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On 2012 in the New Horizons' The PI's Perspective Alan Stern wrote The Kuiper Belt at 20: Paradigm Changes in Our Knowledge of the Solar System (more also archived) which includes:

Most of the known KBOs are just 100 to 300 kilometers across, about one-tenth of Pluto’s diameter. But some are smaller than 100 kilometers across, and some are larger than 300 kilometers across. In fact, there is great diversity among KBOs:

  • Some are red and some are gray;
  • The surfaces of some are covered in water ice, but others (like Pluto) have exotic volatile ices like methane and nitrogen;
  • Many have moons, though none with more known moons than Pluto;
  • Some are highly reflective (like Pluto), others have much darker surfaces;
  • Some have much lower densities than Pluto, meaning they are primarily made of ice. Pluto’s density is so high that we know its interior is about 70% rock in its interior; a few known KBOs are more dense than Pluto, and even rockier!

Question: How many Kuiper Belt objects have moons? How do we know this? What observational techniques have revealed moons of KBOs?

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"Moons" is a misnomer here, the majority of Kuiper Belt Objects (KBOs) with companions are binaries.
Now there are several populations in the Kuiper Belt, characterized by their eccentricities and inclinations. The cold classicals are hereby the most important ones, having mostly circular, low inclination orbits and a high binarity fraction of about 30% (Noll et al. 2008).
Hot classicals, with eccentric and inclined orbits possess a binarity fraction of about or less than 1 in 40.
The other important KBO populations, i.e. the scattered disc, the resonant ones and the centaurs possess negligible binarity. In terms of numbers, the classicals make up about 2/3 of the entire KBO population by numbers, and the cold classicals are about 50-60% of the total classical population.

So in total you'd get a fraction of at least $f_{\rm Moon}\approx 0.66*0.5*0.3\approx10\%$ of KBOs having 'moons'.

In order to find and characterize binary orbits, strange light curves pose a first hint, but only high spatial resolution actually helps to discern the two binary components. As most KBOs are low-mass objects, their binary orbital times are long. Prolonged, 5-15 year snapshot observations with the HST allowed recently to characterize the spin direction, for example (Grundy et al. 2019).

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