Asteroid Impact Hazard Warning from the Near-Earth Object Surveyor Mission

NASA’s Near-Earth Object Surveyor mission, scheduled for launch in 2027 September, is designed to detect and characterize at least two-thirds of the potentially hazardous asteroids with diameters larger than 140 m in a nominal 5 yr mission. We describe a model to estimate the survey performance using a faster approach than the time domain survey simulator described in Mainzer et al. (2023). This model is applied to explain how the completeness for 5 and 10 yr surveys varies with orbit type and asteroid size and to identify orbits with notably high or low likelihoods of detection. Size alone is an incomplete proxy for impact hazard, so for each asteroid orbit, we also calculate the associated hazard based on the impact velocity and the relative likelihood of impact. We then estimate how effective the mission will be at anticipating impacts as a function of impact energy, finding that a 5 yr mission will identify 87% of potential impacts larger than 100 Mt (Torino-9, “Regional Devastation”). For a 10 yr mission, this increases to 94%. We also show how the distribution of warning time varies with impact energy.


Introduction
Near-Earth objects (NEOs) are a population of minor planets on unstable orbits that come within 1.3 au of the Sun (Binzel et al. 2015).They represent the most accessible population for exploration and sample return, as demonstrated recently by the Hayabusa 2 mission to Ryugu (Watanabe et al. 2017) and the OSIRIS-REx mission to Bennu (Lauretta et al. 2017).Asteroids in Earth-crossing orbits pose an impact hazard, and sufficiently large objects can result in ground damage (e.g., Chelyabinsk; Popova et al. 2013) or even a major extinction event (Alvarez et al. 1980).
To address the hazard posed to Earth by asteroids and comets, NASA's Planetary Defense Coordination Office has undertaken many activities to both discover and characterize the population of impactors, as well as to test potential approaches to mitigation.The Double Asteroid Redirection Test (Daly et al. 2023) successfully impacted a small moon of the near-Earth asteroid Didymos to evaluate the technological capabilities needed to divert a ∼100 m class body.In order to divert an object, however, we must first know that it exists.NASA's Near-Earth Object (NEO) Surveyor mission (Mainzer et al. 2023) is designed to detect two-thirds of all potentially hazardous objects larger than 140 m in diameter by the end of its 5 yr baseline mission and to bring the catalog of known potentially hazardous objects to >90% completeness by the end of a 12 yr total survey lifetime.In this context, potentially hazardous asteroids (PHAs) are defined as those objects larger than 140 m in diameter and with minimum orbital intersection distances (MOIDs; that is, the smallest distance between the asteroid's orbit and Earth's) of less than 0.05 au.
In this work, we construct a model of the expected performance of NEO Surveyor using an approach that is less computationally intensive than full time-domain simulations.This model enables rapid understanding of the dependence of completeness on the mission design, survey parameters, and population models, in addition to providing an independent check of the predictions from the larger and more detailed simulation.We apply the model to estimate the completeness and expected warning time that NEO Surveyor will provide for an impacting object as a function of impact energy.
Section 2 provides a brief overview of the NEO Surveyor mission, as currently designed.Section 3 describes the survey completeness model.This includes the synthetic sample population of PHAs (Section 3.1), the thermal model used to estimate the infrared flux from an asteroid (Section 3.2), the factors determining detectability by NEO Surveyor (Section 3.3), the addition of probabilistic factors such as detector gaps and confusion (Section 3.4), the way in which track formation is modeled (Section 3.5), the derived survey completeness for a given asteroid size (Section 3.6), the extension of completeness to include all sizes and thermal model parameters (Section 3.7), comparison to the full time-domain simulation (Section 3.8), and, finally, the impact of other surveys (Section 3.9).Section 4 provides the mapping to impact hazard and the derivation of warning distributions.Section 5 summarizes the results.Four appendices cover the predicted radiometric sensitivity of NEO Surveyor, derivations for the impact of detector gaps, the dependence of track formation on constituent measurement probabilities, and an expression for the probability of impact.

Mission Overview
The NEO Surveyor mission is currently scheduled to launch in 2027 September.The key parameters relevant to this analysis are listed in Table 1.A given patch of sky that lies within the ecliptic latitude-longitude constraints of the field of regard is visited four times over a period of 6 hr with approximately 150 s of integration time per visit in each of the two simultaneously observed bands.All detected sources are extracted from the sequence of four images, and "tracklets" are formed from the moving-object detections.The same patch of sky is then reobserved at intervals of about 13 days, whenever it lies within the latitude-longitude constraints.Tracklets are then reported to the Minor Planet Center, where two or more consecutive tracklets will be linked to form a track, with accuracy sufficient to enable follow-up detection and orbit determination.More details on the mission design and survey strategy can be found in Mainzer et al. (2023).
The sensitivity of the instrument is characterized by the 5σ detection threshold in each of the NC1 and NC2 bands as a function of ecliptic latitude and longitude (Appendix A).The primary driver of sensitivity is the emission from local zodiacal dust, which we model using the DIRBE-derived model of the zodiacal cloud (Gorjian et al. 2000).

Survey Completeness Model
Our model estimates the probability that an asteroid in a given elliptical orbit is found by a proposed survey strategy.The probability is calculated as an average over all combinations of Earth and asteroid orbital phases.It is a function of the asteroid size and thermal properties.The integral survey completeness is then estimated by repeating this process over a sample of orbits drawn from a population of synthetic NEO orbits and averaging over distributions of thermal properties and asteroid diameter.The impact of other surveys on completeness is discussed in Section 3.9.

Asteroid Sample Population
The synthetic population of 802,000 NEOs created by Granvik et al. (2018) was used as the input.The MOID was calculated for each case (using the moid.m function of Santilli's ASTROTIK), and the subpopulation of 144,000 PHA orbits with MOID < 0.05 au was extracted, ignoring the size limit of D > 140 m that is typically used, to allow us to quantify the full range of potential impactors.A sample of 10,000 orbits from this PHA population is shown in Figure 1.The boundaries of the region to the left and right occur where there is grazing contact between the Earth and asteroid orbits.
We assume that the population diameter density distribution follows a power law with index −2.5;i.e., the number of objects with a diameter between D − δ/2 and D + δ/2 is proportional to D −2.5 δ (or an index of −1.5 for the corresponding cumulative distribution).For diameters of 70-1600 m, this is consistent with the distribution of absolute H magnitudes in the Granvik sample, as well as Grav et al. (2023) and Nesvorný et al. (2023), assuming a 100 m diameter object has H = 22.7 (Granvik et al. 2018; based on a geometric albedo p V of 0.14).We also assume that the size is uncorrelated with the orbit parameters, although recent models suggest this may not be the case in practice (Nesvorný et al. 2023).

Asteroid Thermal Model
We used the Near-Earth Asteroid Thermal Model (NEATM) (Harris 1998) to derive the infrared flux emitted by an asteroid.In this prescription, the asteroid is represented by a sphere of diameter d ast with a maximum surface temperature of T max that falls to 0 K according to cos 1/4 f, where f is the angle from the subsolar point, and where A is the Bond albedo that determines the fraction of solar flux S that is absorbed, η is the "beaming parameter," ε is the mid-IR emissivity, and σ is the Stefan-Boltzmann constant.
The beaming parameter is used to empirically capture the behavior introduced by surface thermal properties such as thermal inertia and porosity, as well as any nonsphericity in the body's shape (Hansen 1977;Harris & Drube 2016).The NEATM has been the model of choice to infer NEO properties from a number of surveys (IRAS, Tedesco et al. 2002;Akari, Usui et al. 2011;Spitzer, Trilling et al. 2016;Gustafsson et al. 2019).Multiwavelength measurements from the NEOWISE survey (Mainzer et al. 2011) have been used to estimate η and A for a sample of 428 NEOs.There is a significant range covered by these values: the distribution of (1 -A)/η spans a factor of approximately 5, corresponding to a factor of 1.5 in T max .Mainzer et al. (2011) determined that the beaming parameter exhibits a weak dependence on the phase angle α (units: degrees) of the solar illumination: For our nominal thermal model, we adopt this relationship for η, a mean Bond albedo of 0.05, and mid-IR emissivity of ε = 0.9 (Mainzer et al. 2011), but in Section 3.7, we average the results over the distribution of (1 -A)/η.We calculate the flux predicted in the two bands, NC1 and NC2, by integrating over the contributions from the emitting surface of the sphere for asteroids at varying range and solar elongation angle.Figure 2 shows the results for a 140 m diameter asteroid in the NC2 band with the nominal thermal model parameters.The flux in the longer-wavelength NC2 band is generally much higher than for NC1, as illustrated by the spectral energy distribution curves shown in Mainzer et al. (2023) (Figure 2).
We note that the NEATM has limitations.At illumination phase angles >65°, the fast rotator model produces better size estimates (Mommert et al. 2018).The beaming parameter η may also correlate with size (Harris & Drube 2016;Hung et al. 2022).These refinements are not included in the current analysis.

Detectability
Four constraints must be satisfied simultaneously for an asteroid to be detected in any single visit (approximately 150 s of integration).
1.The line of sight from NEO Surveyor to the asteroid must lie within the survey ecliptic longitude constraint.2. The line of sight must lie within the survey ecliptic latitude constraint.3. The projected angular velocity on the sky must be above the minimum value (required to distinguish it from stationary background objects) and below the maximum value (to avoid the signal-to-noise ratio, SNR, loss from excessive streaking during an exposure).4. The flux from the asteroid must exceed the 5σ detection threshold for the specified ecliptic latitude and longitude (see Appendix A).The combined SNR from the NC1 and NC2 bands is used.
The asteroid flux at the point of observation depends on the size of the asteroid, the distance from the Sun, the distance from the asteroid to the spacecraft, and the Sun-asteroidspacecraft phase angle.It is computationally convenient to precalculate the maximum range at which an asteroid can be detected as a function of size, ecliptic latitude, and longitude.Figure 4 shows an example of the detection criteria applied to an example NEO orbit.The orbit has semimajor axis a = 0.93 au, eccentricity e = 0.78, inclination i = 0°, longitude of ascending node = 0°, and argument of perihelion ω = 215°.We use this zero-inclination case to illustrate the subsequent analysis steps we undertake, as they will be applied to the full sample of PHAs drawn from the Granvik population.The green arcs represent the portions of the asteroid orbit in which detection is possible from the instantaneous Earth/NEOS location shown.
In this analysis, the positions of the Earth and the NEO in their respective orbits are represented by their mean anomaly.The detectable range of asteroid mean anomaly depends on the mean anomaly of the Earth and, by extension, NEO Surveyor, which is located at the L1 Lagrange point, approximately 0.01 au from the Earth.Figure 5 shows the detectability for the example NEO orbit depicted in Figure 4, scanning over all mean anomalies for both the Earth and the NEO.The colors in the plot encode three of the four detection criteria listed above: blue shows when the latitude constraint is met, red is for the longitude constraint, and green is for sensitivity.Overlapped areas have the color of the appropriate red, green, and blue combination, meaning that any region where all parameters are met will be colored white.For the zero-inclination example orbit, the latitude constraint is met throughout the plane.The specific geometry shown in Figure 4 is represented by a vertical cut through the plot at MA Earth = 275°.The projected angular velocity criterion is rarely a constraint for the NEO orbits examined, and, while included in the analysis, it is not represented in the plot.

Probabilistic Detection, Detector Gaps, and Confusion
In the previous section, we assumed that tracklet formation was successful if the four binary detection criteria were met.In practice, missed detections, confusion, and gaps between the detector focal planes introduce a probabilistic element to tracklet formation that will reduce the overall efficiency.
Missed detections can occur if the NEO image falls on bad pixels or is coincident with a cosmic-ray event on the detector.This possibility is minimized by coadding six dithered exposures (∼25 s each) to form an integration of ∼150 s.Confusion with other objects can result in the data processing system incorrectly associating one or more of the four detections, preventing the formation of the tracklet.For simplicity, these two effects are represented in this analysis with a fixed probability per tracklet; in reality, the potential for confusion will vary with the density of background main-belt asteroids (detailed simulation predicts that this is a minor contribution to missed detections, however).
The NEO Surveyor instrument has a mosaic of four focal plane arrays in each of the two bands.An NEO image that falls in a gap between adjacent focal planes is not detected.In Appendix B, we show that the resulting probability of tracklet detection depends on the angular velocity of the NEO.We use P(no missed detections) P(no confusion) = 0.9 and the prescription for P(no gap) derived in Appendix B.
There is also the possibility that very fast-moving objects can "outrun" the progression of the survey on the sky and thereby elude detection.Because this analysis implicitly assumes that the field of regard is covered instantaneously, rather than being tiled out over time (75°of solar longitude over 6 days), this survey-escape mechanism is not covered here, although it is included in the project's time-domain simulation (Mainzer et al. 2023).

Observing Cadence and Track Formation
At any given time, the locations of the Earth and the NEO are represented by a single point in the mean anomaly versus mean anomaly domain of Figure 5.As the bodies proceed along their orbits, a straight-line locus is traced out with a gradient equal to the ratio of the orbital periods, T Earth /T NEO , as illustrated in Figure 5.In this case, the mean anomaly for each body is zero at the start of the mission, and the locus starts at the origin, but all phasings are considered.After each orbit, the mean anomaly wraps from 360°to 0°.The locus continues for 5 yr of mission time; i.e., it traverses the horizontal extent of the plot five times.
A given patch of sky is reobserved every 13 days, as long as it remains within the solar latitude-longitude constraints.This 13 day observing cadence is represented by the magenta circles, but only those occurring in the white regions that also satisfy the range and angular velocity criteria can result in a tracklet.If two consecutive observations fall in a white region, then we assume that the two resulting tracklets could be linked to form a track, subject to the tracklet detection probabilities of the previous section.The first such window for the phasing depicted in Figure 5 occurs in the upper right, almost 1 yr into the mission.Following the wraps, we see that there are further windows later in the 5 yr period in the lower right of the plot.
The probability of obtaining at least one track for this phasing is An expression for P(no track) is derived in Appendix C as a function of the constituent tracklet probabilities over the duration of the survey.The probability of finding an NEO with orbits characterized by elements (a, e, i, ω) over a 5 yr mission is obtained by taking the average of P(track formed) over all permutations of orbital phasings {MA Earth , MA NEO } at the start of the mission (i.e., where the red locus starts in the plane of Figure 5) and over the phasing of the observation within the 13 day period (i.e., where the magenta circles fall along the locus).The numerical calculation was performed with MA Earth , MA NEO = 1°, 2°, 3°, K 359°and observation phasing increments of 1°in the Earth mean anomaly interval.Finer resolution adds little to the fidelity of the output.The result obtained for the example orbit in Figure 5 is 99.5%.Two additional orbits are shown in Figure 6.The first is for an NEO orbit with a period very close to 1 yr.While the white, detectable region is large, the lack of diversity in the overlapped red locus over 5 yr results in only a 66% probability of creating a track.The NEO orbit in Figure 6(b) has an inclination of 60°.The solar latitude constraint is significant, and the white, detectable region spans only a small fraction of phase space.The probability of creating a track is 53%.
These values represent the probability of finding a 140 m diameter NEO in the specified orbit over a 5 yr survey assuming the nominal set of thermal model parameters.Corresponding values for a 10 yr survey are obtained in the same way by extending the red line locus to 10 horizontal wraps.

NEO Surveyor Completeness
We repeat the calculation of the previous section for a sample of orbits drawn randomly from the synthetic PHA population described in Section 3.1.Figure 7 shows the results obtained for a sample of 10,000 orbits for both 5 and 10 yr surveys.Each dot represents the probability of finding a 140 m diameter NEO in an orbit averaged over all possible phasings as described in Section 3.5.
The mean probability over the 10,000 samples gives the mission completeness for 140 m diameter NEOs.The 5 yr survey has a completeness of 57% for objects in this size bin.This increases to 77% for the 10 yr survey; cumulative completeness values (i.e., for all objects larger than 140 m) will be higher, as larger objects are more likely to exceed the detection threshold.The coloration highlights where objects are missed.In addition to the expected increase in completeness with survey time, there are three trends/structures apparent in the distribution of completeness: (1) a deficit for specific values of semimajor radius (e.g., 1.0, 1.3, and 1.6 au), (2) reduced completeness along the low-and high-eccentricity boundaries of the distribution (most apparent for 2-2.5 au on the 5 yr plot), and (3) a trend toward lower completeness at high semimajor axis and high eccentricity.Each of these is explained below.
The deficits corresponding to the vertical stripes in Figure 7 result from orbital resonance between NEO Surveyor at 1 au and the target asteroid.This is illustrated in Figure 6(a) for the case of an a = 1 au NEO orbit.The deficits at 1.0, 1.3, 1.6, 2.1, and 2.5 au correspond to resonances of 1:1, 2:3, 2:1, 3:1, and 4:1, respectively.For example, an object with a period of 1 yr, lying behind the Sun as seen from NEO Surveyor, will remain close to the Sun and unobservable, particularly if the orbit has low eccentricity.In these configurations, the object does not come close to the Earth, however, and an impact can only occur after sufficient time has passed to evolve the relative orbital phasing.In Figure 6(a), the red line must advance until it intersects one of the points of closest approach indicated by a black cross and in the meantime must pass through a white region of detectability.This behavior is included in the analysis of warning time in Section 4.3.
The two remaining features are best explained by Figure 8.The color scale represents the fraction of a year in which a 140 m NEO at a fixed point in space is detectable by NEO Surveyor.The peak of 25% occurs for an asteroid at a distance of 1 au from the Sun.Objects within 0.71 au of the Sun are not detectable because of the 45°solar longitude constraint.The 140 m objects outside 1.5 au are not detectable because of a combination of the sensitivity and the 120°maximum solar longitude constraint.A set of orbits with a = 1.75 au are overlaid in red.For this semimajor radius, the potentially  hazardous orbits (MOID < 0.05 au) start at e ∼ 0.4.This orbit mostly passes through the outer part of the ring, corresponding to a lower probability of detection when averaging over all orbital phasings.NEOs with 0.4 < e < 0.6 spend more time in the higher-likelihood region.For e > 0.6, the NEO perihelion lies in the inner, undetectable region, and these orbits are only potentially detectable for the short period of time in which they cross the ring.This explains the reduced completeness at the low-and high-e boundaries of the PHA domain in Figure 7.The general reduction in completeness toward high a and high e arises because objects in these orbits spend a smaller fraction of time within the detectable region.

Cumulative Completeness over Asteroid Size and Thermal
Model Distributions In the previous section, we derived the survey completeness for PHAs with 140 m diameter and the nominal thermal model parameters.The next step is to perform a weighted average over the distributions of diameter and thermal parameters to obtain the cumulative completeness.Figure 9 illustrates the process.The cumulative completeness for objects larger than 140 m after a 5 yr survey is estimated to be 0.76.

Comparison to Time-domain Simulation
The NEO Surveyor project maintains a detailed time-domain simulation to predict mission performance.The NEO Surveyor Survey Simulator (NSS) is described by Mainzer et al. (2023).Each pointing of the survey is simulated using a solar system reference population, with the locations of tens of thousands of moving objects propagated over the duration of the mission.The reference population comprises both known objects (e.g., main-belt asteroids) and a synthetic set of NEOs.
The approach described in this paper is complementary to the detailed simulation.It is less computationally intensive, allowing rapid trades and sensitivity analyses.While lacking the fidelity of the detailed model, it nevertheless shows very good agreement, as illustrated in Figure 10, providing crossvalidation.We note that this comparison excludes preexisting discoveries by other surveys.It is also well suited to studies of hazard level and warning, as described in Section 4.

Impact of Existing and Other Future Surveys
The analysis of the previous sections addressed the completeness expected for the NEO Surveyor mission in isolation.Existing surveys (e.g., Spacewatch, the Catalina Sky Survey, PanSTARRS, NEOWISE) have already discovered many thousands of objects.Grav et al. (2023) performed a detailed analysis and estimated that approximately 40% of near-Earth asteroids with diameters larger than 140 m have already been found.A version of Figure 10 including this contribution is shown in Mainzer et al. (2023; their Figure 13).
The ground-based Vera C. Rubin Observatory, due to come online in 2025, will execute the 10 yr Legacy Survey of Space and Time (LSST), which is expected to make a major contribution to the inventory of known asteroids (Ivezić et al. 2019).The NEO Surveyor and LSST are complementary, with the former primarily looking toward the Sun in the thermal infrared and the latter looking away from the Sun in the visible.With the addition of appropriate statistics for observing efficiency and survey cadence, the analysis described here could be extended to the LSST to provide a prediction of the joint performance, inevitably superior to either considered alone.

Hazard Level
The hazard of asteroid impacts on the Earth is characterized by a combination of impact probability and impact energy (Gehrels 1995, and references therein).In this section, we estimate both of these quantities for the objects in the synthetic PHA population.Combining this with the survey completeness calculations of the previous sections, we obtain an answer to the questions "If an asteroid impacts the Earth, what is the probability that it was previously found by the NEO Surveyor?" and "How much warning was there?"Both vary with the impact energy.

Impact Probability
The two orbits in Figure 11 have the same MOID, but intuitively, the in-plane orbit has a higher impact probability than the inclined case.
In Appendix D, we use geometry and a gravitational focusing factor to derive an expression for the impact probability: where k is a constant of proportionality, r Earth is the Earth's radius, T Earth and T ast are the orbital periods, v Earth and v ast are the orbital velocities at MOID separated by angle θ, and v esc is the escape velocity from the Earth's surface.As expected, the probability of impact scales with the Earth cross-sectional area and becomes large when θ is small, consistent with the picture in Figure 11.
In Figure 12, a sample of 1000 orbits is drawn from the Granvik PHA subpopulation with MOID < 0.01 au.For each orbit, the relative probability of impact is calculated using The Earth orbit is the blue circle.Red ellipses are zero-inclination NEO orbits with a = 1.75 au and e = 0, 0.1.0.2, K 0.9.Red dots are marked at intervals of 5°in mean anomaly.
Equation (5) and represented by the logarithmic color scale in Figures 12(a) and (b).The orbits with the highest impact probability are the low-inclination, "tangentially grazing" orbits at the boundaries of the PHA distribution in eccentricity versus semimajor radius domain.We also show the velocity difference of the objects at the time of impact, which increases with inclination as the out-of-plane motion contributes to the velocity.

Impact Energy and Hazard Completeness
In the absence of the Earth's gravity, the relative speed at MOID is given by 2 and plotted in Figures 12(c) and (d).High-inclination orbits have higher impact speed but are both less prevalent and less likely to cause an impact.The impact energy is estimated simply as the kinetic energy of a sphere with density ρ: We make no attempt to partition this into the energy dissipated in the atmosphere on entry versus the ground impact, which will depend on the angle of entry, mass, and composition, governed by the processes of ablation and fragmentation (Register et al. 2017).The energy is expressed in megatons of TNT, where 1 Mt = 4.184 × 10 15 J.The asteroid density is set to 2600 kg m −3 (Collins et al. 2005), although we note that measured densities for objects visited by spacecraft span a significant range: 1190 kg m −3 for Bennu (Goossens et al. 2021), 1190 kg m −3 for Ryugu (Watanabe et al. 2019), 1900 kg m −3 for Itokawa (Fujiwara et al. 2006), and 2670 kg m −3 for Eros (Yeomans et al. 2000).The influence of a different density on the results below can be estimated by sliding the distributions along the x-axis by the appropriate amount; e.g., halving the density corresponds to a shift of approximately −0.3 in log 10 (impact energy).
In Figure 13, we combine the relative impact probability (Figures 12(a) and (b)) with the impact energy calculated from the impact speed in Figures 12(c) and (d) using Equation ( 6) and the NEO Surveyor 5 yr completeness (Section 3.7) for a 140 m diameter asteroid in each orbit.Objects with a higher impact probability tend to be more complete, since they are more likely to have a semimajor axis close to 1 au with low eccentricity.The 140 m objects with the highest impact energies have low completeness because they tend to have high eccentricity.The upward curvature toward lower impact energies is a result of gravitational focusing.
Figure 13 also illustrates the importance of including the impact velocity, in addition to the size, when assessing the hazard.The impact energy of a given asteroid mass spans over an order of magnitude.
In Figure 14(a), we extend the size range to include objects with diameters spanning 35-1600 m in multiplicative steps of 2 .The relative abundance of small objects relative to large objects is reflected in the diminishing probability from left to right, based on an underlying diameter distribution power-law index of −2.5 and accounting for the logarithmic spacing.The resulting slope is given by (1 + β) / 3, where β is the powerlaw index of the diameter distribution.In contrast to the fixedsize case of Figure 13, potential high-energy impacts are much more likely to have been previously found with NEO Surveyor than the more prevalent low-energy impacts.At a given impact    energy, low-mass, high-impact-speed events are less likely than high-mass, low-impact-speed events.Fortunately, these latter objects are easier to detect.The corresponding relative impact probability and discovery probability are shown in Figures 14(b) and (c).
Figure 15 shows the probability of discovery for impacts exceeding a given threshold energy.The NEO Surveyor 5 yr curve is obtained by integrating the product of the distributions in Figures 14(b) and (c).Also shown is the curve for a 10 yr NEO Surveyor mission.For example, for impacts of 100 Mt or larger, the probability that the object would be discovered by NEO Surveyor increases from 87% (5 yr survey) to 94% (10 yr survey).This analysis is for NEO Surveyor alone and does not include objects found by other surveys.

Warning Time
To estimate the warning time for an object of a given size and orbit, we can start from the point of impact in the NEO detectability versus mean anomalies domain (e.g., Figure 5) and trace back in time to determine the history of the track formation opportunities that preceded it.A single track spanning 13 days is unlikely to be sufficient to establish a high probability of impact but would trigger additional observations using both ground-based telescopes and the "Targeted Follow-Up" mode of NEO Surveyor to more precisely determine the orbit parameters.For this analysis, we assume that the clock for the warning time starts at the first track detection by NEO Surveyor.
Since each tracklet-formation opportunity is probabilistic in nature, a Monte Carlo simulation was used to determine the median warning time for each combination of asteroid size and orbit.This process was performed for the 1000 orbits with the shortest MOIDs in the Granvik sample, with asteroid diameters  of 35, 50, 70, 100, 140, ... 1600 m.The impact energy and impact probability were calculated for each case, and the results were binned into half-decade intervals of impact energy.Within each bin, the 10th, 50th, and 90th percentiles in warning time were determined, accounting for the impact probability.
Figure 16(a) shows the results for an impact occurring coincident with the end of a 10 yr survey.The warning in this case cannot exceed 10 yr.For a 100 Mt impact, 10% of cases have a warning of 9 1/2 yr or more, and 50% have at least a 7 yr warning.The high-energy impacts tend to be from large objects that are more easily detected, so the warning time is longer than for low-energy impacts.Figure 16(b) shows how the probability of there being no warning increases as the impacts become smaller.For a 100 Mt impact, the probability of no warning is ∼20%.
For an impact occurring Q yr after the end of the survey, the percentiles shown in Figure 16(a) can simply be increased by Q.For example, a 100 Mt impact 7 yr after the end of the survey has a median warning time of 7 + 7 = 14 yr.The exception is those cases with no warning (Figure 16(b)), which continue to be unwarned by NEO Surveyor, no matter how far in the future they occur, under the assumption that no new observations are obtained after the conclusion of the 10 yr survey.

Summary
We have described a survey completeness model for predicting the performance of the NEO Surveyor mission.The model uses the sky pointing constraints, instrument sensitivity, and observing cadence to predict the probability of finding an asteroid of given size and thermal properties in a specified elliptical orbit, averaged over all phasings.The process is repeated for different sizes, thermal parameters, and orbits drawn from a parent synthetic population to estimate the integral survey completeness.Results are obtained orders of magnitude faster than for the higher-fidelity, frame-by-frame time-domain simulation.
The survey completeness model was used to characterize the objects missed in 5 and 10 yr surveys, which fell into two broad categories: those at or near orbital resonance with the Earth and those with large semimajor radii that spend little time close to the Earth's orbit.
Both the detectability of an asteroid and the impact hazard depend on combinations of the orbit shape and the object diameter, resulting in strong correlations.High-inclination orbits tend to have high impact energy but are much less likely than those closer to the ecliptic plane, both because there are fewer objects in those orbits and because the probability of impact for each object is lower.The high-probability impactors at a given energy are generally higher-mass, slower-moving objects, which makes them easier to detect.
The model was also applied to estimate the probability that future impactors will have been previously detected by NEO Surveyor.A 5 yr survey will identify 87% of potential impacts larger than 100 Mt (Torino-9, "Regional Devastation").For a 10 yr mission, this increases to 94%.The distribution of expected warning time was also characterized as function of the impact energy.
earth ast ^The first impact opportunity in the orbital mean anomaly domain arises at the location labeled with subscript 1, with a grazing impact at point X 1 .The last opportunity occurs at the location labeled with subscript 2, with impact at X 2 .The probability that both the Earth and the asteroid are present simultaneously in this interval is given by where T Earth and T ast are the orbital periods, and t C1C2 and t X1X2 are the times for the Earth and asteroid to move between the first and last impact locations.This result does not include the focusing attraction of the Earth's gravitational field.Jones & Poole (2007) show that the effective impact cross section of the Earth is increased by a factor + ( ) , where v esc is the escape velocity from the surface of the Earth, approximately 11 km s −1 .With this modification, Equation (D2) becomes

Figure 1 .
Figure 1.Parameters for a sample of 10,000 potentially hazardous orbits drawn from the Granvik sample with MOID < 0.05 au.

Figure 2 .
Figure 2. (a) Observing geometry.(b) Asteroid flux for 140 m diameter as a function of solar longitude and range for the NC2 band, as predicted by the nominal NEATM model.The gray regions are excluded by the NEO Surveyor observing constraints.

Figure 3
Figure3represents the results as contours in the ecliptic plane for five different asteroid diameters.The diameters progress in multiples of two and include the canonical 140 m threshold for a PHA.The strange shape of the innermost surface for a 70 m diameter is a result of the phase angle being close to 180°for an asteroid between the Earth and the Sun, resulting in only a thin crescent of emission with less flux than if the asteroid is moved beyond the Sun.Figure4shows an example of the detection criteria applied to an example NEO orbit.The orbit has semimajor axis a = 0.93 au, eccentricity e = 0.78, inclination i = 0°, longitude of ascending node = 0°, and argument of perihelion ω = 215°.We use this zero-inclination case to illustrate the subsequent analysis steps we undertake, as they will be applied to the full sample of PHAs drawn from the Granvik population.The green arcs represent the portions of the asteroid orbit in which detection is possible from the instantaneous Earth/NEOS location shown.In this analysis, the positions of the Earth and the NEO in their respective orbits are represented by their mean anomaly.The detectable range of asteroid mean anomaly depends on the mean anomaly of the Earth and, by extension, NEO Surveyor, which is located at the L1 Lagrange point, approximately 0.01 au from the Earth.Figure5shows the detectability for the example NEO orbit depicted in Figure4, scanning over all mean anomalies for both the Earth and the NEO.The colors in the plot encode three of the four detection criteria listed above: blue shows when the latitude constraint is met, red is for the longitude constraint, and green is for sensitivity.Overlapped areas have the color of the appropriate red, green, and blue combination, meaning that any region where all parameters are

Figure 3 .
Figure 3. Maximum detectable range in the ecliptic plane for nominal thermal model parameters.At a point on one of the contours, the flux from an asteroid of the specified diameter, with the nominal thermal model parameters, is equal to the 5σ detection threshold for the ecliptic latitude/longitude of the line of sight from the telescope to the target.The gray shaded regions are excluded because the solar elongation angle is less than 45°or greater than 120°.

Figure 4 .
Figure 4. Detectability of a 140 m diameter NEO with NEO Surveyor.The Earth orbit is in blue, and the coplanar asteroid orbit is in red.The black dots denote 5°intervals of mean anomaly, with the first few values indicated.The green region corresponds to the detection sensitivity criterion; the yellow areas indicate the solar longitude constraints.Thick green lines show the portion of the asteroid orbit detectable from the NEO Surveyor location shown.

Figure 5 .
Figure 5. NEO detectability as a function of Earth and asteroid mean anomaly for the orbit shown in Figure 4.The basic detection criteria (excluding angular velocity) are encoded by color and labeled when satisfied.The locus of mean anomalies for a 5 yr mission starting at 0, 0 is shown by the red lines, with 13 day observation opportunities at the magenta circles.The points of closest approach are marked with a black cross.

Figure 7 .
Figure 7. Survey completeness for 10,000 PHA orbits with a 140 m diameter PHA and nominal thermal parameters for (a) 5 yr survey duration and (b) 10 yr survey.

Figure 8 .
Figure 8. Annually averaged spatial response of NEO Surveyor.The color scale gives the fraction of time a given point in the ecliptic plane is detectable.The Earth orbit is the blue circle.Red ellipses are zero-inclination NEO orbits with a = 1.75 au and e = 0, 0.1.0.2, K 0.9.Red dots are marked at intervals of 5°in mean anomaly.

Figure 9 .
Figure 9. Calculation of integral completeness over the thermal model and diameter distributions.(a) Distribution of population with respect to thermal parameter (a combination of the Bond albedo, A, and beaming parameter, η, from Section 3.2) and diameter (power-law index −2.5),including marginal profiles.Distributions are normalized such that the population integral for objects larger than 140 m is unity.(b) 5 yr survey completeness for NEOs as a function of thermal model and diameter.The contour levels (0.1, 0.2, K 0.9) follow an interpolation of the model runs at the black dots.(c) Distribution of detected objects from the product of (a) and (b).(d) Detected objects averaged over the thermal model parameters (red curve) compared to parent population distribution (blue curve).The net completeness for diameters larger than 140 m is 0.76 (ratio of areas to the right of the dashed line).

Figure 10 .
Figure 10.Survey completeness for NEOs larger than 140 m vs. time, as predicted by the detailed NSS time-domain simulation (blue; from Mainzer et al. 2023) and this model (red).

Figure 11 .
Figure 11.Two NEO orbits with the same orbital elements, except the left panel has inclination = 0°, and the right panel has inclination = 60°.

Figure 12 .
Figure 12.Sample of 1000 potentially hazardous orbits with MOID < 0.01 au.(a) and (b) Relative probability of impact, logarithmic color scale.(c) and (d) Impact speed.

Figure 13 .
Figure 13.NEO Surveyor 5 yr survey completeness as a function of hazard for a 140 m diameter asteroid in a random sample of 1000 PHA orbits.

Figure 14 .
Figure 14.(a) 5 yr survey completeness as a function of hazard for asteroid diameters of 35-1600 m.Each dot represents the completeness for a given PHA orbit and asteroid diameter.The dashed line is an approximate delineation between objects that are found by NEO Surveyor and those that are not.(b) Relative probability of impact per half-decade of impact energy.(c) Probability of prior discovery by 5 yr NEO Surveyor survey for each half-decade of impact energy.

Figure 15 .
Figure 15.Probability of prior discovery of impactor with energy exceeding the value on the x-axis.The wavy structure in the curves is an artifact from the discrete sampling of the asteroid diameter distribution (Figure 14(a)).

Figure 20
Figure 20 shows the lower end of the MOID distribution for the Granvik synthetic PHA population of 144,000 orbits.The probability density is approximately constant for values less

Figure 20 .
Figure 20.Histogram of MOID for the 144,000 orbits in the Granvik PHA synthetic population.