Some physics about black holes (1/2)

Some physical properties of a black hole
In this page the classical macroscopic physical properties of black holes are discussed. Altough black holes are extreme physical phenomena which require considerations which belong to the realm of Relativistic Physics or Quantum Physics, the classical physical approach is sufficient to derive some interesting properties of a black hole.

Logarithmic plots
The properties are presented in plots on logaritmic scales to capture the entire range of values of physical properties a black hole can have, from a quantum mechanical to a cosmological scale.
A black hole can be very small, comparable with the size of an electron. Or very large, the size of an entire star system.
The minimum mass of a black hole is 2.177 x 10-8 kg. The maximum mass is probably limited to about 10 - 60 billion Solar mass. According to models developed by Inayoshi and Haiman in 2016, the efficiency of the accretion process is believed to drop significantly for such large black holes. In the immensely large accretion disk mass begins to flow outwards and star formation has set in, which hampers the feeding process of the huge black hole.
The heaviest known black hole is known as TON 618, it is a quasar, some 18 billion light years from here. It has a mass of about 65 billion Solar mass, that is about a twentieth of the mass of the entire Milky Way. It nevertheless seems to exceed the mass limit as proposed by Inayoshi and Haiman.
The smallest black hole has a lifetime of about 10-44 seconds. The life time of a black hole of one Solar mass is 2 x 1067 years which is 1.6 x 1057 times the current age of our universe.

Black hole
A black hole is considered as a clump of matter which is collapsing under its own mass or gravity into an infinitely tiny point of space. There is no force left to counter balance the gravity. This tiny point is called a singularity because all our conceptions about space and time seem to have broken down.
Because of the fact that matter has been compressed into a tiny point the gravity in the vicinity of such a singularity has become extremely strong. So a relative tiny space has emerged around this singularity in which the gravity is that strong that the escape velocity has exceeded the speed of light. The distance to that singularity at which the escape velocity is equal to the speed of light is called the Schwarzschild radius. This radius is proportional to the mass of the singularity. So the singularity with its spatial sphere with a diameter of two times the Schwarzschild radius manifests itself as a dark sphere from which light cannot escape: a black hole. From a far distance however the gravital influence of a black hole with mass M is indistinguishable from that of a normal star with mass M.

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Physical constants

Figure 1

The Schwarzschild radius of the UMBH (ultra massive black hole) TON 618 of about 1300 AU (180 light hours) is incredible.
The radius of Neptune's orbit is 30.1 AU. Neptune is our most outer planet. The size of our entire solar system is dwarfed by the size of TON 618.

According to Inayoshi and Haiman (The Astrophysical Journal, Vol 828, No 2, Sep 2016) a black hole cannot grow much larger than TON 618. The required gas supply rate to let an UMBH grow even larger is estimated at about 1000 solar mass per year. The accretion process of such UMBH's becomes hampered by the strong outflow of of hot gas jets. It is even expected that in the immense large accretion desk surounding such an UMBH star forming might set in.

I guess an UMBH can still gain size when it swallows whole stars or black holes. However an accretion disk mainly contains gas, so I guess gas is the main diet for black holes. Statistically, stars and black holes make up a very small portion of its diet.
A merger between two UMBH's or two quasars might be a groundbreaking event. It will result in an UMBH with around twice the size and mass. However it is probably a rare event and cannot be considered as a sustainable mechanism to have UMBH's grow in size by orders of magnitude.


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Figure 7

Black holes will turn into white holes
Every black hole loses mass due to Hawking radiation. In fact all the mass it has swallowed will eventually be released as photon energy. The evaporation rate is depending on its mass. This plot shows that the evaporation rate of a black hole, due to Hawking radiation, is higher for small black holes with less mass. The lower the mass the higher the evaporation rate. Consequently, the lower the mass the more intense and energetic the emitted radiation.

A short-lived micro white hole of 1.0 metric Tons
A black hole which has a luminosity which is equal to LSun has a mass of 1.0 metric Tons and is 108 times smaller than a proton. However, it has a lifetime of about 100 nanoseconds. The evaporation rate is astronomically high: 10 metric Mega Tons per second. So it seems that when black holes are starting to be observable as a white hole they will be be observed as a burst of Gamma radiation with rapidly increasing intensity and energy over a time period of only 100 nanoseconds.

The luminosity of the smallest back hole / white hole
The smallest black hole, which is possible theoretically, has a mass of 2.177 x 10-8 kg. But it is the brightest one with a luminosity of 2.36 x 1048 Watt. But because of its extremely short lifetime of 8.68 x 10-40 seconds it will only release 2.05 x 109 Joule. That is what a one gigawatt powerplant would produce in two seconds.
So the swan song of this micro black hole is a 8.68 x 10-40 seconds burst of Gamma radiation, with a total energy of about 1.28 x 1022 MeV.

Personal reminder:
It requires some research on my side to find out whether there are models which describe the possble radiation spectra of such Gamma radiation bursts. In other words what is the expected energy distribution of the Gamma photons which are emitted during the burst? And what are the factors which might influence such a distribution?

Another fact what confuses me has to do with the calculated photon energy:
h·ν = h·c3/(4·G·M)
The mass (M) of our micro black hole is 2.177 x 10-8 kg.
So the calculated photon energy is 3.073 x 109 Joule
We know that the luminosity is 2.36 x 1048 Watt.
That means that it takes 1.3 x 10-39 seconds to emit that 3.073 x 109 Joule, which is, however, longer than the remaining 8,68 x 10-40 seconds of lifetime.
We should however keep in mind that in these very last moments of 10-40 seconds both the emitted energy and therefore luminosity is increasing in an extremely fast rate. The classical mathematical relationship between energy and luminosity (E = L x Δt) is probably not that straightforward.
The luminosity is derived from the evaporation rate: dM/dt. The energy, in this consideration, is the emitted photon energy whose frequency of radiation can be calculated with expression 4 shown at the top of this page. That is another reason why a description of the relationship between emitted energy and luminosity needs a more sophisticated approach.
Although a black hole of mass equal to the Planck mass is a rather extreme case, this case seems nevertheless relevant to me. After all it is the final phase of any black hole which is no longer fed.

These formulas seem also to suggest that for such a small micro black hole of absolute minimum mass there is no energy distribution of emitted Gamma photons, only one extremely high energy Gamma photon is emitted. How about conservation of momentum, I might ask, for which at least two photons are required?

To put this huge amount of energy in perspective we have to calculate how much matter represents the same amount of energy.
The rest energy of a proton: Ep,0 = mp·c2 = 1.504 x 10-10 Joule.
When this rest energy is expressed in eV instead of Joule: 938.6 MeV.
So 1.28 x 1022 MeV of energy is equivalent to the total rest mass of 1.36 x 1019 protons (which is about 22 microgram).

It is to be expected that a fraction of the emitted Gamma photons result in the production of matter and antimatter through proton-antiproton and electron-antielectron pair production for example. After a short while (nanoseconds?) matter and antimatter might recombine resulting in a secondary burst of Gamma radiation.

Personal speculation:
It would not surprise me if this process of pair production is repeated multiple times resulting in multiple generations of Gamma ray bursts with decreasing intensity and energy (over a time period of about several nanoseconds). When the energy of each Gamma photon has dropped below a threshold value of about 1 MeV the pair production will stop. If however all previous generations of Gamma Ray burst have resulted in pair production it will probably mean that the black hole dissipation event will eventually result in a Gamma ray burst with a photon energy of 1 MeV or less. We will probably never be able to detect the previous generations of Gamma ray bursts.
But there might be a condition which can influence the recombination process. Maybe there is a possiblity that some of the produced particles can leak outwards.

In this page four properties of a black hole have been discussed: its mass, size, lifetime and its luminosity.
In the NEXT PAGE a nomogram is presented to enable a coherent evaluation of eleven properties:
  1. Mass
  2. Evaporation rate. The rate at which a black hole its mass due to Hawking radiation
  3. Size, its Schwarzschild radius RS
  4. Lifetime
  5. the frequency of the emitted Hawking radiation which can range from low radio frequencies up to very high frequency gamma radiation.
  6. Surface temperature
  7. Luminosity
  8. Gravitational acceleration at distance RS, which is a measure of the strength of the gravitational force
  9. Gravitational tidal effects at distance RS
  10. Time to free fall from distance RS
  11. Effective mass density

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