Abstract
Quark nuggets are theoretical objects composed of approximately equal numbers of up, down, and strange quarks. They are also called strangelets, nuclearites, AQNs, slets, Macros, and MQNs. Quark nuggets are a candidate for dark matter, which has been a mystery for decades despite constituting ~ 85% of the universe’s mass. Most previous models of quark nuggets have assumed no intrinsic magnetic field; however, Tatsumi found that quark nuggets may exist in magnetars as a ferromagnetic liquid with a magnetic field B_{S} = 10^{12±1} T. We apply that result to quark nuggets, a darkmatter candidate consistent with the Standard Model, and report results of analytic calculations and simulations that show they spin up and emit electromagnetic radiation at ~ 10^{4} to ~ 10^{9} Hz after passage through planetary environments. The results depend strongly on the value of B_{o}, which is a parameter to guide and interpret observations. A proposed sensor system with three satellites at 51,000 km altitude illustrates the feasibility of using radiofrequency emissions to detect 0.003 to 1,600 MQNs, depending on B_{o,} during a 5 year mission.
Introduction
About 85%^{1} of the universe’s mass does not interact strongly with light; it is called dark matter, is distributed in a halo throughout a galaxy^{2}. Identifying the nature of dark matter is currently one of the biggest challenges in science. As reviewed most recently by Salucci^{3}, extensive searches for a subatomic particle that would be consistent with dark matter have yet to detect anything above background signals.
Most models assume dark matter interacts with normal matter only through gravity and the weak interaction. However, detailed analysis of the accumulating data on galaxies of different types suggest dark matter and normal luminous matter interact somewhat more strongly, on the time scale of the age of the Universe^{3}. Magnetized quark nuggets (MQNs)^{4} are an emerging candidate for dark matter that quantitatively meets the traditional interaction requirements for dark matter and still interacts with normal matter on the time scale of the age of the Universe through the magnetic force. In this paper we investigate a new method for detecting MQNs and measuring their properties.
Quarks are the basic building blocks of protons, neutrons, and many other particles in the Standard Model of Particle Physics^{5}. Macroscopic quark nuggets^{6}, which are also called strangelets^{7}, nuclearites^{8}, AQNs^{9}, slets^{10}, and Macros^{11} are theoretically predicted objects composed of up, down, and strange quarks in essentially equal numbers. A brief summary of quarknugget research^{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 chargetomass ratio, formation, stability, and detection has been updated from Ref.^{16} and is provided for convenience as Supplementary Note: Quarknugget research summary.
Most previous models of quark nuggets have assumed negligible selfmagnetic field. However, Tatsumi^{15} explored the internal state of quarknugget cores in magnetars and found that quark nuggets may exist as a ferromagnetic liquid with a surface magnetic field B_{S} = 10^{12±1} T. Although his calculations used the MIT bag model with its wellknown limitations^{20}, his conclusions can and should be tested. We have applied his ferromagneticfluid model of quark nuggets in magnetars to magnetized quark nuggets (MQNs)^{4,16} dark matter and extend those results to calculate how MQNs rotate and radiate radio frequency (RF) emissions during and after interaction with normal matter. We also explore how the RF emissions can enable detection of MQNs.
As done in Ref.^{4}, we will use B_{o} as a key parameter. The value of B_{o} is related to the mean value <B_{S}> of the surface magnetic field through.
If MQN mass density ρ_{QN} = 10^{18} kg/m^{3} and the density of dark matter ρ_{DM} = 1.6 × 10^{8} kg/m^{3} at time t ≈ 65 μs (when the temperature T ≈ 100 MeV in accord with standard ΛCDM cosmology), then B_{o} = <B_{S}> . If better values of ρ_{QN}, and ρ_{DM} are found, then the corresponding values of <B_{S}> can be calculated with Eq. (1) from those better values and from B_{o} determined by observations.
Previous papers on MQNs showed:

1.
selfmagnetic field aggregates MQNs with baryon number A = 1 into MQNs with a broad mass distribution^{4} that is characterized by the value of B_{o} and typically has A between ~ 10^{3} and 10^{37},

2.
aggregation dominates decay by weak interaction so massive MQNs can form and remain magnetically stabilized in the early universe even though they have not been observed in particle accelerators^{4},

3.
the selfmagnetic field forms a magnetopause that strongly enhances the interaction cross section of a MQN with a surrounding plasma^{16} that is primarily sustained by radiation and electron impact ionization in the high temperature plasma formed by normal matter stagnating against the magnetopause,

4.
B_{o} > 3 × 10^{12} T is excluded^{4} by the lack of observed deeply penetrating impacts that deposit > MegatonTNT equivalent energy per km, so Tatsumi’s range of B_{S} is reduced to 1 × 10^{11} T ≤ B_{o} ≤ 3 × 10^{12} T,

5.
MQNs satisfy criteria for dark matter even with baryon’s interacting with MQNs through the selfmagnetic field^{4}, and

6.
the unexcluded range of B_{o}, the low (~ 7 × 10^{–22} kg m^{−3}) density of local dark matter, net incident velocity of ~ 250 km/s, and the high average mass of MQNs constrain the flux of MQNs of all masses to between 10^{–7} and 4 × 10^{–15} m^{−2} y^{−1} sr^{−1} and constrain the flux for MQN masses > 1 kg to between 3 × 10^{–14} and 2 × 10^{–17} m^{−2} y^{−1} sr^{−1}, so very large area detectors or very long observation times are required^{4} to detect MQNs.
In this paper, we investigate the possibility of detecting MQNs interacting with the largest accessible target: Earth, with its magnetosphere to 10 Earth radii. We show MQNs necessarily experience a net torque while passing through matter, can spin up to kHz to GHz frequencies, and emit sufficient narrowband electromagnetic radiation that could be detected in additional tests of the MQN hypothesis for dark matter.
Detection is also complicated by the magnetopause plasma (hot ionized gas) shielding some radiofrequency (RF) emissions. Spacecraft reentering the atmosphere cannot communicate with ground stations because the surrounding plasma shields the emissions until the spacecraft slows down. The same blackout effect prevents RF emissions from MQNs from being detected during the highvelocity interaction of MQMs with matter in the troposphere or ionosphere.
However, MQNs that exit the atmosphere can be detected by their narrowband, timevarying RF radiation, with frequency equal to the rotation frequency. In addition, MQN detection rate should be strongly correlated with the direction of darkmatter flux into the detector aperture. That preferred direction is determined by Earth’s velocity about the galactic center and, consequently, through the darkmatter halo. These characteristics should permit their detection and differentiation from background RF.
Results
In this section, we show (1) when MQNs interact with plasma, their rotational velocity increases as their translational velocity decreases, (2) MQNs passing through Earth’s atmosphere on a flyby trajectory produce sufficient RF emissions to be detected by satellites but not groundbased sensors, and (3) the frequency, RF power, and event rate depend strongly on the value of the B_{o} parameter in Eq. (1).
Rotational spinup
Plasma is an ionized state of matter with sufficient electron number density n_{e} at temperature T_{e} for the electrons and ions to behave as a quasineutral fluid. Quantitatively, that means there is at least one electron–ion pair in a spherical volume of radius equal to a Debye shielding length λ_{D},
in which the permittivity of free space ε_{o} = 8.854 × 10^{–12} F/m, the Boltzmann constant k_{B} = 1.38 × 10^{–23} J/K, and electron charge e = 1.6 × 10^{–19} C. The Debye length is also the distance over which the thermal energy of the electrons permits charge separation to occur and defines electrical quasineutrality of a plasma^{36} versus an ensemble of electrons and ions which behave as particles. The distinction is important for MQNs. A plasma impacting the magnetic field of a MQN acts as a conducting fluid and compresses the magnetic field to form a magnetopause^{16}. A comoving electron–ion pair impacting the magnetic field will separate because the forces from their opposite charges deflect them in opposite directions and will create a local electric field between them. The combined force from their electric E and magnetic B fields causes them to move in the same E × B direction at the same velocity^{36} and through the magnetic field, instead of strongly compressing it. In contrast, conducting plasma shorts out the electric field and prevents particle penetration.
A MQN moving through a plasma experiences a greatlyenhanced slowing down force^{16} through its magnetopause, which is the magnetic structure formed by particle pressure from a plasma stream balancing magnetic field pressure around a magnetic dipole. For example, the solar wind forms a magnetopause with Earth’s magnetic field. Since the particles’ mean free path for collisions is much larger than the Larmor radius in the magnetic field, the physics of Earth’s magnetopause is collisionless and applicable to the very smallscale lengths of a quarknugget’s magnetopause.
As derived in Ref.^{16}, the cross section σ_{m} for momentum transfer by the magnetopause effect is
for magnetopause radius r_{m}, MQN radius r_{QN}, MQN speed v, and mass density of surrounding matter ρ_{x}. The total force F_{o} exerted by the plasma on the quark nugget is approximately.
Equations (3) and (4) let us calculate the decelerating force on the MQN and its translational velocity during passage through matter^{16}.
In this paper, we extend the dynamics to calculate the torque and rotational velocity of the MQN passing through matter. Papagiannis^{37} showed that the solar wind, which has mass density ρ_{x} ≈ 10^{–20} kg/m^{3} and velocity v ≈ 3.5 × 10^{5} m/s, exerts a torque T (N⋅m) on Earth (radius r_{o} = 6.37 × 10^{6} m and magnetic field B_{o} = 3. × 10^{–5} T) as a function of the angle χ between the magnetic axis and the normal to both the magnetic axis and the direction of the solar wind. His semiempirical result is expressed in MKS units as
in which C_{2} = 1,400 with units of \(Ns\left( {kg} \right)^{  0.5} m^{  1.5} T^{  1}\). Papagiannis validated the expression for the angles χ within 0.61 radians of 0 and within 0.61 radians of π (i.e. –0.61 ≤ χ ≤ + 0.61, –3.14 ≤ χ ≤ –2.53, and + 2.53 ≤ χ ≤ + 3.14, as illustrated in Fig. 1). By symmetry of the magnetic field, the torque is 0 at χ = 0 and χ = ± π/2.
Since Papagiannis was only considering Earth, he limited his calculations to within 0.61 radians of the normal to the plasma velocity. Rigorously reproducing and extending his computational results to the larger angles required for MQN rotation is beyond the scope of this paper. Therefore, we extend his results by observing the amplitude of the torque is symmetric about χ = –3π/4, –π/4, π/4 and 3π/4, as shown in Fig. 1 by solid blue lines, and approximate the rest of the torque function by extrapolation of the tan(χ) function in Eq. (5), as shown with dotted blue lines in Fig. 1. The resulting functions F_{χ} and T, shown in Eq. (6), replace Eq. (5).
The torque is negligible for Earth but is very large for a quark nugget. The rate of change of angular velocity ω for MQN with mass m_{QN}, moment of inertia I_{mom} = 0.4 m_{QN} r_{QN}^{2} experiencing torque T is.
Equations (2)–(7) were solved for the angular velocity versus time. The interaction produces a velocitydependent and angledependent torque that causes MQNs to oscillate initially about an equilibrium. Since the quark nugget slows down as it passes through ionized matter, the decreasing forward velocity decreases the torque with time, so the timeaveraged torque in one halfcycle is greater than the opposing timeaveraged torque in the next halfcycle. The amplitude of the oscillation necessarily grows, as shown in Fig. 2. Once the angular momentum is sufficient to give continuous rotation, the net torque continually accelerates the angular motion to produce a rapidlyrotating quark nugget. As shown in Fig. 2, MHz frequencies are quickly achieved even with a 0.1 kg quark nugget moving through 1 kg/m^{3} density air at 250 km/s. For smaller or larger masses, the resulting angular acceleration and velocity are respectively larger or smaller.
Several other approximations for the torque in the intervals shown with dotted lines in Fig. 1 gave the same frequency within 5%, which is within the uncertainty of the magnetic field parameter B_{o}.
Equilibrium frequency and radiated power
Rotating magnetic dipoles emit electromagnetic radiation in the far field with power per steradian^{38} given by
in SI units, with Z_{o} = 377 Ω, ω is angular frequency, and c is the speed of light in vacuum. The magnetic dipole moment m_{m} = 4π B_{o} r_{QN}^{3}/μ_{o}, and angle of rotation χ is the angle between the velocity of the incoming plasma and the magnetic moment. The total power radiated^{38} is
The spinup process strongly depends on the details of the surrounding material mass and MQN velocity along the path of the MQN, the MQN mass, and surface magnetic field B_{o}. In spite of these complexities, we find that the spinup time is very much less than the MQN transit time through the region of highest torque, and the lower limit of final rotation frequency can be adequately estimated by assuming the energy gained per cycle equals the energy radiated per cycle:
in which the torque T is given by Eq. (6) and the radiated power P is given by Eq. (9). Combining Eqs. (6)–(10) gives
In the simulations discussed below, we solve Eqs. (3) and (4) for position and velocity v(t) calculated along a trajectory through the atmosphere to the position of maximum density ρ_{x}, where we solve Eq. (11) for the lowerlimit to the maximum frequency ω_{max}.
Attenuation of RF power by magnetopause plasma
The surrounding magnetopause plasma has a characteristic plasma frequency
in which n_{e} = the local electron number density, e = the electron charge, and m_{e} = the electron mass. The characteristic efold length for attenuating the radiated power for frequency ω < ω_{pe} is \(k_{atten}^{  1}\):
In practice, the scale length 0.5c/ω_{pe} is approximately 0.5% of the ion Larmor radius, which is approximately the minimum thickness of the magnetopause boundary. Therefore, magnetopause plasma strongly absorbs RF energy for frequencies less than the plasma frequency.
Since plasmas strongly absorb electromagnetic radiation with frequency less than the plasma frequency, which varies with solar activity, practical detection of MQNs by their RF emissions is limited to ≥ 0.03 MHz in the solarwind plasma near Earth orbit, to ≥ 0.4 MHz in the magnetosphere, and to ≥ 40 MHz in the ionosphere. The equilibrium frequency of the MQN spinup process and the high density of the troposphere means, in practice, all RF emissions in the troposphere are strongly shielded. Therefore, we will focus on detecting MQNs in the magnetosphere after they have transited Earth’s atmosphere, as illustrated in Fig. 3.
The middle MQN trajectory in Fig. 3 represents a direct impact^{16}. The bottom trajectory represents a MQN that is gravitationally captured and does not exit into the magnetopause. The top trajectory represents a MQN that spins up during transit and is detected by satellite S1. Satellite S2 would not detect these MQNs since they have not passed through sufficient matter to spin up. Therefore, appropriate differences in event rates as a function of satellite position would support detection of dark matter.
Trajectories of quark nuggets are from a preferred direction in Fig. 3 because the velocity of the solar system about the galactic center and through the halo of dark matter nearly dominates the random velocity of quark nuggets, as discussed in a subsequent section.
Radiofrequency background from MQNs distributed throughout the galaxy
Darkmatter is distributed throughout the galaxy^{2,3}. Therefore, MQN dark matter in interstellar space might interact with the local plasma density and emit RF radiation that fills the universe over billions of years. The resulting RF might be detectable as a galactic background near Earth. Olbers’ Paradox on why the night sky is dark^{39} addresses the same phenomenon for photons from stars. Evaluating the potential for detecting and interpreting this radiation is complex. A detailed analysis is beyond the scope of this paper, which focuses on detection of nearEarth MQNs. However, the following preliminary analysis shows that the MQN hypothesis cannot be tested by observations of galactic RF background near Earth.
The plasma density and temperature in interstellar space vary greatly:

1.
a minimum of ~ 10^{–4} to ~ 10^{–2} particles/cm^{3} in Hot Ionized Media (HIM) at T_{e} ~ 10^{6} to 10^{7} K composing 20% to 70% of interstellar space,

2.
to ~ 0.2 to 0.5 particles/cm^{3} in Warm Ionized Media (WIM) at T_{e} ~ 8,000 K composing 20–50% of space,

3.
to higher mass densities represented by ~ 10^{2} to 10^{6} particles/cm^{3} in molecular clouds at T_{e} ~ 10 to 20 K composing < 1% of space^{40}.
As discussed above, matter has to act like a fluid plasma (instead of an ensemble of isolated charged particles) to form a magnetopause. Quantitatively, the scale length λ_{D} for charge separation in Eq. (2) has to be less than the magnetopause radius r_{m} defined in Eq. (3), so
For example, using the midrange value of B_{o} = 2.0 × 10^{12} T and ρ_{QN} = 1 × 10^{18} kg/m^{3}, the MQN mass has to be greater than 10^{7} kg for a magnetopause to form in HIM. The equilibrium rotation frequency for such massive MQNs is much less than 30 MHz and the corresponding RF is absorbed by the solarwind plasma near Earth.
The corresponding threshold MQN mass for forming a magnetopause in WIM is 10 kg. However, the higher density plasma in the WIM slows those MQNs, so their RF is still less than 30 kHz, so it is also absorbed in the solarwind plasma near Earth.
In general, the larger mass density of the rest of interstellar media produce even lower frequency RF, which is absorbed even further from Earth. We analyzed these effects for the full range of interstellar plasma conditions^{40} and MQN mass distributions^{4} as a function of the B_{o} parameter. We found wherever MQN dark matter can produce RF emissions, the emissions are either absorbed in the surrounding plasma or in the solarwind plasma near Earth. The RF is not detectable near Earth and cannot be used to test the MQN hypothesis. Therefore, we focus on detecting MQNs passing very near Earth and radiating well above the 30kHz cutoff frequency of the solarwind plasma.
Discriminating MQN events from background
Theoretical profiles of dark matter halos are guided by astrophysical observations, which are consistent with a mass density of dark matter near Earth of about 7 × 10^{–22} kg/m^{3} ± 70%^{3,14}. This extremely low mass density and the very broad mass distributions^{4} mean that the flux of MQNs is very low. Detecting them over the full range of notexcluded values of B_{o} requires interaction volumes of at least planetary size and requires a means of reliably subtracting background. The orientation of the sensed MQN flux with respect to the direction of inflowing dark matter provides one such opportunity.
Models differ in their degree of selfinteraction. Computer simulations^{41} covering a substantial portion of these models indicate dark matter occupies a halo within and around the galaxy’s ordinary matter and has a Maxwellianlike, isotropic velocity distribution. Although simulations predict the velocity distribution of dark matter varies somewhat with the selfinteraction model, the most probable, isotropic speed is ~ 220 km/s with a fullwidthathalfmaximum of ~ 275 km/s.
The solar system moves through this highspeed darkmatter halo in its ~ 250 km/s motion about the galactic center. The direction of this motion is towards the star Vega, which has celestial coordinates: right ascension 18 h 36 m 56.33635 s, declination + 38° 47′ 01.2802″. In addition, Earth moves around the Sun at ~ 30 km/s. The vector sum of these two velocities gives the net velocity of Earth through dark matter, and the negative of this vector sum is the velocity of dark matter relative to Earth, shown in Fig. 4 for the position of Earth on the first day of each month.
A sensor on Earth should detect the most events per hour when it is sensitive to the flux of dark matter from the direction of Vega and much less when Earth shields the detector from the flux. A satellite in orbit about Earth would encounter a higher flux of MQNs when it is not shielded by Earth and when it is within range of MQNs that have transited through enough matter to spin up, as illustrated by S1 in Fig. 3.
The darkmatter velocity distribution has a streaming component U_{s} relative to the sensor and an isotropic component U_{iso} relative to the galactic center. The isotropic component smooths the transition between the directly exposed and Earthshielded conditions. U_{iso} can be adequately approximated for our purposes as the highest probability thermal speed \(U_{iso}=\sqrt{\frac{2kT}{M}}\) of a 3D Maxwellian velocity distribution of dark matter with mass M and with temperature T, where k is the Boltzman constant. Simulations^{41} indicate that U_{s} ≈ U_{iso}. Therefore, we calculated the detection rate as a function of the sensor's orientation on Earth with respect to Vega and S = U_{s}/U_{iso} using the method developed by Cai et al.^{42}. The results are shown in Fig. 5.
If this variation with respect to Vega’s position is observed, i.e. the event rate for satellite S2 in Fig. 3 is appropriately and systematically less than the event rate for S1, the result would be convincing evidence of having detected MQN dark matter.
Including MQNs from all directions
The flux in Fig. 5 is normalized to F_{θ=0}, the total number of events m^{−2} y^{−1} sr^{−1} for a detection surface facing directly into the streaming velocity U_{s}, and was approximated from the mass distributions in Ref.^{4}, assuming the mean incoming velocity U_{s} = 2.5 × 10^{5} m/s and assuming the effect of U_{iso} ≠ 0 on the flux is negligible to first order for θ = 0. Assuming cylindrical symmetry in azimuthal direction φ and integrating the curve in Fig. 5 over solid angle with dΩ = sinθ dθ dφ gives the correction factor for estimating the event rate in m^{−2} y^{−1} for MQNs incident from all directions
The results of the simulation for θ = 0 in subsequent sections are multiplied by 5.56 to estimate the event rate for MQNs from all directions.
Simulating MQNs flying by Earth
Consider the simple case of a quarknugget with a trajectory parallel to a tangent to Earth, as illustrated by the three trajectories in Fig. 3. Satellites sense MQNs after they have transited the highest density matter along their trajectories and experienced the corresponding torque, as described in Eq. (6), to produce the maximum frequency and radiated power. After they pass into the magnetosphere, they can be detected since their emissions above ~ 0.4 MHz are no longer shielded by the higherdensity plasma of the ionosphere.
In our simulations, Earth’s atmosphere is divided into increments ∆h of altitude h. MQN trajectories are characterized by their minimum altitude for 0 < h ≤ 9 r_{e}, for r_{e} = 6.378 × 10^{6} m, which is one Earth radius. For each increment ∆h, test MQNs with masses consistent with the mass distributions of Ref.^{4} are injected from the right in Fig. 3 with initial velocity in the x direction v_{x} = U_{s} = 2.5 × 10^{5} m/s. Their positions and velocities are calculated under the combined effects of gravity and magnetopause interaction. For each test particle, the maximum torque encountered in its trajectory, i.e. the torque in Eq. (6) at the maximum value of the product of total velocity and the square root of the local mass density, is used to calculate the equilibrium frequency from Eq. (11) and the corresponding RF power from Eq. (9).
Characteristic times τ_{up} = ω_{max} I_{mom}/T_{max} for spin up and τ_{down} = 0.5 I_{mom} ω_{max}^{2}/P_{max} for spin down by radiation loss are calculated. Since we find τ_{up} is much less than transit time through the atmosphere, ω_{max} is the lower limit to the maximum frequency, from Eq. (11). The value of τ_{down} helps determine the detectability of each representative MQN. Characteristic efold times for spin up and spin down are included in Supplementary Results: Representative Data Tables for Sensor Design. Values of τ_{down} vary from a minimum of 2.4 × 10^{3} s to a maximum of 1.9 × 10^{6} s and provide adequate time for detection.
The cylindrically symmetric cross sectional area A_{h} associated with the altitude increment ∆h and altitude h above Earth radius r_{e} is illustrated in Fig. 3 and is given by
MQNs with final velocity exceeding lowEarth orbital velocity ≥ 7,400 m/s escape the RFabsorbing ionosphere and will be recorded by a satellitebased sensor.
Atmospheric density as a function of altitude h = r – r_{e} for MQNs at radius r and Earth radius r_{e} were derived from the literature and fit with the following equations:
For radius r below the magnetosphere^{43}, i.e. 0 > r – r_{e} > 2.873 × 10^{5} m:
and for radius r in the magnetosphere^{44}, i.e. 2.873 × 10^{5} m > r – r_{e} > 10 r_{e}:
Mass density in Earth’s magnetosphere depends strongly on solar activity and varies greatly. The data in Ref.^{44} were averaged to produce Eq. (18), which should be adequate to estimate the annual event rate.
Our simulations show MQNs radiating between 10^{–28} W and 10^{+13} W and at frequencies between 0.35 MHz and ~ 2 GHz. Very high frequencies are associated with negligible RF power, and very high powers are associated with frequencies that are shielded by the magnetopause plasma. Results for RF power as a function of frequency are shown in Fig. 6 for events radiating at more than 1 μW and at frequencies more than 100 kHz. Results are shown for four representative and nonexcluded values of B_{o}.
As shown in Fig. 6, highest power emissions occur at the lowest frequencies and highest values of B_{o}. These originate from the most massive MQNs penetrating the troposphere, but there are very few of them. The map associated with B_{o} = 1.5 × 10^{12} T is common to the maps of all B_{o} values. The differences represent aggregation runaway as discussed in Ref.^{4}.
Figure 6 represents events by frequency and RF power but does not indicate the expected number of events per year. For each test MQN, the effective target area, given by Eq. (16), was multiplied by the corresponding number flux from Ref.^{4} and by the 5.56 factor from Eq. (15), and summed over all simulated events with RF power greater than a sensor’s detection threshold to estimate the number of events that might be observable per year as a function of detection threshold. The results are shown in Fig. 7.
At 1 nW threshold, the number of events per year that might be detectable in space out to 10 r_{e} is ~ 30,000, ~ 8, ~ 0.8, and ~ 0.01 for B_{o} = 1.5 × 10^{12} T, 2.0 × 10^{12} T, 2.5 × 10^{12} T, and 3.0 × 10^{12} T, respectively. For 1 μW detection threshold, the number events per year drops to ~ 10,000, ~ 7, ~ 0.07, and ~ 0.003 for B_{o} = 1.5 × 10^{12} T, 2.0 × 10^{12} T, 2.5 × 10^{12} T, and 3.0 × 10^{12} T, respectively.
The number of events per year decreases so strongly with increasing B_{o} because larger values of B_{o} cause faster aggregation of MQNs in the early universe and, consequently, larger mass MQNs^{4}. Since the mass per unit volume of dark matter is constrained by observations to be ~ 7 × 10^{–22} kg/m^{3}, the number density of MQNs decreases for increasing mass and increasing B_{o}. So the flux of MQNs and detection rate decrease for increasing B_{o}.
Detailed results for MQNs with RF power greater than 1 nW and with sufficient flux to be in the most probable 80% of events are provided in Supplementary Results: Representative Data Tables for Sensor Design. The information should be useful for designing sensors for detecting MQNs.
Baseline sensor system
A realistic sensor system is, of course, essential for testing the MQN darkmatter hypothesis. A convenient coincidence of frequency range and emerging technology enable a practical sensor. We outline one such baseline system and compute the corresponding event rate in this section.
As shown in Fig. 3, the system has three satellites equally spaced in a circular orbit with inclination 38.783° (to match MQN flux) at 51,000 km altitude where (1) the background plasma density is sufficiently low to permit good RF propagation in the intended detection band of 10^{5} to 10^{6} Hz, (2) radiation damage from electrons in the outer Van Allen belt is minimized, and (3) coverage by three satellites is acceptable. The Interplanetary Monitoring Platform IMP6 (Explorer 43) spacecraft^{45,46} is the reference architecture for the sensor system’s spacecraft. IMP6 was a 16sided drum, 1.8m long by 1.35m diameter, having four 46m long monopole antennas operating in pairs as 91 m long dipoles, and spinning at 5.4 revolutions per minute to scan space. Our baseline design is the same architecture but with 350 m long dipoles.
The sensor is based on the standard radar equation, formulated as transmitreceive equation:
where P_{r} is the received power, f is the frequency in Hz, c is the speed of light, G_{r} is the antenna gain, P_{s} is the source power, R is the distance, and G_{s} is the source gain, which will be taken to be unity to represent the time averaged value. Lossless dipoles up to a halfwavelength long have gains G_{r} < 1.6 and directivities D_{r} from 1.5 to 1.6. The weak signals require amplification in the receiver. The noise power P_{N} and the signal to noise ratio S/N for an amplified receiver are
where Boltzmann’s constant k_{B} = 1.38 × 10^{–23} J/K, T_{noise} is the absolute radiation noise temperature, T_{a} is the preamplifier noise temperature, and △ f is the receiver resolution bandwidth.
Solving Eq. (21) for the range R gives
Since many measurements will be made on each MQN as it transits through detection range, we can operate at signal/noise ratio S/N = 1 and use signal averaging and pattern recognition to reliably detect the signal.
The background noise temperature T_{noise} is a major factor in determining sensor performance. IMP6 measured the galactic background noise in the magnetosphere between 354 km and 206,000 km altitude and between 130 and 2,600 kHz. To determine noise temperatures from the noise power measurements of Brown^{45}, we equate his values for spectral brightness B to the brightness in the RayleighJeans law for blackbody radiation and solve for the noise temperature T_{noise}
The results are shown in Table 1. The frequency range of interest encompasses a temperature maximum of 24 million degrees (MK) near 0.8 MHz. Ground based measurements at the poles during solar minima by Cane^{47} are also shown in Table 1 for comparison and are consistent with our analysis of Brown’s^{45} data.
Researchers accustomed to working at much higher frequencies, e.g. in satellite communications or radio astronomy, may find these values of T_{noise} to be unreasonably high. However, T_{noise} is decreasing from 0.8 MHz to 3.0 MHz in Table 1; it continues to decrease with increasing frequency to agree with T_{noise} observed in those disciplines.
Fitting the data with the minimum of two polynomials reproduces the table for 0.2 MHz ≤ f ≤ 3.0 MHz within ± 1% and provides a useful function for estimating sensor performance:
For these low frequencies f (MHz), the noise temperature is millions of degrees Kelvin. From the denominator under the squareroot term in Eq. (22), we see that variation of range with antenna gain is small if
Since noise temperature T_{a} can be less than 100 K (which does not require a cryogenic amplifier) for the frequencies of interest, 25 MK > T_{noise} > 3 MK in Table 1, and D_{r} ~ 1.5, Eq. (23) gives 6 × 10^{–6} < G_{r} < 5 × 10^{–5} before the range becomes very sensitive to antenna gain. For source powers of 1·w, antenna gains of 0.001 to 1, radiation noise temperatures of 10 million K for the present case, amplifier noise temperatures of 100 K, receiver bandwidths of 1 Hz, and signal to noise ratios of unity, one gets ranges of 14,954 km and 15,740 km at 0.5 MHz. Therefore, range is very insensitive to antenna gain, and antenna gains that are well below unity are acceptable when the radiation temperature is very high.
The antenna gain G_{r} depends on the dimensions and the termination impedance of the dipole antenna. Tang, Tieng, and Guna’s theory^{48} was used to design the antenna subsystem. The antenna gain G_{r} as a function of frequency is shown in Fig. 8 for four terminating impedances.
For a wide range of terminating impedances, G_{r} easily satisfies Eq. (25) and the simple dipole is suitable for the baseline system. The optimal dipole design will likely incorporate resistive and/or possibly reactive loading at discrete positions along its length to achieve the best bandwidth and gain response if the additional complexity does not preclude deployment in space.
The other variables in Eq. (22) are source gain G_{s} and signaltonoise ratio (S/N). Both are set to 1.0. For multiple measurements like the satellites will be recording, detections with S/N = 1 are routine, and detections have been demonstrated for S/N as low as 0.1.
We found two options that apparently meet key requirements for the receiver: Δf = 1 Hz and a million simultaneously recorded frequencies. Assuming that such compact and lowpower, groundbased electronic hardware can be made into spacequalified hardware, they provide proofofprinciple of receiver feasibility. The Tektronix RSA 306B Real Time Spectrum Analyzer system with Tektronix proprietary software on a separate computer monitors 10^{6} frequencies every 2 s with the required bandwidth Δf = 1 Hz and outputs the results to a laptop computer. In addition, the N210 USRP unit from National Instruments, which we have been using for autonomous data collection with our custom software for 254 frequencies every 25 μs, can meet the requirements by replacing the Field Programmable Gate Arrays (FPGAs) currently used for processing fast Fourier transforms with General Purpose Computing on Graphics Processing Units (GPGPUs).
In both cases, processing the data on 10^{6} simultaneous frequencies to find the unique signature of an MQN and extract the desired information will be a significant challenge. A combination of onboard processing with GPGPUs to select data for download and groundbased postprocessing should be sufficient but has yet to be demonstrated.
Estimated MQN detection rates
Although the nonexcluded range for B_{o} at the time of this publication is 1 × 10^{11} T ≤ B_{o} ≤ 3 × 10^{12} T, recent results being prepared for publication reduce the nonexcluded range to 1.5 × 10^{12} T ≤ B_{o} ≤ 3 × 10^{12} T. If B_{o} < 1.5 × 10^{12} T is included, the predicted MQN detection rate will be even larger and will shift to higher frequencies.
The number of events expected with the threesatellite sensor system described in this section was calculated for four values of B_{o} in the more limited range. The calculation includes (1) computed number, frequency, and RF power of flythrough MQNs, (2) the measured RF background temperature as a function of frequency, (3) the sensor range as a function of MQN frequency and RF power from Eq. (22), and (4) the fraction of the area monitored by the three satellites at the 51,000km altitude of their orbit as a function of MQN frequency and RF power. The number of MQNs that should be detected in five years of observations for 0.1–1.1 MHz RF as a function of B_{o}:

1,620+ /− 40 if B_{o} = 1.5 × 10^{12} T,

8.1+ /− 2.8 if B_{o} = 2.0 × 10^{12} T,

0.05+ /− 0.2 if B_{o} = 2.5 × 10^{12} T, and

0.003+/− 0.6 if B_{o} = 3.0 × 10^{12} T.
Since quark nuggets are also baryons, their magnetic field may be similar to that of protons, which have the equivalent magnetic field B_{o} between 0.9 × 10^{12} T and 2.2 × 10^{12} T.
The case for B_{o} = 1.5 × 10^{12} T is quite different because the corresponding MQN mass is much smaller, so the flux is much larger and the frequencies are much higher. Approximately 7% of events are between 0.1 and 1.0 MHz, 26% are between 1 and 3 MHz, 54% are between 3 and 10 MHz, and 13% are from 10 to 200 MHz. The system should be optimized for 0.1 to 1.0 MHz. If the RF background permits some additional measurements between 1 and 200 MHz would explore the special case of B_{o} ~ × 10^{12} T, unless additional observations in the near future exclude that case.
Obtainable measurements and information
MQNs transiting through the magnetosphere, as shown in Fig. 3, provides a frequency source f_{0}(t) moving at constant nonrelativistic velocity v in a straight line. A stationary observer at distance r(t) records the observed frequency versus time. The closest approach occurs at time t_{0} and at distance r_{0}. The observed frequency f (t) will be
Total Doppler shifts are expected to be about that 1.5% of f_{0}. The source frequency is expected to decrease with time. During the brief periods of detectability, the source frequency should change linearly with time by amounts comparable to or less than the Doppler shifts.
Figure 9 shows what can be extracted from measurements of frequency versus time from the moving source. The velocity v_{0} is determined from the asymptotic limit lines. The slope of these lines determines the rate of change of frequency of the source. The location of the inflection point determines f_{0} and t_{0}, and the slope of the central asymptote determines (v_{0}/r_{0})f_{0} – df_{0}/dt from which r_{0} is determined with the other measured variables.
Because signal strengths are not likely to be measurable in the far asymptotic regions, it is important that good data be observed in the regions of maximum d^{2}f/dt^{2}, which occur at distances less than 2r_{0}. Optimal fits of data to Eq. (26) will provide estimates of the variables of interest.
Thus, from the measurement it is possible to estimate t_{0}, f_{0}, df_{0}/dt, r_{0}, v_{0}, and dv_{0}/dt and analyze the data to get two measurements of MQN mass, one from f_{0} and df_{0}/dt and one from v_{0} and dv_{0}/dt. With enough data the MQN mass distribution can be constructed and compared with the predictions in Ref.^{4} and be used to refine Tatsumi’s value of the surface magnetic field in magnetar cores.
Discussion
We have shown that MQNs transiting through ionized matter experience a torque on their magnetopause and spin up to high frequencies, ranging from kHz to GHz, depending on MQN mass and velocity, mass density of the surrounding matter, and the B_{o} parameter. Rotating MQNs radiate. If the radiation is above the plasma frequency of the surrounding matter, the RF radiation will propagate and can, in principle, be used to detect MQN dark matter at a substantial distance.
We did examine the possibility of using groundbased sensors to detect MQNs transiting the magnetosphere and radiating at > 40 MHz, the cutoff frequency of the ionosphere. The event rate is at most 0.3, 0.02, 0.0004, and 0.00007 per year even if 2π steradians solid angle can be observed. A spacebased system is more promising.
Our results have identified requirements for MQN detection systems and a baseline system of three satellites that should test the MQN hypothesis for dark matter by detecting between 1,600 and 0.003 MQNs for B_{o} between 1.5 × 10^{12} T and 3.0 × 10^{12} T, respectively. Recording their Doppler shifted frequencies during their passage by a sensor provides two different measurements of the MQN mass, so a mass distribution can be obtained in time.
The pattern of trajectories with respect to the direction of Vega, frequency, and rate of change of frequency would provide strong evidence of MQNs and eventually characterize the actual mass distribution of MQNs to compare with the predictions of Ref.^{4}.
Very low mass MQNs will have frequencies of up to 2 GHz, but their negligible RF power precludes their being detected. Very high mass MQNs will emit RF power of up to 10^{13} W, but their negligible flux makes their detection unlikely in a year of observation. As shown in Supplementary Results: Representative Data Tables for Sensor Design, frequencies between 100 and 800 kHz are the most likely frequencies to be observed from MQNs.
Discerning quarknugget signals from all humancaused and naturally occurring, nonMQN RF is facilitated by their characteristics: (1) narrowband RF, unlike RF from lightning and other discharges, (2) continuous emissions, unlike pulsing radars, (3) relatively unmodulated in frequency and amplitude, unlike communication RF, (4) moving at ~ 200 km/s, unlike all human sources, and (5) initially increasing in frequency to a maximum and then slowly decreasing, unlike magnetosonic waves.
Finally, we note that solar and planetary atmospheres can be even largerarea, but less accessible, targets for RFemitting MQNs.
Data availability
All final analyzed data generated during this study are included in this published article with its Supplements.
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Acknowledgements
We gratefully acknowledge S. V. Greene for first suggesting that quark nuggets might explain the geophysical evidence that initiated this research (she generously declined to be a coauthor), Albuquerque Academy for hosting preliminary experiments, Robert Nellums for reviewing portions of the work, and Jesse A. Rosen for editing and improving this paper. This work was supported by VanDevender Enterprises, LLC.
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J.P.V. was lead physicist and principal investigator. He developed computer programs to calculate the quarknugget mass distribution, analyzed the results, wrote the paper, prepared the figures, and revised the paper to incorporate the improvements from the other authors and reviewers. C.J.B. analyzed the detectability of the RF emissions from MQNs and provided valuable comments on the paper. C.C. analyzed the expected distribution of signals as a function of time of year, time of day, and sensor position with a realistic MQN velocity distribution, including the motion of Earth through the darkmatter halo towards the star Vega. A.P.V. redteamed (provided critical, independent, review of) the analysis that avoided unjustified or erroneous conclusions. B.A.U. contributed important suggestions on planets to include and discussion of the relevant plasma physics of the magnetopause interaction.
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VanDevender, J.P., Buchenauer, C.J., Cai, C. et al. Radio frequency emissions from darkmattercandidate magnetized quark nuggets interacting with matter. Sci Rep 10, 13756 (2020). https://doi.org/10.1038/s41598020707183
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