and data are shared as the pMCs dock. The current design for the pMC is a round disk approximately 1 meter in diameter and 10 cm thick, with the assembled MC spacecraft visualized as stacked pMC units. One pMC would carry the primary and secondary mirrors, a second the science package, optical communications package and star tracker sensors, and so on. In-space assembly need not be rushed. The paper mentions a time period of several months as needed to complete the operation.
The 28-year cruise phase ends in the region of 550 AU, with two of the five or six MC spacecraft now maneuvering to track the primary optical axis of the exoplanet host star, which is the line connecting the center of the star to the center of the Sun. The host star is thus a key navigational resource which will be used to determine the precise position of the exoplanet under study. Interestingly, motion in the image plane has to be accounted for – this is due to the effect of the wobble of the Sun caused by gas giants in our Solar System. Such wobbles are hugely helpful for those using radial velocity methods to study planets around other stars. Here they become a complicating factor in extracting the data the mission will need to construct its exoplanet imagery.
The disposition of the spacecraft at 550 AU is likewise interesting. All of the MC spacecraft are, as the acronym makes clear, capable of conducting the mission. It now becomes necessary to subtract the Sun’s coronal light from the incoming data, which is accomplished by having one of the spacecraft follow an inertial path down the center of the spiral trajectory the other craft will follow (the other craft all move in a noninertial frame to make it possible to acquire the SGL photons). Having one craft on an inertial path means it sees no exoplanet photons, and thus its coronal image can be subtracted from the data gathered by the other four craft. The inertial path spacecraft also acts as a local reference frame that can be used for navigation.
Image: A meter-class telescope with a coronagraph to block solar light, placed in the strong interference region of the solar gravitational lens (SGL), is capable of imaging an exoplanet at a distance of up to 30 parsecs with a few 10 km-scale resolution on its surface. The picture shows results of a simulation of the effects of the SGL on an Earth-like exoplanet image. Left: original RGB color image with (1024×1024) pixels; center: image blurred by the SGL, sampled at an SNR of ~103 per color channel, or overall SNR of 3×103; right: the result of image deconvolution. Credit: Turyshev et al., “Direct Multipixel Imaging and Spectroscopy of an Exoplanet with a Solar Gravity Lens Mission,” Final Report NASA Innovative Advanced Concepts Phase II.
The spacecraft are moving at more than 20 AU per year and have up to five years between 550 and 650 AU to lock onto the primary optical axis of the exoplanet host star. As the craft reach 650 AU, the optical axis of the host star becomes what the authors call a ‘navigational steppingstone’ toward locating the image of the exoplanet, which once acquired begins a science phase lasting in the area of ten years.
The details of image acquisition are themselves fascinating and as you would imagine, complex – I send you to the paper for more. My focus today is the novelty of the architecture here. If we can assemble a mission capable spacecraft (and indeed a small fleet of these) out of the smaller pMC units, we reduce the size of sail needed for the perihelion acceleration phase and make it possible to achieve payload sizes for missions far beyond the heliosphere that would not otherwise be possible. We build this out of a known base; in-space assembly and autonomous docking have been demonstrated, and technologies for assembly operations continue to be refined. NASA’s On-Orbit Autonomous Assembly from Nanosatellites and CubeSat Proximity Operations Demonstration mission are examples of this ongoing research.
What a long and winding path it is to extend the human presence via robotic probe ever further from our planet. This paper examines technologies needed to advance this movement, and again I point to the ongoing Interstellar Probe study at JHU/APL as another rich source for current and projected thinking about the needed technologies. In the case of the SGL mission, what is being proposed could have a major impact on the search for life elsewhere in the universe. Imagine a green and blue exoplanet seen with weather patterns, oceans, continents and rich spectral data on its atmosphere.
But I come back to that mission architecture and the idea of self-assembly. As the authors write:
We realize that this architecture fundamentally changes how space exploration could be conducted. One can imagine small- to medium-scale spacecraft on fast-traveling scouting missions on quick cadence cycles that are then followed by flagship-class space vehicles. The proposed mission architecture leverages a global technology base driven by miniaturization and integration, and other technologies that are coming into fruition, including composite materials based on hierarchical structures, edge-computing platforms, small-scale power generation, and storage. These advances have had an effect on the small spacecraft industry with the development of a worldwide CubeSat and nanosat ecosystem that have continually demonstrated increasing functionality in missions…
We’ll continue to track robotic self-assembly and autonomy issues with great interest. I’m convinced the concept opens up mission possibilities we’ve yet to imagine.
The paper is Helvajian, “Mission Architecture to Reach and Operate at the Focal Region of the Solar Gravitational Lens,” Journal of Spacecraft and Rockets. Published online 1 February 2023 (full text). For earlier Centauri Dreams articles on the SGL mission, see JPL Work on a Gravitational Lensing Mission, Good News for a Gravitational Focus Mission and
Solar Gravitational Lens: Sailcraft and In-Flight Assembly.
by Paul Gilster | Feb 3, 2023 | Exotic Physics |
“‘This,’ said I at length, to the old man — ‘this can be nothing else than the great whirlpool of the Maelström’… The ordinary accounts of this vortex had by no means prepared me for what I saw. That of Jonas Ramus, which is perhaps the most circumstantial of any, cannot impart the faintest conception either of the magnificence, or of the horror of the scene — or of the wild bewildering sense of the novel which confounds the beholder.” So wrote Edgar Allen Poe in 1841 in a short story called “A Descent into The Maelström,” reckoned by some to be an early instance of science fiction. In today’s essay, Adam Crowl explores another kind of whirlpool, armed with the tools of mathematics to take the deepest plunge imaginable, into the maw of a supermassive black hole. Adam’s always fascinating musings can be followed on his Crowlspace site.
by Adam Crowl
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 Galactic Core Super-Massive Black Hole (SMBH), otherwise known as Sagittarius A* (pronounced as ‘Sagittarius A Star’), by the GRAVITY instrument have determined its mass and distance to new levels of accuracy:
Ro = 8,275 parsecs (+\-) 9.3 parsecs and a mass of (in 106 Msol) 4.297 +\- 0.013.
Image: The galactic centre in infrared. Credit: NASA.
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.
Image: From Tales of Mystery and Imagination, by Edgar Allan Poe, with illustrations by Arthur Rackham (1935).
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 a 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 (rI) all the way to the centre 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 straightforward:
? is the angular distance traveled, with a range from negative infinity to zero, by convention. I’ve plotted r against ? here:
The Red Circle at r = 3 M 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 = 5.95M 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:
with ? running from an angle ? to 0. Then the Proper-Time ? of the inspiral trajectory is:
The above equation is true for any black-hole, spinning or stationary. For a stationary Black Hole, rI = 6M, so the equation simplifies to:
But what is M? It’s the “geometrised” mass of the Black Hole, which is derived by muliplying the mass by G/c2. 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 ? = (5.95/6) ? (?) to ? = 0, the total time is ? = 1291.14M. 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.
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 / m2 (Pressure) = E / m3 (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:
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* 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:
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:
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…
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.
As we conclude, let’s check back in with Edgar Allan Poe, who knew a few things about terrifying plunges himself. In “Descent into the Maelstrom,” he gives us a look into what might be considered a 19th Century conception of a black hole and the journey into its bizarre interior:
“Looking about me upon the wide waste of liquid ebony on which we were thus borne, I perceived that our boat was not the only object in the embrace of the whirl. Both above and below us were visible fragments of vessels, large masses of building timber and trunks of trees, with many smaller articles, such as pieces of house furniture, broken boxes, barrels and staves. I have already described the unnatural curiosity which had taken the place of my original terrors. It appeared to grow upon me as I drew nearer and nearer to my dreadful doom. I now began to watch, with a strange interest, the numerous things that floated in our company. I must have been delirious — for I even sought amusement in speculating upon the relative velocities of their several descents toward the foam below. ‘This fir tree,’ I found myself at one time saying, ‘will certainly be the next thing that takes the awful plunge and disappears,’ — and then I was disappointed to find that the wreck of a Dutch merchant ship overtook it and went down before. At length, after making several guesses of this nature, and being deceived in all — this fact — the fact of my invariable miscalculation — set me upon a train of reflection that made my limbs again tremble, and my heart beat heavily once more.”
References
Mummery, A. & Balbus, S. “Inspirals from the innermost stable circular orbit of Kerr black holes: Exact solutions and universal radial flow,” Physical Review Letters 129, 161101 (12 October 2022).
https://doi.org/10.48550/arXiv.2209.03579
Fragione, G. and Loeb, A., “An Upper Limit on the Spin of SgrA* Based on Stellar Orbits in Its Vicinity” (2020) ApJL 901 L32
DOI 10.3847/2041-8213/abb9b4
