The following are detectable with radar at Arecibo, if visible (+0 to +37 degrees declination):
The ability to detect an asteroid with radar is dependent on many factors: the properties of the transmitting and receiving system(s), the properties of the asteroid, and the geometry of the observation. The emitted signal depends on the power output of the transmitter and the antenna gain of the transmitting station. The signal weakens with the distance to the target by the inverse-square law. The signal intercepted by the target depends on the target diameter, and the signal reflected depends on the radar reflectivity (radar albedo) of the target. The signal weakens again by the inverse-square law as it returns. The recorded signal then depends on the effective collecting area of the receiving antenna.
The corresponding amount of recorded noise depends on system temperature of the receiver across the bandwidth of the echo due to the rotation of the asteroid. The bandwidth of the echo is directly proportional to the target diameter and inversely proportional to the rotation period and sub-rdar latitude. Therefore, viewing a large target with a short rotation period along the equator produces a wide echo bandwidth due to the size and rotational velocity of the target along the observer line of sight, while viewing a small target with a long rotation period pole-on produces a narrow echo bandwidth. With this in mind, observing slowly rotating objects at high sub-radar latitudes can significantly boost the observed signal-to-noise ratio.
In summary, radar detectability depends on:
Transmitting Station: transmitter power, antenna gain
Target: diameter, rotation period, radar albedo
Geometry: distance to target, sub-radar latitude
Receiving Station: effective collecting area, system (noise) temperature
Detectability, namely the signal-to-noise ratio, improves as the following increase:
[TX Power]*[Gain]*[Collecting Area]*[Radar Albedo]*[Diameter^(3/2)]*[Rotation Period^(1/2)]
Detectability improves as the following decrease:
Note that the signal strength depends on the fourth power of the distance due to the inverse-square law for radiating the signal to the target and the echo returning to Earth. Distance between Earth and the target is the dominant factor in determining whether an object can be detected with radar. To compensate for the distance to the target, you need a lot of transmitted power, a high-gain antenna, and a big dish to collect the signal with a low-noise receiver, which is where Arecibo comes in!
To reduce so many variables down to a readable plot, we must make a few assumptions. For the transmitter, we use current values: transmitter power of 350 kW and antenna gain (sensitivity) of 7 K/Jy. In an ideal world, the transmitter power would be close to 1 MW with an antenna gain of 10 K/Jy, but klystron/generator failures and warping of the primary reflector during Hurricane Maria have reduced these values substantially. For the receiver, we use the effective area of the telescope (proportional to the antenna gain) and a system temperature of 24 K. For the target, often we only have an estimate of its absolute magnitude H ahead of observations, but we translate H to a diameter by assuming a visual albedo of 0.2. We also assume a radar albedo of 0.1. The sub-radar latitude is rarely known in advance, so we assume an equatorial view. This leaves distance to the target, diameter, and rotation period as the remaining variables, which are used in the figure above to illustrate how detectable an object is with the Arecibo planetary radar system.
The background data points in the figure are near-Earth asteroids in the Asteroid Lightcurve Database (LCDB; Warner et al., 2009, and updates) with quality factors of 3 (black; best) and 2 (gray; good). The red lines indicate regions of radar detectability from Arecibo with the assumptions listed in the legend at the upper right. The uppermost line represents any object with an echo bandwidth of 200 Hz. This is a convenient envelope that roughly encompasses the entire near-Earth asteroid population and helps to constrain the other lines in the figure. Each kinked line represents the smallest objects that can be detected at, in this example, a signal-to-noise ratio (SNR) of 20 in one hour (60 minutes) of observing time at the labeled distance from Earth. This is a rough SNR limit for a confident detection with radar leading to precise radar astrometry (in line-of-sight velocity/Doppler and distance/range). Higher SNRs of 100 or so are typically required for radar imaging rather than Doppler and ranging astrometry. While the lines themselves indicate the smallest objects with different rotation periods that are detectable at that distance from Earth, they also indicate that any object that plots to the right of the kinked red lines is detectable at that distance from Earth. For a given size and signal, an object with a longer rotation period is more easily detected because the echo bandwidth covers a narrower swath of receiver noise, so the effective SNR increases for longer rotation periods. This is what makes small asteroids with rapid rotations so difficult to observe with radar; the signal is spread over a wider bandwidth of noise that reduces the SNR.
Why is there a kink? This relates to why slowing an object down to zero rotation (an infinite rotation period) does not make the signal-to-noise infinite. The strength of the echo signal does not depend on the rotation state of the object, but the noise does! The integration time afforded to a radar observations is typically limited to the round-trip time for light to reach and return from the target minus the time required to switch from transmitting to receiving. The inverse of the integration time is the finest frequency resolution possible for the observation. If an object rotates slowly enough, its echo bandwidth can be less than the frequency resolution resulting in an unresolved echo. Since the noise component relates to the thermal noise of the receiver across the echo bandwidth, eventually the noise bandwidth reaches a lower limit of twice the frequency resolution. Tracing one of the red lines to smaller sizes with longer rotation periods, the echo bandwidth will eventually become unresolved. At this point, the SNR is no longer a function of the rotation period. In fact, it means that there is no rotation period that allows smaller objects to produce the desired SNR! Hence, the kink that makes the red line vertical. One way to mitigate this issue is to run a bistatic radar observation where the transmitter and receiver are not co-located, such as when Arecibo transmits and the Green Bank Telescope receives the echo. In a bistatic setup, the integration time becomes arbitrary because the transmitter and receiver are always on and no longer need to switch back and forth. Longer integration times allow for finer frequency resolutions and push the kink in the lines further down and to the left to smaller objects and longer rotation periods. Of course, one must also adjust the figure to use the effective area of Green Bank and the system temperature of their receiver. Fun fact: for objects about 2 or 3 lunar distances away, an Arecibo/Green Bank bistatic setup has about the same signal-to-noise as an Arecibo monostatic setup because the increased integration time cancels out the reduced receiving area of the smaller Green Bank Telescope (and you get better frequency resolution)!
So far, we have ignored visibility when considering how detectable an object is with the Arecibo planetary radar system. Because of its fixed primary reflector and limited mobility in the zenith-angle direction, Arecibo can only observe objects between declinations of roughly +0 to +37 degrees and within 19.7 degrees of zenith (> 70.3 degrees of elevation) for up to 2.75 hours (at zenith, +18.5 degrees declination) and less time at other declinations. Therefore, just because an object is close enough to Earth to be detectable with radar, it does not mean the object is observable from Arecibo! Be sure to check the radar detectability of an object at the distance it will be from Earth when it is within the Arecibo declination range. Also, because of the limited visibility and maximum track length of 2.5 hours, the farthest target that Arecibo and receive an echo from is less than 10 AU away. This limits Arecibo to observations of Saturn's rings and Titan as the most-distant bodies detectable with the Arecibo planetary radar system (but only when the Saturn system is at northern declinations during half of its heliocentric orbit). Even if Arecibo could recieve echoes from further away than Saturn, the signal-to-noise diminishes extremely quickly due to the fourth power of the distance. What about an object 24 hours round-trip time away? That would be about 10 times farther away than the Saturn system and 10,000 times fainter, so... nope!