Radar and Optical Observations and Physical Modeling of Binary Near-Earth Asteroid 2018 EB

Radar techniques involve the use of radio waves to obtain information about distant objects. For the case of planetary radar, we transmit a circularly polarized wave for the duration of the round-trip time between the observer and the target, and then we switch to receive the echo for the same duration (less the time required to switch from transmit to receive). Each transmit and receive cycle is called a "run." The reflected wave returns in the same (SC) and opposite (OC) senses of circular polarization as the transmitted signal. The data collected during one run are processed with discrete Fourier transforms of a certain length that define the Doppler frequency resolution. If we use the entire receive duration of a run to resolve the data, then we obtain a single measurement, or "look," per run. Alternatively, we can process the data more coarsely and have multiple looks per run. Data from multiple runs are combined into incoherent sums to reduce the fractional noise fluctuation by $\sqrt{{\sum }_{\mathrm{runs}}{N}_{\mathrm{looks}}}$. More details on observational and data reduction techniques for radar can be found in Magri et al. (2007).

We used two observing setups: Doppler-only (continuous wave, CW) and binary phase-coded (BPC) waveforms. Our processed data have the form of echo power spectra and delay-Doppler images. The received echo is characterized by its bandwidth in Doppler frequency and, for the case of BPC waveforms, spread in time delay (range). The Doppler broadening of the echo is a function of the asteroid's rotation period, diameter, and spin axis orientation with respect to the radar line of sight (Ostro 1993):

$$\begin{eqnarray}&&B=\displaystyle \frac{4\pi D}{\lambda P}\cos (\delta ),\end{eqnarray} \tag{ 1 }$$

where B is the bandwidth, D is the object's maximum breadth in the plane of the sky perpendicular to the spin vector, λ is the radar wavelength, P is the rotation period, and δ is the subradar latitude.

We observed 2018 EB at Goldstone (8560 MHz, 3.5 cm) on 1 day in April and at Arecibo (2380 MHz, 12.6 cm) and Goldstone on 5 days in October of 2018. Table 1 summarizes the observations. We started observing on April 7 at Goldstone, 5 weeks after the object was discovered by NEOWISE and 3 days after its closest approach at 0.027 au. Arecibo had equipment issues at that time, and observations there were not possible. We did not know any physical properties except its diameter, ∼240 m, based on a preliminary estimate by the NEOWISE team (later published in Masiero et al. 2020). The asteroid was expected to be a relatively weak target at Goldstone with signal-to-noise ratios (S/Ns) sufficient for obtaining echo power spectra and ranging observations but not high-resolution delay-Doppler imaging. We observed 2018 EB again in October at ∼0.40–0.45 au from Earth. The asteroid had traversed ∼100° across the sky since April. Arecibo was back online at this time, and the S/Ns were sufficient for high-resolution imaging.

Figure 1 shows the echo power spectrum and delay-Doppler images obtained on the first (and only) observing day in April at Goldstone. There is a narrow spike with a bandwidth of ∼1 Hz in the echo power spectrum when processed with 0.125 Hz and 0.25 Hz resolutions. There is also a radar-bright pixel in front of the primary echo in the 0.25 μs (37.5 m) and 0.125 μs (18.75 m) radar images. The spike and the bright pixel are located at negative frequencies relative to the primary echo. The bright pixel shifted slightly toward the primary's echo between the times when the images in Figure 1 were obtained, which indicates an object that moved away from Earth with a negative Doppler shift. These are the classic signatures of a binary system (Margot et al. 2002, 2015).

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We obtained significantly more data at Arecibo and Goldstone in October that show more orbital motion by the satellite and rotation by the primary. Data obtained at Arecibo achieved significantly stronger S/Ns and finer range resolutions than data obtained at Goldstone in April. Figure 2 shows all the echo power spectra from both observatories, and Figure 3 shows delay-Doppler images obtained at Arecibo in October. We used these data to assess the bandwidth of the primary on each day. We estimated bandwidths visually by measuring the width of the OC echoes where the signal was at least three standard deviations (3σ) above the noise level (Table 4). Such bandwidth estimates are less affected by noise than if they were measured at 0σ and are conservative lower bounds for the same measured bandwidth. All bandwidth measurements from Goldstone were converted from the X band to the S band by scaling them by the ratio of the observing frequencies: 2380 MHz/8560 MHz = 0.278. The average bandwidth was 2.0 ± 0.2 Hz in the Arecibo data. The bandwidth increased between April 7 and October 5, and both Goldstone and Arecibo data showed an increase from October 5 to 7. The bandwidth could change due to an elongated shape viewed at different orientations, a change in subradar latitude, or both.

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The S/Ns at Arecibo were ∼60 per run and strong enough for 0.1 μs (15 m) resolution imaging. Figure 3 shows daily collages of Arecibo radar images of the primary. Each panel contains about ∼7 minutes of data integration. At this resolution, 2018 EB appears to be a relatively featureless object. Visual inspection does not show any obvious surface features that could be used to estimate the rotation period. There were some hints of slight asymmetry in the shape of the leading edge in the first and sixth frames on October 5 and the first three frames on October 6, but the time delay (range) resolution, the extent of the imaging, and the S/Ns were not sufficient to reveal a clear trend. The S/Ns deteriorated toward the end of each track because the object was moving toward the edge of the Arecibo dish, where the system was less sensitive.

We measured the size of the primary based on the available delay-Doppler images. We refer to the number of continuous, bright pixels in the range as the "visible extent." We measured the visible extents by counting contiguous pixels that are at least 3σ above the noise level, and we summarize the results in Table 4. The average visible extent for the October radar images was ∼160 m. The April 7 0.25 μs image suggests a visible extent of at least ∼110 m.

We used the bandwidths and visible extents from Table 4 to obtain a zeroth-order period estimate for the primary based on Equation (1). We make the simplistic assumption that the visible extent is a proxy for the radius, given that this object appears roughly spheroidal in the images. We obtained periods in the realm of 4 hr for an equatorial line of sight. If the average subradar latitude was ∼45°, the period would be ∼30% shorter (∼3 hr).

Table 2 summarizes the photometric observations used in this analysis, and the lightcurves are shown in Figure 4. 2018 EB was a challenging target for photometry because it only became brighter than 20th magnitude during the close approaches in 2018 April and October, when it was moving rapidly across the sky. Joe Pollock (personal communication) was the first to obtain a lightcurve on April 5 from the PROMPT-3 telescope in Chile. The lightcurve was almost flat during ∼2 hr of observations, implying either an object with low shape elongation, a nearly pole-on view, or a period longer than a few hours. Warner & Stephens (2019a) published two more lightcurves obtained on April 8 and 9 from the Palmer Divide Station. The lightcurves had on-sky positions 38° and 46° from the initial photometry on April 5. The April 8 lightcurve was only about 1 hr long and obtained through a band of clouds. The photometry showed a significant scatter with no recognizable trend. The lightcurve from April 9 covered 6 hr and appeared to have ∼0.4 mag amplitude. The last lightcurve was obtained on October 18 from the 1 m aperture telescope at the LCO site in Chile. The object was very faint, ∼20 mag, and <40° from the waxing gibbous Moon. The lightcurve was slightly under 4 hr long and showed less than 0.2 mag amplitude. Borisov et al. (2021) also report a lightcurve of 2018 EB using <2 hr of data on October 6. This lightcurve coincided in time with radar observations from Arecibo and Goldstone and looks remarkably similar to our April 9 lightcurve. Because of the short time span and overlap with existing observations, we did not use these data in our analysis.

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We remeasured the April 9 lightcurve based on raw imaging data from Warner & Stephens (2019a) because the asteroid was moving very rapidly against the stellar background, and special care was needed when selecting the stars for relative photometry. We used seven sets of 10 background stars with consecutive fields being linked by three to four stars that were common to both fields. For the final 11 frames of the April 9 lightcurve, only five background stars were used due to a lack of suitable stars. Consequently, the data quality deteriorated significantly toward the end of the observing run.

2018 EB already had several rotation periods published by the time we started this analysis. Warner & Stephens (2019a) reported periods of 6.32 hr and 3.16 hr based on a preliminary photometry reduction. However, our radar data do not support these two values. A radar-estimated diameter of ∼320 m in combination with a 6.32 hr period would produce a ∼30% narrower Doppler frequency bandwidth than reported in Table 4 (see Equation (1)). Alternatively, a ∼320 m-sized object in combination with a 3.16 hr period would produce ∼30% wider bandwidths than reported in Table 4. Our radar line of sight would have to be ∼45° on all observing days in order to produce the observed ∼2 Hz bandwidths. A similar argument applies to the even more rapid period of 2.600 ± 0.437 hr reported in Borisov et al. (2021).

All lightcurves used in this analysis are shown in Figure 4, and their raw data are listed in Table 3. We initially attempted to fit the April 9 lightcurve with the second-order Fourier series in Matlab, and we obtained a range of rotation periods from 4.1 to 4.8 hr. This was consistent with the radar-derived period based on the bandwidth of the primary (Section 2.1). These preliminary period estimates were useful cross-checks for the period obtained from the shape modeling analysis described in Section 4.

We obtained broadband colors of 2018 EB with the Sloan Digital Sky Survey g', i' and z' filters on 2018 October 19 using the SOAR Telescope Goodman Spectrograph and Imager (Table 2). Conditions were mostly clear, with moderate winds (30 km hr⁻¹) and below average seeing (around 2''). All exposures were 15 s with the telescope tracked at sidereal rates. In the $g^{\prime} ,i^{\prime}$, and $z^{\prime}$ filters, a total of 14, 20, and 26 images were taken. The exposures were taken in alternating sequences of $g^{\prime} -i^{\prime} -z^{\prime}$. Images were processed using standard flat-field and bias-correction techniques. The photometry on each image was measured using the Photometry Pipeline (Mommert 2017) and a fixed aperture radius of 6 pixels (18) for the $g^{\prime}$- and $i^{\prime}$-band images and 5 pixels (15) for the $z^{\prime}$ band. Photometric calibration on each frame was achieved by referencing to ∼five on-chip field stars with approximately solar-like colors from the SkyMapper DR1 catalog (Wolf et al. 2018).

The images for photometry were taken over a span of about 30 minutes, a nonnegligible fraction of the estimated ∼4 hr rotation period for the primary but not long enough to improve period determination. We use images with the highest signal-to-noise, taken in the $i^{\prime}$ band, to account for lightcurve variability when computing colors from nonsimultaneous images. The full sequence of 20 $i^{\prime}$-band images was linearly fit, showing a small ∼0.05 mag change over about 30 minutes. This functional fit to the $i^{\prime}$-band data enabled computation of $g^{\prime} -i^{\prime}$ and $i^{\prime} -z^{\prime}$ colors at the times corresponding to each individual $g^{\prime}$- and $z^{\prime}$-band image. The final colors, $g^{\prime} -i^{\prime}$ = 0.72 ± 0.05 and $i^{\prime} -z^{\prime}$ = 0.04 ± 0.10, thus represent the weighted mean of all individual measurements with uncertainties presented as the standard deviation of all values for a given color. These reflectance values were then compared to each classification in the Bus taxonomic system (Bus & Binzel 2002). The formal best fit is the Xk-type (rms = 0.02). The envelopes for other less likely taxonomic types are shown in Figure 5, including S-type (rms = 0.07) and C-type (rms = 0.09).

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The satellite was first detected during relatively coarse-resolution delay-Doppler imaging in April at Goldstone (Figure 1). In October, Arecibo achieved S/Ns 12 times stronger than those at Goldstone in April, and the echo from the satellite was clearly visible. Figure 6 shows the daily sums that display the motion of the satellite in its orbit about the primary. The maximum projected separation in range between the satellite and the primary centers of mass increased from ∼300 m to ∼390 m between October 6 and 7 and is consistent with a radar line of sight that was approaching the mutual orbit plane. The satellite was also detected in the echo power spectra and imaging data at Goldstone from October 5 to 9. The S/Ns at Goldstone were sufficient for detection of the satellite, but we did not obtain any detailed imaging. The satellite went into a radar eclipse downrange from the primary on October 7 while we observed from Goldstone. Observations at Arecibo ended just as the satellite was approaching its maximum separation behind the primary, but Goldstone continued to observe for another 3 hr. We detected the satellite echo in CW observations until about 10:30 UTC (Figure 7), but the echo disappeared until ∼12:00 UTC from 75 m imaging data (Figure 8). We initially attributed this to low S/Ns, but we later realized that the more likely explanation was a radar eclipse.

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Figure 9 shows echoes from the satellite obtained at Arecibo on October 5–7. We processed the data so that they have one look per run and thus the finest possible Doppler frequency resolution of 0.029 Hz on October 5 and 0.031 Hz resolution on October 6–7. We summed 20 runs, spanning ∼28 minutes, to create a single delay-Doppler image. We used a second-order polynomial, p₀ + p₁ t + p₂ t², where t is time and p_i are the coefficients, to correct for the satellite's motion in range and Doppler frequency when summing the data from different runs. In this shift-and-stack technique, the polynomial accounts for both the drift due to orbital motion and the small drift due to the imperfect ephemeris of the system. Naidu et al. (2020) used ${p}_{0}+{p}_{1}{\sin }({p}_{2}t+{p}_{3})$ to correct for orbital motion of the satellite of (65803) Didymos, but given the short data arcs and the additional drift due to ephemeris corrections, we found it sufficient to use a polynomial.

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We estimated the visible extent and Doppler bandwidth for the satellite by counting contiguous pixels above 3σ signal strength (Table 4). The uncertainties in Table 4 include our considerations of the delay-Doppler imaging resolution, image quality, and number of looks. The average visible extent on October 7 (∼66 m) appears to be slightly less than observed on October 5 and 6 (∼80 m), hinting that the object could be somewhat elongated. We also searched for bandwidth changes in the individual images on October 5 and 7 because there was substantial orbital motion between the first and last images. However, the S/Ns dropped significantly from the first to the last frame on each day as 2018 EB was setting at Arecibo. We can report only tentative evidence that the bandwidth narrowed by one Doppler frequency bin on October 7, but the S/Ns in the frames are weak, so a search for bandwidth changes in the satellite was inconclusive.

If we double the mean visible extent for the satellite from Table 4, then we obtain a zeroth-order diameter estimate of ∼150 m. It is unlikely that the volume of the satellite is much larger than implied by this diameter, but the volume could be smaller if the shape is flattened, like the satellites of (66391) Moshup (1999 KW4) (Ostro et al. 2006), (153591) 2001 SN263 (Becker et al. 2015), and Didymos (Daly et al. 2023). We thus assign an uncertainty of ${150}_{-45}^{+15}$ m on the equivalent diameter for the satellite. These error bars account for the data resolution and S/Ns and are necessarily subjective. More precise estimates would require shape modeling, but the images do not have sufficient rotational coverage, S/Ns, or resolution. A satellite with a diameter of ∼150 m is relatively large for a primary ∼320 m in diameter, but systems with D_s /D_p ∼ 0.5 have been observed before: (66063) 1998 RO1 (Pravec et al. 2016), (85938) 1999 DJ4 (Pravec et al. 2006), (385186) 1994 AW1 (Richardson et al. 2015), (494658) 2000 UG11 (Pravec et al. 2006), 2003 SS84 (Nolan et al. 2003a), and (410777) 2009 FD (Naidu et al. 2015b). We used Equation (1), the mean bandwidths, and the estimated diameters from Table 4 in order to constrain the rotation period of the satellite. If we assume that the subradar latitudes were close to equatorial, then this simplified approach suggests that the satellite has a rotation period of 17 ± 4 hr.

For radar observations, the satellite's orbit is typically determined using measurements of range-Doppler separations between the satellite's center of mass (COM) and that of the primary over time (Margot et al. 2002; Ostro et al. 2006; Shepard et al. 2006; Fang et al. 2011). The COM for both objects generally lies near the middle of the trailing edge of their respective radar echoes.

The COMs for the primary and the satellite can be obtained from shape models, and they can also be estimated visually. We did not have a shape model for the satellite due to its small size and unresolved radar images, so we estimated the location of its COM visually. The COM of the primary was estimated via shape model. This shape model was developed at the initial stage of this analysis and used for the orbit fit. However, it became apparent that the orbital fitting part provided a good constraint on the orbital pole and presumably the pole of the primary, which led to reevaluation and another iteration of the entire shape modeling process. We start by describing the orbital fit procedure first, followed by the shape modeling with an understanding that we went through "shape model–orbit fit–shape model" iterations. The differences in the COM estimates for the primary between the starting and final shape models were within the errors assigned to the range and Doppler separations between the satellite's and the primary's COMs listed in Table 5. We assigned uncertainties based on data resolution, S/Ns, and our subjective sense about data quality. The most precise measurements, with errors of 30 m in range and 0.1 Hz in Doppler frequency, were obtained at Arecibo on October 5–7. The data obtained at Goldstone on October 8–9 had significantly larger uncertainties due to the lower S/Ns and coarser resolutions but were still valuable because they sampled different orbital geometries as the asteroid traversed the sky. In the case of 2018 EB, we also observed a radar eclipse, and the timing of this event can also be utilized in orbit fitting.

The orbital fit was a two-step process: the first step initialized the Keplerian orbital elements, and the second step optimized these elements via the downhill simplex method (Press et al. 1992). We used six Keplerian elements to fit the data: semimajor axis a, eccentricity e, inclination i, mean anomaly at the epoch M, argument of periapsis ω, and longitude of the ascending node Ω. The semimajor axis was initialized between 0.40 and 0.55 km in steps of 0.01 km. This range was chosen because of the measured separations in Table 5. The starting orbits were circular, e = 0. The inclination was initialized from 0° to 180° in steps of 5° with respect to the International Celestial Reference Frame (ICRF). The mean anomaly and the node were initialized from 0° to 360° in steps of 10°. The system mass was initialized for a sphere with a radius of 0.15 km and a density between 0.5 and 2.2 g cm⁻³ in steps of 0.1 g cm⁻³. We used an upper bound of 2.2 g cm⁻³ because the NEA binary and triple systems for which densities are available, namely, (66391) Moshup (Ostro et al. 2006), (136617) 1994 CC (Brozović et al. 2011), (153591) 2001 SN263 (Becker et al. 2015), (185851) 2000 DP107 (Naidu et al. 2015a), and (276049) 2002 CE26 (Shepard et al. 2006), all have densities less than 2 g cm⁻³.

Each starting set of orbital elements was converted into a state vector in the Cartesian coordinates (x₀, y₀, z₀, v_x0, v_y0, v_z0) with the COM of the primary at the origin and referenced to the ICRF. The state vector was propagated in time via conic theory (Goodyear 1965) and converted into radar observables, time delay, and Doppler frequency separations from the primary. We also calculated separations between the satellite and the primary in the plane of the sky in order to evaluate conditions for the radar eclipse on October 7. We defined the plane-of-sky separation d as

$$\begin{eqnarray}&&d=\displaystyle \frac{\left|{\vec{\rho }}_{1}\times {\vec{\rho }}_{2}\right|}{{\rho }_{1}{\rho }_{2}},\end{eqnarray} \tag{ 2 }$$

where ${\vec{\rho }}_{1}$ is the position vector from the observer to the primary's center and ${\vec{\rho }}_{2}$ is the position vector from the observer to the satellite's center (Aksnes 1974). The conditions for radar eclipse occur if d < R₁ + R₂, where R₁ and R₂ are radii for the primary and the satellite (assumed 150 m and 75 m), and if $\left|{\vec{\rho }}_{1}\right|\lt \left|{\vec{\rho }}_{2}\right|$. We calculated the separation value d for the start and stop times of the eclipse, 10:30 UTC and 12:00 UTC, and kept all combinations of orbital elements that resulted in d within 50% of R₁ + R₂. This corresponds to 30–40 minutes uncertainty on the start–stop times, which is a conservative 3σ estimate given the weak data in Figure 8. The "eclipse score" for the fit was calculated as (d − (R₁ + R₂))/(R₁ + R₂). For the "perfect" correspondence to the observed start–stop times, we would get zero for both ingress and egress.

A 7D simplex search was used to adjust the initial state vector of the satellite and the nominal system mass, aiming to find a solution with the lowest rms of the residuals. The final state vector was converted back into Keplerian elements. We also converted pole directions from R.A. and decl. to the ecliptic longitude and latitude. We kept solutions with e < 0.2, 0.4 < a < 0.55 km, and an rms of the residuals ≤1. Figures 10 and 11 show the final distribution of the orbital parameters for 950 candidates. The possible orbit poles occupy a narrow swath across the sky. The prograde poles have a noticeable clumping of candidates around the longitude and latitude (90°, 45°). We further constrained the candidate region by selecting for the solutions that have an overall rms of the residuals of ≤0.7 and an eclipse score of ≤0.2. The final 116 candidates marked in orange in Figure 10 are all in the prograde direction. Furthermore, 87% are in the region of 80°–100° in longitude and 40°–50° in latitude. Table 5 shows delay and Doppler residuals for the best solution: orbit pole at λ = 93° and β = 48°, a = 0.50 km, e = 0.15, system mass M = 2.03 × 10¹⁰ kg, and P = 16.85 hr. We used the bounds set by the final 116 candidates to write a more general orbit solution: $a={0.50}_{-0.01}^{+0.04}$ km, e = 0.15 ± 0.04, $M={2.03}_{-0.08}^{+0.52}\times {10}^{10}$ kg, and $P={16.85}_{-0.26}^{+0.33}$ hr. We note that the nominal orbit has a significant eccentricity, like the orbits of (164121) 2003 YT1 (Nolan et al. 2004), (311066) 2004 DC (Taylor et al. 2008), and (153958) 2002 AM31 (Taylor et al. 2013). No solutions were found for e ≤ 0.05, and the residuals were significantly worse for the solutions where we forced e ∼ 0.1. Eccentricity introduces a possibility of apsidal precession for the oblique shape of the primary. However, we cannot determine the precession rate because the radar data arc is too short (October 5–9), and the data are too coarse. With the current uncertainty on the orbit period, we do not know how many full revolutions have occurred between April and October.

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Our modeling data set contained weighted contributions from lightcurves, echo power spectra, and delay-Doppler images. The radar data consisted of 23 Goldstone echo power spectra, four Arecibo echo power spectra, and 48 delay-Doppler images from Arecibo. We removed the satellite echoes from the echo power spectra by interpolating between the adjacent Doppler bins. The satellite was removed from the imaging data by using masking files consisting of binary pixel values: zeros and ones. If the image pixel is multiplied by a masking pixel with a value of 0, it does not contribute to the fit. The OC power spectra and delay-Doppler images obtained at Arecibo on October 5–7 contained up to 7 minutes of rotation by the primary. Our intention was to average as little rotation as possible in order to keep the features sharp while maintaining good noise statistics and reasonably strong S/Ns. If we assume that the primary's rotation period is ∼4 hr, then 7 minutes corresponds to ∼10° of rotation.

The Goldstone echo power spectra, although weaker than the Arecibo data, were important for the modeling because they constrained the bandwidth of the primary and the pole direction on 3 days without Arecibo observations: April 7, October 8, and October 9. On April 7, we summed ∼7 minutes of CW data, equivalent to about 10° of rotation. For CW data acquired at Goldstone in October, we summed ∼20 minutes of data due to low S/Ns, which introduced ∼30° of rotational smear. To help model data with this much rotation, we used Shape to synthesize five instantaneous echo power spectra from the model separated in time by only a few minutes (e.g., 240 s on October 6). These spectra were then averaged to obtain a single, rotationally smeared model spectrum that was compared with the data.

The primary appeared to be a symmetric object without pronounced surface features. As such, a sphere was a reasonable starting approximation for its shape. We sampled diameters between 0.28 and 0.36 km, with increments of 0.02 km. We conducted a grid search across the entire sky in order to constrain the pole direction. The ecliptic longitudes (λ) and latitudes (β) were sampled uniformly in the steps of 5°. We sampled rotation rates from 1800° to 2500° day⁻¹ (3.5–4.8 hr) in steps of 20° day⁻¹ based on the period constraints we discussed in Sections 2.1 and 2.2. The fit quality was evaluated based on χ² value for each (fixed) combination of size, period, and pole direction. The initial shape modeling data set contained delay-Doppler and Doppler-only data weighted so that they contributed 64% and 36% to the χ² value, respectively. We did not use lightcurves at this stage of the modeling because the shapes were approximated as spheres. The preliminary results showed a preference toward diameters D > 0.30 km and periods 3.9 hr < P < 4.5 hr. All size and period combinations showed a χ² minimum in the pole region represented by the dark swaths in Figure 12. This region encompassed the orbital pole constraint from Figure 10.

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In the second stage of modeling, we used the eighth-degree spherical harmonics to represent the shape of the primary. Given that the first stage of modeling did not significantly constrain the pole direction, and considering that it is reasonable to assume that the satellite's orbit pole corresponds to the pole of the primary, we adopted 10 poles from the orange region in Figure 11 as candidates: (50°, 35°), (60°, 40°), (70°, 40°), (70°, 45°), (80°, 45°), (90°, 45°), (90°, 50°), (100°, 45°), (110°, 40°), and (120°, 35°). We initialized spin rates from 1300° to 4600° day⁻¹ in steps of 20° day⁻¹ (1.9–6.6 hr period). We used a wider period search than we did initially because we now included the lightcurves, and we wanted to test more rigorously if slower or more rapid periods are possible. Figure 13 shows how the χ² varied with the spin rate. We show individual χ² contributions from the CW data because these data cover the entire radar observing campaign (April 7, October 5–9), and they are less noisy than the delay-Doppler images. We note that the χ² remains relatively flat for a large span of spin rates. The preferred rotation rate occurs around 2000° day⁻¹ (P ∼ 4.3 hr), similar to our findings from Sections 2.1 and 2.2. The overall χ² with contributions from CW, delay-Doppler images, and lightcurves shows a similar trend. The inclusion of lightcurves resulted in the steeper slope in χ², selecting for less rapid rotation rates. Nolan et al. (2013) pointed out that the standard statistical definition of a 1σ error (e.g., Bevington & Robinson 2003) based on the curvature of the calculated χ² produces "unreasonably large values" because the fit cannot distinguish between contributions from the signal and from the background noise. Thus, we visually inspected the various models and found that 1σ errors (fits noticeably less good than the best-fit model) roughly corresponded to a 10% increase in χ². We adopted ${2000}_{-{260}^{^\circ }}^{+{280}^{^\circ }}\,$ day⁻¹ $({4.3}_{-0.5}^{+0.6}\,\,\mathrm{hr})$ as a possible range of spin rates/periods. We also show in the last panel of Figure 13 that the overall χ² remained almost flat with respect to our 10 candidate poles. This means that we can select any of these candidate poles as "nominal." While it is not unusual to have a relatively unconstrained or prograde/retrograde ambiguous pole (Hudson & Ostro 1994; Busch et al. 2007; Brozović et al. 2011), the rotation rate is usually constrained better than ∼14%, as we have here. 2018 EB was too small and too slow for the radar data to show obvious bandwidth changes for different period considerations, and the delay-Doppler images did not have any obvious features to track. Lightcurves were also able to accommodate a wide range of spin rates due to either noisy (April 9) or relatively flat lightcurves (April 5 and October 18). The absence of a clear period signature prevented any detailed shape modeling. However, a ∼14% constraint was sufficient to estimate the object's dimensions and to obtain general outlines of the shape. The size estimate based on the shape model is still better than the visual assessment and important for determination of the bulk density and radar and optical albedos.

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We converted the 50 best spherical harmonics models into vertex shape models with 512 vertices, 1020 triangular facets, and an effective resolution of ∼27 m. All shape and spin parameters were allowed to adjust at this point. We selected one of the models (spin rate of 2030° day⁻¹, pole direction λ = 70°, β = 50°) as "nominal" based on its overall χ² value and our subjective preference during a visual inspection. The nominal pole was adjusted from the initialized value of λ = 70° and β = 45°. We used all 50 vertex model finalists to calculate the 3D shape uncertainties.

Figure 14 shows the Goldstone and Arecibo OC echo power spectra and their respective fits. The fits match the bandwidths and the spectral shapes. Figure 15 shows delay-Doppler data, fits, and plane-of-sky views of the nominal model. The model reproduces the bandwidths, visible extents, and echo shapes within reasonable limits given the low S/Ns of the data. The radar line of sight was ∼42° off the equator on October 5, which shrank the bandwidth by ∼26% with respect to an equatorial orientation. Hence, the echo occupied fewer pixels, and this made the data more difficult to model. The radar line of sight was ∼18° off the equator on October 7. We used the nominal spin rate (2030° day⁻¹) to search for common features at the same longitudes in the images obtained on different days. However, although there is some overlap in coverage across multiple days, the images had different S/Ns, and we could not find common features. Figure 4 shows the lightcurves and their respective fits for the nominal pole (λ = 70°, β = 50°) and a period of 4.26 hr. The fit reproduced the nearly flat lightcurve on April 5 and a ∼6 hr lightcurve with pronounced variations on April 9. The model also matched the small-amplitude lightcurve on October 18.

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Figure 16 shows principal-axis views of our preferred vertex model. The model resolution (∼27 m) was coarser than the delay-Doppler imaging resolution of 15 m, but it was sufficient to describe the data at hand. We also had a relatively large uncertainty on the rotation period, which prevented any detailed modeling of the shape features. The radar images and lightcurves provided nearly complete coverage. The overall shape is somewhat more subdued and more oblate than the toplike shapes of other binaries such as Moshup (Ostro et al. 2006). However, not all primaries modeled to date have a prominent equatorial bulge. For example, the primary of 2002 CE26 had a relatively rounded appearance (Shepard et al. 2006), and the shape of Didymos, as observed by the DART spacecraft, also appeared oblate with a less pronounced equatorial bulge (Daly et al. 2023). The 2018 EB model has principal-axis dimensions of X = 0.34 ± 0.04 km, Y = 0.33 ± 0.04 km, and Z = 0.29 ± 0.04 km and an equivalent diameter of 0.30 ± 0.04 km. We define the equivalent diameter as the diameter of a sphere that has the same volume as the model. Our equivalent diameter is somewhat larger than the NEOWISE estimate of D = 0.23 ± 0.07 km (Masiero et al. 2020), but the diameters are consistent within their uncertainties. Size variations among the final 50 vertex shape models were much smaller than the uncertainties listed in Table 6. We adopted ∼10% uncertainties on the X and Y principal dimensions to account for the data resolution and unmodeled sources of errors. We also checked these uncertainties by manually scaling the size of each axis. Shape models estimated from inversion of radar data are known to have larger uncertainties on the short principal axis (Hudson & Ostro 1999), especially when the observations are confined to nearly equatorial subradar latitudes. However, we observed 2018 EB at −42° in subradar latitude on October 5 and at +4° on October 9. This relatively large range of subradar latitudes allowed us to adopt the Z-axis uncertainty of 15%. Lightcurves from April 5 and 9 and October 11 also contributed to the constraints on the axis ratios.

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Table 6. Physical Properties of the Primary

Parameter	Value
Pole direction
λ	70°
β	50°
Principal axes
X	0.34 ± 0.04 km
Y	0.33 ± 0.04 km
Z	0.29 ± 0.04 km
Axis ratios
X/Y	1.03 ± 0.17
Y/Z	1.14 ± 0.21
Equivalent diameter	0.30 ± 0.04 km
Surface area	0.30 ± 0.08 km2
Volume	0.015 ± 0.006 km3
DEEVE
X	0.33 ± 0.04 km
Y	0.33 ± 0.04 km
Z	0.26 ± 0.04 km
Rotation period	${4.3}_{-0.5}^{+0.6}$ hr
Optical albedo	pV = 0.028 ± 0.016

Note. Physical parameters obtained from shape modeling. The prograde model was treated as nominal based on the orbital fit preference for the prograde pole solution. We adopt the orange region in Figure 10 as the pole uncertainty. Uncertainties in the physical dimensions were determined based on statistical variations among 50 shape model candidates and by considering the imaging data resolution. "DEEVE" stands for dynamically equivalent, equal volume ellipsoid. The optical albedo was calculated from the absolute magnitude 21.9 ± 0.5 and the effective diameter of the system (D_eff = 0.33 ± 0.05 km; see Section 5).

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The nominal model of the primary has a small elongation of ∼3%, although within the uncertainties, models with higher elongations are possible. In the analysis above, we noted that the echo bandwidth varied by ∼12% as the object rotated during imaging. However, the S/Ns were relatively weak and could have contributed to the bandwidth variations. The 12% corresponds to only two Doppler frequency bins in the images. We also note that the April 5 lightcurve was almost flat during 2 hr of observations. We initially did not know if this was due to low shape elongation, a nearly pole-on view, or a period longer than a few hours. After the shape modeling, we realized that the lightcurve was obtained in a nearly equatorial view from the Earth (+15° body-fixed latitude) with an almost fully lit surface (<15% of the surface was shadowed). Thus, this lightcurve is consistent with a low elongation of the primary.

Margot et al. (2002) and Pravec et al. (2006) reported that at least 16% of NEAs larger than ∼200 m in diameter are binary systems. Since completion of the upgrade at Arecibo in 1999, the Arecibo and Goldstone radars have observed 79 binaries and four triples out of 634 NEAs with H < 22. Two additional binary systems, (410777) 2009 FD and 2015 TD144, have H > 22. As noted in Benner et al. (2016), besides transmit power, radar S/Ns are proportional to P^1/2 and D^3/2, where P is the rotation period and D is the diameter, so small, rapidly rotating satellites are more difficult to detect than larger, slowly rotating ones in synchronous orbits about their primaries. The satellite can also escape notice if the Doppler resolution in the echo power spectra or delay-Doppler images is too coarse or too fine with respect to the satellite's bandwidth. A prominent example was Dimorphos, which in 2003 was only seen in Goldstone data when we processed the data at a finer frequency resolution well after the observations concluded (Naidu et al. 2020). At the other extreme, there are almost certainly some rapidly spinning secondaries that we failed to detect due to low S/Ns (Margot et al. 2015). Hence, the statistics we discuss below represent lower bounds on the abundances of multiple systems in the NEA population.

Table 8 lists known NEA binary and triple systems, estimated sizes of their components, estimated periods, and constraints on semimajor axes. The overall number of known multiple systems, observed either optically or with radar, is 86. The number of radar-observed binary NEAs with H < 22 relative to all radar-observed NEAs in this category places a lower bound of ∼13% on the abundance of binaries. Triple asteroids such as (153591) 2001 SN263, (136617) 1994 CC, (3122) Florence, and (348400) 2005 JF21 comprise <1% of the NEA population with H < 22 detected by radar. Comparable-mass binaries such as (69230) Hermes, 1994 CJ1, (190166) 2005 UP156, and 2017 YE5 (Margot et al. 2003; Taylor et al. 2014, 2020; Virkki et al. 2022) comprise <1% and appear to be as rare as triple systems.

About 180 NEAs with an absolute magnitude of 21 ≤ H ≤ 23 have been observed with radar, and we observed 10 binaries in this sample. Taken at face value, binaries comprise at least ∼6% of this small NEA population. 2018 EB is the first binary system to date with H > 21 that has been studied in detail. Among 11 binaries with H > 21, 2015 TD144 appears to have the smallest primary, with a diameter of only ∼90 m, and one of the smallest satellites, which is perhaps a few tens of meters in diameter (P. Taylor, personal communication). One equal-mass binary is known in this population of small objects: 1994 CJ1 (H = 21.4; Taylor et al. 2014). To date, there are no known triple systems with H > 21. Discoveries of binaries with H > 21 have the same observing biases as discussed earlier, but usually fewer observing dates are available, which reduces the probability of detecting a satellite.

The next opportunity to observe 2018 EB with radar at S/Ns comparable to those discussed here will occur in 2059 April and October. The asteroid will approach Earth at 0.034 au on April 4 and 0.058 au on October 6. The system will remain below 22nd magnitude until 2058, but it will brighten to magnitudes ∼16.3 and ∼18.5 in 2059 April and October. The rotation period of the primary could be investigated in greater detail, and mutual events could be detected with ∼1 m aperture telescopes in April.