Current wide-field surveys discover ∼15 Jupiter-family comets (JFCs) each year, typically identified via visual detection of a dust coma or tail. The same surveys also discover many asteroids that have distant JFC-like orbits, but with no reported activity. We observed asteroids on Jupiter-crossing orbits beyond the depth of typical survey imaging using the 2.5 m Isaac Newton Telescope. We used deep imaging to observe 16 asteroids in this region, plus seven known comets for comparison. Three asteroids (2011 WM183, (669525) 2012 XO144, and 2020 RX133) showed surface brightness profiles consistent with low-level activity, equating to ∼19% of our total sample. We note that 2020 RX133 is a Jupiter Trojan. When we considered the heliocentric distance range of the asteroids at the time when they showed activity, this fraction increased to 33% of the targets in the 3.16 au ≤ Rh ≤ 4.56 au region, and therefore it is possible to infer that at least ∼30 asteroids with TJ ≤ 3.05 and in the 4.05 au < a < 5.05 au parameter space may potentially exhibit low-level activity. We also estimated nuclear radii for the three active targets of rn = 1.8 ± 0.2 km, rn ≤ 0.8 km, and rn ≤ 0.5 km for (669525) 2012 XO144, 2011 WM183, and 2020 RX133, respectively. The median color index for the observed asteroids is (g − r)PS1 = 0.52 ± 0.13, aligning with those expected for D-type asteroids
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