New Constraints on Macroscopic Dark Matter Using Radar Meteor Detectors

Meteors can be categorized by how bright they appear [24]. Generally, the larger and heavier a meteoroid is, the more spectacular the event. If it is particularly bright, it is called a fireball. Larger meteoroids may survive the atmosphere, reaching the Earth’s surface, and are called meteorites. Phrases like visual meteors, photographic meteors, and radio/radar meteors indicate the particular detection technique used [24]. Different techniques are most efficient for different ranges of meteoroid masses [24, 34]. Meteoroids of the smallest masses, like those we compare to here, are best detected with radar. The typical mass range detected by meteor radar systems is roughly $10^{-10}$ g to $1$ g, corresponding to a size range of roughly $10^{-7}\textrm{ cm}^{2}\textrm{ to }1\textrm{ cm}^{2}$ (sporadic meteors, defined below, have densities less than $2\textrm{ g cm}^{-3}$) [34, 33, 35], so these meteoroids are small as macroscopic objects but large compared to nuclei.

Meteors are also divided into classes based on their origin. Shower meteors are observed when Earth passes through the trails of dusty debris left by short-period comets [36]. Meteors from a particular shower appear to originate from a single direction called the radiant. Sporadic meteors, or sporadics, are produced by a diffuse, roughly isotropic background of meteoroids of cometary and asteroidal origin [36]. Sporadics greatly outnumber all known shower meteors and are the major contributors to the total mass influx into the atmosphere, with the flux dominated by smaller meteoroids. For the remainder of this paper, we focus on the sporadic meteors, as the DM flux in the solar neighborhood is assumed to be roughly isotropic, like the sporadic flux.

As shown in Fig. 1, radio waves can be reflected by different regions of ionization in the vicinity of the meteoroid body or along its trail in the atmosphere. Head echoes are radio reflections from the immediate, roughly spherical region of ionization surrounding the meteoroid [33]. Head echoes are strongly Doppler-shifted, as the ionized region appears to move at the meteoroid’s velocity. While such ionization lasts for as long as the meteoroid is ablating, the observed duration of the head echoes is restricted by the spatial width of the radar beam. As high-power, narrow-beam radar systems are typically used to detect head echoes, the observed durations range from a few to tens of milliseconds, though meteoroids can ablate for seconds or longer [33]. Trail echoes (or specular meteor echoes) are radio reflections from the quasi-stationary, roughly cylindrical ionization trail left behind as a meteoroid passes through the atmosphere. These echoes may last for up to hundreds of seconds depending on the altitude, the ionization density, atmospheric diffusion rates, electron re-attachment and recombination rates, and other atmospheric effects like wind and turbulence [24, 37, 38]. Trail echoes reflect at or near the transmitted frequency, as the quasi-stationary trail is subject only to wind motion, resulting in small Doppler shifts in the received signal.

A radar system has a transmitting antenna that broadcasts a radio signal into a volume, and a receiving antenna that monitors for echoes. In monostatic radar systems, a single antenna may perform both roles or the two antennas may be located in the same area. In bistatic radar systems, the two antennas may be spatially separated by tens to thousands of kilometers, enabling the radar system to determine the meteor orbit. Novel multistatic systems with multiple antennas for both the transmitter and the receiver are also being developed [39]. We focus on monostatic radar systems.

When a meteoroid passes through the active volume, radio waves from a particular radar system may reflect off the head or trail ionization deposits (if large enough), depending on the geometry of the meteor velocity vector and the wave vector (k-vector) of the radio waves [24]. If the two vectors are aligned (when the meteoroid is directly approaching), radio waves will reflect off only the head ionization. For other orientations, radio waves will also reflect off the trail ionization. Monostatic radar systems are best able to detect trail echoes when the meteoroid velocity is nearly perpendicular to the wave vector (specular reflection). For such trail-echo systems, the zenith angle of the meteor and the elevation angle of the radar signal with respect to the receiver (the complement of the zenith angle) are the same. For head-echo systems, no such specular assumption is made because of the spherical shape of the radar target, and therefore reflections can be obtained at a fixed observation zenith angle for varying arrival zenith angles. Also, for meteoroids of the same size and mass, head echoes are typically weaker than trail echoes when measured by the same radar system, with the power difference being a few tens of decibels (see Ref. [24], page 33).

Radar systems typically do not detect both head and trail echoes. Since head echoes are fainter than trail echoes, the transmitting systems for head-echo radar typically have high peak transmission power and a narrow-beam antenna pattern. Modern high-power, large-aperture radar systems, like the 1 MW (monostatic) Shigaraki Middle and Upper Atmosphere radar (SMUR) in Japan, are able to detect very faint head echoes throughout the meteoric region in the atmosphere [33].

On the other hand, trail-echo radar systems, like the University of Colorado Boulder’s 32 kW (monostatic) Antarctic Meteor (CUAM) radar, have relatively lower transmitting power but a wide-beam, all-sky illumination pattern [40]. These systems are cheaper to run, so the sky can be monitored for meteor trails continuously. Due to the lower operating frequencies, these systems are also more appropriate for measurement of thermospheric winds (the CUAM radar system was built for this purpose) [37, 38]. The receiving antennas of the CUAM radar system also have interferometric capabilities that allow determination of the angle-of-arrival to a few degrees [40]. Because of the lower transmitting power, they probe a different range of meteoroid masses than head-echo radar systems.

We consider elastic scattering of a DM particle with mass $m_{\chi}$ and initial velocity $v_{\chi i}\approx 300$ km/s with a nucleus (typically nitrogen or oxygen) in the atmosphere, with mass $m_{A}$ and velocity $v_{Ai}\approx 0$ in comparison to $v_{\chi}$. After a collision, the DM final velocity $v_{\chi f}$ is

where $\theta_{CM}$ is the recoil angle in the center of momentum frame, while all other quantities are expressed in the lab frame [41]. For $m_{\chi}\gg m_{A}$, the DM final velocity is

After each collision, the DM velocity is reduced by a negligible amount, so that many collisions are required for the DM to deposit its energy. We average over the scattering-angle distribution by setting $\cos{\theta_{CM}}=0$ [20]. Therefore, after $N$ successive collisions with atmospheric nuclei, the DM final velocity is reduced to

This equation allows us to model the evolution of the DM velocity in the atmosphere. Rearranging, we can also get the maximum number of collisions that will leave the DM above some final velocity $v_{\chi f}$:

Since $v_{\chi i}$ appears inside the logarithm, the number of scattering events is insensitive to even large changes in the assumed initial velocity, and hence also to the assumed DM initial velocity distribution.

There is a maximum scattering angle in the lab frame for elastic collisions between two particles when one is initially at rest [41]. For a DM collision with an atmospheric nucleus at rest (with $m_{\chi}\gg m_{A}$), the maximum DM scattering angle is

making the DM trajectory nearly straight.

The next step is to relate $N$ to the path length in the atmosphere, $L$, and the cross section. At constant density,

where $n_{A}$ is the atmospheric number density, $\rho_{A}$ is the atmospheric mass density, $m_{A}$ is the average nuclear mass, and $\sigma_{\chi}$ is the DM scattering cross section. The velocity of a DM particle evolves as

To take into account the varying density of the atmosphere, we use the isothermal atmospheric model (see Appendix A for more details) for the mass density of the atmosphere, $\rho_{A}$, as a function of altitude, $h$:

where $\rho_{0}\simeq 1.3\times 10^{-3}\textrm{ g cm}^{-3}$ is the density at sea level, and $H\simeq 7\textrm{ km}$ is the scale height assuming an atmosphere composed of approximately $80\%$ nitrogen and $20\%$ oxygen (the relative composition of the atmosphere at meteoric altitudes and below is fairly constant).

When the density variation is taken into account, the quantity $L\rho_{A}$ is replaced by the integrated mass column density $X(h)$. For objects traveling downward to an altitude $h$, the mass column density for the isothermal model can be expressed simply as

where $X_{0}=\rho_{0}H\simeq 910\;\text{g}\text{ cm}^{-2}$ is the approximate vertical column density at sea level.

For simplicity, we restrict ourselves to DM particles entering at zenith angles $0\leq\theta\leq 60\degree$. This corresponds to a solid angle of $\Omega=\pi$ sr, half the sky above the horizon. When an object passes through the atmosphere at an angle $\theta$, the column density increases approximately by a factor of $\sec{\theta}$ (ignoring Earth’s curvature):

The final velocity of such a DM particle as a function of altitude $h$ is then

and the kinetic energy at a particular altitude is

We note that this formula, derived for collisions with individual particles, was shown in Ref. [42] to be equivalent to the result in the fluid regime. In the limit of large DM mass, this also agrees with the continuous energy loss formalism discussed in Ref. [43].

Figure 2 shows the region in the DM mass and cross-section plane where DM deposits most of its energy within 70–130 km, the altitude range where most meteors are detected. For large values of the reduced cross section ($\sigma_{\chi}/m_{\chi}$), DM loses nearly all of its energy high in the atmosphere. The reduced cross section at which DM loses 90% of its initial energy above an altitude $h$ is

The rate (technically per unit length, not time) at which a DM particle deposits energy into the atmosphere can be found by differentiating the expression for energy with respect to distance. Let the DM trajectory be from $z=-\infty$ to $z=0$ (corresponding to sea level, $h=0$; see Fig. 1 for an illustrative diagram), so that $\differential{h}/\differential{z}=-\cos{\theta}$. Then, the energy deposited in the atmosphere $(E_{\textrm{atm}})$ per unit DM path length is

Figure 3 shows the DM energy deposition rate in the atmosphere, which has a shape reminiscent of a Bragg peak that describes the energy loss rate of charged particles in matter [44], though the reasons are different. As a DM particle passes through the atmosphere, it encounters an exponentially increasing number of scatterers in its path, increasing the energy deposition rate. Eventually, the DM is slowed enough so that the energy deposition rate peaks at an altitude $h_{\textrm{peak}}$, found by maximizing the rate with respect to the distance travelled, after which it rapidly loses energy before coming nearly to rest,

This roughly corresponds to the altitude at which the total mass of nuclei the DM particle has scattered is comparable to the DM mass (see Ref. [45] for an equivalent derivation of the peak altitude). For a given mass, the maximum energy deposition rate is independent of cross section (see the two peaks in Fig. 3 for $m_{\chi}=10^{-4}$ g),

with the DM particle losing approximately $63\%$ of its initial energy before reaching $h_{\text{peak}}$ because

Once DM has lost nearly all of its energy, its dynamics are controlled by thermal scattering and gravity.

The DM velocities, and hence also the nuclear recoil velocities, are $\approx 300$ km/s on average, well above the speed of sound in air. In principle, this could cause formation of a hydrodynamic shock [45], though we have not explored this. If a shock is formed, we expect that it would increase the ionization rate through atom-atom collisions, increasing the radar detectability.

Following the literature on the diffusion of electrons produced along a meteor trajectory (see, e.g., Refs. [24, 47]), we assume that the recoiling nucleus slows down to thermal velocities of the atmosphere $v_{A}^{\textrm{th}}$, instantaneously creating an ionization deposit with an initial radius of order the atomic atmospheric mean free path $r_{0}\sim\lambda_{A}$. The number of ionized electrons per unit track length is called the electron line density, and is defined as

where $n_{e}$ is the ionization density.

The initial ionization density follows a Gaussian profile, $n_{e}(r,t=0)\propto\exp(-\frac{r^{2}}{r_{0}^{2}})$. Plugging this into Eq. 23 determines the proportionality constant in terms of $q_{e}$:

Because ions are heavier than electrons and therefore are less efficient radiators, we ignore the distribution of ions for radar detection of the ionization deposits.

The initial radius is roughly equal to the mean free path, which varies with altitude,

where $\lambda_{0}\simeq 10^{-6}$ cm is the atomic mean free path at sea level. At meteoric altitudes, $70$ km $\leq h\leq$ $130$ km,

To be conservative in our calculations of the numbers of electron-ion pairs produced by DM, we assume that the initial recoil is of a neutral atom and that all electron-ion pairs are created by subsequent atomic collisions. We have verified with a GEANT4 [48] simulation that if a nitrogen atom at 3 keV (a typical recoil energy) is injected into the atmosphere at meteoric altitudes, nearly all of its energy is converted into ionization of the gas. (We get very similar results with injected ions instead of atoms.) This enables us to relate the electron line density along the DM trajectory to the energy deposition rate as

where $\expectationvalue{I}$ is the average energy required to create an electron-ion pair in the atmosphere, which depends primarily on the first ionization energies of oxygen and nitrogen atoms ($13.62$ eV and $14.53$ eV, respectively [49]).

Figure 4 shows that the range of DM mass and cross section that can produce significant ionization does not depend strongly on the choice of $\expectationvalue{I}$. The experimentally determined value of the average energy required for a moving charged particle to create an electron-ion pair in air is $\simeq 34$ eV, i.e., 2–3 times larger than the first ionization potential of the target atom [50].

The dominant process in the dilution of the initial ionization density is ambipolar atmospheric diffusion, during which electrons and ions in the electrically neutral gas move out and the trail expands radially at thermal velocities of the surrounding environment [24]. The initial radius expands after some time $t$ as

where $D_{a}$ is the atmospheric ambipolar diffusion coefficient, which increases roughly exponentially with altitude. At meteoric altitudes, the diffusion coefficient is approximately [47, 24]

By solving the radial diffusion equation (equivalently, the 2D random-walk problem),

with the initial distribution given by Eq. (24), the ionization number density is a Gaussian function of time and radial distance away from the ionization trail [47, 24],

As above, the prefactor of $q_{e}$ is required to ensure that Eq. (23) is automatically satisfied.

The total number of electrons $N_{e}$ along the DM trajectory is the integral over the trail length,

Through observation and modeling of meteor trails, it is found that radar echoes from dense trails (with high ionization density) decay faster than predicted with only diffusion effects. For a more realistic treatment, one must also consider electron attachment to neutral molecules and atoms, which we take into account. We neglect some smaller corrections: electron-ion recombination (a small effect at meteoric altitudes [24]), turbulent diffusion, and anomalous diffusion due to field irregularities [24, 51].

The atmospheric molecular number density is $n_{A}=\rho_{A}/m_{A}\approx 2.5\times 10^{19}\;e^{-h/H}\textrm{ cm}^{-3}$ (assuming an isothermal atmosphere composed of approximately $80\%$ nitrogen and $20\%$ oxygen) at meteoric altitudes. This is much bigger than the ionization density $n_{e}$ characteristic of meteor echoes ($\lesssim 10^{11}\,\textrm{cm}^{-3}$, though the exact values can vary significantly based on the size and composition of the meteoroids, the entry angle, and the altitude at which the meteor appears) [24]. Therefore, apart from ambipolar diffusion, electron attachment to neutral molecules and atoms dominates the dilution of the ionization density. To be sure, we have confirmed that $n_{e}\ll n_{A}$ in all the cases we consider.

The ionization lifetime, defined as the average time taken for electrons to be captured by neutral molecules and atoms in the atmosphere, is $\tau\equiv(\beta_{e}n_{A})^{-1}$, where $\beta_{e}$ is the electron attachment rate. Taking attachment into account, the line density varies with time as

While the ionization lifetime in the atmosphere at different altitudes is not well known and depends inversely on the ionization density, experimental data show that $1\textrm{ s}\lesssim\tau\lesssim 10\textrm{ s}$ for altitudes $70\text{ km}\lesssim h\lesssim 90\text{ km}$ for an ionization density of $10^{6}$ cm^-3 [52]. Using $\tau=10$ s for $h=90$ km,

With this value of the attachment rate, $\tau\approx 0.6\textrm{ s}$ for $h=70$ km. Our results are not sensitive to the choice of $\beta_{e}$; changing it by even a factor of two has a negligible effect. For meteoric altitudes, we then write the ionization lifetime as a function of altitude,

where the exponential is large. Including this correction to Eq. (31), the ionization density can then be written as

Figure 5 shows the ionization density as a function of radial distance from the DM trajectory for different times after the instantaneous (at $t=0$) formation of the ionization deposit with initial radius $r_{0}$. Note that both underdense echoes (echoes from ionization deposits that radio waves can penetrate) and overdense echoes (from deposits that radio waves cannot penetrate) can be detected, and the horizontal line here is simply for reference. These two regimes are discussed in detail in the next section.

For a monostatic radar system, the RCS of an ionized target can be inferred from the power received by the receiver using the radar equation:

where $\lambda$ is the wavelength of the interrogating radio wave. The transmitter is characterized by its power ($P_{t}$), antenna gain ($G_{t}$), and its range (distance; $R_{t}$). The receiver is characterized similarly, where $P_{r}$ includes thermal noise and the scattered signal. Essentially, the RCS is the ratio of received power to transmitted power, once antenna effects and distance losses have been accounted for.

The term RCS, with dimensionless units of decibels per square meter (dBsm), can also refer to the logarithm of $\sigma_{\textrm{RCS}}$ relative to a square meter, so that $\sigma_{\textrm{RCS}}=10^{4}\textrm{ cm}^{2}$ corresponds to $0$ dBsm:

We neglect polarization effects, i.e., the orientation of the incident electric field with respect to the target. For both radar systems considered here, the transmitter and receiver are circularly polarized in the same plane and are co-located (for SMUR, they are the same antenna). For a head-echo system, we neglect polarization effects because the reflector is approximately spherical, as discussed below. For a trail-echo system, we neglect them because polarization loss will occur only for the most inclined arrival directions, where the flux is minimal.

The dielectric constant $\kappa$ of an ionization deposit with electron number density $n_{e}$ is [24]

where $r_{e}\approx 2.8\times 10^{-13}$ cm is the classical electron radius, $\nu$ is the radio frequency (36.17 MHz for the CUAM radar and 46.5 MHz for the SMUR radar), and

is the plasma frequency. If $\nu>\nu_{p}$, then $\kappa>0$ so that the incident radio wave penetrates the ionization deposit and electrons scatter the radio wave independently. The resulting echo is called an underdense echo. If $\nu<\nu_{p}$, then $\kappa<0$ and the radio wave does not penetrate the ionization column, as secondary collisions between electrons become important. In this case, the electrons in the ionization deposit oscillate collectively at the radio frequency and hence reflect the wave back like a metallic conductor. The resulting echo is called an overdense echo. The transition between underdense and overdense echoes is defined by a critical ionization density, $n_{c}$, through Eq. (40). For the radar systems we consider, $n_{c}\approx 10^{9}\,\textrm{cm}^{-3}$ as both use similar radio wavelengths. Setting the ionization density in Eq. (36) equal to $n_{c}$, we get

where $r_{c}$ is the critical radius of the ionization column within which $\kappa\leq 0$. This radius defines the boundary of the ionization column from where the radio wave is totally reflected for overdense echoes.

The two radio scattering regions (underdense and overdense) are useful because they are each described by simple analytical expressions using relevant quantities. We emphasize that both types of echoes are detectable. The boundary between the two regions is typically described by a transitional value $q_{\textrm{tr}}$ of the electron line density found by setting $r_{c}^{2}=r_{0}^{2}+4D_{a}t=\lambda^{2}/4\pi^{2}$ such that $r_{c}$ bounds a volume large and dense enough to attenuate the incident wave by $1/e$ [24]. Ignoring electron attachment effects ($\tau\to\infty$) — which is valid for low values of $q_{e}$ (see Fig. 7) — we get

which is independent of the incident radio frequency, and is used as a standard benchmark value for the over/under dense transition, even though in practice this transition is frequency dependent (see Ref. [24], page 215).

For overdense echoes, we can define an effective ionization radius $r_{p}$ (also called the plasma radius), taking the maximum of the critical radius $r_{c}$ defined above. Inverting Eq. (41), we get the critical radius as a function of time,

Figure 6 shows that the critical radius grows to a maximum value $r_{c}^{\textrm{max}}\equiv r_{p}$ and is

where $W(\eta)$ is the Lambert-W or product-log function defined by $\eta=W(\eta)e^{W(\eta)}$.

Note that if electron attachment effects are ignored, the radius of the ionization deposit, found in a similar way using Eq. (31), equivalently by taking the limit $\tau\to\infty$ of Eq. (44) above, is simplified to

Figure 7 shows that the size of the radius is suppressed for large electron line densities when attachment effects are considered along with ambipolar diffusion.

Figure 8 compares the effective radius for overdense scattering with and without attachment as a function of altitude for a particular DM candidate. While we plot the radius for very large values of the line density to illustrate the effect of attachment clearly, it is to be noted that very large values of $q_{e}$ ($\gg 10^{16}\textrm{ cm}^{-1}$) are not physical [24], as the ionization density must be less than the atmospheric number density. Note that, although it is not written as a function of altitude, $q_{e}$ implicitly depends on altitude through its dependence on both atmospheric density and the DM velocity. This produces the altitude dependence in Fig. 8.

Meteor head echoes are characterized by radio reflections off spherical ionization distributions, surrounding the meteoroid and co-moving with it [54, 55, 56]. Therefore, we describe the ionization deposit in the immediate neighborhood of a DM particle as a spherical Gaussian distribution in Eq. (36) centered on the particle. Although the actual shape of the ionization deposit may be more ellipsoidal (with the electron density falling off more quickly behind the parent particle), the assumption of a spherical deposit with a Gaussian profile (for the number density of electrons) is good, as demonstrated by simulations and empirical measurements [57, 58]. For initial velocities $v_{\chi i}\gtrsim 70\textrm{ km/s}$, the deviation from spherical geometry becomes more pronounced, but this has a small effect on the radar cross section.

The head ionization is a near perfect electrically conducting sphere, with the RCS characterized by the size of the effective radius of the sphere $r_{p}$ relative to the radio wavelength $\lambda$ [58]. If $r_{p}\gg\lambda$ (optical scattering), the RCS approaches a constant value equal to the cross sectional area of the head plasma,

If $r_{p}\ll\lambda$ (Rayleigh scattering), the RCS is

If $r_{p}\sim\lambda$ (Mie scattering), the RCS oscillates about the average value $\pi r_{p}^{2}$. For simplicity, we take the average value as the RCS for Mie scattering.

In underdense trails, electrons scatter independently but coherently. The scattering cross section of a free electron for monostatic radar systems is $\sigma_{e}=4\pi r_{e}^{2}$ [24]. The total power received at the receiving antenna can be calculated by adding the contributions to the electric field vector from all the electrons in a line element of the trail where they scatter in phase [24].

When the diameter of the cylindrical trail is small in comparison to the radio wavelength $\lambda$, the total quasi-instantaneous power received from all electrons from some finite section of the trail can be expressed as [24]

where $R_{0}$ is the minimum range along the trail, and electron attachment effects are ignored (valid for underdense trails with low ionization densities) [24]. The majority of the received power comes from the first Fresnel zone around the point of closest approach to the radar station [24]. The power from additional Fresnel zones alternate in phase and largely cancel out. The received power at the moment of ionization formation ($t=0$) is scaled due to the finite initial width of the column. Taking ambipolar diffusion into account, the echo power decays as

Note that as shown in Fig. 7, electron attachment effects are negligible for underdense echoes ($q_{e}\leq q_{\textrm{tr}}$).

The maximum power registered at the receiver is $P_{r}=P_{r}(t=0)$, which occurs when the meteor leaves the first Fresnel zone. Since the range varies slowly near $R_{0}$ — the point of closest approach to the radar station — and the length of the section of trail contributing to the majority of the received power is small compared to the range, $R\simeq R_{0}$ [24]. For a meteor entering the atmosphere at zenith angle $\theta\neq 0$ and altitude $h$, $R_{0}=h/\sin{\theta}$. Replacing Eq. (48) in Eq. (37), the maximum underdense trail-echo RCS (for backscatter) is then (for $q_{e}\leq q_{\textrm{tr}}$),

The ionization trail through the atmosphere would be perfectly cylindrical if the DM velocity were infinite (assuming the trail has already expanded to the initial radius $r_{0}$) and if the atmospheric density was uniform. Since DM velocity is finite but larger than the radial diffusion velocities of electrons ($v_{A}^{\textrm{th}}\sim 0.5\textrm{ km/s}$ [24]) by 2–3 orders of magnitude (the typical DM velocity is $v_{\chi}\sim 300\textrm{ km/s}$), the trail is more conical than cylindrical. Because both the initial radius $r_{0}$ and the diffusion coefficient $D_{a}$ exponentially decrease with altitude, the conical ionization trail but has an exponential taper [24]. Note that these details are also true for ionization trails produced by meteors, which have typical velocities of 70–130 km/s.

For overdense trails, the ionization is a near perfect electrically conducting cylinder of radius $(r_{p}\gg\lambda)$ [24], in which case the RCS is

for incident spherical waves from a source at a perpendicular distance $R_{0}$ [24]. The power delivered by scattered waves to the receiver is then given by Eq. (37).

The equations for the radar cross section $\sigma_{RCS}$ depend on the DM mass and cross section through the electron line density and the plasma radius, given in Eqs. (27) and (44), respectively. Using these, we convert the DM velocity distribution for a particular DM mass and cross section into an RCS distribution. Note that both the altitude and the zenith angle must be specified to compute the RCS for both head echoes and trail echoes; while the equations for trail echoes depend explicitly on the altitude and zenith angle, those for head echoes depend indirectly on these quantities via the energy deposition rate given in Eq. (17).

For the DM velocity distribution at the top of the atmosphere $f(v_{\chi i})$, we use the Standard Halo Model as parameterized in Ref. [59], with a velocity dispersion of $270\textrm{ km/s}$, and accounting for the motion of the Sun around the Galaxy. Meteor radars (like the SMUR and CUAM radars [33, 40]) are typically tuned to scan for meteors within the altitude range of 70–130 km (where most meteors are detected), with entry velocities of 11–70 km/s at 130 km altitude [60]. To account for the radar sensitivity to altitude and velocity, for each DM mass and cross section we use Eq. (14) to calculate the maximum entry velocity at the top of the atmosphere such that the DM slows down to meteoric velocities at 130 km. This restricts the velocity distribution at the top of the atmosphere for each DM candidate (for details, see Appendix B).

For each DM mass and cross section, we then convert the restricted velocity distribution to a detected RCS (defined by the maximum RCS produced by DM within the meteoric altitude range) distribution for DM that enters the atmosphere at a particular zenith angle. We consider only incoming DM particles with zenith angles $\theta\leq 60\degree$, where the curvature of the Earth can be neglected and the atmospheric mass column density in Eq. (11) is minimally corrected. Finally, by summing the detected RCS distributions over allowed zenith angles, we get the RCS spectrum for each DM candidate.

Figure 9 shows example spectra for different DM masses and cross sections (integrated over zenith angles), compared to the SMUR and CUAM data [33, 40]. For head echoes detectable by the SMUR radar system $(\lambda\sim 645\textrm{ cm})$, the RCS varies from $-50\textrm{ to }30\textrm{ dBsm}$ [33]. For trail echoes detectable by the CUAM radar system $(\lambda\sim 829\textrm{ cm})$ [40], we assume that the RCS varies from $20\textrm{ to }70\textrm{ dBsm}$ based on our calculations using raw trail-echo count data from a week of observations by the CUAM radar system, which involves some uncertainties that are small relative to our precision goals.

To set our limits, we conservatively allow that all observed echoes could be DM signals, even though they are likely all backgrounds due to meteors. We rule out a DM mass and cross section if the total number of DM events in any unit-RCS bin is significantly larger than the observed number of meteors in that bin. The rest of this subsection describes the details of setting this limit.

For trail echoes, we restrict the data to meteors with $\theta\leq 60\degree$, matching the restriction on DM mentioned in Sec. VI.1 (relaxing the restriction would increase the fluxes of both DM and meteors by a factor of $\lesssim 2$ and would have little effect on the signal-to-background ratio). For head echoes, we are unable to make this cut, as information on the zenith angle is unavailable, but we still only consider DM with $\theta\leq 60\degree$ (making our results for head echoes conservative). Our analysis is thus only sensitive to DM arriving from a solid angle of $\Omega=\pi$ sr, or 1/4 of a full sphere; integrating the incoming flux over $\cos\theta\,d\cos\theta$ also to account for the component of the flux perpendicular to the surface gives 3/16 of a full sphere. As a result, the flux of DM (per unit velocity) our analysis is sensitive to is

where $f(v_{\chi})$ is the fraction of DM particles at velocity $v_{\chi}$, and $\rho_{\chi}\approx 0.3\ \textrm{GeV}\textrm{ cm}^{-3}$ is the DM mass density at Earth’s position [1]. For $m_{\chi}=1\ \textrm{g}$ and $v_{\chi}=300\ \textrm{km/s}$, $\derivative*{\phi_{\chi}}{v_{\chi}}\simeq 3\times 10^{-7}\ \textrm{km}^{-2}\ \textrm{hr}^{-1}\textrm{(km/s)}^{-1}$.

Because the energy deposition in the atmosphere depends on the DM velocity $v_{\chi}$, we can convert the velocity spectrum of Eq. (52) into an RCS spectrum, using the equations of the previous two sections. $\phi_{\chi}$ then represents the total flux in a given RCS range. We assume that the number of detected DM particles per unit RCS follows a Poisson distribution, for which the PDF is given by

where $\mu_{\chi}=\phi_{\chi}\times A_{\textrm{det}}\times T_{\textrm{obs}}$ is the expected number of radar echoes produced by DM per unit RCS for a radar with effective detector area $A_{\textrm{det}}$ and total observation duration $T_{\textrm{obs}}$. We exclude a given DM mass and cross section by conservatively requiring that the DM spectrum never be higher than the observed meteor spectrum at 95% CL. That is, DM is ruled out if for any RCS, the CDF, $P(n\leq N_{m})\leq 0.05$, where $N_{m}$ is the observed meteor count per unit RCS.

The detector area, also called the equivalent radar collection area, depends on the radar antenna gain pattern and can be expressed as a function of the RCS, $A_{\textrm{det}}=A_{\textrm{det}}(\textrm{RCS})$ [33]. For the SMUR radar system, $A_{\textrm{det}}$ ranges from $1$ km² for RCS of $-50$ dBsm to $10^{3}$ km² for $30$ dBsm [33]; for the head-echo data, we use $T_{\textrm{obs}}=33$ hr [33]. For the CUAM radar system, we calculated $A_{\textrm{det}}\simeq 3\times 10^{4}$ km² over the entire RCS range due to the all-sky nature of the antenna beam; for the trail-echo data we use, $T_{\textrm{obs}}=118$ hr.

Figure 10 shows the constraints for trail echoes detectable by the CUAM radar. In the left panel, we show the estimated boundaries for the constraints using the minimum detectable RCS (lower edge), the minimum energy for DM to survive to below 130 km altitude (left edge), the largest mass that still has a large enough flux for potential detectable events (right edge), and the maximum reduced cross section $(\sigma_{\chi}/m_{\chi})$ for which DM loses nearly all of its energy above 130 km altitude (ceiling). In the right panel, we overlay on top of the estimated boundaries the actual constraints for DM using trail-echo data, and a few descriptive lines that explain the boundaries of the exclusion region. We show the corresponding plot for head echoes detectable by the SMUR radar in App. C.

Figure 11 shows the radar-derived exclusion regions in the plane of DM mass and cross section. To calculate the regions, we find the detected RCS spectrum for each DM candidate as shown in Fig. 9 and plot the DM mass and cross section in the plane if the spectrum exceeds the meteor data. For current constraints, the velocity distribution for each candidate is restricted so that DM velocity is within the meteoric velocity range at 130 km altitude, as described in Sec. VI.1 and App. B.

For the projected sensitivities, we use the full initial-velocity range of $\sim$ 11–800 km/s, where $11\textrm{ km/s}$ is the escape velocity from Earth and $\sim 800\textrm{ km/s}$ is approximately the sum of the escape velocity from the Galaxy $(\sim 550\textrm{ km/s})$ and the velocity of the Sun $(\sim 220\textrm{ km/s})$. For meteor velocities $v>70\textrm{ km/s}$, we assume that $N_{m}\approx 0$ because of the extremely low flux of such meteors, but defer to future experiments to conclusively test this. (More details about the meteor and DM velocity distributions are given in Appendix B.) If future experiments are sensitive to very fast-moving meteors, our limits could improve by orders of magnitude in both mass and cross section, even with the same exposure.

We also show prior DM constraints. The light gray region is excluded by observations of the Milky Way satellite population [61] and cooling of Galactic gas clouds [62], while the dark gray region is excluded by observations of long-lived white dwarfs ([63], but see also Ref. [64], a reanalysis that produces a substantially smaller region by using a different density profile and more conservative treatment of a thermonuclear runaway). Relatively large cross sections have been probed by experiments sensitive to interactions in the lower atmosphere: particle detectors on a satellite, “Skylab” [15], and searches for optical flashes with the Desert Fireball Network, “Fireballs” [23]. Relatively small cross sections have been probed by a variety of shallow and deep underground dark-matter or repurposed experiments: “Chicago” [16], “DAMA” [65, 62], “DEAP-3600” [17], “Ohya” [15], and “Mica” [11]. The “Humans” region is constrained by null observations of unique human injuries/death by DM [66]. Future sensitivities (not shown) to ultraheavy DM have been derived based on collisions with stars [67], signals in IceCube [68], and tracks in quartz [69]. Ref. [70] also places constraints from collisions with stars, but at masses above the range we show.

Though our limits overlap with cosmological constraints, our results are complementary in several ways: they are independent of cosmological models or assumptions; they explicitly focus on composite DM, which is not typically the focus of cosmological studies; and they have the advantage of differential sensitivity to the DM mass and cross section. In addition, the sensitivity of our approach can be significantly improved.