Near-Earth Object Observations using Synthetic Tracking

Synthetic tracking (ST) has emerged as a potent technique for observing fast-moving near-Earth objects (NEOs), offering enhanced detection sensitivity and astrometric accuracy by avoiding trailing loss. This approach also empowers small telescopes to use prolonged integration times to achieve high sensitivity for NEO surveys and follow-up observations. In this study, we present the outcomes of ST observations conducted with Pomona College’s 1 m telescope at the Table Mountain Facility and JPL’s robotic telescopes at the Sierra Remote Observatory. The results showcase astrometric accuracy statistics comparable to stellar astrometry, irrespective of an object’s rate of motion, and the capability to detect faint asteroids beyond 20.5th magnitude using 11 inch telescopes. Furthermore, we detail the technical aspects of data processing, including the correction of differential chromatic refraction in the atmosphere and accurate timing for image stacking, which contribute to achieving precise astrometry. We also provide compelling examples that showcase the robustness of ST even when asteroids closely approach stars or bright satellites cause disturbances. Moreover, we illustrate the proficiency of ST in recovering NEO candidates with highly uncertain ephemerides. As a glimpse of the potential of NEO surveys utilizing small robotic telescopes with ST, we present significant statistics from our NEO survey conducted for testing purposes. These findings underscore the promise and effectiveness of ST as a powerful tool for observing fast-moving NEOs, offering valuable insights into their trajectories and characteristics. Overall, the adoption of ST stands to revolutionize fast-moving NEO observations for planetary defense and studying these celestial bodies.


Introduction
Observing near-Earth Objects (NEOs) holds significant importance for planetary defense, solar system formation studies, and resource mining applications.While meter-size or smaller NEOs harmlessly disintegrate in the Earth's atmosphere, larger ones can cause devastating damage.The US Congress mandated NASA to find NEOs larger than 140 m with at least 90% completeness due to the potential regional devastation caused by such impacts (NSTC 2023).NASA's Near-Earth Object Observations programs (NEOO) fund projects to discover, track, and characterize NEOs in response to this mandate.
Currently, we have identified approximately 40% of NEOs larger than 140 m, leaving about 15,000 NEOs to be discovered (NSTC 2023).Current surveys, such as the Catalina Sky Survey (CSS) and PanSTARRS, are producing more than 3000 NEOs per year (https://cneos.jpl.nasa.gov/stats/siteall.html) with about 500 larger than 140 m.While the detection rate has been steadily increasing, finding 90% of the NEOs of size 140 m or larger can easily take an extra 20 years.Fortunately, the upcoming Rubin Telescope and NEO Surveyor Mission are expected to accelerate the discovery process (NSTC 2023).
However, we cannot be optimistic because NEOs smaller than 140 m can still be very hazardous and the frequency for smaller asteroids to impact Earth is much higher than that of larger asteroids (NSTC 2023).The incident of the Chelyabinsk meteor (Brumfiel 2013) measuring about 20 m underscores the need to detect potential threats from NEOs larger than 10 meters.Therefore, NEOO seeks to inventory all the NEOs that could post a threat or serve as potential mission targets.NEOs smaller than 140 m constitute a much larger population (Tricarico 2017) with the vast majority of their threats remaining unknown (NSTC 2023) because their smaller sizes require closer proximity to Earth to be sufficiently bright for observation.The associated trailing loss from the faster motion rate becomes a substantial hurdle for surveying small hazardous NEOs.Synthetic tracking (ST) is a powerful technique designed to detect fast-moving NEOs and perform follow-up observations (Shao et al. 2014;Zhai et al. 2014;Heinze et al. 2015).
Enabled by CMOS cameras and modern GPUs, ST takes multiple short-exposure images to avoid trailing loss associated with traditional long exposure (∼ 30-second) CCD images and integrates these short-exposure images in post-processing using GPUs.CMOS cameras can read large format frames (∼60 Mpixel) at high frame rates with read noise of only about 1e per read 1 .Such a low read noise means even during dark times near the new moon, taking frames at 1 Hz, the read noise is still lower than the sky background noise for an 11-inch telescope.ST avoids trailing loss using a high frame rate (short exposure time) to make NEO motion negligible compared with the size of the point-spread-function (PSF), which is typically 2 arcsec.For surveying NEO, a 1 Hz frame rate is usually sufficient to avoid trailing loss because NEOs would unlikely move more than a typical PSF size of 2 arcsec during 1 sec2 .
A single short exposure image in general does not suffice for detecting new NEOs, therefore, we need to integrate many frames (of order 100) to improve the signal-to-noise ratio (S/N) in post-processing.For follow-up observations, this task can be readily carried out because we know approximately the rate of motion, so in post-processing, we can stack up the images according to the motion to track the target.Even though the rate of motion may not be very accurate in case the ephemeris is off, the effort to find the best tracking can be made by adjusting the tracking with a least-squares fitting.For detecting new NEOs, this effort of post-processing integration is very large because we need to search over a large set of trial velocities, which is typically a 100×100 velocity grid for us.To speed up this, ST uses modern GPUs that offer thousands of processors at a low cost.For example, using the Nvidia V100 GPUs, we can keep up with real-time processing for our NEO survey experiments using an exposure time of 5 seconds.ST has demonstrated success in detecting small NEOs of ∼ 10 m.These objects tend to move fast (> 0.5 arcsec/sec) and often elude surveys like PanSTARRS and CCS due to the excessive trailing loss.With the capability of integrating a long time (many frames), ST empowers small telescopes to detect faint objects, a feat unattainable without this technique.
The flexibility of ST post-processing has many advantages over the traditional long-exposure approach.ST can track both the target and stars, thus, producing more accurate astrometry than the traditional approach that has to deal with centroiding streaked objects leading to degraded precision as the rate increases (Vereš et al. 2012).We have demonstrated 10 mas level NEO accuracy using ST (Zhai et al. 2018) with typically better than 10 mas astrometric solutions.To achieve 10 mas level NEO astrometry, we found it necessary to correct the differential chromatic refraction (DCR) effect of the atmosphere to account for the wavelength dependency of air refraction.
In addition, we found ST robust against star confusion in performing follow-up observations, where we can exclude the frames where the NEO gets very close to a star.
The chance of confusion increases with the rate of motion, so traditionally it would be hard to avoid the contamination of the streaked stars when tracking fast-moving NEOs without using ST.Another advantage of using ST is its proficiency in recovering NEOs with highly uncertain ephemerides, where the significant rate errors would make the traditional approach fail to track these NEOs.This paper presents the results and data processing from using ST for NEO observation.
The paper is organized as follows: In Section 2, we describe the instrument, operations, and data processing involved in using ST to observe NEOs.In Section 3, we present results showcasing the advantages of employing ST.Finally, we conclude with an outlook on the future of ST in NEO observation.

NEO Observation using Synthetic Tracking
ST necessitates the capture of images with an exposure time short enough to prevent significant NEO motion relative to the size of the PSF.However, this must be balanced with the potential increase of read noise from reading out images too rapidly.Consequently, determining the ideal exposure time and the number of frames becomes a critical decision, influenced by the system's hardware configuration and the prevailing sky background level.
In this section, we offer a comprehensive overview of our instrumentation and elaborate on the operational strategies, with a specific emphasis on the meticulous design of observation cadence.

Instrument Description
Our NEO observations use a total of three systems, each with distinct key parameters outlined in Table 1.The first system comprises Pomona Coellge's 40 inch telescope located at the Table Mountain Facility (TMF).This Cassegrain telescope features a 1 m f/2 primary mirror with a 30 cm secondary mirror, resulting an effective focal length of 9.6 meter for the imaging system.A Photometrics 95B Prime sCMOS detector is installed at the Cassegrain focus with a pixel array size of 1608×1608.The 11µm pixel corresponds to a scale of 0.226 ′′ per pixel enabling a critical sampling of PSF for our best seeing conditions of 1.5 as at the  TMF.The field of view (FOV) is 6 ′ ×6 ′ .This system does not have any refractive elements, thus its field distortion is insensitive to color making astrometric calibration easier.We have used it to achieve 10 mas level NEO astrometry (Zhai et al. 2018).
We have built two additional robotic telescope systems using commercial off-the-shelf (COTS) telescopes from Celestron located at the Sierra Remote Observatory (SRO).One system (SRO1) consists of three 11-inch RASA telescopes at f/2.2 arranged with offsets in Declination, giving approximately a total FOV of 6.6 deg × 3.3 deg.We use SRO1 to survey NEOs nominally.The other system (SRO2) has a single 14-inch RASA telescope for follow-up observations.We use both the ZWO and QHY 600 Mpixel CMOS cameras using the Sony IMX 455 Chip, which has a pixel size of 3.76µm giving pixel scales of 1.26as and 0.98as respectively for the SRO1 11-inch and SRO2 14-inch telescopes.The relevant parameters are listed in Table 1.

Configuring Science Observations
For operation, we want to maximize the instrument S/N in determining the exposure time and number of frames to acquire.To minimize trailing loss, the exposure time should be as short as possible.However, for the same amount of integration time, using shorter exposure increases the number of reads, thus the read noise.We now discuss how to choose an appropriate exposure time to balance the trailing loss and the total amount of noise.
The total background noise per pixel can be modeled as the RSS (root-sum-squares) of the read noise, dark current, and background illumination: where σ rn is the standard deviation of read noise, ∆t is the exposure time, I bg is the sky background, and I dark is the detector dark current.It is convenient to define a time scale τ 2 for the variance of the read noise to be the same as that of the noise from the background illumination plus the dark current as We can factor where the second factor shows the contribution of read noise to the total noise, which increases as we shorten the exposure time ∆t.When τ 2 ≪ ∆t, we are background noise limited, the total noise σ n only increases slowly when shortening exposure time ∆t.When ∆t is not much larger than τ 2 , the read noise factor 1 + τ 2 /∆t becomes sensitive to the variation of ∆t.
Fig. 1 illustrates the relationship between trailing loss and streak length (for detailed derivations, refer to Appendix A).The rule of thumb for using ST to observe NEOs is to set an appropriate exposure time ∆t so that even the most swiftly moving NEOs within the scope of interest do not result in streaks spanning more than one PSF.This constraint effectively contains trailing loss to below 12%.
It is useful to introduce a scale for rate of motion as F W HM/τ 2 , corresponding to a streak length of the PSF's FWHM for exposure time ∆t = τ 2 .If the rate range of interest is much less than F W HM/τ 2 , we then can easily choose an exposure time ∆t to be larger than τ 2 for pixel noise to be background noise limited and simultaneously having very little trailing loss.This is the typically the case for using a CMOS camera to observe NEOs because typical CMOS cameras have only 1-2 e read noise when operating in rolling shutter mode (for example, see https://www.photometrics.com/products/prime-family/prime95b).
For Pomona College 40 inch telescope at TMF, the τ 2 is less than 0.3s.Assuming PSF FWHM is 2 as, F W HM/τ 2 ∼ 6.7as/sec is much higher than typical NEO's sky rate.
To optimize the sensitivity, we include the trailing loss and read noise factor together to define a detection sensitivity S(v, ∆t) as Fig. 2 displays contours of constant values of S(v, ∆t) as a function of ∆t and v in units of ∆t/τ 2 and FWHM/τ 2 respectively.For a given rate of motion range, there is an optimal exposure time marked by the red dashed line.When the rate of motion < FWHM/τ 2 , we have have a pretty good sensitivity of 0.7 with an optimal choice of exposure time ∆t/τ 2 ∼ 1.5.If rate of motion < 0.3 FWHM/τ 2 , this can be improved to 0.85.As an example, considering again our telescope at TMF with τ 2 = 0.3s and FWHM = 2 as giving the unit for velocity FWHM / τ 2 ∼ 2/0.3 = 6.7(as/s).If we are interested in NEOs moving as fast as 1as/sec, or 0.15 in units of FWHM/τ 2 , we have a range of exposure times would achieve better than 0.85 detection sensitivity regarding trailing loss and read noise trade The red dashed lines represent optimal exposure time for a given velocity.
Discussion above is mainly for observing faint objects, especially for discovering new NEOs, where we have ignored the photon noises from the target itself.In case of observing a bright target, whose photon noise is much higher than the total read noises of all the relevant pixels within a PSF, we in general only need to consider the trailing loss to use an exposure time so that v max ∆t < FWHM.
After choosing an exposure time, the second factor to consider is the total integration time T = N f ∆t, or the number of frames N f to integrate for the final SNR to be sufficient for detection or achieving certain astrometric accuracy.For example, at SRO, we require a detection threshold of 7.5 for a low false positive rate of 2% per camera field (Zhai et al. 2014).The 40 inch telescope at TMF is mainly used for follow-up observations providing highly accurate astrometry.We have been targeting at better than 100 mas accuracy, which for a PSF size of 2 arcsec, this means the SNR would need to be at least 13 in view of uncertainties of centroiding is ∼ 0.64 FWHM/SNR (Zhai et al. 2014).Using results in Appendix A, we have the total SNR where have introduced total number of photoelectrons N bg = T 10 −0.4(m b −m 0 ) a 2 and N target = T 10 −0.4(mt−m 0 ) for the sky background and target, and total dark counts N dark = T I dark with m 0 being the telescope system zero point (stellar magnitude giving 1 photoelectron/sec), a being the pixel scale, m b being the background brightness per pixel measured in magnitude, and m t being the target brightness.In general, using ST, we operate with S(∆t, v) > 0.7 for most of the NEO observations.

Survey and Follow-up Observation Cadence
We have been experimenting with the SRO1 system to detect new NEOs.We use a 5-second exposure and integrate 100 frames to reach a detection limiting magnitude of about 20.5 for dark nights near new moon (see the example of the last subsection).
The SRO1 system has a combined FOV of about 20 sqdeg from the three RASA 11-inch telescopes.On average, we spent about 700 seconds per pointing, which includes slew time, refocusing time (every 8 pointings), and an extra waiting time for the synchronization of the three telescopes especially the extra 800-900 msec dead time that the ZWO camera has between 5-sec frames while QHY cameras do not have this dead time.We scan along the RA four consecutive FOVs and then repeat the scan for confirmation.Repeating scan is operationally inefficient and we are working on a software capability to use the SRO2 system to do follow-up observations upon a detection from SRO1.This triggered follow-up allows SRO1 to scan the sky at a rate two times faster (without the burden of the revisit).
We regularly perform follow-up observations for NEO candidates from the Minor Planet Center's confirmation page (NEOCP) using the system at the TMF.Because the telescope is sufficiently large, the frame is dominated by sky noise, i.e., τ 2 ∼ 0.3s is small relative to ∆t, which we typically use 1s, 2s, and 3s exposures and integrate.We usually integrate 300 or 600 frames depending on the brightness and rate of the target.
The 14-inch telescope system (SRO2) performs follow-up observations for candidates with large uncertainties in their ephemerides.These candidates are not suitable for the TMF 40 inch telescope to follow due to the small FOV.For NEO candidates from SRO1, we use also 5-second exposure and a 100 frame integration.Our SRO1 system uses S/N threshold of 7.5 to survey NEOs.The larger collecting area of 14-inch (versus 11 inch) and better imaging quality gives us an improvement factor of about 1.6 in S/N, thus SRO2 can reliably confirm the candidates with an S/N of 1.6×7.5 = 12 unless they are false detections.This telescope has been also used to confirm objects from the Zwicky Transit Facility (ZTF) and NEO candidates from the NEOCP.We are developing software to fully automate the operation of SRO2 to schedule and perform follow-up observations, as well as processing and submitting the data.

Calibration
For calibration, we generate a mean dark frame, a flat field response, and a list of bad pixels.The mean dark frame is estimated by averaging multiple dark frames taken with the same exposure time as the science data.The flat field response, which physically is the product of the relative pixel quantum efficiency and optical throughput, can be measured by observing the twilight sky.The flat field response can be computed by taking an average over multiple measurements and then normalized so that the mean response over the whole field is 1.We generate a list of bad pixels by applying a noise level upper limit threshold, a dark level upper limit threshold, and a lower limit threshold for flat field response.
The twilight flat field calibration can be performed in two ways.One setup is to take the measurements when the twilight light is much stronger than any stars in the field so that photons from stars in the field can be ignored relative to the sky background.To avoid saturation, we typically use a very short exposure (no more than 0.1s) to keep the pixel light level for the twilight sky at about half full-well counts.Turning off the tracking to let stars drift in the field helps because the trailing loss further reduces the star lights relative to the twilight sky background.We usually take hundreds of frames and it is straight forward to take an average over these frames and perform a normalization to yield a flat field response.However, this approach requires the experiment to be carried out in a very limited time window during twilight.
In case of missing the desired twilight time window, an alternative approach for flat field response calibration can be employed.This approach involves activating sidereal tracking and deliberately shifting the pointing to capture multiple sky background images with stars at different pixel locations in the field.This diversification guarantees that each pixel has multiple opportunities to exclusively capture the sky background free of star-generated photons.Subsequently, the data is processed by first eliminating pixel data where star signals are detected.After scaling the sky background of each sky image to the same level, an average can be computed for each pixel across the image set, exclusively considering instances when the pixel registers the sky background without any star signals.
The data processing is slightly more involved, but we gain the flexibility of when to take the data.This approach works even when the sky background is not high, where a longer integration can be implemented as needed.

Data Processing
The framework and procedural stages of data processing have been outlined in Zhai et al. (2014).For follow-up observation data processing, Zhai et al. (2018) provides a thorough description of how to generate astrometry for observing known NEOs.Here we give an overview of the data processing, highlight how ST identifies targets, and detail in generating highly accurate astrometry by correcting the DCR effect of the atmosphere as well as accounting for accurate timing when stacking up frames.

An overview of synthetic tracking data processing
In the contrast to conventional asteroid detection data processing (Rabinowitz 1991;Stokes et al. 2000;Denneau el al. 2013), ST works on a set of short-exposure images, which we call a "datacube" because of the extra time dimension in addition to the camera frame's row and column dimension.The goal of data processing is to 1) identify the stars in the field and find an astrometric solution to map sky and pixel coordinates; 2) detect significant signals (search mode) or identify target (follow-up); 3) estimate astrometry and photometry for the detected objects or follow-up target.
The data processing consists of three major steps: 1. Preprocessing, where we apply calibration data, remove cosmic ray events, and re-register frames to get data ready; 2. Star field processing, where we estimate sky background, identify stars in the field, and match stars against a catalog; 3. Target processing, where we identify the target and estimate its location, rate of motion, and photometry.
Preprocessing is instrument dependent and generally requires subtracting a mean dark frame of the same exposure time from each frame and then dividing each frame by a flat field response to account for the throughput and QE variation over the field (see subsection 2.2.3).While a well-tracking system may not need re-registration, a re-registration is needed for our systems, which could drift more than 10 arcsec during the course of an integration.
Re-registration can be done by estimating offsets between frames by estimating positions of one or a few bright stars in each frame or cross-correlating Fourier transforms of each frame.We then remove the cosmic ray events and bad pixel signals by setting the values at these pixels to a background value.Cosmic ray events are identified as signal spikes above random noise level localized in both temporal and spatial dimension.
Star field processing first detects stars in the field by co-adding all the frames with stars well-aligned after the frame re-registration.A least-squares fitting to a Gaussian or matching algorithm (Padgett et al. 1997) identifies stars in the field by matching similar triangles formed by triplets of stars at the vertices from both the field and the catalog, where the shapes of triangles are determined by the relative distances between the stars.
To reduce the computer time for matching stars, we usually need to know in advance the approximate location of the field in the sky, the pixel scale, and the size of the FOV to look up a star catalog, which is the Gaia Data Release 2 (DR2) (Gaia Collaboration et al. 2016) for our data processing.To avoid excessive combinations of triplets of stars, this process starts with a small subset of the brightest stars in the field.A pair of correctly matched triangles in the field and catalog gives an affine transformation between the pixel and sky coordinates.The affine transformation from correctly matched triangles should transform other stars in the field to sky positions close to their catalog positions.This is typically used as a criterion for validating a star matching; a large percentage of stars in the field should be matched with the catalog.Using the matched stars, we can solve for the mapping (the astrometric solution) between the pixel coordinate and the position in the plane of the sky as an affine transformation, and thus the right ascension (RA) and declination (Dec).Because of non-ideal optics, we often need to go beyond the affine transformation to use two-dimensional lower order polynomials to model the field distortions for more accurate astrometric solutions (see Zhai et al. (2018) for details).For our TMF system with only a FOV of 6', an affine transformation is sufficient for 10 mas accuracy.A 3rd order polynomial is needed to achieve 5 mas accuracy.For SRO1 and SRO2, a 3rd order polynomial is sufficient for achieving 50 mas astrometric solution.
Target processing encompasses identifying a specific follow-up target or searching for new objects.In general, we first removed star signals to by setting pixels near detected stars to zero assuming we have estimated and subtracted the sky background (Zhai et al. 2014), so that we deal with frames with noises and signals from the target or objects to be detected.For a follow-up target with a known sky rate of motion, we can stack up images to track the target.We also apply a spatial kernel matching the PSF to improve S/N.The target is located by finding the pixel that has the highest S/N in the expected region of the field.Sometimes, the target is too faint or the ephemeris has uncertainties larger than what was estimated, human intervention is needed to help identify the object in the field.

Search for NEOs using GPUs
The advantage of ST for NEO search and recovery is the improved sensitivity from avoiding the trailing loss at the price of a large amount of computation for processing the short exposure data cubes.For example, our SRO1 system typically uses a 5 second exposure time and integrates 100 frames.The camera frame size is 61 MPix giving a data cube of size about 12.2 GB stored in raw data as unsigned 16-bit integers.During data processing, the data are stored as 32-bit floating point numbers, which means 24.4 GB of memory.Our velocity range of interest is ±0.63as/sec (0.5 pixel/sec for a pixel scale of 1.26 as) over both RA and Dec and we use a 100×100 grid to cover this range with a grid spacing of 0.0126 as/sec.This means that our maximum rate error is about ±0.0063as/sec.
For 500-second integration, the maximum streak length along a row or column due to this rate error is about 3.2 as or 2.5 pixels.Since our best PSF has a FWHM of about 2.5 pixels, the trailing loss due to digital tracking error is less than 12% as shown in Fig. 1.The amount of computation is 61 × 10 6 × 100 × 100 × 100 ∼ 6.1 × 10 13 FLOPS per data cube.
Fortunately, modern GPUs like a Tesla V100 allow us to process data in real-time; a single Tesla V100 with 32 GB memory can perform the search in about 440 seconds.
The performance is not limited by the GPU's processing speed but by the memory bandwidth, especially how to efficiently use the cache memory.We note that the velocity grid spacing is determined so that the rate error due to discretization only causes a streak (in post-processing) of no more than 1 PSF per integration.A typical velocity grid spacing is then 2 PSF per integration time.For our SRO1 system, the velocity grid spacing is We note that the range of rate for searching a moving object is only limited by the total amount of computation needed.With multiple GPUs, it is possible to search over even larger range of rate to detect for example earth orbiting objects.

Reduce systematic astrometric errors
Gaia's unprecedented accuracy allows us to push NEO astrometry to 10 mas (Zhai et al. 2018).Highly accurate astrometry requires properly handling systematic astrometric errors such as the DCR effect, star confusion, and timing error.

Difference Chromatic Refraction (DCR) Effect
Unless observing at the zenith, the light rays detected are bent by the atmosphere due to refraction.Because the index of air refraction depends on the wavelength of light ∼ 1/λ 2 (Ciddor 2002), the atmospheric refraction bend more the blue light than red light.This introduces a systematic error in astrometry, the DCR effect, if the target and reference objects have different spectra.If a narrowband filter is applied, the DCR effect becomes much less because the variation of atmospheric refraction is significantly reduced by limited passband.However, to detect as much photon as possible to improve S/N, we typically use broadband or clear filters.
DCR effects can be modeled using an air refraction model (Ciddor 2002) where θ z is the zenith angle, complementary to the elevation angle, ϕ z is the parallactic angle between the zenith and celestial pole from the center of the field.In general, we do not have spectral information of NEO candidates from the NEOCP, so we assume a solar spectrum for them assuming they reflect sun light uniformly across the band as a leading order approximation with C tar ≈ 0.85 (estimated using Fig. 3 in Andrae et al. (2018) assuming an effective temperature of 5800 K for solar spectrum).
As an example, when we use a clear filter, without any DCR correction, the astrometric residuals for a field observing Feria ( 76) with a clear filter are displayed in the left plot in Fig. 3, where the dominant astrometric residuals are along the direction of zenith from the field.Using Eq. ( 7) to correct the DCR effect for a target with solar spectral type, we significantly reduced the RMS of the residuals from more than 40 mas to about 15 mas and the directions of residuals appear random.As a comparison, if we apply a Sloan i-band  showing consistency with Gaia DR2 an RMS of 5 mas using a third order polynomial field distortion model.

Target position estimation and confusion elimination
Star confusion is another source of systematic errors for astrometry.We exclude the frames where the NEO gets close to a star whose light could affect the centroiding of the NEO.When excluding frames to avoid star confusion, we need to derive astrometry based on the timestamps of the frames that do not have confusion.For our TMF system, we are confident that our timing accuracy is better than 10 msec, which was confirmed by the small (< 0.1 as) astrometric residuals from observing a GPS satellite (C11) relative to the ephemerides from Project Pluto (https://www.projectpluto.com/) and the results from the International Asteroid Warning Network (IAWN) 2019 XS timing campaign (Farnocchia et al. 2022) that we participated in.

Results
In this section, we present results from our instrument on the Pomona 40-inch telescope at the TMF and robotic telescopes at the SRO.Our instrument at TMF (654) has consistently produced accurate astrometric measurements and our robotic telescopes at SRO has been able to detect faint NEOs at about mag = 20.5.

Astrometric Precision using Synthetic Tracking
Synthetic tracking avoids streaked images by having exposures short enough so that the moving object does not streak in individual images, allowing us to achieve NEO astrometry with accuracy similar to stellar astrometry.We have been regularly observing NEOs from .    .The DCR correction is necessary for achieving 10 mas astrometry consistency with the Gaia DR2 catalog as shown in Fig. 3 and Fig. 5 when using a clear filter.We also found that the DCR correction would give consistent astrometry for bright NEAs like Freia (76).The fact that the DCR correction makes broadband astrometry much closer to an i-band filter with lower astrometric residuals as shown in the right plot in Fig. 3 validates our approach of DCR correction.We assume solar spectrum for NEOs by default.However, NEO spectra can deviate from the solar spectrum, so improvements can be made by measuring the color of the NEOs.To our knowledge, vast majority of the NEO observations do not correct the DCR effects, therefore it is possible for these astrometric measurements to be biased due to DCR effects depending on average elevations of observations; measurements with DCR corrections like ours only appear to be biased along Dec because DCR effects along RA can be positive or negative depending on the sign of hour angles, thus not introducing an overall bias.It would be interesting to further understand this bias by comparing all the observations with and without DCR corrections and the filters applied.

Detection Sensitvity
Using ST to survey fast-moving NEOs avoids the trailing loss, and thus reaches detection sensitivity as if we were tracking the targets.This enables small telescopes to observe and detect NEOs using long integrations, which would not be possible if the trailing loss degrades the S/N as we integrate long (see Fig. 2 for the degraded sensitivity when ∆t increases beyond the optimal exposure time).We have been experimenting with our 11-inch telescope system SRO1 (U68) to survey NEOs with a 5-second exposure time and an integration of 100 frames, giving a limiting magnitude of ∼ 20.5 for clear dark nights.deg/day, (-7.6, 8.1) deg/day along (RA, Dec) with apparent magnitude of 20.5 mag.If using an exposure time of 30 s, the streak length would be 13.8 arcsec.Assuming 2 as FWHM for PSF, the trailing loss would be a factor ∼ 13.8/(1.5* 2) 4.6, so more than 1.5 stellar magnitude fainter.(For PanSTARRS, this would even more as the integration time is 45 s and the PSF is more compact than 2 as).We detected this object as an SNR about 8.3 enabled by ST.

Robustness against Star Confusion
Confusion occurs when a NEO moves very close to a star.If tracking the NEO, the stars streak.In case of confusion, the streak of a star would overlap with the tracked NEO.This is an extra burden for operations to avoid confusion, which can be challenging if observing a dense field.ST is robust against this confusion because we take short    to measure the size of PSF and the streak length instead of using 4σ g , we summarize our results for convenient use as the follows.
For short streak length L < 3 FWHM, an approximate reduction factor of SNR is given by For long streak length L > 3 FWHM, sensitivity reduction factor of SNR is given by ) passband used for observations and this dependency is captured by parameters a and b.To maximize photon usage for sensitivity, we use the full band of the CMOS ("clear filter") by default.We observed the Freia (76) asteroid using a clear filter and also a Sloan i-band filter.Without and DCR correction, we have residuals as displayed in the left plot in Fig. 3 where we can see errors shown as a two-dimensional vector tend to align with the direction allows us to determine coefficients a and b empirically.For example, for our "clear band," a ≈-168 mas and b ≈ 20 mas.In contrast to the clear filter, the residuals shown in Fig. 4 for the i-band filter is hard to identify and the color dependency is hard to see suggesting that a and b for i-band is smaller than 10 mas.We also display the astrometric residuals with i-band filter and the DCR effects are much smaller buried in the random noises as shown in the right plot in Fig. 19.

Fig. 1 .
Fig. 1.-Detection sensitivity with trailing loss as function of the streak length measured in FWHM of PSF.

Fig. 2 .
Fig.2.This is consistent with our discussions regarding the regime where the read noise is much lower than the background noise.Trailing Loss Factor vs Exposure and Rate of Motion NEO search mode or when recovering follow-up targets with large uncertainties in ephemerides, we need to use GPUs to perform shift/add over a grid of velocities covering the rate of interest to detect signals above an S/N threshold, which we use 7.5 to avoid false positives (Zhai et al. 2014).The detected signals are clustered in a 4-d space (2-d position and 2-d velocity) to keep only the position and velocity with the highest S/N.The last step in target processing is a least-squares fitting (moving PSF fitting) using a model PSF to fit the intensities of the moving object in the datacube to refine the positions and velocities of the detected signals 500 sec ≈0.01 pix/s.The velocity grid is ± 50 in both RA and Dec giving a range of rate of ±0.63as/s.When we recover an NEO whose ephemeris becomes highly uncertain, we need to search in the neighborhood of the expected location according to the ephemeris and to cover at least a region of the 3-σ uncertainty of the ephemeris.The velocity grid to search should cover around the projected rate of motion also covering at least the 3-σ uncertainty of the rate.We use SRO2 to do NEO recovery.Because 0.63 as/s ≈15 deg/day, for a newly discovered object without follow-up for 1-2 nights, we typically can recover these objects without trouble because typically the position errors are less than 10 deg and rate errors are less than 10 deg/day.Since SRO2 has a FOV size of 2.6 deg × 1.7 deg and for recovery, the uncertainties is along the track of the NEO, thus we only need a one-dimension (instead of 2-d) search, so the computation load is much less for recovering an object than the general NEO search.
and the spectra of target and reference objects (Stone 1996).For 10 mas accuracy, we found it sufficient to use a simple empirical model based on color defined as difference of Gaia magnitudes in blue and red passbands (Andrae et al. 2018) as discussed in appendix B. The DCR correction in RA and Dec between reference color C ref = (B − R) ref and target color C tar = (B − R) tar is expressed as

Fig. 3 .
Fig. 3.-Astrometric solution residuals using a 3rd order polynomial to fit field distortions with the Gaia DR2 catalog to show differential chromatic refraction (DCR) effect with a clear filter.Left plot shows residuals without DCR correct and right plot shows residuals after correcting DCR effect using a simple quadratic color model.

Fig. 4 .
Fig. 4.-Astrometric residuals of a field observing asteroid Freia (76) using an i-band filter the DCR effects are too small to identify.
The centroiding error due to star confusion in units of the FWHM of the PSF is estimated as the gradient of the intensity of the star at the NEO's location relative to the NEO peak intensity divided by its FWHM ∼ |grad(I star (at N EO))|/ (I NEO (peak)/F W HM ).Based on this estimation, we exclude frames that could lead to centroiding errors above a threshold, e.g.0.01, which should be determined by accuracy requirement assuming this corresponds to a centroiding error ∼ 0.01 FWHM.The "moving-PSF" fitting is done for frames without confusion.The general cost function for the least-squares fitting of "moving PSF" to the timestamps from non-real-time operating systems can have errors due to indeterministic runtime behaviors.Hardware timing can be achieved by using a GPS clock to trigger the start of the exposure at a preset time or by letting a GPS clock record the signals generated by a camera upon the completion of a frame.For example, our Photometrics Prime 95B used at TMF can accept a trigger signal from a Meinberg GPS clock to initiate the exposure of frames.Camera reference manual should be referred to interpret timestamps correctly.For example, CMOS cameras have both global shutter and rolling shutter modes and there could be dead time between frames.We usually operate CMOS cameras in the rolling shutter mode for high frame rate and low read noise.Rolling shutter mode delays the exposure time window of each row of the image by a small constant time offset relative to the previous row in the order of readout.It is important to account for this small time delay between the consecutive rows because we usually have the timestamps for reading out the first or last row, but the target is observed at some row in between.This delay is 19.6 usec per row for our Photometrics camera at TMF.A useful test for understanding the details of timing is to use a GPS clock to trigger both the camera and an LED light and examine the recorded frames.Using this test, we found our Photometrics camera has an extra delay of 50 msec for the first frame to start after the trigger signal for the camera.

Fig. 8 .
Fig. 8.-Dstribution of standard deviations of the measurements for each target (right), where the black dashed line represent an estimated value of the 1-σ uncertainty of the astrometric measurements (left) and the dependency of residual standard deviations versus rate of motion (right).

FigFig
Fig. 9.-Rate of motion versus apparent magnitudes of the NEOs from the confirmation page during 2021/10 to 2022/04.

Fig
Fig. 14.-An example of NEO (2022 UA28) discovered by JPL's robotic telescope at SRO (U68) Left image shows integration tracking 2022 UA28 (at the center) and right image is the integration tracking sidereal with the NEO track marked as the green dashed line, where the trailing loss makes NEO signal buried in noises..

3. 4 .
Recovery of Candidates with Highly Uncertain EphemeridesFast-moving NEOs need timely follow-up observations after the initial detection.NEOs may be discovered without timely follow-up observations due to the unavailability of follow-up facilities, poor weather/day-light conditions for observation, or latency in data processing.Fast-moving objects tend to develop a large uncertainty quickly due to the propagation of errors.Because most of the follow-up facilities do not have large FOVs and the vast majority of the facilities rely on tracking the object to avoid trailing loss, it is quite hard to recover an object that has larger than 1 deg angular uncertainties in the sky position.Fortunately, our SRO2 system with ST has good capability to recover NEO candidates with relatively large uncertainties because ST does not require accurate knowledge of the rate of motion and our 4.47 sqdeg FOV is capable of searching efficiently a large portion of the sky.To illustrate this capability, Table3summarizes four examples of recovery using SRO2, where we show the large uncertainties of the ephemerides derived from the initial discovery in the second column and the last column is the uncertainties after we recover the object.For example, 2021 TZ13 was discovered on 20211010, based on only 4 observations, the ephemerides are highly uncertain with error 1-5 deg.Our SRO2 recovered this object, which otherwise would have been lost.Other examples of recovery are for 2022MD3, 2023 BH5, 2023 BE6.
Fig. 19.-Astrometric errors due to DCR effect is approximately a linear function of color in left plot for a clear filter; the dependency is not significant for i-band filter as shown in the right plot.

Table 1 :
System Parameters

Table 2 :
JPL Horizons NEO Ephemeris Uncertainties and the Residual Biases

Table 3 :
NEO that are recovered succesfully