Kurzgesagt – In a Nutshell 

Sources - Last Thing to Ever Happen


We thank Dr. Matt Caplan for his help with this script.

– After a messy birth our universe was a sleepy baby, warm and dark, filled with swirling clouds of hot hydrogen and helium. The story of creation is a story of this gas and where it will end up. Shortly after, the universe got busy making the first generation of stars. They were massive and lived violent lives, forging new elements, only to release most of them when they blew up. 


Hydrogen and helium nuclei (as well as very small amounts of nuclei of other light elements) were created minutes after the Big Bang, when the universe was hot and dense everywhere, in a process called “Big Bang nucleosynthesis”.

#NASA (retrieved 2023): Universe – Basics: The Universe History

https://universe.nasa.gov/universe/basics/
 Quote: “One second after the big bang, the universe consisted of an extremely hot (18 billion degrees Fahrenheit or 10 billion degrees Celsius) primordial soup of light and particles. In the following minutes, an era called nucleosynthesis, protons and neutrons collided and produced the earliest elements – hydrogen, helium, and traces of lithium and beryllium. After five minutes, most of today’s helium had formed, and the universe had expanded and cooled enough that further element formation stopped. At this point, though, the universe was still too hot for the atomic nuclei of these elements to catch electrons and form complete atoms. The cosmos was opaque because a vast number of electrons created a sort of fog that scattered light.”

Some 380,000 years after the Big Bang, hydrogen and helium nuclei combined with electrons to form the first neutral atoms. This gave rise to giant clouds of neutral gas, made mostly of hydrogen, that hundreds of millions of years later collapsed and gave rise to the first stars. Heavier chemical elements were only created in the cores of the stars via nuclear fusion reactions. These first stars were very massive and lived very short lives, ending their days in violent supernova explosions that spread the newly created heavier chemical elements through the universe.


#Grochala, W. (2015): First there was hydrogen. Nature Chemistry, Vol. 7

https://www.nature.com/articles/nchem.2186#

Quote: “The history of hydrogen — the element that fills the world as we know it — consists of a most dramatic set of events. Hydrogen and helium atoms emerged a measly 379,000 years after the Big Bang. As the hot, dense plasma of protons, electrons and photons that was the universe began to cool and expand, electrons and protons gathered to form atoms. Four hundred million years later stars — such as our very own Sun — evolved from gravitationally collapsed clouds of hydrogen gas, providing the heat necessary to sustain life in an otherwise giant, freezing, cosmic abyss at 2.7 kelvin. The third colossal breakthrough in hydrogen history came some 4.4 billion years ago, when the temperature on Earth dropped below 100 °C and dihydrogen oxide began to condense at its surface, allowing the emergence of life in the new aqueous environment.”


#Space Telescope Science Institute (2019): What Were the First Stars Like?

https://webbtelescope.org/contents/articles/what-were-the-first-stars-like

Quote: “How massive were the first stars?

Short answer: Probably really big, but maybe not.

Although astronomers are quite certain what the first stars were made of, they are less sure how massive they were. Strangely, one constraint on the mass of metal-free stars comes from what we don’t see: Because we haven’t observed any metal-free stars, we are fairly certain that there could not have been many small ones. Small stars like the Sun last for billions of years. If small Population III stars were common, we should have detected some by now.

Extremely large stars, on the other hand, burn their fuel very quickly. A star 60 times the mass of the Sun lasts less than one million years. If all of the first generation of stars were extremely massive, none would still exist, and it would make sense that we have not seen any.”

#Space Telescope Science Institute (2019): What Were the First Stars Like?

https://webbtelescope.org/contents/articles/what-were-the-first-stars-like

Quote: “What were the first stars made of?

Short answer: Hydrogen and helium (and tiny amounts of lithium). That’s it.

Astronomers know that the first stars, officially known as Population III stars, must have been made almost solely of hydrogen and helium—the elements that formed as a direct result of the big bang. They would have contained none of the heavier elements like carbon, nitrogen, oxygen, and iron that are found in stars shining today. In other words, Population III stars were metal-free. (Astronomers refer to any element heavier than helium as a metal.)


This might seem like a bold statement given that we have not actually observed any metal-free stars. But as with all scientific claims, it is based on evidence and reasoning: We know from observations, experiments, and calculations that only hydrogen and helium (and minute amounts of lithium) were formed directly after the big bang. The only way that heavier elements like carbon, oxygen, and iron can form is by fusion of lighter elements in the cores of stars. So until the first stars began to form them, none of these elements existed in the universe. The first stars must have been metal-free.”



The synthesis of new chemical elements in the cores of the stars by means of nuclear fusion is called "stellar nucleosynthesis" (as opposed to “Big Bang nucleosynthesis” referred to above). Very simplified, hydrogen is fused into helium and, depending on the type of star helium is later fused into other heavier elements (carbon, neon, oxygen, silicon and more) up to iron. 


#Prantzos, N. & Ekström, S. (2011): Stellar Nucleosynthesis. Encyclopedia of Astrobiology.

https://link.springer.com/referenceworkentry/10.1007/978-3-642-11274-4_1084 

Quote: “Stellar nucleosynthesis is the process involving nuclear reactions through which fresh atomic nuclei are synthesized from pre-existing nuclei or nucleons. The first stage of nucleosynthesis occurred in the hot, early Universe, with the production of H, He, and traces of Li-7 (primordial nucleosynthesis). In the present-day Universe nucleosynthesis occurs through: (1) thermonuclear reactions in stellar interiors and explosions (building nuclei up to the Fe-peak), (2) neutron captures in stellar interiors and explosions (building nuclei above the Fe-peak), and (3) spallation reactions in the interstellar medium, whereby light nuclei (Li, Be, and B) are produced by fragmentation of heavier ones (C, N, and O).

(...) 

Searching for the sites of nuclear reactions which shaped the cosmic abundances has been the main theme of nuclear astrophysics in the past half century. It is now well established that:

– The light isotopes of H and He, along with 10–30% of the fragile 7Li, have been produced in the hot early Universe by thermonuclear reactions between neutrons and protons.

– All the elements between C and the Fe-peak have been produced by thermonuclear reactions inside stars, either during their quiescent evolutionary stages or during the violent explosions (supernovae) that mark the deaths of some stars.

– Elements heavier than those of the Fe-peak have been produced by neutron captures in stars, either in low neutron densities and long timescales (s- elements) or in high neutron densities and short timescales (relements); a minor fraction of those heavy elements has been produced by photodisintegration of the heavy isotopes in supernova explosions (p-isotopes).

– Finally, the light and fragile isotopes of Li, Be, and B are not produced in stellar interiors (they are rather destroyed in high temperatures), but by spallation reactions, with high energy cosmic ray particles removing nucleons from the abundant C, N, and O nuclei of the interstellar medium.

(...)"

– Countless stars were born and refined the gas available in the universe, cycling matter around, each generation giving most of its gas and fresh elements to the next. 


Before we go on, it might be useful to have a quick look over the stellar life cycle since the fate of a star is determined by its mass.

 In a very simplified way, the destiny of the different types of stars could be summarized as follows.


#Chandra X-ray Center, at the Smithsonian Astrophysical Observatory (retrieved 2023): A Guide to Understanding Stellar Evolution

https://chandra.si.edu/stellarev/

Quote: “The evolutionary processes of stars depend upon their initial mass. Mid-sized stars eject planetary nebulae, leaving a white dwarf core remnant. More massive stars explode as supernovae, leaving neutron stars or black holes at the centers of the supernovae remnants. The elements that were created within the cores of the first stars were ejected into space where they intermingled with the surrounding interstellar medium. This medium — the gas and dust between the stars — provides the raw material for the formation of new generations of stars. Eventually, these elements became incorporated into large clouds of gas and dust that condensed and formed protostars. And so the cycle of stellar formation and destruction continues. Each new generation further enriches the interstellar medium with heavy elements that become incorporated into the next generation. We are just beginning to understand stellar formation and destruction —and how the Sun, Solar System and life on Earth are connected to this never-ending cycle.”

#Max Planck Institute for Astronomy (2022): Astronomers identify the ancient heart of the Milky Way galaxy

https://www.mpia.de/news/science/2022-19-ancient-heart

Quote: “More generally, for almost all stars, there is a “building style” that allows a general verdict on age: a star’s so-called metallicity, defined as the amount of chemical elements heavier than helium that the star’s atmosphere contains. Such elements, which astronomers call “metals”, are produced inside stars through nuclear fusion and released near or at the end of a star’s life – some when a low-mass star’s atmosphere disperses, the heavier elements more violently when a high-mass star explodes as a supernova. In this way, each generation of stars “seeds” the interstellar gas from which the next generation of stars is formed, and generally, each generation will have a higher metallicity than the rest.”


#Space Telescope Science Institute (2019): What Were the First Stars Like?

https://webbtelescope.org/contents/articles/what-were-the-first-stars-like

Quote: “What happened to the first stars?

Short answer: They burned out, exploded, or collapsed.

Based on what we know about physics, we know that if any of the earliest stars were smaller than 0.9 times the mass of the Sun, they should still be shining today. But if they were very massive, they would have had extremely short lifetimes, burning up their fuel quickly and dying within a few million years of forming. What exactly happened to them would depend on their mass.


Computer models show that stars greater than 10 and less than about 140 times the mass of the Sun would have ended in supernova explosions, the metals that formed in their cores blown out and dispersed into the surrounding universe. The remains of these stars would have collapsed to form neutron stars or black holes.


Larger metal-free stars, up to about 300 solar masses, might have exploded in a strange type of supernova. These explosion would have blasted every bit of the star out in all directions, leaving nothing behind—no neutron star or black hole. The only signs of these stars would be found in the chemical composition of later stars that formed from their exploded remains.


Even larger metal-free stars would have collapsed directly to form black holes, taking everything with them without even exploding first. While these stars would not have contributed matter to form new stars, they may have influenced galaxies in other ways. It may be that these black holes were the seeds of supermassive black holes found at the centers of galaxies today.”



– But not all gas is returned.  Every time a new generation of stars forms, they also make more and more red dwarfs that burn slowly and live for trillions of years. When they die, they don’t give their gas back to the universe but turn into white dwarfs. So red dwarfs lock up more gas forever.


As mentioned above, when red dwarfs die they turn directly into white dwarfs, without going through a red giant/planetary nebula phase, and therefore without giving any of their gas back into space:

#Chandra X-ray Center, at the Smithsonian Astrophysical Observatory (retrieved 2023): A Guide to Understanding Stellar Evolution

https://chandra.si.edu/stellarev

Red dwarfs also live the longest of all stars, with lifespans reaching trillions of years: 


#Adams, F.C. et al. (2005): M dwarfs: planet formation and long term evolution Adams. Astronomische Nachrichten / Astronomical Notes Vol. 326 (10)

https://www.researchgate.net/publication/229696585_M_dwarfs_Planet_formation_and_long_term_evolution

Quote: “The second part of this paper describes stellar evolution calculations for M dwarfs, which live far longer than the current age of the universe. These diminutive stellar objects remain convective over most of their lives, continue to burn hydrogen for trillions of years, and do not experience red giant phases in their old age. Instead, red dwarfs turn into blue dwarfs and finally white dwarfs.

(...)

Although it has long been known that these small stars will live for much longer than the current age of the universe, these stellar evolution calculations reveal some surprises. For example, the star with initial mass M = 0.1 M remains nearly fully convective for 5.74 trillion years. As a result, the star has access to almost all of its nuclear fuel for almost all of its lifetime. Whereas a 1.0 M star only burns about 10 percent of its hydrogen on the main sequence, this star, with 10 percent of a solar mass, burns nearly all of its hydrogen and thus has about the same main sequence fuel supply as the Sun.

(...)

One of the most interesting findings of this work is that small red dwarfs do not become red giants in their post-mainsequence phases. Instead they remain physically small and grow hotter to become blue dwarfs. Eventually, of course, they run out of nuclear fuel and are destined to slowly fade away as white dwarfs.


#NASA (retrieved 2023): Type of Stars

https://universe.nasa.gov/stars/types/#otp_red_dwarfs

Quote: “Red dwarfs are the smallest main sequence stars – just a fraction of the Sun’s size and mass. They’re also the coolest, and appear more orange in color than red. When a red dwarf produces helium via fusion in its core, the released energy brings material to the star’s surface, where it cools and sinks back down, taking along a fresh supply of hydrogen to the core. Because of this constant churning, red dwarfs can steadily burn through their entire supply of hydrogen over trillions of years without changing their internal structures, unlike other stars. Scientists think some low-mass red dwarfs, those with just a third of the Sun’s mass, have life spans longer than the current age of the universe, up to about 14 trillion years. Red dwarfs are also born in much greater numbers than more massive stars. Because of that, and because they live so long, red dwarfs make up around 75% of the Milky Way galaxy’s stellar population.”



– Some more gas is locked forever in other remains of dead stars: neutron stars and black holes. Which is bad, as it reduces the material for new stars.


#Adams, F.C. & Laughlin, G. (1996): A Dying Universe: The Long Term Fate and Evolution of Astrophysical Objects. Reviews of Modern Physics, Vol. 69 (2)

https://arxiv.org/pdf/astro-ph/9701131.pdf

Quote: “When ordinary star formation and conventional stellar evolution have ceased, all of the remaining stellar objects will be in the form of brown dwarfs, white dwarfs, neutron stars, and black holes.



– Today the universe is a great home for us and will remain so for billions of years. But most of the gas has been used up or trapped. Over 90% of all the stars that will ever be born have been born already. 


Scientists have found that half of all the stars that have ever existed were created between 9 and 11 billion years ago, with the other half created since then. That means that the rate at which new stars are born has dropped off considerably and that, if such a  trend continues, 95 percent of all the stars that will ever be born in the universe have been born already:


#Sobral, D. et al. (2021): A large Hα survey at z = 2.23, 1.47, 0.84 & 0.40: the 11 Gyr evolution of star-forming galaxies from HiZELS. Monthly Notices of the Royal Astronomical Society, Vol. 428 (2)

https://academic.oup.com/mnras/article/428/2/1128/1000290 

Quote: “The star formation activity over the last ∼11 Gyr is responsible for producing ∼95 per cent of the total stellar mass density observed locally, with half of that being assembled in 2 Gyr between z = 1.2 and 2.2, and the other half in 8 Gyr (since z < 1.2). If the star formation rate density continues to decline with time in the same way as seen in the past ∼11 Gyr, then the stellar mass density of the Universe will reach a maximum which is only 5 per cent higher than the present-day value.”



– Eventually almost all the stars will be red dwarfs slowly dying.


It is thought that after some 100 trillion years, the “stelliferous era” (“star-forming age”) will come to an end:

#Busha, M.T. et al. (2003): Future Evolution of Cosmic Structure in an Accelerating Universe. The Astrophysical Journal, Vol. 596 (2)
https://iopscience.iop.org/article/10.1086/378043/meta

As mentioned above, the longest lived stars are red dwarfs, which can live up to several trillion years, i.e. about 1000 times longer than sunlike stars. Therefore, there will be a time in the universe (between the birth of the last star that will ever be born and the death of the last star) in which basically all existing stars will be red dwarfs dying.



The End of Everything – but not quite


– In a few trillion years the cosmic gas will finally have run out. About 88% of the mass of every galaxy will be white dwarfs, 2% neutron stars and black holes, and about 10% gas giants and sad brown dwarf losers.


The “final mass function” of all the stars in the universe has been calculated here:


#Adams, F.C. & Laughlin, G. (1996): A Dying Universe: The Long Term Fate and Evolution of Astrophysical Objects. Reviews of Modern Physics, Vol. 69 (2)

https://arxiv.org/pdf/astro-ph/9701131.pdf

Quote: “When ordinary star formation and conventional stellar evolution have ceased, all of the remaining stellar objects will be in the form of brown dwarfs, white dwarfs, neutron stars, and black holes. One way to characterize the stellar content of the universe at this epoch is by the mass distribution of these objects; we refer to this distribution as the ‘‘final mass function’’ or FMF.”

As the figures in (2.23) above indicate, the fraction of brown dwarfs will be 0.097 (i.e. about 10%), the fraction of white dwarfs will be 0.88 (88%) and the fraction of neutron stars will be 0.024 (about 2%).

 In this study, the label “brown dwarf” includes all substellar objects (i.e. all objects that lack enough mass to start ignition and become stars). This includes white dwarfs but also all gas giant planets like Jupiter:

Quote: “For a given initial mass function, we must find the final masses mF of the degenerate objects resulting from the progenitor stars with a given mass m. For the brown dwarf range of progenitor masses, m < mH, stellar objects do not evolve through nuclear processes and hence mF = m. Here, the scale mH ≈ 0.08 is the minimum stellar mass required for hydrogen burning to take place.”


In the same way, the label “neutron star” also includes black holes:

Quote: "The above values for NNS, and MNS were obtained under the assumption that all stars m > mSN ∼ 8 produce neutron stars. In reality, a portion of these high mass stars may collapse to form black holes instead, but this complication does not materially affect the basic picture described above."



– White dwarfs are the geriatric version of a star, not much bigger than earth but on average as massive as half our sun, some even much more. This makes them the third densest objects in the universe, after neutron stars and black holes. About a 1,000,000 times denser than the sun today. 


The corpse of a star can be a white dwarf, a neutron star or a black hole. These are the densest objects in the universe, since they contain a mass similar to that of a star compressed in a much smaller volume. In order of magnitude, their densities of a star like the Sun, a typical white dwarf and a typical neutron star are:


#NASA: “Imagine the Universe – Density Comparisons”

https://imagine.gsfc.nasa.gov/science/toolbox/cool_star_fact1.html 

Quote: “The density of a neutron star is 7 x1014g/cm3. Here are some other densities for comparison.


The Sun - 1.41 g/cm3

White Dwarf - 2 x 106 g/cm3 “


As for black holes, their density can vary a lot and it is also difficult to define precisely for technical reasons. However, defining their “effective density” as their total mass divided by the volume enclosed by the event horizon, one finds that the density of a black hole as massive as the Sun is of the order of 1016 gr/cm3: 


#Toth, Viktor T: “Hawking Radiation Calculator” (used 2023)

https://www.vttoth.com/CMS/hawking-radiation-calculator

–Since they used to be active stars, their surface can be as hot as 150,000 degrees. White dwarfs are dim, hot, dense spheres that don’t do anything anymore. 


A well-known example is Messier, also known as "The Ring Nebula", a remnant of a sun-like star with a white dwarf as its center that has a surface temperature of 120,000 °C


#NAS (2017): Hubble’s Messier Cataloge. Messier 57 (The Ring Nebula).

https://www.nasa.gov/feature/goddard/2017/messier-57-the-ring-nebula 

Quote: “M57, or the Ring Nebula, is a planetary nebula, the glowing remains of a sun-like star. The tiny white dot in the center of the nebula is the star’s hot core, called a white dwarf. M57 is about 2,000 light-years away in the constellation Lyra, and is best observed during August. Discovered by the French astronomer Antoine Darquier de Pellepoix in 1779, the Ring Nebula has an apparent magnitude of 8.8 and can be spotted with moderately sized telescopes.”


#NASA (2009): The Ring Nebula

https://www.nasa.gov/mission_pages/hubble/servicing/SM4/multimedia/wfpc/ring.html 

Quote: “The gradations of color illustrate how the gas glows because it is bathed in ultraviolet radiation from the remnant central star, whose surface temperature is a white-hot 216,000 degrees Fahrenheit (120,000 degrees Celsius).”


#Cool Cosmos at IPAC (retrieved 2023): How hot is the Sun?

https://coolcosmos.ipac.caltech.edu/ask/7-How-hot-is-the-Sun- 

Quote: “The temperature at the surface of the Sun is about 10,000 Fahrenheit (5,600 Celsius). The temperature rises from the surface of the Sun inward towards the very hot center of the Sun where it reaches about 27,000,000 Fahrenheit (15,000,000 Celsius). The temperature of the Sun also rises from the surface outward into the Solar atmosphere. The uppermost layer of the Solar atmosphere, called the corona, reaches temperatures of millions of degrees. The corona is the bright halo of light that can be seen during a total Solar eclipse.”



– But eventually even white dwarfs will die because they are slowly losing their heat – it takes at least 10 trillion years, more than 700 times longer than the current age of the universe. As they do their cooling down, the universe around them will irreversibly grow dark, as more and more white dwarfs burn out, and turn into dead husks: black dwarfs. Spheres of death, as cold as space itself, invisible against the dark backdrop.



When they form white dwarfs are very hot objects, but as time goes by they will eventually lose their heat. The cooling of an object is an asymptotic process, which means that it takes an infinite time for the object to cool down to 0 kelvin. So the “cooling time” for a white dwarf is not uniquely determined unless we choose a final temperature.

The physics controlling their cooling down is complicated, but a rough estimate for their temperature over time is given by the following expression:

#Maoz, Dan (2016): “Astrophysics in a Nutshell”. Princeton University Press.
 (from chapter 4.2.3.3, “White-Dwarf Cooling”)
https://press.princeton.edu/books/hardcover/9780691164793/astrophysics-in-a-nutshell

(Where T is the temperature in kelvin and t is the elapsed time in years.)

This simple formula should be understood as only giving very rough orders of magnitude, but it still gives us an idea of how the cooling process works. Using it we find the following (rounded) values for the cooling times of a white dwarf:


which are similar to the ones found in other sources:


#Barrow, John and Tipler, Frank (1986): The Anthropic Cosmological Principle, Oxford University Press.

https://global.oup.com/academic/product/the-anthropic-cosmological-principle-9780192821478?cc=de&lang=en&


Hot bodies emit light, and their color depends on their temperature. White dwarfs have a blue-white color because they are very hot, but as they cool down they will gradually become dimmer and dimmer.

For the human eye, an object start to “shine” (to be visible) at temperatures between 500K and 1000K:


#Jeffery, D: “Blackbody spectra”. Lecture notes. University of Nevada, Las Vegas; Department of Physics and Astronomy (retrieved 2023)
https://www.physics.unlv.edu/~jeffery/astro/blackbody/blackbody_spectra_1.html 

So white dwarfs will start to “look black” once they have cooled down to temperatures of the order of a few hundred kelvin. According to the simple expression used above, these temperatures are reached after a time of the order of trillions of years.



– Over trillions and trillions of years, every object in every galaxy is eventually either ejected into the void or its orbit decays and it will fall into the central black hole and be destroyed. In about a quintillion years, all galaxies have evaporated… 


Eventually, most individual objects inside a galaxy will be ejected even if the galaxy exerts a gravitational pull on them. This happens because, over eons, stars will randomly collide against each other and these collisions will end up pushing most of the stars out the galaxy (while the rest will be eventually fall into the central black hole). This process is known as “stellar evaporation” and is expected to happen in a timescale between quintillions (1018) and sextillions (1021) of years:


#Adams, F.C. & Laughlin, G. (1996): A Dying Universe: The Long Term Fate and Evolution of Astrophysical Objects. Reviews of Modern Physics, Vol. 69 (2)

https://arxiv.org/pdf/astro-ph/9701131.pdf

Quote: “The galaxy itself evolves through the competing processes of orbital decay of orbits via gravitational radiation and the evaporation of stars into the intergalactic medium via stellar encounters. Stellar evaporation is the dominant process and most of the stars will leave the system at a time η ~ 19. Some fraction (we roughly estimate ~0.01–0.10) of the galaxy is left behind in its central black hole.”


[The parameter η in the reference above refers to the logarithmic time in years, as defined in eq. (1.1) of that reference I.e. η = 19 equals a timescale of 1019 years.]


The estimated timescale for galaxies to fully evaporate varies across sources and depends if one considers an individual galaxy or a whole cluster of galaxies, but it’s of the order of quintillion-sextillion years:


#John, J.D. & Tipler, F.J. (1986): The Anthropic Cosmological Principle. Oxford University Press.

https://global.oup.com/academic/product/the-anthropic-cosmological-principle-9780192821478?cc=de&lang=en&


– … and every object is on its own, in the center of its own observable universe, emptiness as far as can be seen in any direction, traveling through black nothingness. Still, there are things that will happen. 


The term "observable universe" doesn’t mean that new universes are being formed. What will happen in the far future is that the universe will be expanding at an exponential rate due to the push of dark energy, and all distances will double in size every 12 billion years or so:

#Baez, J. (2016): The End of the Universe. Department of Mathematics, University of California Riverside.
 https://math.ucr.edu/home/baez/end.html

Quote: “As the universe expands these things eventually spread out to the point where each one is completely alone in the vastness of space. If dark energy works as we expect, the distance between things that aren't gravitationally bound to each other will double every 12 billion years.”


This scale for the doubling time of the size of the universe (~ 1010 years) is billions of times smaller than the age of the universe at the moment we’re now considering (~ 1019 years), which means that the universe will have increased in size by the unimaginable large factor of 2 to the power of 1 billion:


21,000,000,000 ~ 101,000,000,000


And all gravitationally isolated objects will be separated from each other by an equally tremendous distance. 


This means that all objects will be so far away from each other that they’ll no longer be able to interact with one another in any way. So, for all practical purposes, each object will be isolated in its own “observable universe”.


#Caplan, M.E. (2020): Black dwarf supernova in the far future. Monthly Notices of the Royal Astronomical Society, Vol. 497 (4)

https://academic.oup.com/mnras/article/497/4/4357/5884975

Quote: “Instead, to get a sense of the cosmological conditions where black dwarf supernova are found, consider that the cosmic event horizon for each comoving observer in a de Sitter universe is found at reh = c/H ≈ 1010 light-years (Melia 2007; Neat 2019), so after becoming gravitationally unbound, we may expect all objects to recede beyond their mutual cosmic event horizons on a time-scale similar to the current age of the universe (Adams & Laughlin 1997). We therefore expect that every degenerate remnant in the universe will be causally disconnected from every other degenerate remnant, which will all see each other red-shifted to infinity long before any transients considered in this work may occur (Maoz 2016; Davis & Lineweaver 2004).”



– Black holes are dying. Slowly. They’ll fizzle away by emitting Hawking radiation until they are so small that they die in a final flash of light. This will take about a googol years, 10 to the power of 100 years, until the last super massive black hole dies. 


All black holes slowly “evaporate” and lose mass by emitting quantum particles, a process known as “Hawking radiation”:


#NASA (2023): 10 Questions You Might Have About Black Holes

https://solarsystem.nasa.gov/news/1068/10-questions-you-might-have-about-black-holes/

Quote: “10. Can black holes get smaller?

Yes. The late physicist Stephen Hawking proposed that while black holes get bigger by eating material, they also slowly shrink because they are losing tiny amounts of energy called "Hawking radiation."

Hawking radiation occurs because empty space, or the vacuum, is not really empty. It is actually a sea of particles continually popping into and out of existence. Hawking showed that if a pair of such particles is created near a black hole, there is a chance that one of them will be pulled into the black hole before it is destroyed. In this event, its partner will escape into space. The energy for this comes from the black hole, so the black hole slowly loses energy, and mass, by this process.

Eventually, in theory, black holes will evaporate through Hawking radiation. But it would take much longer than the entire age of the universe for most black holes we know about to significantly evaporate. Black holes, even the ones around a few times the mass of the Sun, will be around for a really, really long time!”


The time it takes for a black hole to completely evaporate is amazingly large and it increases with the mass of the black hole. For the most massive black holes in the universe (the supermassive black holes residing at the centers of galaxies, with typical masses of the order of tens of billions of solar masses), this time is of the order of 1 googol years, or 10100 years:


#Redd, N.T. (2021): The Beginning to the End of the Universe: How black holes die. https://www.astronomy.com/science/the-beginning-to-the-end-of-the-universe-how-black-holes-die/

Quote: “But don’t expect a black hole to disappear any time soon. It takes a shockingly long time for a black hole to shed all of its mass as energy via Hawking radiation. It would take 10100 years, or a googol, for a supermassive black hole to fully disappear. “The entire age of the universe [is] a fraction of [the time] it would take,” says Priyamvada Natarajan, a researcher at Yale University who probes the nature of black holes. “As far as we’re concerned, it is eternity.”

#Toth, V.T. (retrieved 2023): Hawking Radiation Calculator
https://www.vttoth.com/CMS/hawking-radiation-calculator

– A black dwarf is a sphere the size of earth, as massive as a star but almost as cold as absolute zero. 


As explained above, a black dwarf is simply a white dwarf that has cooled down to very low temperatures – an extremely slow process that takes between trillions and quadrillions of years depending on the final temperature we focus on. So strictly speaking, it is a theoretical stellar object that we have not yet observed, because the universe is not old enough for them to have come into existence yet:  


#Chandra X-ray Observatory (retrieved 2023): Q&A: Miscellaneous X-ray Sources

https://chandra.harvard.edu/resources/faq/sources/misc_sources/misc_sources-9.html

Quote: “A black dwarf that evolved from a white dwarf should be a solid, crystal object, kind of like a very dense form of diamond. Although astronomers expect that these objects will eventually form, none so far exist because the universe isn't old enough for them to have formed (the evolution of low-mass stars that form white dwarfs is relatively slow). A brown dwarf should also eventually form into a solid object, but we are not sure if the resulting body is also called a black dwarf.”


As they are just cold white dwarfs, they basically have the same mass and size of white dwarfs, i.e. as explained above, they are objects with the mass of a star and with a size about that of the Earth:

#NASA (retrieved 2023): Imagine the Universe! The Life Cycles of Stars

https://imagine.gsfc.nasa.gov/educators/lifecycles/LC_main3.html
 Quote: “A star like our Sun will become a white dwarf when it has exhausted its nuclear fuel. Near the end of its nuclear burning stage, such a star expels most of its outer material (creating a planetary nebula) until only the hot (T > 100,000 K) core remains, which then settles down to become a young white dwarf. A typical white dwarf is half as massive as the Sun, yet only slightly bigger than the Earth. This makes white dwarfs one of the densest forms of matter, surpassed only by neutron stars. White dwarfs have no way to keep themselves hot (unless they accrete matter from other closeby stars); therefore, they cool down over the course of many billions of years. Eventually, such stars cool completely and become black dwarfs.”



– Stars stay alive because of their intense heat in their cores


A star is an object with a huge mass, and as such is subject to the inward pull of gravity, which wants to “compress” the star. The reason why a living star does not collapse under its own weight lies in the nuclear fusion reactions that take place at its core. These reactions generate huge amounts of energy that push outwards and counterbalance the inner pull of gravity:


#NASA (retrieved 2023): Stars 

https://science.nasa.gov/astrophysics/focus-areas/how-do-stars-form-and-evolve

Quote: “Stars are fueled by the nuclear fusion of hydrogen to form helium deep in their interiors. The outflow of energy from the central regions of the star provides the pressure necessary to keep the star from collapsing under its own weight, and the energy by which it shines.”


 – so why do black dwarfs not collapse into a black hole? What keeps them together? Deep inside a black dwarf, matter is squeezed to densities millions of times greater than anything we see on earth. The pressure is so great that electrons can’t combine with nuclei to form atoms. Instead matter is weird, degenerate: the nuclei are compressed by the weight of the star, locked into a rigid lattice, while the electrons form a plasma around them. 


As mentioned above, apart from the large difference in temperature, the internal structure of a black dwarf is very similar to that of a white dwarf. In these objects, atomic nuclei and electrons are as compressed as allowed by the laws of quantum mechanics. (It is true that in neutron stars matter is compressed even further, but these objects lack electrons and, roughly speaking, can be considered as a huge atomic nucleus.)

In white/black dwarfs, the atomic nuclei form a lattice and the electrons take the form of a gas, or “plasma”, that lives within that lattice:

#NASA (retrieved 2023): Imagine the Universe! – White Dwarf Stars

https://imagine.gsfc.nasa.gov/science/objects/dwarfs2.html 

Quote: “What's inside a white dwarf? Because a white dwarf is not able to create internal pressure (e.g. from the release of energy from fusion, because fusion has ceased), gravity compacts the matter inward until even the electrons that compose a white dwarf's atoms are smashed together. In normal circumstances, identical electrons (those with the same "spin") are not allowed to occupy the same energy level. Since there are only two ways an electron can spin, only two electrons can occupy a single energy level. This is what's known in physics as the Pauli Exclusion Principle. In a normal gas, this isn't a problem because there aren't enough electrons floating around to fill up all the energy levels completely. But in a white dwarf, the density is much higher, and all of the electrons are much closer together. This is referred to as a "degenerate" gas, meaning that all the energy levels in its atoms are filled up with electrons. For gravity to compress the white dwarf further, it must force electrons where they cannot go. Once a star is degenerate, gravity cannot compress it any more, because quantum mechanics dictates that there is no more available space to be taken up. So our white dwarf survives, not by internal fusion, but by quantum mechanical principles that prevent its complete collapse.


Degenerate matter has other unusual properties. For example, the more massive a white dwarf is, the smaller it is. This is because the more mass a white dwarf has, the more its electrons must squeeze together to maintain enough outward pressure to support the extra mass. However, there is a limit on the amount of mass a white dwarf can have. Subrahmanyan Chandrasekhar discovered this limit to be 1.4 times the mass of the Sun. This is appropriately known as the "Chandrasekhar limit."


With a surface gravity of 100,000 times that of Earth, the atmosphere of a white dwarf is very strange. The heavier atoms in its atmosphere sink, and the lighter ones remain at the surface. Some white dwarfs have almost pure hydrogen or helium atmospheres, the lightest of elements. Also, gravity pulls the atmosphere close around it in a very thin layer. If this occurred on Earth, the top of the atmosphere would be below the tops of skyscrapers.


Scientists hypothesize that there is a crust 50 km thick below the atmosphere of many white dwarfs. At the bottom of this crust is a crystalline lattice of carbon and oxygen atoms. Since a diamond is just crystallized carbon, one might make the comparison between a cool carbon/oxygen white dwarf and a diamond.”



– And these electrons hold the star together. We are simplifying, but imagine matter as a subway train and electrons as passengers. If there are empty seats passengers spread out because they care a lot about their personal space.  But as a black dwarf is so incredibly dense, this is like squishing more and more passengers into our train. Gravity is pushing in, trying to collapse it. The Passengers are forced to sit and stand close together, which they hate. And so the passengers, our electrons try to push out against gravity as hard as they can. 


It is the pressure exerted by these “maximally compressed electrons” that holds the star together and prevents it from collapsing under its own weight. The technical name of the outward force exerted by this degenerate gas of electrons is called “electron degeneracy pressure”:

#Swinburne University of Technology (retrieved 2023): Electron Degeneracy Pressure. The SAO Encyclopedia of Astronomy
https://astronomy.swin.edu.au/cosmos/E/electron+degeneracy+pressure 

Quote: “The Pauli exclusion principle states that no two electrons with the same spin can occupy the same energy state in the same volume. Once the lowest energy level is filled, the other electrons are forced into higher and higher energy states resulting in them travelling at progressively faster speeds. These fast moving electrons create a pressure (electron degeneracy pressure) which is capable of supporting a star!


In particular, electron degeneracy pressure is what supports white dwarfs against gravitational collapse, and the Chandrasekhar limit (the maximum mass a white dwarf can attain) arises naturally due to the physics of electron degeneracy. As the mass of a white dwarf approaches the Chandrasekhar limit, gravity attempts to squeeze the star into a smaller volume, forcing electrons to occupy higher energy states and attain faster velocities. At the Chandrasekhar limit, the pressure exerted by the electrons travelling at close to the speed of light becomes insufficient to support the star, and the white dwarf collapses into a much denser state.


Electron degeneracy occurs at densities of about 106 kg/m3.



- Quantum mechanics ruins everything. Simplifying a lot: when particles get close enough, sometimes they can jump at each other and fuse together. A process called ‘quantum tunneling’. This happens constantly in stars because of their intense heat. It is one of the key reasons stars can fuse elements into new ones.


Nuclear fusion is the process by which two atomic nuclei approach each other and get “glued” into a larger atomic nucleus. But atomic nuclei have the same electric charge, so at short distances they experience an extremely strong electrostatic repulsion.

In the cores of stars, the fusion process is helped by the large temperatures, which give the nuclei a very high kinetic energy which helps them to get close to one another despite the electrostatic repulsion. However, even the high temperatures of the core of a star aren’t enough to surmount the electrostatic repulsion. If nuclear fusion happens at all, it is because of a process called “quantum tunneling”: an exclusively quantum mechanism that allows a particle to get through a potential barrier even if it doesn’t have enough energy to do so classically (i.e. non quantum-mechanically):


#Princeton University, Department of Astrophysical Sciences (retrieved 2023): The physics of fusion in stars”. Lecture Notes AST403/PHY402, Stars and Star Formation

https://www.astro.princeton.edu/~gk/A403/fusion.pdf
 Quote: “The strong force binds nucleons (protons and neutrons) in nuclei but has a limited range, of order one fermi: 1 fm ≡ 10−13 cm = 10−15 m, so a fermi is also a femtometer. At separation r, the electrostatic energy between nuclei of charges Z1e and Z2e is ≈ 1.5Z1Z2 MeV fm/r, whereas thermal energies are ~ kBT = 1.4(T/107 K) keV. Since the Boltzmann distribution falls off exponentially at E ≫ kBT, and T ≈ 1.58 × 107 K at the center of the Sun, the probability that two colliding protons could approach within 1 fm would be ~ e−670 ~ 10−290 if classical physics applied. But quantum-mechanical tunneling allows the protons to go “under” the Coulomb barrier with a probability that is much larger than this, though still exponentially suppressed.”


– But it also happens at a temperature near absolute zero. Just, well, mind numbingly slowly.


But the same process of nuclear fusion via quantum tunneling at high temperatures (thermonuclear fusion) can also happen at low temperatures if the density is sufficiently high. In this case quantum tunneling and nuclear fusion can also take place, only at a much lower rate. Such a process is known as “pycnonuclear fusion”:


#Afanasjev, A.V. et al. (2003): Pycnonuclear Reactions. Wissenschaftlich-Technische Berichte

FZR-401

https://inis.iaea.org/collection/NCLCollectionStore/_Public/35/095/35095061.pdf 

Quote: “Fusion reactions inside compact astrophysical objects can be divided into thermonuclear fusion and pycnonuclear processes. While thermonuclear fusion takes place in relatively hot and dilute plasmas inside stars with only the high-energetic components of the velocity distribution being important, pycnonuclear fusion happens at rather high densities where mostly low-energetic nuclei contribute to the fusion process.”


In a black dwarf, pycnonuclear fusion is expected to slowly fuse the atomic nuclei in the star lattice all the way up to iron, i.e. basically the same thing that massive stars achieve by means of thermonuclear fusion, but only in the astonishing timescale of ~101000 or 101500 years instead of the usual ~106 years that massive stars need to fuse chemical elements up to iron:

#Caplan, M.E. (2020): Black dwarf supernova in the far future. Monthly Notices of the Royal Astronomical Society, Vol. 497 (4)

https://academic.oup.com/mnras/article/497/4/4357/5884975

Quote: “Pycnonuclear fusion reactions, driven by quantum tunnelling of adjacent nuclei on the crystal lattice within black dwarfs, will tend to process the matter towards iron-56 (which is possibly the ground state of baryonic matter; Page 1992). Dyson (1979) estimates a tunnelling time-scale of


T = eST0, (1)

where T0 is a characteristic nuclear time-scale (ℏ/mpc2 ≈ 10−25 s) and S is an action integral approximated for fusion by 


S ≈ 30A1/2Z5/6, (2)

with A and Z as the mass and charge of the fusion product, respectively. Dyson (1979) obtains these expressions by approximating the action S ≈ (8MUd2/ℏ2)1/2 using mean barrier height U and width d for a tunnelling particle of mass M. This barrier is then taken to be the Coulomb barrier screened over a distance d = Z−1/3(ℏ/me2) with a reduced mass M = Amp/4 for two A/2 and Z/2 nuclei. This treatment excludes the sensitive density dependence for pycnonuclear fusion, which occurs most quickly at higher densities (Schramm & Koonin 1990; Afanasjev et al. 2012; Meisel et al. 2018). For silicon-28 nuclei fusing to iron group elements, Dyson’s scheme gives an approximate 101500 yr time-scale for tunnelling to process black dwarfs to iron-56


– Here, in this lone black dwarf something fantastic occurs. Nothing happens for a trillion years. Nothing at all. Can you imagine that? But then! A single fusion reaction: two carbon nuclei combine by quantum tunneling to become magnesium! Another 100 trillion years pass. It happens again! Then nothing for another bazillion years. Oh! Two oxygen nuclei combine into silicon! 


These timescales for particular pycnonuclear fusion processes cannot be found in the published literature. They have been calculated and communicated to us by Prof. Matt Caplan, nuclear physicist from Illinois State University and author of the work on black dwarf decay upon which a big part of this script is based:

#Caplan, M.E. (2020): Black dwarf supernova in the far future. Monthly Notices of the Royal Astronomical Society, Vol. 497 (4)

https://academic.oup.com/mnras/article/497/4/4357/5884975

Abstract: In the far future, long after star formation has ceased, the universe will be populated by sparse degenerate remnants, mostly white dwarfs, though their ultimate fate is an open question. These white dwarfs will cool and freeze solid into black dwarfs while pycnonuclear fusion will slowly process their composition to iron-56. However, due to the declining electron fraction, the Chandrasekhar limit of these stars will be decreasing and will eventually be below that of the most massive black dwarfs. As such, isolated dwarf stars with masses greater than ∼1.2 M⊙ will collapse in the far future due to the slow accumulation of iron-56 in their cores. If proton decay does not occur, then this is the ultimate fate of about 1021 stars, approximately 1 percent of all stars in the observable universe. We present calculations of the internal structure of black dwarfs with iron cores as a model for progenitors. From pycnonuclear fusion rates, we estimate their lifetime and thus delay time to be 101100 yr. We speculate that high-mass black dwarf supernovae resemble accretion induced collapse of O/Ne/Mg white dwarfs while later low mass transients will be similar to stripped-envelope core-collapse supernova, and may be the last interesting astrophysical transients to occur prior to heat death.



– Remember the breathtaking amount of time it took for a supermassive black hole to evaporate? This is a brief moment in comparison to what is going on here. 


The typical timescale for a black dwarf to decay is roughly of the order of 101000 years:

#Caplan, M.E. Black dwarf supernova in the far future. 2020.

https://academic.oup.com/mnras/article/497/4/4357/5884975

Quote: “From pycnonuclear fusion rates, we estimate their lifetime and thus delay time to be 101100 yr.”


which is a whopping 10900 times longer (a 1 followed by 900 zeroes – a number so large that there is no name for it) than the typical timescale of evaporation of a supermassive black hole.


–The difference between a second and trillions of years has lost all meaning.

 A curious property of these timescales is that they are so huge that the units we choose to measure them become largely irrelevant. In order of magnitude, one year is:


1 year ~ 107 seconds. 


So if we express 101000 years in seconds, we get:

101000 years ~ 101017 seconds

But we can also say that this is roughly equal to:


101000 years ~ 101000 seconds


since the relative error in the exponent is 17/1000 = 1.7%


The same happens if we express 101000 years in trillions of years:


1 year = 10–12 trillion years

Which means that:


101000 years = 10988 trillion years ~ 101000 trillion years


since the relative error in the exponent will again be of the order of 1%.


So these timescales are so huge that, for any practical purpose, we get that:


101000 years ~ 101000 seconds ~ 101000 trillion years


This is so because the difference between a year and a second (a factor of 107) or between a year and a trillion years (a factor of 1012) is very extremely tiny compared to the prefactor (101000). This is the precise meaning of the statement that the difference between a second and a trillion years has lost all meaning.



– When silicon nuclei fuse, they form Nickel-56. Nickel-56 is radioactive, which means it is unstable. And when it eventually decays and turns into iron, it emits two positrons – antimatter electrons. And these two positrons, find two electrons and annihilate them and themselves. Which is a problem. Remember how the uncomfortable electrons produce the pressure to hold the star together? Destroying an electron means fewer friends to help them hold up the star. Losing an electron does not give them more space to scratch their butts, it just makes gravity squeeze harder, the walls closing in on those that remain. In the case of the most massive black dwarfs this is catastrophic. Eventually, one by one, the black dwarf turns into a sphere of iron. And every time this happens, more electrons get annihilated.


As explained above, a white or black dwarf is subject to an inner pull due to gravity, which is counterbalanced by the outward push exerted by electron pressure. Electrons are nearly massless, so basically all the gravitational pull experienced by the star is due to the amount of nucleons (protons and neutrons) that it contains. Therefore, a key parameter when assessing the stability of a white/black dwarf is the electron-to-nucleon ratio, or “electron fraction”, usually denoted Ye:


Ye = number of electrons / (number of protons + number of neutrons)


For a “typical” white/black dwarf made of carbon-12 (6 protons, 6 neutrons and 6 electrons), oxygen-16 (8 protons, 8 neutrons and 8 electrons) or silicon-28 (14 protons, 14 neutrons and 14 electrons), the number of electrons will always be 1/2 the number of protons plus nucleons, so Ye = 0.5.

 However, if Ye becomes too low, it will mean that there are too few electrons per nucleon and that, at some point, the gravitational pull exerted by the nucleons will be able to overcome the outward pressure exerted by the electrons. And given enough time, this is what will happen in a black dwarf. The process is as follows.


As our black dwarf goes on fusing elements via pycnonuclear fusion, more and more nuclei will be transmuted into heavier ones. But this process doesn’t go indefinitely. When two silicon-28 nuclei fuse, they produce nickel-56, which is made of 28 protons and 28 neutrons. But this isotope of nickel is unstable, and quickly decays to cobalt-56 (27 protons and 29 neutrons) by transforming one proton into a neutron and emitting a positron. But cobalt-56 is also unstable, and further decays by the same positron-emitting nuclear process into iron-56 (26 protons and 30 neutrons).


Once these positrons are produced in the interior of the black dwarf, they will swiftly meet one of the electrons inside the star. But positrons and electrons are antiparticles of one another, so when they meet they annihilate each other in a flash of energy. This means that, as more and more iron is produced, more and more electrons are annihilated, which reduces the total electron fraction of the star and increases the gravitational pull relative to the electron pressure:

#Caplan, M.E. (2020): Black dwarf supernova in the far future. Monthly Notices of the Royal Astronomical Society, Vol. 497 (4)

https://academic.oup.com/mnras/article/497/4/4357/5884975

Quote: “Finally, once the core has largely burnt to silicon-28 (or a similar nuclide), we expect it will fuse to produce nickel-56 (or a similar iron group element) which can then decay through positron emissions to iron-56, with Ye = 0.464. The annihilation of two electrons from these reactions slowly deleptonizes the star and reduces the core Ye, resulting in contraction which increases core pressure.”



– At least 101000 years pass without anything visible happening in the entire universe.  That number means basically forever, but not quite. And then, finally, the last thing to ever happen in the universe happens. The black dwarf has lost one too many electrons. It can no longer support its immense mass and goes into an uncontrolled collapse - a supernova. It first implodes and then explodes as bright as a galaxy and fills the empty universe with light again! A beautiful moment nobody will get to enjoy. And then, as quickly as it began, it is over. 


As explained above, the subsequent chain of extremely slow pycnonuclear reactions taking place inside a black dwarf progressively produces positrons. These annihilate electrons, reducing the total number of electrons inside the star and therefore diminishing the electron pressure relative to the gravitational pull exerted by protons and neutrons.

After a time of the order of 101000 years, the number of electrons inside the black dwarf becomes too low to counterbalance the gravitational pull. When this occurs, the star collapses under its own weight and –just as it happens when the core of a massive star collapses–, the whole star undergoes a violent supernova explosion:

#M E Caplan, Black dwarf supernova in the far future. 2020.

https://academic.oup.com/mnras/article/497/4/4357/5884975

Quote: “If proton decay does not occur then in the far future we expect approximately 1 percent of all stars today, about 1021 stars, to collapse and explode in supernova beginning in approximately 101100 yr and lasting no more than about 1032,000 yr. At such advanced time, it is difficult to imagine any other astrophysical processes occurring, which may make black dwarf supernova the last transients to occur in our universe prior to heat death.”