Limits on Magnetized Quark-Nugget Dark Matter from Episodic Natural Events

Abstract

A quark nugget is a hypothetical dark-matter candidate composed of approximately equal numbers of up, down, and strange quarks. Most models of quark nuggets do not include effects of their intrinsic magnetic field. However, Tatsumi used a mathematically tractable approximation of the Standard Model of Particle Physics and found that the cores of magnetar pulsars may be quark nuggets in a ferromagnetic liquid state with surface magnetic field B_o = 10^12±1 T. We have applied that result to quark-nugget dark matter. Previous work addressed the formation and aggregation of magnetized quark nuggets (MQNs) into a broad and magnetically stabilized mass distribution before they could decay and addressed their interaction with normal matter through their magnetopause, losing translational velocity while gaining rotational velocity and radiating electromagnetic energy. The two orders of magnitude uncertainty in Tatsumi’s estimate for B_o precludes the practical design of systematic experiments to detect MQNs through their predicted interaction with matter. In this paper, we examine episodic events consistent with a unique signature of MQNs. If they are indeed caused by MQNs, they constrain the most likely values of B_o to 1.65 × 10¹² T +/− 21% and support the design of definitive tests of the MQN dark-matter hypothesis.

1. Introduction

The great majority of mass in the Universe is non-luminous material called dark matter [1]. Gravity from dark matter literally holds galaxies together [2]. The nature of dark matter has been studied for decades but remains one of the most puzzling mysteries in science [3]. Most dark-matter candidates are assumed to interact with normal matter only through gravity, but stronger interactions are consistent with requirements for dark matter if their effective interaction time is billions of years [3]; Magnetized Quark Nuggets (MQNs) are one such candidate for dark matter. Previously published work [4] shows the primordial origin of MQNs and their compatibility with requirements [5] of dark matter. Their origin in the very early Universe was in the magnetic aggregation of Λ⁰ particles (consisting of one up, one down, and one strange quark) into a broad mass distribution of stable ferromagnetic MQNs before they could decay. After t ≈ 66 μs after the big bang, mean MQN mass is between ~10⁻⁶ kg and ~10⁴ kg, depending on the surface magnetic field B_o. The corresponding mass distribution is sufficient for MQNs to meet the requirements of dark matter in the subsequent processes, including those that determine the Large Scale Structure (LSS) of the Universe and the Cosmic Microwave Background (CMB).

For the last four decades, searches for dark-matter candidates have focused on particles beyond the Standard Model of Particle Physics [6]. MQNs are composed of Standard Model quarks. However, mathematical techniques for applying the Model in the ~90-MeV-energy scale of MQNs rely on approximations. In addition, the key result that leads to MQNs requires a Quantum Chromo Dynamics (QCD) fine-structure constant α_c ≈ 4 at this energy scale; the value of α_c at this energy scale is not known. To the extent that the calculations can be performed, MQNs are consistent with the Standard Model and do not require a new particle Beyond the Standard Model (BSM). Therefore, MQNs have been somewhat controversial as dark-matter candidates.

In the case of this paper, the controversy may be mitigated because we are not claiming discovery. The paper does find that the theory of MQNs is consistent with a reported observation. However, the event is very rare, is not reproducible, was not recorded by multiple observers, and cannot be quantitatively validated after the fact by anyone else. Consequently, the evidence does not meet today’s standard for a discovery.

Quarks are components of many particles in the Standard Model of Particle Physics. Ensembles of strange, up, and down quarks (in approximately equal numbers) are called quark nuggets and are thought to be stable at sufficiently large masses [7] and qualify as candidates for dark matter. Quark nuggets are also called strangelets [8], nuclearites [9], Axion Quark Nuggets (AQNs) [10], slets [11], Macros [5], and MQNs [4]. A brief summary of four decades of research [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] on quark-nugget charge-to-mass ratio, formation, stability, and detection is provided for convenience and completeness as Appendix A: Quark-Nugget Research Summary.

Most models of quark nuggets do not include effects of their intrinsic magnetic field. However, Xu [26] has shown that the low electron density, permitted in stable quark nuggets, limits surface magnetic fields from ordinary electron ferromagnetism to ~2 × 10⁷ T. Tatsumi [27] examined ferromagnetism from a One Gluon Exchange interaction and concluded that the surface magnetic field could be sufficient to explain the ~10¹² T magnetic fields inferred for magnetar cores. Since the result depends on the currently unknown value of α_c at the 90 MeV energy scale, the result needs to be confirmed with relevant observations and/or advances in QCD calculations.

Tatsumi’s result has recently been applied to quark-nugget dark matter. By definition of ferromagnetic, the lowest energy state in Tatsumi’s ferromagnetic liquid is with magnetic dipoles aligned. The individual quark nuggets are formed with baryon number A = 1, as are neutrons and protons, and may have spin and a corresponding surface magnetic field similar to that of protons and neutrons. However, unlike protons and neutrons, ferromagnetic dipoles of quark nuggets (MQNs) align upon aggregation and maintain the surface magnetic field. Their magnetic field at substantial distances is large because their aggregated size is large, not because their intrinsic magnetization (and corresponding surface magnetic field) is necessarily larger than that of other baryons.

Previous work [4] addressed the formation and aggregation of magnetized quark nuggets (MQNs) in the early Universe into a broad and magnetically stabilized mass distribution with baryon number A between ~10³ and 10³⁷ before they could decay by the weak interaction; addressed the compatibility of MQNs with the requirements of dark matter; and addressed their interaction with normal matter through their magnetopause [28], while losing translational velocity, gaining rotational velocity, and radiating electromagnetic energy [36].

Electromagnetic energy accumulated in the Universe from MQNs is unfortunately not detectable because the plasma in most of the Universe is too low density to establish the magnetopause effect and the electromagnetic radiation from the rest of interstellar space is too low frequency to propagate through the solar-wind plasma and reach Earth. However, MQNs transiting through the plasma and gas around Earth spin up to MHz frequencies and should be detectable as they exit the magnetosphere [36]. Predicted event rates are strongly dependent on the MQN magnetic field B_o.

Uncertainty in Tatsumi’s estimate of B_o = 10^12±1 T is too large to design a system to systematically look for MQNs. In this paper, we examine one type of episodic event consistent with a unique signature of MQNs, i.e., an MQN impacting Earth on a nearly tangential trajectory, penetrating the ground for a portion of its path, and emerging where it can be observed. We calculate what would be observed and compare the results with extant observations. Such episodic events are impossible to predict or reproduce and fall short of the standard for evidence of discovery in physics. Therefore, we are not asserting the discovery of MQNs but are examining consistency with MQNs and the resulting constraints on B_o. The results are useful for designing systematic tests of the MQN dark-matter hypothesis.

Without a reasonably small uncertainty in B_o, success in fielding systematic experiments is unlikely. We attempted such an experiment by instrumenting a 30 sq-km area of the Great Salt Lake in Utah, USA, and looked for acoustic signals from MQN impacts. No impacts were observed in 2200 hours of recording. Subsequent theory [4] explained the null result and showed that the predicted mass distribution of MQNs means that impacts are very rare. Even a planet-sized detector is marginal. Consequently, we have turned to observations of episodic, naturally occurring events to narrow the uncertainty in B_o.

Reference 4 shows the most important unknown in the theory of MQNs is the surface magnetic field parameter B_o, which quantifies the uncertainty in the distribution of MQN mass and the event rate. The mean of the surface magnetic field <B_S> is related to B_o in reference [4], through

$$<B_S>=\left(\frac{{\rho }_{QN}}{{10}^{18}(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)}\right)\left(\frac{{\rho }_{DM\_T=100\mathrm{M}\mathrm{e}\mathrm{v}}}{1.6\times {10}^8(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)}\right)B_o$$

(1)

In Equation (1), ρ_QN is the MQN mass density, and ρ_DM is the density of dark matter at time t ~65 μs, when the temperature T in the early Universe was ~100 MeV [37]. If better values of ρ_QN, ρ_DM, and B_o are determined by observations, then a more accurate value of <B_S> can be calculated with Equation (1).

In this paper, we show that a nearly tangential impact and transit through a chord of Earth by an MQN provides a unique signature: a magnetically levitated mass of greater than nuclear density that ionizes and excites the atmosphere for many minutes. We compare the results of analytic calculations and computer simulations of such an event to the observations on 6 August 1868, published by M. Fitzgerald [38] in the Proceedings of the Royal Society, the premier scientific publication at that time. The event’s unusual characteristics are consistent with an extremely rare, nearly tangential, MQN impact. Similar events have been reported elsewhere [39], yet Fitzgerald’s report is the best documented and only scientifically published event we have found, making it the most suitable for comparison with theory.

As noted above, Tatsumi [27] estimated that magnetar cores have B_o = 10^12±1 T. The results reported in this paper and reference 4 constrain the most likely values of B_o to 1.65 × 10¹² T +/− 21% and will permit the design of a systematic experiment to test the MQN hypothesis.

2. Materials and Methods

This study used:

• Analytic methods presented in detail in the results section.
• Computational simulations coupling the Rotating Magnetic Machinery module and the Nonlinear Plasticity Solid Mechanics module of the 3D, finite-element, COMSOL Multiphysics code [40]. Details are included in Appendix B: COMSOL Simulation of Rotating Magnetized Sphere Interaction with Plastically Deformable Conductor.
• Original field work at the location reported by Fitzgerald [38] in County Donegal, Ireland, is documented in Appendix C: Field Investigation of Fitzgerald’s Report to Royal Society. The GPS locations are included to facilitate replication, subject to acquiring permission from the property owners listed in Acknowledgements. Radiocarbon dating was conducted by Beta Analytic Inc. 4985 SW 74th Court, Miami, FL 33155, USA.

3. Results

3.1. Nearly Tangential Impact and Transit of MQNs through Earth

Figure 1 illustrates three MQN impacts on an idealized portion of a spherical Earth.

Only massive MQNs have enough momentum to penetrate significant distances through water or rock. Therefore, we focus on MQNs between 10⁵ kg (maximum mass associated with B_o ~1.3 × 10¹² T) and 10¹⁰ kg (maximum mass associated with B_o ~3 × 10¹² T), and use these extremes to illustrate each calculation.

3.2. Slowing Down in Passage through a Portion of Earth

Hypervelocity MQNs ionize surrounding matter through a shock wave and interact with that matter through a magnetopause in the same way that Earth interacts with the solar wind through its magnetopause. The equations governing the interaction are derived in reference [28] and are summarized in Equations (2) through (6) for convenience.

The force equation for a high-velocity body with instantaneous radius r_m, and velocity v, moving through a fluid of density ρ_p with a drag coefficient K ~1, is

$$F_e\approx K\pi r_m^2{\rho }_p{v}^2$$

(2)

The geometric radius r_QN of the MQN of mass m and mass density ρ_QN is

$$r_{QN}={\left(\frac{3m}{4\pi {\rho }_{QN}}\right)}^{\frac{1}{3}}$$

(3)

However, MQNs have a velocity-dependent interaction radius [28] equal to the radius of their magnetopause.

$$r_m\approx {\left(\frac{2B_o^2}{{\mu }_{0}K{\rho }_p{v}^2}\right)}^{\frac{1}{6}}{r}_{QN}$$

(4)

MQNs with mass 10⁵ kg have r_QN ~4 × 10⁻⁵ m. For B_o ~1.3 × 10¹² T and v = 250 km/s, an MQN passing through water of density 1000 kg/m³ has the magnetopause radius r_m ~0.025 m. The corresponding values for mass m = 10⁹ kg with B_o ~3 × 10¹² T are r_QN ~9 × 10⁻⁴ m and r_m ~0.71 m. Although their nuclear density makes these massive MQNs physically small, their large magnetic fields and high velocities make their interaction radius and cross section very large, even in solid-density matter.

The interaction radius of an MQN varies as velocity v^−1/3 in Equation (4). Including that velocity dependence in the calculation with initial velocity v_o gives velocity v as a function of distance x:

$$v=v_o{\left(1-\frac{x}{x_{\max }}\right)}^{\frac{3}{2}}$$

(5)

in which x_max is the stopping distance for an MQN:

$$x_{\max }=\left(\frac{3m}{2\pi r_{QN}^2}\right){\left(\frac{{\mu }_o{v}_o^2}{2K^2{\rho }_p^2{B}_o^2}\right)}^{\frac{1}{3}}$$

(6)

MQNs with mass 10⁵ kg, B_o ~1.3 × 10¹² T, and velocity v_o ~250 km/s have x_max ~241 km passing through water. The corresponding value for mass of 10⁹ kg with B_o ~3 × 10¹² T is x_max ~3000 km.

From the geometry in Figure 1, the angle of incidence θ₂ for trajectories that emerge from Earth with negligible velocity is given by

$${\theta }_2={\cos }^{-1}\left(\frac{x_{\max }}{2r_{Earth}}\right)$$

(7)

in which r_Earth is the radius of the Earth, or ~6.38 × 10⁶ m. For π/2 > θ > θ₂, MQNs emerge from Earth with velocity

$$v_{exit}=v_o{\left(1-\frac{x_{exit}}{x_{\max }}\right)}^{\frac{3}{2}}\mathrm{for}{x}_{exit}=2r_{Earth}\cos (\theta )$$

(8)

Integrating Equation (5) gives the transit time t_exit

$$t_{exit}=\left(\frac{2x_{\max }}{v_o}\right)\left({\left(\frac{x_{\max }}{x_{\max }-x_{exit}}\right)}^{\frac{1}{2}}-1\right)$$

(9)

Transit time t_exit is strongly dependent on (x_max − x_exit) and is infinite at x_max = x_exit. A 10⁵ kg MQN with B_o = 1.3 × 10¹² T, initial velocity v_o = 250 km/s, and incidence angle θ₂ = 88.91817° penetrates a distance x_exit = 240.09 km of water and emerges in t_exit ~54 s with v_exit = 10 m/s. If the incidence angle θ₂ = 88.92278°, then x_exit = 239.89 km, transit time t_exit ~24 s, and v_exit = 100 m/s.

During the transit, the MQN is falling towards the center of Earth with acceleration g = 9.8 m/s², which is not considered in Equations (7) through (9). The deviation from the straight-line approximation is δr ~$\frac{1}{2}g{t}_{exit}^2$ and the corresponding fractional error in path length introduced by neglecting gravity is

$$\delta \sim \frac{\delta r}{\sqrt{x_{exit}^2+{(\delta r)}^2}}$$

(10)

For the example in the previous paragraph, fractional error in path length because of gravity is δ ~0.06 for v_exit = 10 m/s and is δ ~0.012 for v_exit ~100 m/s. In general, fractional error decreases with increasing v_exit, decreasing B_o, decreasing MQN mass, and increasing mass density of material transited (granite with ρ_p = 2300 kg/m³ or water ρ_p = 1000 kg/m³).

If the MQN slows to <<10 m/s, gravity dominates its motion, and it does not emerge from the ground or water. Fast objects are difficult to perceive, especially if an observer is not primed to expect an event. A human observer requires ~0.25 s to perceive an object as a thing [41]. If the object is about 1 m in diameter and moving at > 100 m/s, the object will have moved > 25 m before cognitive acquisition and may be out of range before the observer can process the image well enough to be confident of what was seen. Therefore, we suggest that only MQNs emerging with 10 m/s ≤ v_exit ≤ 100 m/s are likely to be reported by human observers.

Rearranging Equation (8) gives the incidence angle θ_vexit for a given v_exit:

$${\theta }_{vexit}={\cos }^{-1}\left(\left(\frac{x_{\max }}{2r_{Earth}}\right)\left(1-{\left(\frac{v_{exit}}{v_o}\right)}^{\frac{2}{3}}\right)\right)$$

(11)

For v_exit = 10 m/s and 100 m/s, the corresponding impact angles are, respectively, θ₁₀ and θ₁₀₀, which will be used in estimating the event rates for directly observable MQN events.

3.3. Estimated Event Rates

The number of events per year on Earth is estimated as follows:

• Earth is moving about the galactic center, in the direction of the star Vega, and through the dark-matter halo with a velocity of ~230 km/s [42]. Therefore, dark matter streams into the Earth frame of reference with mean streaming velocity ~230 km/s.
• Dark matter in the halo also has a nearly Maxwellian velocity distribution with mean velocity of ~230 km/s, so the ratio of streaming velocity to Maxwellian velocity is approximately 1 [42].
• Approximating the velocity of dark matter streaming from the direction of Vega as ~230 km/s, we calculate the cross section A_10-100 for transiting a chord through Earth and emerging with velocity between v₁₀ = 10 m/s and v₁₀₀ = 100 m/s:

$$A_{10-100}=2\pi r_{Earth}^2\left(\sin {\theta }_{100}-\sin {\theta }_{10}\right)$$

(12)

More generally, the cross section for MQNs impacting between θ_min and θ_max is

$$A=2\pi r_{Earth}^2{\left(\sin {\theta }_{\max }-\sin {\theta }_{\min }\right)}_{}$$

(13)

4.

MQNs can have masses between 10⁻²³ kg and 10¹⁰ kg [4]. We approximate such a large range by (1) associating the flux of all MQNs that have mass between 10ⁱ kg and 10ⁱ⁺¹ kg with a representative mass 10^i+0.5 kg (which we call the representative decadal mass) for −23 ≤ i ≤ 10; (2) calculating the behavior of each decadal-mass MQN; (3) assuming all the MQNs in that decadal range behave the same way. The associated number flux is called the decadal flux F_{m_decade} (number N/y/m²/sr) and was computed [4] as a function of B_o from simulations of the aggregation of quark nuggets from their formation in the early Universe and evolution to the present era.

5.

For A_{10-100_m_decade}, defined as the A_10-100 appropriate to a decadal mass m, the number of events per year per steradian for MQNs streaming from the direction of Vega and emerging with velocity between 10 and 100 m/s is F_{m_decade} A_{10-100_m_decade}, summed over all decadal masses m.

6.

For random velocity, approximately equal to streaming velocity, reference [36] shows that 5.56 sr is the effective solid angle that generalizes the streaming result to include MQNs from all directions.

7.

Therefore, the total number of events per year somewhere on Earth with v_exit between 10 and 100 m/s is

$$N=5.56{\displaystyle \sum _{m\_decade}(F_{m\_decade}{A}_{10-100\_m\_decade}})$$

(14)

In Section 3.6, we will show that MQNs transiting Earth along a chord spin up to MHz frequencies and emit substantial radio frequency (RF) power. If and only if the RF is not absorbed by the surrounding plasma, these MQNs can be detected by their RF emissions propagating around Earth in the waveguide between the ground and the ionosphere. Their cross section is given by Equation (13) with θ_max = π/2 and θ_min = θ₂ from Equation (7).

Figure 2 shows the estimated number of events per year somewhere on Earth as a function of B_o. Two modes of transit (through granite or water) and both potential modes of detection (human or radiofrequency) are considered.

Figure 2 shows that detection on or near water is much more likely than detection deep within continents and shows the event rate is strongly dependent on parameter B_o. Unless the RF is absorbed by the surrounding plasma, the detection of MQNs by RF instruments (sensitive to any velocity) is much more likely than the detection of MQNs at 10 to 100 m/s, observable by human observers; however, records of human observations span centuries. Even one reliable report of an MQN event with the characteristics of a nearly tangential transit would suggest low values of B_o and a mechanism that enhances the density of dark matter inside the solar system compared to that of interstellar space, as briefly described in Section 4.5.

3.4. Rotation at Megahertz Frequencies

MQNs interact with matter through its magnetopause. The shape of the magnetopause depends on the angle between the MQN velocity and the MQN magnetic moment, as illustrated in Figure 3.

Time-dependent asymmetry of the magnetopause produces a velocity-dependent and angle-dependent torque on the MQN and causes the MQN to oscillate initially about an equilibrium [36], as shown in Figure 4. Since the quark nugget slows down as it passes through ionized matter, the decreasing forward velocity reduces the torque with time, so the time-averaged torque in one half-cycle is greater than the opposing time-averaged torque in the next half-cycle. The amplitude of the oscillation necessarily grows. Once the angular momentum is sufficient to give continuous rotation, the net torque continually accelerates the angular motion to produce a rapidly rotating quark nugget. As shown in Figure 4, MHz frequencies are quickly achieved, even with a 10⁶ kg quark nugget moving through granite (2300 kg/m³ density matter) by the time v has slowed to 220 km/s.

As developed in reference [36] and summarized in Equations (15) through (19) for convenience, hypervelocity MQNs transiting through matter experience a torque that causes them to rotate with a frequency that depends on B_o, MQN mass m, MQN velocity v, and density ρ_p of the surrounding material. Rotating magnetic dipoles radiate at power P, where

$$P=\frac{Z_o}{12\pi }{\left(\frac{\omega }{c}\right)}^4{m}_m^2$$

(15)

in SI units, with Z_o = 377 Ω, ω = angular frequency, and c = the speed of light in vacuum. Magnetic dipole moment m_m = 4π B_o r_QN³/μ_o.

Assuming the energy loss per cycle from electromagnetic radiation to the surrounding plasma is just balanced by the energy gain per cycle from the torque T,

$${\displaystyle \underset{0}{\overset{\frac{2\pi }{\omega }}{\int }}Tdt}=\frac{2\pi P}{{\omega }^2}$$

(16)

in which

$$\begin{array}{l}T=C_2{\rho }_p^{0.5}v{B}_o{r}_{QN}^3{F}_{\chi }\\ F_{\chi }=MIN(ABS(\tan \chi ),ABS(\cot \chi ))\frac{\tan \chi }{ABS(\tan \chi )}\end{array}$$

(17)

The constant C₂ = 1400 with units of N s kg^−0.5 m^−1.5 T⁻¹, and the angle of rotation χ is the angle between the velocity of the incoming plasma and the magnetic moment.

The rate of change of angular velocity ω for MQN of mass m, with moment of inertia I_mom = 0.4 m r_QN², and experiencing torque T, is

$$\frac{d\omega }{dt}=\frac{T}{I_{mom}}$$

(18)

Combining Equations (15) through (18) gives

$${\displaystyle \underset{0}{\overset{\frac{2\pi }{\omega }}{\int }}\frac{v(t)F_{\chi }(\chi (\omega t))}{{\omega }^2}dt=\frac{5.54\times {10}^{-22}{B}_o{r}_{QN}^3}{{\rho }_x^{0.5}}}$$

(19)

which is solved numerically for the equilibrium angular velocity ω or frequency f = ω/(2π).

After emerging from dense matter, MQN rotational frequency decreases as rotational energy is radiated away with power P. Since P varies as ω⁴ and the rotational energy varies as ω², the frequency as a function of time is not exponential. Solving for ω(t) gives

$$\begin{array}{l}{C}_{RF}=\frac{1.08\times {10}^{43}}{B_o^2{m}^{\frac{1}{3}}}\\ \omega (t)={\omega }_0\sqrt{\frac{C_{RF}}{{\omega }_0^{2}t+C_{RF}}}\end{array}$$

(20)

Representative results for slowing down during transit through water and granite and for parameters of greatest interest are shown in Appendix D: Tables of MQN Interactions with Water and Granite.

The tables show RF frequencies at emergence from water or granite are weakly dependent on B_o but strongly dependent on mass and range from 7 MHz to 0.3 MHz for mass m between 10⁵ kg and 10¹⁰ kg, respectively. Rotational energy ranges from ~0.1 MJ to ~1000 MJ and equals ~10⁻¹¹ times the translational energy at impact. RF power emission at emergence ranges from ~4 GW to 22 TW.

The tables also show that the RF power, calculated with Equations (14) and (19) at t = 1200 seconds after emergence, ranges from ~6 MW to ~200 GW for the most massive MQNs with B_o between 1.3 × 10¹² T and 3.0 × 10¹² T, respectively.

Event rates in Figure 2 assume MQNs have the same flux as interstellar dark matter. However, the MQN flux may exceed that of interstellar space. Some MQNs passing through a portion of the solar photosphere are slowed to less than escape velocity from the solar system. Some of these are subsequently deflected by the net gravity of the planets so that they cannot return to the Sun and be absorbed. They accumulate. Therefore, this aerocapture process can enhance the local flux of MQNs and make Figure 2 a worst-case scenario. Enhancement factor, multiplied by observation time, would have to be >> 1000 for MQNs with nearly tangential trajectories to be observed.

Since RF detection occurs in real time, Figure 2 shows that the predicted event rate is too low to use RF to observe MQNs with nearly tangential trajectories, even if the RF is not absorbed by the plasma surrounding the MQN. Therefore, RF detection is best done in space [36] where the cross section is much larger and RF cutoff can be avoided.

In contrast, direct observations by human observers can cover centuries and may be recorded for us to analyze. The next section explores the observables in such an event to compare with observations published in the Proceedings of the Royal Society regarding an event on 6 August 1868 [38].

3.5. Simulations of a Rotating MQN with Plastically Deformable Conducting Witness Plate

The force between a rotating magnetized sphere and a plastically deformable conducting material was simulated by coupling the Rotating Magnetic Machinery module and the Nonlinear Plasticity Solid Mechanics module of the 3D, finite-element, COMSOL Multiphysics code [40]. The geometry is illustrated in Figure 5.

Details of the simulation are provided in Appendix B: COMSOL simulation of rotating magnetized sphere interaction with plastically deformable conductor.

A small-scale experiment validated the COMSOL force calculation. A 3 mm or 6 mm thick copper plate was placed below a rotating spherical magnet and suspended with a calibrated spring to measure the force on the plate as a function of 1) separation between the center of the sphere and the front of the plate and 2) rotational frequency of the sphere. The measured force agreed with the force computed by the COMSOL simulation to within 10%. The agreement validates the computational method with the rotating coordinate system in the COMSOL module.

The computational mesh cannot resolve the microscopic diameter of an MQN. Therefore, we approximate the MQN with a 0.1 m radius, magnetized sphere with surface magnetic induction B = 17,000 T, which corresponds to an MQN with mass m = 7 × 10⁷ kg and B_o = 1 × 10¹² T.

Simulations with different values of electrical conductivity σ and frequencies f showed that the dynamics of the problem scales with the electromagnetic skin depth λ:

$$\lambda =\frac{1}{\sqrt{\pi {\mu }_0\sigma f}}$$

(21)

Simulations converged best with low frequency. Consequently, we used frequency f = 10 Hz and varied the conductivity σ to explore the time-averaged force as a function of λ. The scaling with λ let us apply the results to higher frequencies and more realistic values of σ. Distance between the center of the sphere at z = 0 and the surface of the simulated peat was z_p = −0.3 m. Results are shown in Figure 6.

The levitating force is strongly dependent on the skin depth and decreases rapidly with increasing skin depth. A skin depth of 0.05 m, corresponding to σ = 10⁷ S/m and f = 10 Hz for σf = 10⁸ SHz/m, was chosen for the simulation of plastic deformation.

The radius of the magnetized sphere was set at 0.1 m and its magnetic induction field was set at 2085 T, corresponding to an MQN with mass ~9 × 10⁷ kg and B_o = 1 × 10¹² T. The radius of the rotating volume in the COMSOL mesh was set at r = 0.2, and the front surface of the 4 × 4 × 2 m deformable conductor was at r = 0.3 m.

For a time-averaged force of 10⁷ N in the z-direction (the direction opposing gravity), the maximum magnetic induction in the peat is 18.5 T and the maximum induced current density is 3 × 10⁸ A/m². The time-averaged forces were −1.1 × 10⁷ N, −0.3 × 10⁷ N, and −0.05 × 10⁷ N in the z-, x-, and y-directions, respectively.

Deformation of the material as a function of time from the integrated simulation is shown in Figure 7 as contour plots for four times and two orthogonal directions.

A rotating magnetic dipole is equivalent to two oscillating current loops oriented at 90° and with currents 90° out of phase. The linear superpositions of the two magnetic fields induce currents that are almost independent of orientation but produce a net force in the direction (x) perpendicular to the axis of rotation (y). That explains the shape of the deformations in Figure 7a,b.

The simulation has many limitations. The MQN cannot be realistically resolved, so the gradient in the B field is not exact. The rotating sphere cannot move downward as material is displaced, so the deformation stops when the applied stress reaches equilibrium with the strain specified in the stress–strain curve. The yield strength of the material is a constant, independent of the degree of compression and of flow of liquid driven by the J × B force in the material, so the deformation rate is only qualitative. Consequently, these simulations provide only semi-quantitative results to compare with observations.

Despite these limitations, these simulations clearly show that a rapidly rotating magnetized sphere with sufficient mass and sufficient magnetic dipole field, such as a massive MQN, will create a hole in a plastically deformable conducting material by currents induced in its interior. They also show the sphere will experience a force that moves it through the material to create a trench in conducting, deformable material.

In addition, the results in Figure 6 clearly indicate that the levitating force decreases rapidly with increasing skin depth, which varies inversely with the square root of frequency. Therefore, the MQN height above conducting material decreases as the rotational energy is depleted by RF emissions and by the resistive dissipation of current induced in the peat.

The RF power emissions are predicted to be megawatts to many gigawatts and are certainly sufficient to ionize and excite the surrounding air. Thermal and magnetohydrodynamic motion and mixing of the air around an MQN complicates simple estimates of the shape of the luminosity; however, Equation (15) shows that the radiated power varies as the fourth power of the frequency, so the local electric field varies as the square of the frequency. Therefore, the ionization and excitation of air will diminish faster than the height above the peat.

These simulations assume induced current flows in the conducting medium. If the surface electric field, from the rate of change of magnetic induction, is sufficient to form a plasma at the air–ground interface, then magnetic pressure will deform the material as if λ ~0.

3.6. Comparison with M. Fitzgerald’s Report to the Royal Society

M. Fitzgerald’s report [37] to the Royal Society describes an event that occurred on 6 August 1868. The reported observations are consistent with the characteristics of a nearly tangential MQN impact as developed in this paper: a luminous orb with clear skies overhead, persisting for much longer than weather-related ball lightning [44], moving slowly across and into a plastically deformable conducting medium (peatland), decreasing in diameter with time, and creating trenches in the ground. Since it is difficult to obtain copies of proceedings more than a century old, Fitzgerald’s description is quoted from the published report in Appendix C: Field Investigation of M. Fitzgerald’s Report to Royal Society. In summary, a ~0.60 m diameter, light-emitting orb was observed travelling at ~1 m/s over and into a peat bog in County Donegal, Ireland, for ~20 min. During that time, its diameter decreased to ~0.08 m; it displaced over 10⁵ kg of water-saturated peat; it produced approximately 1 m wide trenches in the peat.

Although we initially dismissed his report as incredible, we eventually realized that peat only grows a few centimeters per century [45], so the features he described must still be visible if the reported event actually occurred. His report was sufficiently detailed to enable an investigation, which we conducted from 2004 through 2006.

Fitzgerald’s reported features and our corresponding findings are summarized by the following:

• An approximately 6.4 m square hole described by Fitzgerald on the course from the crown of the ridge to the south of Meenawilligan, towards the town of Church Hill. We found a 6.4 m square hole 0.7 m deep along that course.
• An approximately 180 m distance reported to the next deformation. We found the deformation had been partially destroyed by draining of the field for sheep grazing. If this deformation were still the reported 100 m length, the southern end would be 175 ± 2 m from the hole.
• An approximately 100 m long, 1.2 m deep, and 1 m wide trench. As stated above, this deformation has been truncated by the owner having drained the field. The remaining trench is currently 63 ± 1 m long, 0.2 ± 0.05 m deep (soft to 0.8 ± 0.05 m), and 1.2 ± 0.1 m wide. Carbon dating of peat inside and outside the trench confirms a disturbance occurred, consistent with the report.
• Unspecified distance to the third excavation. We found the distance to be 5 ± 0.3 m.
• Curved trench formed when the stream bank was “torn away” for 25 m and dumped into the stream. We found the remaining curved trench to be 25 ± 1 m long and 1.4 ± 0.1 m deep. The 1863 Ordnance Survey map does not show the stream diversion that Fitzgerald reported as occurring on 6 August 1868. Therefore, the event happened after 1863. Fitzgerald’s submission to the Royal Society is dated 20 March 1878, so the event occurred before 1868. Therefore, the event is independently dated between 1863 and 1878.
• Cave in the stream bank directly opposite the end of the “torn away” bank. We found the cave at that position. It is currently 0.45 ± 0.08 m wide, 0.3 ± 0.06 m high, and 0.5 ± 0.1 m deep. However, its proximity to the water line raises the possibility that its origin was flowing water and not the event Fitzgerald reports.

The extant features support Fitzgerald’s account. This part of Ireland was, and still is, sparsely populated and did not have local newspapers that might have recorded an unusual sound or tremor. Therefore, no contemporaneous and independent eyewitness report confirms his account.

The vegetation is dominated by members of the heath (Ericaceae) and sedge (Cyperaceae) families. Surface features would become obscured to some degree by the growth of vegetation and debris carried in by wind, reducing the time a depression could be observed from a distance. A nearby crater, attributed [46] to a vertical impact of a 10 kg MQN in 1985, has changed little in the past 35 years, which is consistent with the current state of the features Fitzgerald attributed to the 1868 event.

The size of the trenches and the yield strength of peat gives a downward force of >10⁷ N, which implies a rotating, magnetically levitated mass of ~10⁶ kg. That mass and the volume of the ~0.08 m diameter luminosity implies a mass density >10⁹ kg/m³, which is inconsistent with normal matter. Its levitation implies an extremely large magnetic dipole rotating at >1 MHz to levitate the large mass.

The repulsive force of electromagnetic induction by an MQN, as derived in this paper, are consistent with the levitation of an MQN core and the displacement of the peat in its path. The yield strength of peat was measured and found to be 530 kN/m² ± 23%, which is the value used in the simulations of plastic deformation above. Electrical conductivity σ within 0.2 m of the surface was measured to be 22 mS/m ± 30%, which is consistent with published values for peatlands [47], as follows: 25 mS/m near the surface and ~380 mS/m at up to 2 m depth.

Table A2. Representative examples are given for MQNs with impact velocity of 250 km/s transiting through granite as a function of B_o and MQN mass m_qn.

Bo (T)	1.3 × 1012	1.5 × 1012	2 × 1012	2.5 × 1012	3 × 1012	3 × 1012
mqn (kg)	3.2 × 105	3.2 × 105	3.2 × 106	3.2 × 106	3.2 × 106	3.2 × 109
rQN (m) for ρQN = 1018 kg/m3	4.2 × 10−5	4.2 × 10−5	9.1 × 10−5	9.1 × 10−5	9.1 × 10−5	9.1 × 10−4
Magnetopause radius rm (m)	2.2 × 10−2	2.3 × 10−2	5.4 × 10−2	5.8 × 10−2	6.2 × 10−2	6.2 × 10−1
Flux for MQN decadal mass (N/y/m2/sr)	1.4 × 10−15	1.4 × 10−15	1.5 × 10−16	4.3 × 10−18	2.7 × 10−19	1.3 × 10−19
xmax (m)	138,434	125,839	223,837	192,900	170,824	1,709,027
x10 m/s (m)	138,272	125,692	223,575	192,674	170,624	1,707,028
x100 m/s (m)	137,683	125,156	222,622	191,852	169,897	1,699,749
θ2 (°)	89.37838	89.43494	88.99486	89.13380	89.23293	82.30288
θ10 (°) for vexit = 10 m/s	89.37911	89.43560	88.99604	89.13481	89.23383	82.31194
θ100 (°) for vexit = 100 m/s	89.38176	89.43801	89.00032	89.13850	89.23710	82.34492
texit (s) for vexit = 10 m/s	3.1 × 101	2.8 × 101	5.1 × 101	4.4 × 101	3.9 × 101	3.9 × 102
texit (s) for vexit = 100 m/s	1.4 × 101	1.3 × 101	2.3 × 101	1.9 × 101	1.7 × 101	1.7 × 102
δ fractional error for vexit = 10 m/s	3.5 × 10−2	3.1 × 10−2	5.6 × 10−2	4.8 × 10−2	4.3 × 10−2	3.9 × 10−1
δ fractional error for vexit = 100 m/s	6.9 × 10−3	6.3 × 10−3	1.1 × 10−2	9.6 × 10−3	8.5 × 10−3	8.5 × 10−2
Cross section for all vexit	1.5 × 1010	1.2 × 1010	3.9 × 1010	2.9 × 1010	2.3 × 1010	2.3 × 1012
Cross section for vexit = 10 to 100 m/s	1.3 × 108	1.1 × 108	3.3 × 108	2.5 × 108	1.9 × 108	2.0 × 1010
Total number per year	1.2 × 10−4	9.7 × 10−5	3.2 × 10−5	6.9 × 10−7	3.4 × 10−8	1.7 × 10−6
Number per year for 10 to 100 m/s vexit	9.9 × 10−7	8.2 × 10−7	2.7 × 10−7	5.9 × 10−9	2.9 × 10−10	1.4 × 10−8
Frequency (MHz)	9.0 × 100	8.9 × 100	4.0 × 100	3.9 × 100	3.9 × 100	3.9 × 10−1
Rotational energy (J)	2.2 × 105	2.2 × 105	2.1 × 106	2.0 × 106	2.0 × 106	1.9 × 109
RF power (MW)	1.2 × 104	1.5 × 104	1.1 × 105	1.6 × 105	2.2 × 105	2.2 × 107
RF power (MW) after 1200 s	6.7 × 100	5.1 × 100	6.1 × 101	4.0 × 101	2.8 × 101	2.3 × 105

in Appendix D (Tables of MQN Interactions with Water and Granite) gives 3 to 9 MHz for the frequency of a massive MQN when it first emerges from the ground. The corresponding skin depth λ ~1.1 to 2.0 m for σ = 22 mS/m and λ ~0.27 to 0.47 m for σ = 385 mS/m. As shown in Figure 6, these values of λ are too large to produce the reported levitation and deformation if the induced current is flowing through the peat, as assumed in the simulations.

However, the large electric field from the rate of change of the magnetic induction should cause the surface to flash over and form a plasma on top of the peat, similar to how the air-water interface in pulsed power devices flash over. (Montoya, R. and Danneskiold, J. Five seconds at F/16, with a broken camera. Sandia Lab News (7 June 2018). https://www.sandia.gov/news/publications/labnews/articles/2018/08-06/Randy.html. (accessed on 24 July 2020).) If so, then the effective skin depth λ ~0 and the deformation are consistent with the frequencies calculated for the MQN. Computationally or experimentally simulating such a process would be extremely difficult, so surface flashover is consistent with relevant experience but has not been confirmed in the field.

Since the field with the second deformation (the remaining 63 m of trench) has not been drained, its slope (10%) is almost the same as it was in 1868. The simplicity of this particular deformation allows us to estimate the minimum mass of the core of this object from the yield strength and the volume. The product of the minimum pressure P, which we equate to the measured yield strength 530 ± 120 kN/m², times the volume change ∆V, is the minimum work required to compress the trench (1.4 m wide, 1.2 m deep, and 100 m long) and is ~10⁸ joules. The energy for this work came purely from gravitational energy as the “globe of fire” descended the slope from the beginning of the trench to its terminus. The corresponding mass is m ~10⁸/(gh) ~10⁶ kg, where g is the acceleration of gravity in m/s² and h is the change in distance toward Earth’s center in meters and agrees with the mass estimated from the yield strength and trench diameter.

If the induced currents are flowing in a plasma on the surface, the pressure equals the magnetic pressure P_m produced by the magnetic field B:

$$P_m=\frac{B^2}{2{\mu }_0}$$

(22)

Since the pressure supports a mass m against the acceleration g of gravity, P_m = mg/(πr²) for r = the half-width of the trench and μ_o = 4 π × 10⁻⁷ H/m. The value of B spatially and temporarily averaged over the effective area πr² is ~5.4 T and is consistent with the spatial and temporal maximum value of B = 18.5 T from the simulation.

Fitzgerald reports that the diameter of the luminous ball diminished from ~0.6 m diameter at the beginning to ~0.08 m at the end of the event; however, our survey shows that the width and depth of the depressions in the peat were about the same at the beginning and end of the event. Therefore, the core of the object remained unchanged, with diameter <0.08 m, volume of <3 × 10⁻⁴ m³, and density of >10⁶/(3 × 10⁻⁴) or >3 × 10⁹ kg/m³—at least 200,000 times the density of normal matter. Matter does not exist with density between 3 × 10⁹ kg/m³ and nuclear density. Such a large mass density implies nuclear density matter and is consistent with the ~10¹⁸ kg/m³ mass density of MQNs.

Based on

Table A1. Representative examples are given for MQNs with impact velocity of 250 km/s transiting through water as a function of B_o and MQN mass m_qn.

Bo (T)	1.3 × 1012	1.5 × 1012	2 × 1012	2.5 × 1012	3 × 1012	3 × 1012
mqn (kg)	3.2 × 105	3.2 × 105	3.2 × 106	3.2 × 106	3.2 × 106	3.2 × 109
rQN (m) for ρQN = 1018 kg/m3	4.2 × 10−5	4.2 × 10−5	9.1 × 10−5	9.1 × 10−5	9.1 × 10−5	9.1 × 10−4
Magnetopause radius rm (m)	2.5 × 10−2	2.6 × 10−2	6.2 × 10−2	6.7 × 10−2	7.1 × 10−2	7.1 × 10−1
Flux for MQN decadal mass (N/y/m2/sr)	1.4 × 10−15	1.4 × 10−15	1.5 × 10−16	4.3 × 10−18	2.7 × 10−19	1.3 × 10−19
xmax (m)	241,197	219,252	389,995	336,093	297,630	2,977,672
x10 m/s (m)	240,914	218,995	389,539	335,700	297,282	2,974,189
x100 m/s (m)	239,887	218,062	387,878	334,268	296,014	2,961,506
θ2 (°)	88.91690	89.01545	88.24855	88.49068	88.66344	76.50504
θ10 (°) for vexit = 10 m/s	88.91817	89.01660	88.25059	88.49245	88.66500	76.52112
θ100 (°) for vexit = 100 m/s	88.92278	89.02080	88.25806	88.49888	88.67070	76.57968
texit (s) for vexit = 10 m/s	5.4 × 101	5.0 × 101	8.8 × 101	7.6 × 101	6.7 × 101	6.7 × 102
texit (s) for vexit = 100 m/s	2.4 × 101	2.2 × 101	3.9 × 101	3.4 × 101	3.0 × 101	3.0 × 102
δ fractional error for vexit = 10 m/s	6.0 × 10−2	5.5 × 10−2	9.7 × 10−2	8.4 × 10−2	7.4 × 10−2	6.0 × 10−1
δ fractional error for vexit = 100 m/s	1.2 × 10−2	1.1 × 10−2	1.9 × 10−2	1.7 × 10−2	1.5 × 10−2	1.5 × 10−1
Cross section for all vexit	4.6 × 1010	3.8 × 1010	1.2 × 1011	8.9 × 1010	7.0 × 1010	7.1 × 1012
Cross section for vexit = 10 to 100 m/s	3.9 × 108	3.2 × 108	1.0 × 109	7.5 × 108	5.9 × 108	6.1 × 1010
Total number per year	3.5 × 10−4	2.9 × 10−4	9.7 × 10−5	2.1 × 10−6	1.0 × 10−7	5.2 × 10−6
Number per year for 10 to 100 m/s vexit	3.0 × 10−6	2.5 × 10−6	8.2 × 10−7	1.8 × 10−8	8.7 × 10−10	4.5 × 10−8
Frequency (MHz)	7.0 × 100	7.0 × 100	3.2 × 100	3.1 × 100	3.0 × 100	3.1 × 10−1
Rotational energy (J)	1.4 × 105	1.3 × 105	1.3 × 106	1.2 × 106	1.2 × 106	1.2 × 109
RF power (MW)	4.4 × 103	5.6 × 103	4.3 × 104	6.2 × 104	8.3 × 104	8.4 × 106
RF power (MW) after 1200 s	6.5 × 100	5.0 × 100	6.0 × 101	3.9 × 101	2.8 × 101	2.0 × 105

in Appendix D (Tables of MQN Interactions with Water and Granite) and on the MQN’s motion to the northeast, the impact must have been >125 km southwest of the 1868 observations for impact velocity ~250 km/s. That location is deep in the Atlantic Ocean, >80 km from land, and too far away to be heard by Fitzgerald. Therefore, Table A1 in Appendix D (Tables of MQN Interactions with Water and Granite) applies and indicates the impact was necessarily >200 km from land and is unlikely to be found.

Uncertainties in the current distribution and in applying the quasi-static and uncompressed yield strength to such a dynamic process leads us to assign +/− an order of magnitude error bar to the mass estimate. The resulting 10^6+/−1 kg mass corresponds to the maximum mass in the mass distributions [4] for B_o = 1.65 × 10¹² T +/− 21%. For comparison, the magnetic moments and mass densities of protons and neutrons, which are also baryons, correspond to magnetic fields B_o = 1.5 × 10¹² T and 2.5 × 10¹² T, respectively, in reasonable agreement with our value for MQNs. The smaller range is a considerable improvement over Tatsumi’s 10^12+/−1 T estimate and permits the design of a systematic test of the MQN dark-matter hypothesis.

4. Discussion

The principal result of this paper is reducing uncertainty in the key parameter of the MQN theory to B_o = 1.65 × 10¹² T +/− 21%. The result depends on (1) Fitzgerald having accurately reported what he observed, (2) the event being caused by a nearly tangential MQN impact as we have calculated, and (3) the absence of a more likely explanation.

4.1. Fitzgerald’s Accuracy

We have done what due diligence is possible on Fitzgerald’s qualifications as a reliable observer. The County records indicate he was the assistant surveyor for County Donegal during this period. That was a responsible position and is consistent with the detailed report to the Royal Society. We also know that the Royal Society had sufficient confidence in his report to have the president of the Society read it into the proceedings. We have no reason to doubt his integrity.

4.2. Consistency with MQN Impact

The details allowed us to find all the secondary observations, i.e., the reported deformations in the peat bog. Carbon 14 analysis, the ordnance survey map of 1860 (prior to the reported event), and the quantitative agreement between our measurements and Fitzgerald’s reported measurements, all support consistency with a nearly tangential MQN impact, as we have calculated.

4.3. Alternative Explanations

We have attempted to identify other possible causes of the secondary observations. Surface features produced by water erosion of peat and settling under gravity are well documented in the geomorphological literature. Sinuous water-cut gullies dissect peat bogs into complex patterns of haggs (residual masses) and groughs (the gullies) [48]. Mass movement processes involving the flow or slide of water-saturated peat cause major disruptions to bog surfaces and can extend for several hundreds of meters downslope [49]. The features described by Fitzgerald are unlike any of the peat erosional features previously reported and cannot be ascribed to conventional geomorphological processes.

One of us (Professor Peter Wilson) is a geomorphologist specializing in peat bogs. He has investigated the four structures Fitzgerald’s eyewitness account connected to the event and concludes that the literature on the geomorphology of peat bogs and his forty years of field work in peat bogs have not suggested any other possible causes for three of the four structures. We conclude that the fourth (the cave) was too close to the stream to preclude its being formed by water flow.

Neutron-stars have the right mass density. However, the gravitational force that holds them together is too weak to sustain the required ~10⁶ kg mass. In addition, pulsar magnetic fields [50] are two orders of magnitude less than those of magnetars [51], upon which the MQN model has been constructed.

Primordial Mini-Black Holes (MBHs) [52] have been proposed to explain luminous and levitating events attributed to anomalous non-weather-related ball lightning [53]. Although an MBH does not have a magnetic field that could levitate it, the net shape of the event horizon from the combined gravitational fields of Earth and a black hole can, in principle, direct evaporating particles downward to provide thrust and levitate the mass [53]. However, the lifetime of a 10⁶ kg MBH is not consistent with the 20-min event reported by Fitzgerald, and the explosion equivalent to 10 million one-megaton hydrogen bombs characteristic of the final evaporation [52] of an MBH was not observed.

We have also sought alternative explanations from others. Given the eyewitness account and given that our simulations show the MQN hypothesis is consistent with Fitzgerald’s eyewitness report, the question becomes can something else reside above and in the peat for 20 min, leave meter-scale (depth and width) structures in the peat, and have a glowing light associated with it. We have presented these results to about a thousand people in university colloquia and in contributed and invited talks at meetings of the American Physical Society and Institute of Electrical and Electronic Engineers in the USA, UK, Russia, and India. None of the audience members have suggested an alternative explanation.

4.4. Limitations to the Evidence

Even though the three criteria identified in the first paragraph of the Discussion are arguably satisfied, the primary event is the “globe of fire” itself and was seen only by Fitzgerald. The Fitzgerald event is not quite singular. Two other similar primary events qualitatively consistent (i.e., meter-scale, luminous, long-lasting, quasi-spherical, and rotating, as observed by the angular momentum imparted to the surrounding water) with nearly tangential MQN impacts were reported by Soviet Navy Captains at sea in 1962 and 1966 [39]. However, the Fitzgerald event is the only one contemporaneously published in the scientific journal of its day and the only one with detailed secondary observables that can be verified by anyone after the event.

We conclude that the primary event is very rare, is not reproducible, was not recorded by multiple observers, and cannot be quantitatively validated after the fact by anyone else. Consequently, the evidence does not meet today’s standard for a discovery.

4.5. Significance

However, we also conclude that we can tentatively accept his report and use it to narrow the uncertainty in B_o since it is consistent with an MQN event, and a more likely explanation has not been found. The resulting uncertainty in B_o can be used to design an experiment to systematically test the MQN hypothesis within the constrained range of B_o. If MQNs are found as predicted, then the acceptance of the Fitzgerald event as an MQN event will have been validated. If nothing is found, then the experiment will have been another null experiment placing a limit of the mass distribution of MQNs characterized by the tested values of B_o, just as all the single-mass quark-nugget experiments to date have been null experiments that placed a flux limit on that single mass.

The results have one additional consequence. Predicted event rates in Figure 2 are so small that even one observed event is consistent with the major portion of dark matter being composed of MQNs. Observing more than one per century suggests a substantial enhancement factor for dark-matter density inside the solar system compared to that of interstellar space. Nuclear-density MQNs are indestructible and can survive passage through the solar photosphere. The magnetopause interaction [28] with matter in the solar photosphere and subsequent deflection by the combined gravity of the planets offer the possibility of enhancing the flux of MQN dark matter within the solar system. Computing an accurate enhancement factor for MQN impacts on Earth requires detailed Monte Carlo simulations beyond the scope of this paper and strongly depends on B_o, the radial profile of mass density in the solar photosphere, the velocity distribution of dark matter in interstellar space, and scattering of MQNs by planetary gravity.

In contrast to other candidates for dark matter, MQNs are baryons and, therefore, are consistent [4] with the Standard Model for particles and fields. Well known physics can guide additional experiments and observations [36] to test the MQN hypothesis for dark matter, invent ways for collecting useful MQNs, and develop applications for an indestructible source of ~10¹² tesla magnetic fields.

5. Conclusions

The two orders of magnitude uncertainty in Tatsumi’s estimate for B_o precludes the practical design of systematic experiments to detect MQNs through their predicted interaction with matter. In this paper, we theoretically examined the signature of a new class of episodic events consistent with a unique signature of MQNs and reported the results of field investigations of one published event consistent with that signature. Tentatively accepting that the event was indeed caused by MQNs constrains the most likely values of B_o to 1.65 × 10¹² T +/− 21%, which can be used to design a systematic test of the MQN dark-matter hypothesis.