Before we close this section we point out that others have applied the concept of gravitational repulsion in various circumstances. Dickau, Kauffmann and Robertson employ it to solve the problem of the accelerating expansion of the universe (in preparation). [77] investigate gravitational repulsion in the Einstein-zero-mass scalar theory. [78] discuss gravitational repulsion in an expanding ball of dust. [79] considered gravitational repulsion in the Kerr–Newman anti-de Sitter spacetime. [80] looked into gravitational repulsion in the Reissner-Nordström and Schwarzschild spacetimes. [81] show that the emission of gravitational waves leads to a repulsive gravitational force that diminishes with time but never disappears. They speculate that the repulsive force may be related to the observed expansion of the Universe. [82] point out that gravitational repulsion could appear in satellite experiments with beams of relativistic particles subject to very precise time measurements. [83] points out that gravitational repulsion occurs in geodesics in a quash-spherical spacetime. [84] investigates a number of aspects of the phenomenon of gravitational repulsion in static sources of the Reissner–Nordström field. [85] study gravitational repulsion in the Kerr and Kerr-Newman fields.

We now ask whether according to our theory an AGN can be the source of an ultra high energy proton of E = 10 20   eV . To accomplish this task we employ the above equation. On the right side of the equation the only unknown quantity is M , the mass of the supermassive black hole in the presumed AGN source. But, we do know the range of masses, which is: 10 6 M ⊙ ≲ M SMBH ≲ 10 10 M ⊙ [86]-[96]. So we can calculate the minimum and maximum distance of an AGN, that could possibly be responsible for producing our ultra high energy proton of 10²⁰ eV.

If the AGN mass is 10 6 M ⊙ then d = 2.175 × 10 6   Gpc . If the AGN mass is 10 10 M ⊙ then its distance is d = 2.175 × 10 10   Gpc . Both of these distances are far greater than the maximum distance such a ultra high energy proton can travel in intergalactic space, which is only 100 Mpc [16][17][97][98]. This maximum distance is due to the Greisen–Zatsepin–Kuzmin (GZK) effect, whereby ultra high energy protons are transformed into pions by interactions with CMB photons [99]-[103]. We conclude: AGNs can not be the source of ultra high energy cosmic rays according to our theory.

In [13] we concluded that blazars, which are a subset of AGN, can not be the source of ultra high energy neutrinos. Instead we concluded that stellar mass bodies must be the source of these particles. Now that we have shown that according to our theory AGN are also not the source of ultra high energy cosmic ray protons, we ask could stellar mass bodies be the source of these particles? Following the procedure in the previous section we calculate the minimum and maximum distance of stellar mass objects to see, if they fall under the GZK limit of 100 Mpc.

As in [13] we base our analysis on the following: 1) The exterior gravitational field of a spherically symmetric non rotating star is the Schwarzschild metric. 2) F to M spectral type stars emit sporadically high energy protons [104]-[107]. 3) Therefore the minimum mass of stellar mass bodies that emit high energy protons is the minimum mass of M stars: 0.075 M ⊙ [108]-[110]. 4) The maximum mass of stellar mass bodies that emit high energy protons is the maximum mass of F stars: 1.7 M ⊙ [111][112].

Inserting the minimum mass of M stars of 0.075 M ⊙ into the Equation (3) yields: 163.1 Mpc. Inserting the maximum mass of F stars: 1.7 M ⊙ yields: 3696 Mpc. These values are larger than the GZK limit of 100 Mpc. Consequently, we conclude that main sequence stars can not be the source of ultra high energy protons.

Since we know the maximum distance of the source of ultra high energy cosmic ray protons, d = 100   Mpc , we can rearrange Equation 3 so that it yields the maximum mass of the source of these particles.

Our ultra high energy proton of 10²⁰ eV with n = 2.27181 × 10 22 and d = 100   Mpc or 3.086 × 10²⁶ cm the above equation yields: 0.046 M ⊙ or 48 Jupiter masses. This is the mass of a brown dwarf, whose masses lie in the range 0.013 to 0.075 M ⊙ , which is 13 to 75 Jupiter masses [113]-[118].

Brown dwarfs are expected to produce energetic protons through magnetic activity analogous to stellar flares. This is because many brown dwarfs possess the same properties that are listed in the introduction for main sequence stars which lead to solar flare type events [119]-[123].

We turn to investigating the range of brown dwarfs masses that are the sources of ultra high energy cosmic protons. To accomplish this task we rearrange Equation 3 or 4.

With d = 100   Mpc or 3.086 × 10²⁶ cm for the maximum mass of a brown dwarf, 0.075 M ⊙ , we obtain: n = 1.39312 × 10 22 and for the minimum mass of a brown dwarf, 0.013 M ⊙ , n = 8.03724 × 10 22 , whereby n is the distance of the source in Schwarzschild radii, α . Now our task is to convert n into E , the measured proton energy. The most convenient relationship to employ is [13]:

where m is the proton rest mass.

Equation (6) was obtained by solving the relationship between the measured proton energy, E and n : E = ( γ − 1 ) m = ( 1 1 − ( 1 − 1 n ) 2 − 1 ) m whereby γ = 1 1 − v 2 = 1 1 − ( 1 − α r ) 2 and v 2 = ( d r d t ) 2 = ( 1 − α r ) 2 .

Curve fitting Equation (6) leads to:

with a = 8.805645474 ± 0.000562541 , b = 0.503052028 ± 0.000125021 and c = 0.998548883 ± 6.18829 × 10 − 5 . Inserting the values of n we derived into this equation we obtain for the maximum mass of a brown dwarf E = 7.85714 × 10 19   eV and for the minimum mass of a brown dwarf E = 1.88741 × 10 20   eV . We conclude that brown dwarfs are the source of ultra high energy cosmic protons in this energy range.

Specifically, brown dwarfs provide a seed population of protons of GeV energy, which is the prerequisite for gravitational repulsion to occur according to Inequality 1. Such protons are then accelerated to very high and ultra high energy via gravitational repulsion.

Brown dwarfs may actually be better proton accelerators than solar-type stars. In order to understand why we need to invoke a simple order of magnitude relationship which yields the maximum particle energy attainable in a magnetized accelerator. It can be estimated from the Hillas confinement condition

where E is the induced electric field, Z the particle charge, e the elementary charge, B the magnetic field strength, L the size of the acceleration region, and β = v / c the reconnection speed in units of c [18]. The key feature of this equation is that the maximum particle energy scales linearly with magnetic field strength. While the magnetic field of the sun in active regions is in the range, 0.1 - 0.3 kG and active M-dwarfs, 1 - 4 kG [124]-[129], the brown dwarf range is: 1 - 5 kG [130][131]. Thus, brown dwarfs can have magnetic fields an order of magnitude stronger than the Sun. In addition brown dwarfs rotation periods are in the range, 2-10 hours [122][132]-[136], while the solar value is 25 days. Rapid rotation strengthens the dynamo mechanism, producing stronger magnetospheres and more violent reconnection events. Finally, we note that the observed flare energies from brown dwarfs reach E flare ~ 10 34   -   10 38   erg [123]. These are comparable to or larger than large solar flares. These properties mean that brown dwarfs are expected to produce higher energetic protons than solar-type stars.

We point out that Photo-ionization processes contribute to the ionization of the substellar atmosphere and magnetosphere, thereby increasing the population of free charged particles available for acceleration. However, photo-ionization itself does not accelerate protons to relativistic energies. The production of high-energy seed protons is instead attributed to magnetic processes such as flares, reconnection, and auroral currents, while the subsequent acceleration to very high and ultra high energies is provided by gravitational repulsion in the Schwarzschild field of the source.

Equation (5) shows that if d is smaller the proton energy will be smaller too. So brown dwarfs are also responsible for proton energies < 7.857 × 10¹⁹ eV. However for protons energies > 1.887 × 10²⁰ brown dwarfs are not the source as we make clear in section 3.4.

3.3.1. Energetics Requirement

The observed ultra high energy cosmic ray (UHECR) flux corresponds to a local energy generation rate of approximately

for particles above ~10¹⁹ eV [137][138].

We adopt a fiducial population of 10¹⁵ for the number of brown dwarfs within 100 Mpc. We obtained this value by using the present-day star-to-brown-dwarf number ratio of approximately 4:1 and the average mass per object derived from the full-sky 20 pc census [139], together with the mean stellar mass density of the Local Volume, Ω ⋆ ≃ 0.0044 of the critical density [140], one obtains a brown-dwarf number density of order

Hence the number of brown dwarfs within 100 Mpc is

The time-averaged UHECR power required per brown dwarf is therefore

If only a fraction, f act , of brown dwarfs is active and each active object produces a UHECR yield E evt every Δ t years, then

Thus the energetics requirement can be satisfied if a few percent of brown dwarfs produce ~10³⁷ erg in UHECRs per year-equivalent, or ~10³⁸ erg once per decade.

3.3.2. Seed-Particle Acceleration and the Hillas Limit

In convenient units Equation 8 becomes:

For brown dwarfs with magnetic fields of order 10³ - 5 × 10³ G and characteristic size L ~ 7 × 10 9 cm, which comes from the typical radius of a brown dwarf. Brown dwarfs have radii comparable to Jupiter, typically:

R BD ≈ 0.7   -   1.4 R J with only weak dependence on mass [113][141]-[146]. Using R J = 7.1 × 10 9   cm , a characteristic size for a brown dwarf is therefore L ~ 7 × 10 9   cm .

Thus magnetospheric activity in substellar objects can plausibly produce relativistic seed particles but does not reach the full UHECR energy range. The Hillas limit therefore determines the seed-particle energy, while the subsequent Schwarzschild gravitational repulsion stage provides the final acceleration to the observed energies above 10¹⁹ eV.

3.3.3. Source Density and Approximate Isotropy

Another major constraint on UHECR models is the effective source density required to avoid strong clustering in the observed sky. Analyses of UHECR anisotropy typically require source densities exceeding roughly

[147]-[149] For brown dwarfs the available source density is enormously larger:

Consequently even an extremely small active fraction yields a very large effective source population. The aggregate emission from such a dense population behaves much more like a quasi-continuous emissivity field than like a sparse set of discrete point sources.

This naturally produces a largely smooth UHECR sky with only weak anisotropy tracing the large-scale matter distribution, broadly consistent with current observations [150]-[153].

3.3.4. Propagation Constraints and the GZK Horizon

UHECR propagation distances are limited by interactions with the cosmic microwave background (CMB). Above ~5 × 10¹⁹ eV, protons lose energy through photopion production

which restricts the effective source distance to of order ~100 Mpc [99][100].

Typical attenuation lengths are

Thus the observed UHECR flux must originate predominantly from sources inside the local universe within the GZK horizon. Even within a sphere of radius 100 Mpc, however, the expected number of brown dwarfs is of order

Therefore the local universe contains an enormous reservoir of potential seed-particle injectors,

3.3.5. Replacement of the Single Fixed 100 Mpc Horizon

In the above we have assumed that the GZK horizon is a fixed number of 100 Mpc. However, the attenuation length depends on the proton energy. In this section instead of assuming a single fixed propagation horizon, we treat the relevant attenuation length as an energy and composition dependent quantity D att ( E , A ) in order to show how the inferred source-mass bounds would change. For proton primaries, the attenuation length is of order Gpc at 10¹⁸ - 10¹⁹ eV, decreases to roughly 100 - 200 Mpc near 5 × 10¹⁹ eV, and falls to only ~10 - 50 Mpc by 10²⁰ - 10^20.5 eV due to photopion production on the cosmic microwave background [17][154][155]. For heavier nuclei the propagation horizon is also composition dependent because of photodisintegration processes.

Accordingly, any source-mass bound previously written using a fixed 100 Mpc horizon should be interpreted as

where δ is the distance exponent appearing in the underlying derivation. Thus the inferred bounds shift upward at 10¹⁸ - 10¹⁹ eV and downward at 10²⁰ - 10^20.5 eV relative to the fixed-distance approximation.

3.3.6. Testable Predictions

The brown dwarf seed particle scenario combined with the Schwarzschild gravitational repulsion acceleration stage leads to several observational signatures that can be tested with current ultra high energy cosmic ray datasets.

1) Weak correlation with nearby galaxy catalogs

Because brown dwarfs and planetary-mass objects are ubiquitous constituents of galaxies, the predicted UHECR sky should correlate weakly with the large-scale distribution of nearby galaxies rather than with rare source classes such as powerful AGN or gamma-ray bursts. Cross-correlation analyses using existing galaxy catalogs (e.g., 2MASS, Cosmicflows, or HECATE) therefore provide a direct test of the model.

2) Energy-dependent anisotropy amplitude

As the UHECR energy increases, the propagation horizon decreases due to interactions with the cosmic microwave background. In a model where the effective number of contributing sources inside the horizon is very large, the anisotropy amplitude should scale approximately with the inverse square root of the number of contributing sources,

where D att ( E ) is the energy-dependent attenuation length. The model therefore predicts a gradual increase in anisotropy amplitude toward the highest energies.

The acceleration mechanism considered here is spherically symmetric and does not depend on anisotropies in the interior structure of the source or in the cosmological background. Gravitational repulsion in the Schwarzschild field depends only on the mass of the source and the radial motion of the particle. Therefore, anisotropy enters only at the observational level through the spatial distribution of sources and propagation effects, not through the acceleration process itself. This distinguishes the present model from jet-based or shock-acceleration scenarios, in which anisotropic geometries play a central role in particle acceleration.

3) Absence of strong point-source clustering

Because the candidate source population is extremely numerous, the observed UHECR sky should appear as the superposition of many weak sources rather than a few dominant objects. Consequently the model predicts little or no statistically significant point-source clustering even at the highest energies.

4) Characteristic seed-acceleration scale

Magnetospheric processes in substellar objects accelerate charged particles up to a characteristic seed energy determined by the magnetic field strength and size of the object. This scale may imprint a spectral transition separating the seed-acceleration regime from the subsequent gravitational acceleration stage.

Future measurements of the energy spectrum, anisotropy amplitude, and cross-correlations with nearby galaxy catalogs using data from the Pierre Auger Observatory and Telescope Array will therefore provide direct tests of the model.

The highest energy cosmic ray particle ever detected had an energy of 3.2 × 10²⁰ eV. It was detected in 1991 by the Fly’s Eye Experiment in Utah [156]. This ultra high energy cosmic ray particle is referred to as the “Oh-My-God” particle. It was most likely a proton, but the possibility of it being a more massive nucleus can not be excluded. Here we assume it was a proton.

Inserting the energy of this particle in Equation 6 leads to: n = 2.326 × 10 23 , which is more than an order of magnitude beyond the maximum value brown dwarfs can achieve. Inserting this value of n along with d = 100   Mpc into Equation 4 we obtain: M = 0.0045 M ⊙ , which is 4.7 Jupiter masses.

It is thought that such celestial bodies can produce protons in the KeV-MeV range and even up to 100 MeV in extreme conditions [130][157]. The prerequisite for gravitational repulsion, Inequality 1, however, requires the production of protons in the GeV energy range. Our theory therefore predicts that such a planetary mass body can under extreme conditions produce such proton energies. This circumstance explains why there are so extremely few detections [158], of ultra high energy particles with E > 1.887 × 10 20   eV .

The differential ultra high energy cosmic-ray spectrum can be approximated by a broken power law

Observations show spectral indices of approximately γ ≃ 3.3 below the ankle and γ ≃ 2.6 above it. In the present scenario the injected spectrum from the acceleration process is expected to be relatively hard, with an index γ i n j ≈ 2 , consistent with many astrophysical acceleration mechanisms. Propagation effects during intergalactic transport—primarily pair production and photo pion interactions with the cosmic microwave background—steepen the spectrum and produce the observed indices. The ankle at E ≃ 5 × 10 18   eV is interpreted as the transition between the Galactic component and the extragalactic component supplied by substellar seed particle sources. The suppression above ~4 × 10¹⁹ eV arises naturally from the GZK energy-loss process together with the maximum attainable energy in the acceleration mechanism. A shallow dip below the ankle can arise from electron–positron pair production by extragalactic protons interacting with the cosmic microwave background [17][155][159].

Figure 1 shows proton energy as a function of the mass of a brown dwarf source. The top red curve assumes the distance of the source is 100 Mpc, the maximum distance a cosmic ray proton can have according to the GZK effect. The radius of the stellar halo of the Milky Way is between 100 and 150 kpc [163]-[167]. Therefore the maximum distance as observed from the sun, which is 8.2 kpc from the galactic center is: between 108 kpc and 158 kpc. In the figure we just use the mean of these two values: 133 kpc. Therefore any brown dwarf proton source between the red and blue curves is extragalactic in origin.

Figure 1 contains four sections. 1) There can be no cosmic ray protons from brown dwarfs above the red line, which is indicated with the word “Forbidden”. This is due to the GZK effect. 2) Between the red and blue curves the proton sources are extragalactic brown dwarfs. 3) Between the blue curve and black curve

Figure 1. Proton energy as a function of brown dwarf mass.

the proton sources are brown dwarfs located in the galactic halo. 4) Below the black curve the brown dwarf proton sources are located in the disk of the Galaxy.

Figure 2 shows the proton energy as a function of stellar mass. The top red curve assumes the distance of the source is 100 Mpc, the maximum distance a cosmic ray proton can have according to the GZK effect. The radius of the stellar halo of the Milky Way is between 100 and 150 kpc. Therefore the maximum distance as observed from the sun, which is 8.2 kpc from the galactic center is: between 108 kpc and 158 kpc. In the figure we just use the mean of these two values: 133 kpc. Therefore any stellar proton source between the red and blue curves is extragalactic in origin.

Figure 2 contains four sections. 1) There can be no cosmic ray protons from stars above the red line, which is indicated with the word “Forbidden”. This is due to the GZK effect. 2) Between the red and blue curves the proton sources are extragalactic stars. 3) Between the blue curve and black curve the proton sources are stars located in the galactic halo. 4) Below the black curve the stellar proton sources are located in the disk of the Galaxy.

Figure 2. Proton energy as a function of stellar mass.