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

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{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 × 10

^{12}T +/− 21% and support the design of definitive tests of the MQN dark-matter hypothesis.

## 1. Introduction

^{0}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 ~10

^{4}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).

_{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.

^{7}T. Tatsumi [27] examined ferromagnetism from a One Gluon Exchange interaction and concluded that the surface magnetic field could be sufficient to explain the ~10

^{12}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.

^{3}and 10

^{37}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].

_{o}.

_{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.

_{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}.

_{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

_{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).

_{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 × 10

^{12}T +/− 21% and will permit the design of a systematic experiment to test the MQN hypothesis.

## 2. Materials and Methods

- 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

^{5}kg (maximum mass associated with B

_{o}~1.3 × 10

^{12}T) and 10

^{10}kg (maximum mass associated with B

_{o}~3 × 10

^{12}T), and use these extremes to illustrate each calculation.

#### 3.2. Slowing Down in Passage through a Portion of Earth

_{m}, and velocity v, moving through a fluid of density ρ

_{p}with a drag coefficient K ~1, is

_{QN}of the MQN of mass m and mass density ρ

_{QN}is

^{5}kg have r

_{QN}~4 × 10

^{−5}m. For B

_{o}~1.3 × 10

^{12}T and v = 250 km/s, an MQN passing through water of density 1000 kg/m

^{3}has the magnetopause radius r

_{m}~0.025 m. The corresponding values for mass m = 10

^{9}kg with B

_{o}~3 × 10

^{12}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.

^{−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:

_{max}is the stopping distance for an MQN:

^{5}kg, B

_{o}~1.3 × 10

^{12}T, and velocity v

_{o}~250 km/s have x

_{max}~241 km passing through water. The corresponding value for mass of 10

^{9}kg with B

_{o}~3 × 10

^{12}T is x

_{max}~3000 km.

_{2}for trajectories that emerge from Earth with negligible velocity is given by

_{Earth}is the radius of the Earth, or ~6.38 × 10

^{6}m. For π/2 > θ > θ

_{2}, MQNs emerge from Earth with velocity

_{exit}

_{exit}is strongly dependent on (x

_{max}− x

_{exi}

_{t}) and is infinite at x

_{max}= x

_{exi}

_{t}. A 10

^{5}kg MQN with B

_{o}= 1.3 × 10

^{12}T, initial velocity v

_{o}= 250 km/s, and incidence angle θ

_{2}= 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 θ

_{2}= 88.92278°, then x

_{exit}= 239.89 km, transit time t

_{exit}~24 s, and v

_{exit}= 100 m/s.

^{2}, 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

_{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/m

^{3}or water ρ

_{p}= 1000 kg/m

^{3}).

_{exit}≤ 100 m/s are likely to be reported by human observers.

_{vexit}for a given v

_{exit}:

_{exit}= 10 m/s and 100 m/s, the corresponding impact angles are, respectively, θ

_{10}and θ

_{100}, which will be used in estimating the event rates for directly observable MQN events.

#### 3.3. Estimated Event Rates

- 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 v_{10}= 10 m/s and v_{100}= 100 m/s:

_{min}and θ

_{max}is

- 4.
- MQNs can have masses between 10
^{−23}kg and 10^{10}kg [4]. We approximate such a large range by (1) associating the flux of all MQNs that have mass between 10^{i}kg and 10^{i+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/m^{2}/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

_{max}= π/2 and θ

_{min}= θ

_{2}from Equation (7).

_{o}. Two modes of transit (through granite or water) and both potential modes of detection (human or radiofrequency) are considered.

_{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

^{6}kg quark nugget moving through granite (2300 kg/m

^{3}density matter) by the time v has slowed to 220 km/s.

_{o}, MQN mass m, MQN velocity v, and density ρ

_{p}of the surrounding material. Rotating magnetic dipoles radiate at power P, where

_{o}= 377 Ω, ω = angular frequency, and c = the speed of light in vacuum. Magnetic dipole moment m

_{m}= 4π B

_{o}r

_{QN}

^{3}/μ

_{o}.

_{2}= 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.

_{mom}= 0.4 m r

_{QN}

^{2}, and experiencing torque T, is

^{4}and the rotational energy varies as ω

^{2}, the frequency as a function of time is not exponential. Solving for ω(t) gives

_{o}but strongly dependent on mass and range from 7 MHz to 0.3 MHz for mass m between 10

^{5}kg and 10

^{10}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.

_{o}between 1.3 × 10

^{12}T and 3.0 × 10

^{12}T, respectively.

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

^{7}kg and B

_{o}= 1 × 10

^{12}T.

_{p}= −0.3 m. Results are shown in Figure 6.

^{7}S/m and f = 10 Hz for σf = 10

^{8}SHz/m, was chosen for the simulation of plastic deformation.

^{7}kg and B

_{o}= 1 × 10

^{12}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.

^{7}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 × 10

^{8}A/m

^{2}. The time-averaged forces were −1.1 × 10

^{7}N, −0.3 × 10

^{7}N, and −0.05 × 10

^{7}N in the z-, x-, and y-directions, respectively.

**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.

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

^{5}kg of water-saturated peat; it produced approximately 1 m wide trenches in the peat.

- 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.

^{7}N, which implies a rotating, magnetically levitated mass of ~10

^{6}kg. That mass and the volume of the ~0.08 m diameter luminosity implies a mass density >10

^{9}kg/m

^{3}, which is inconsistent with normal matter. Its levitation implies an extremely large magnetic dipole rotating at >1 MHz to levitate the large mass.

^{2}± 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 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.

^{2}, 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 ~10

^{8}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 ~10

^{8}/(gh) ~10

^{6}kg, where g is the acceleration of gravity in m/s

^{2}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.

_{m}produced by the magnetic field B:

_{m}= mg/(πr

^{2}) 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 πr

^{2}is ~5.4 T and is consistent with the spatial and temporal maximum value of B = 18.5 T from the simulation.

^{−4}m

^{3}, and density of >10

^{6}/(3 × 10

^{−4}) or >3 × 10

^{9}kg/m

^{3}—at least 200,000 times the density of normal matter. Matter does not exist with density between 3 × 10

^{9}kg/m

^{3}and nuclear density. Such a large mass density implies nuclear density matter and is consistent with the ~10

^{18}kg/m

^{3}mass density of MQNs.

^{6+/−1}kg mass corresponds to the maximum mass in the mass distributions [4] for B

_{o}= 1.65 × 10

^{12}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 × 10

^{12}T and 2.5 × 10

^{12}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

_{o}= 1.65 × 10

^{12}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

#### 4.2. Consistency with MQN Impact

#### 4.3. Alternative Explanations

^{6}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.

^{6}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.

#### 4.4. Limitations to the Evidence

#### 4.5. Significance

_{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.

_{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.

^{12}tesla magnetic fields.

## 5. Conclusions

_{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 × 10

^{12}T +/− 21%, which can be used to design a systematic test of the MQN dark-matter hypothesis.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Quark-Nugget Research Summary

^{−1/3}for 3 < A < 10

^{5}. In addition to this core charge, they find that there is a large surface charge and a neutralizing cloud of charge to give a net zero electric charge for sufficiently large A. So, quark nuggets with A ≫ 1 are both dark and very difficult to detect with astrophysical observations.

^{−8}and 10

^{20}kg within a plausible but uncertain range of assumed parameters of quantum chromodynamics (QCD) and the MIT bag model with its inherent limitations [16].

^{5}GeV (~2 × 10

^{−22}kg). In 2014, Tulin [24] surveyed additional simulations of increasing sophistication and updated the results of Wandelt, et al. The combined results help establish the allowed range and velocity dependence of the strength parameter and strengthen the case for quark nuggets. In 2015, Burdin, et al. [25] examined all non-accelerator candidates for stable dark matter and also concluded that quark nuggets meet the requirements for dark matter and have not been excluded experimentally. Jacobs, Starkman, and Lynn [5] found that combined Earth-based, astrophysical, and cosmological observations still allow quark nuggets of mass 0.055 to 10

^{14}kg and 2 × 10

^{17}to 4 × 10

^{21}kg to contribute substantially to dark matter. The large mass means the number per unit volume of space is small, so detecting them requires a very large area detector.

^{7}T, which is too small for magnetars and MQNs. Tatsumi [27] has shown that, under some special values of the currently unknown QCD coupling constant at the ~90 MeV energy scale, a One Gluon Exchange interaction may allow quark nuggets to be ferromagnetic with a surface magnetic field of 10

^{12±1}T. Such a large magnetic field is sufficient for magnetar cores and MQNs. For a quark nugget of radius r

_{QN}and a magnetar of radius r

_{s}, the magnetic field scales as (r

_{QN}/r

_{s})

^{3}. Therefore, the surface magnetic field of a magnetar is smaller than 10

^{12}T because r

_{s}> r

_{QN}. Since quark-nugget dark matter is bare, the surface magnetic field of what we wish to detect is 10

^{12±1}T.

_{QN}

^{−3}, the collision cross section is still many orders of magnitude too small to violate the collision requirements [10,21,22,23] for dark matter.

^{16}T, so the large self-field described by Tatsumi should enhance their stability. Ping et al. [30] showed that magnetized quark nuggets should be absolutely stable with the newly developed equivparticle model, so the large self-field described by Tatsumi should ensure that quark nuggets with sufficiently large baryon number will not decay by weak interaction.

## Appendix B. COMSOL Simulation of Rotating Magnetized Sphere Interaction with Plastically Deformable Conductor

**n**×

**A**= 0, in which

**n**is the unit vector normal to the boundary and

**A**is the vector potential, so the magnetic induction vector

**B**lies along the boundary surface everywhere. The 20 m radius is sufficiently large to make the force on the peat insensitive to the position of the boundary.

**Figure A1.**

**Geometry of simulation of a rotating quark nugget.**(

**a**) The overall geometry of the simulation shows the 20 m radius conducting boundary with the cylindrical rotating coordinate system centered on the axis and the 4 × 4 × 2 m thick slab of simulated peat. (

**b**) Close up of the peat slab with its top surface located 0.3 m below the center of the 0.1 m radius, spherical magnet simulating the quark nugget, which is inside the cylindrical 0.2 m radius rotating coordinate system.

**F**=

_{v}**J**×

**B**, in which

**J**is the current density in the peat and

**B**is the magnetic field.

^{3}kg/m

^{3}. Yield stress = 5.30 × 10

^{5}Pa. Poisson ratio = 0.4. Young’s modulus = 2 × 10

^{9}Pa. Isotropic tangent modulus = 1.1 × 10

^{8}Pa.

^{5}Pa is exceeded at an elastic strain of 2.0 × 10

^{−4}. Then the ratio of additional stress to additional strain is the isotropic tangent modulus of 1.1 × 10

^{8}Pa.

_{m}in the fixed frame. Since the meshes at the interface of the rotating and non-rotating frames are not identical, the calculation interpolates the scalar magnetic potential between the non-conforming meshes. If the magnetic vector potential

**A**has to be interpolated across the boundary, then current is not conserved. In principle, applying Ampère’s law only inside the peat slab avoids this problem. Nevertheless, we varied the radius of the rotating coordinate system to assess the degree to which the numerical interpolation technique affects the results. The choice of the radius of the rotating coordinate system affected the magnetic field at the surface of the peat by approximately ±50%, so this solution is not ideal. However, choosing 0.2 m for the radius of the rotating coordinate system mitigates the problem. This choice gives an air gap of 0.1 m between the interface and the magnetized sphere and between the interface and the surface of the peat.

^{8}Hz took 1500 s of computer time for one 2.2 ns period. Calculating the full deformation at simulation time of 0.03 s would take 6 × 10

^{6}hours of computer time, which is prohibitive. Comparison of simulations at 1, 10, and 100 Hz showed that the force on the peat scales with the product of electrical conductivity σ (S/m) and frequency f (Hz), and, therefore, scales with the electromagnetic skin depth λ.

^{7}S/m and f = 10 Hz for σf = 10

^{8}SHz/m, was chosen as the baseline case for these exploratory simulations. The radius of the magnetized sphere was set at 0.1 m and its magnetic induction field was set at 2085 T. The radius of the rotating coordinate system was set at 0.2 m and the front surface of the 4 × 4 × 2 m peat slab was at 0.3 m, as shown in Figure A1.

^{7}N to support the estimated 10

^{6}kg mass in Earth’s gravitational field, varied as the cube of the radius of the quark nugget, as shown in Figure A2.

**Figure A2.**Amplitude of the magnetic induction B in the rotating and magnetized sphere required to produce a time-averaged force of 10

^{7}N on the peat as a function of the radius of the magnetized sphere. The radius of the magnetized sphere was varied between 5 and 150 mm. The mesh size was too large for the calculation to converge for radii less than 5 mm.

_{S}at the surface of the sphere of radius R

_{S}and mass M

_{S}:

^{3}kg

^{−0.5}. The coefficient of determination (R

^{2}) value is 0.98.

## Appendix C. Field Investigation of M. Fitzgerald’s Report to Royal Society

**Figure A3.**

**Extract from the 1863 Ordnance Survey map.**The locations of the “square hole” (a), the most prominent trench (b), triangular channel (c), and cave (d), and the reported path of the “globe of fire” (dotted line) between features are shown. The field with the “20-foot-square” hole (a) has been drained and is lower than the field with the trench (b) and the triangular channel (c). Therefore, we do not know the relative elevation of the hole (a) and the trench (b) in 1868.

- Hole (a): ~6.4 m square depression on the course from the crown of the ridge to the south of Meenawilligan towards the town of Churchill.
- Approximately 180 m to the next depression.
- Straight trench (b): ~100 m long, 1.2 m deep, and 1 m wide.
- Unspecified distance to the third depression.
- Curved trench (c): formed when stream bank was “torn away” for 25 m and dumped into the stream.
- Cave (d): a hole in the stream bank directly opposite the end of the “torn away” bank.

**Figure A4.**The location consistent with Fitzgerald’s “20-foot-square hole,” at which the ball of light first disappeared into the peat. (

**a**) Photo of the “square hole” with dimensions and 0.5 m deep contour (dashed blue line). (

**b**) Contours at three depths are shown: 0 m (solid black line), 0.3 m (dashed black line), and 0.5 m (dashed blue line). The natural feeder drainage flows approximately from the lower right to the upper left. Orientation to north is approximate.

**Figure A6.**Fitzgerald’s “25 m long diversion of the stream” and the current path of the stream, which continues to the left and right of the contour map. (

**a**) Photo of the site as seen from the western end; the channel made by the “globe of fire” is on the right and the stream that was recut by the Council is on the left. (

**b**) Contour map of the site constructed from the survey and field notes. Solid lines: surface level. Long dashes: the bottom of the channel at −1.2 ± 0.25 m level. Short dashes: the bottom of the stream, as cut by the County Council in the 1980s, at the −1.9 ± 0.2 m level. Orientation to the north is approximate.

^{2}for uncompressed peat, and the strength increased with increasing compression. Radiation measurements were taken. Only background radiation was detected.

## Appendix D. Tables of MQN Interactions with Water and Granite

**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}.

B_{o} (T) | 1.3 × 10^{12} | 1.5 × 10^{12} | 2 × 10^{12} | 2.5 × 10^{12} | 3 × 10^{12} | 3 × 10^{12} |
---|---|---|---|---|---|---|

m_{qn} (kg) | 3.2 × 10^{5} | 3.2 × 10^{5} | 3.2 × 10^{6} | 3.2 × 10^{6} | 3.2 × 10^{6} | 3.2 × 10^{9} |

r_{QN} (m) for ρ_{QN} = 10^{18} kg/m^{3} | 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 r_{m} (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/m^{2}/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} |

x_{max} (m) | 241,197 | 219,252 | 389,995 | 336,093 | 297,630 | 2,977,672 |

x_{10} m/s (m) | 240,914 | 218,995 | 389,539 | 335,700 | 297,282 | 2,974,189 |

x_{100} m/s (m) | 239,887 | 218,062 | 387,878 | 334,268 | 296,014 | 2,961,506 |

θ_{2} (°) | 88.91690 | 89.01545 | 88.24855 | 88.49068 | 88.66344 | 76.50504 |

θ_{10} (°) for v_{exit} = 10 m/s | 88.91817 | 89.01660 | 88.25059 | 88.49245 | 88.66500 | 76.52112 |

θ_{100} (°) for v_{exi}_{t} = 100 m/s | 88.92278 | 89.02080 | 88.25806 | 88.49888 | 88.67070 | 76.57968 |

t_{exit} (s) for v_{exit} = 10 m/s | 5.4 × 10^{1} | 5.0 × 10^{1} | 8.8 × 10^{1} | 7.6 × 10^{1} | 6.7 × 10^{1} | 6.7 × 10^{2} |

t_{exit} (s) for v_{exit} = 100 m/s | 2.4 × 10^{1} | 2.2 × 10^{1} | 3.9 × 10^{1} | 3.4 × 10^{1} | 3.0 × 10^{1} | 3.0 × 10^{2} |

δ fractional error for v_{exit} = 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 v_{exit} = 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 v_{exit} | 4.6 × 10^{10} | 3.8 × 10^{10} | 1.2 × 10^{11} | 8.9 × 10^{10} | 7.0 × 10^{10} | 7.1 × 10^{12} |

Cross section for v_{exit} = 10 to 100 m/s | 3.9 × 10^{8} | 3.2 × 10^{8} | 1.0 × 10^{9} | 7.5 × 10^{8} | 5.9 × 10^{8} | 6.1 × 10^{10} |

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 v_{exit} | 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 × 10^{0} | 7.0 × 10^{0} | 3.2 × 10^{0} | 3.1 × 10^{0} | 3.0 × 10^{0} | 3.1 × 10^{−1} |

Rotational energy (J) | 1.4 × 10^{5} | 1.3 × 10^{5} | 1.3 × 10^{6} | 1.2 × 10^{6} | 1.2 × 10^{6} | 1.2 × 10^{9} |

RF power (MW) | 4.4 × 10^{3} | 5.6 × 10^{3} | 4.3 × 10^{4} | 6.2 × 10^{4} | 8.3 × 10^{4} | 8.4 × 10^{6} |

RF power (MW) after 1200 s | 6.5 × 10^{0} | 5.0 × 10^{0} | 6.0 × 10^{1} | 3.9 × 10^{1} | 2.8 × 10^{1} | 2.0 × 10^{5} |

**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}.

B_{o} (T) | 1.3 × 10^{12} | 1.5 × 10^{12} | 2 × 10^{12} | 2.5 × 10^{12} | 3 × 10^{12} | 3 × 10^{12} |
---|---|---|---|---|---|---|

m_{qn} (kg) | 3.2 × 10^{5} | 3.2 × 10^{5} | 3.2 × 10^{6} | 3.2 × 10^{6} | 3.2 × 10^{6} | 3.2 × 10^{9} |

r_{QN} (m) for ρ_{QN} = 10^{18} kg/m^{3} | 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 r_{m} (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} |

x_{max} (m) | 138,434 | 125,839 | 223,837 | 192,900 | 170,824 | 1,709,027 |

x_{10} m/s (m) | 138,272 | 125,692 | 223,575 | 192,674 | 170,624 | 1,707,028 |

x_{100} m/s (m) | 137,683 | 125,156 | 222,622 | 191,852 | 169,897 | 1,699,749 |

θ_{2} (°) | 89.37838 | 89.43494 | 88.99486 | 89.13380 | 89.23293 | 82.30288 |

θ_{10} (°) for v_{exit} = 10 m/s | 89.37911 | 89.43560 | 88.99604 | 89.13481 | 89.23383 | 82.31194 |

θ_{100} (°) for v_{exi}_{t} = 100 m/s | 89.38176 | 89.43801 | 89.00032 | 89.13850 | 89.23710 | 82.34492 |

t_{exit} (s) for v_{exit} = 10 m/s | 3.1 × 10^{1} | 2.8 × 10^{1} | 5.1 × 10^{1} | 4.4 × 10^{1} | 3.9 × 10^{1} | 3.9 × 10^{2} |

t_{exit} (s) for v_{exit} = 100 m/s | 1.4 × 10^{1} | 1.3 × 10^{1} | 2.3 × 10^{1} | 1.9 × 10^{1} | 1.7 × 10^{1} | 1.7 × 10^{2} |

δ fractional error for v_{exit} = 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 v_{exit} = 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 v_{exit} | 1.5 × 10^{10} | 1.2 × 10^{10} | 3.9 × 10^{10} | 2.9 × 10^{10} | 2.3 × 10^{10} | 2.3 × 10^{12} |

Cross section for v_{exit} = 10 to 100 m/s | 1.3 × 10^{8} | 1.1 × 10^{8} | 3.3 × 10^{8} | 2.5 × 10^{8} | 1.9 × 10^{8} | 2.0 × 10^{10} |

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 v_{exit} | 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 × 10^{0} | 8.9 × 10^{0} | 4.0 × 10^{0} | 3.9 × 10^{0} | 3.9 × 10^{0} | 3.9 × 10^{−1} |

Rotational energy (J) | 2.2 × 10^{5} | 2.2 × 10^{5} | 2.1 × 10^{6} | 2.0 × 10^{6} | 2.0 × 10^{6} | 1.9 × 10^{9} |

RF power (MW) | 1.2 × 10^{4} | 1.5 × 10^{4} | 1.1 × 10^{5} | 1.6 × 10^{5} | 2.2 × 10^{5} | 2.2 × 10^{7} |

RF power (MW) after 1200 s | 6.7 × 10^{0} | 5.1 × 10^{0} | 6.1 × 10^{1} | 4.0 × 10^{1} | 2.8 × 10^{1} | 2.3 × 10^{5} |

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**Figure 1.**Three MQN trajectories are shown as red arrows, along with their angle of impact θ with respect to the normal surface of an idealized, not to scale, Earth (blue). The trajectory with impact angel θ

_{1}≪ 90° is a more common radial impact. Nearly tangential trajectories that transit through Earth are represented by trajectories with impact angles between θ

_{2}and θ

_{3.}MQNs on a θ

_{2}trajectory emerge from Earth with negligible velocity after transiting a distance x

_{max}, the maximum range of an MQN. MQNs on a θ

_{3}trajectory emerge from Earth with considerable velocity.

**Figure 2.**Estimated number of events per year somewhere on Earth as a function of B

_{o}for MQNs with 10

^{5}kg ≤ m ≤ 10

^{10}kg. The four solid-line curves correspond to event rates based on interstellar dark-matter density [3,42] of ~7 × 10

^{−22}kg/m

^{3}: MQNs transiting through water and emerging with any velocity v

_{exit}(blue); MQNs transiting through water and emerging with velocity 10 m/s ≤ v

_{exit}≤ 100 m/s (gray); MQNs transiting through granite and emerging with any velocity v

_{exit}(red); MQNs transiting through granite and emerging with velocity 10 m/s ≤ v

_{exit}≤ 100 m/s (black).

**Figure 3.**Cross sectional view of the magnetopause is shown (black line) between an MQN (blue circle) with magnetic moment (purple vector) at an angle of 60° to the velocity of the plasma (yellow arrows) flowing into the rest frame of the MQN. The plasma flow produces a net force (red arrow) centered at the top of the MQN and a corresponding torque vector into the page. The magnetopause is the locus of points at which the plasma pressure (on the left in Figure 3) is balanced by the magnetic pressure of the compressed magnetic field on the right. The complex shape of the magnetopause and resulting torque have been computed by Papagiannis [43] for Earth, illustrated in Figure 3, and extended to the case of MQNs. The effect can be understood by considering that the mean distance between the magnetopause and the MQN on the top half of Figure 3 is less than on the bottom half, which means that the magnetic field is compressed more on the top than on the bottom. Since force is transmitted by the compressed magnetic field, the net force is a push on the top, as shown by the red arrow.

**Figure 4.**Estimated angular velocity in the first 10

^{−4}s for 10

^{6}kg quark nugget with velocity v = 220 km/s, initial angle χ = 0.61 rad, initial angular velocity ω = 0, and passing through matter with mass density of 2300 kg/m

^{3}. Note the initial oscillation about 0 until a full rotation occurs, after which the angular velocity increases rapidly.

**Figure 5.**Geometry of simulation of rotating magnetized sphere above a highly conducting material. Magnetized, rotating sphere (blue) is shown above conducting material (gray). Arrows inside sphere indicate rotation and arrows in conducting material indicate force on the material. Arrows in air (white) indicate magnetic field lines at one moment in time. The axis of rotation is the y-axis, out of the plane of the figure. The magnetic axis of the magnetized sphere is initially in the x-direction and remains in the xz-plane.

**Figure 6.**Components of the time-averaged force between the simulated quark nugget and conducting slab as a function of the electromagnetic skin depth λ. The negative (<0) force F

_{z}(blue) opposes gravity and levitates the rotating magnetized sphere for λ < 0.5 m, with the most negative value for λ < 0.03 m. The force generated by the magnetic field traveling through the deformable conductor in the x-direction, as the magnetized sphere rotates about the y-axis, generates a propulsive force F

_{x}(red). F

_{x}is much less than F

_{z}for small skin depths. The much smaller F

_{y}(black) illustrates ±5% error in the calculation, since symmetry requires F

_{y}= 0.

**Figure 7.**Contours of the hole formed by the rotating the magnetic field of the magnetized sphere in (

**a**) the x-direction and (

**b**) the y-direction for times 2.5 ms (blue), 10 ms (gold), 20 ms (red), and 30 ms (black). The magnetic field sweeps through plastically deformable conducting material, and the displacement of the bottom of the hole is approximately −0.25 m in the x-direction. The same contours for the y-direction, which is along the axis of rotation, show the deformation is symmetric about y = 0, as expected. In both cases, the vertical axis has a different scale from the horizontal axis.

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VanDevender, J.P.; VanDevender, A.P.; Wilson, P.; Hammel, B.F.; McGinley, N.
Limits on Magnetized Quark-Nugget Dark Matter from Episodic Natural Events. *Universe* **2021**, *7*, 35.
https://doi.org/10.3390/universe7020035

**AMA Style**

VanDevender JP, VanDevender AP, Wilson P, Hammel BF, McGinley N.
Limits on Magnetized Quark-Nugget Dark Matter from Episodic Natural Events. *Universe*. 2021; 7(2):35.
https://doi.org/10.3390/universe7020035

**Chicago/Turabian Style**

VanDevender, J. Pace, Aaron P. VanDevender, Peter Wilson, Benjamin F. Hammel, and Niall McGinley.
2021. "Limits on Magnetized Quark-Nugget Dark Matter from Episodic Natural Events" *Universe* 7, no. 2: 35.
https://doi.org/10.3390/universe7020035