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This answer to Does anything orbit the Sun faster than Mercury? explains that while Vulcanoid asteroids may have been plentiful in the past, large ones have currently been ruled out though smaller ones less than about 6 km might still be there. It's difficult to observe them because from Earth it requires pointing close to the Sun, and from spacecraft that are much closer (i.e. Mercury and lower) it's really hot and difficult to do.

I think that the current limits are set by analysis of historical STEREO images, space telescopes in orbit at 1 AU designed to look at and around the Sun. A Search for Vulcanoids with the STEREO Heliospheric Imager

My understanding is that during solar system formation there were asteroids all over the place as things formed and then collided, but as some bodies became large large swaths were swept clean and some bands remained.

Have any theories been put forth explaining why the vulcanoid belt seems to be depleted of large asteroids? Is this band, while quite stable, simply not stable enough? Do some think that it simply was never populated by large asteroids to begin with for some reason?

Question: Where have all the Vulcanoids gone? (sung to the tune of Where have all the flowers gone?)

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The effects of solar radiation are the main suspect for clearing out the vulcanoid region of whatever was there to begin with. Radiation pressure will tend to blow small dust out of the region. Larger objects tend to get cleared out due to the Yarkovsky effect and the YORP effect.

The Yarkovsky effect occurs as a result of temperature variations across a rotating object and the time lag for regions of the object's surface to heat up and cool down as they move into and out of the sunlight. This affects the distribution of the radiated photons (which carry momentum), and over time will act to shift the asteroid's orbit. Vokrouhlický et al. (2000) estimate that this effect would clear kilometre-sized objects out of the vulcanoid region on a timescale of a few billion years.

A further effect that is relevant is the YORP effect (short for Yarkovsky, O'Keefe, Radzievskii and Paddack, see also the question What is the difference between the Yarkovsky effect and YORP effect?). This affects the rotation of irregular objects like asteroids and can result in them spinning up to the point where they break apart. This is thought to be a significant contributor to the population of binary asteroids. In the vulcanoid region, this would be a mechanism for breaking up larger asteroids into fragments small enough for the Yarkovsky effect to rapidly clear them out of the vulcanoid region: Collins (2020) puts this right in the title: "The YORP Effect Can Efficiently Destroy 100 Kilometer Planetesimals At The Inner Edge Of The Solar System".

... The YORP effect destroys Vulcanoids by spinning them up so fast that the gravitational accelerations holding components of the body together are matched by centrifugal accelerations, this causes the body to rotationally fission. i.e break apart. We calculated the timescale of this fission process for a parent Vulcanoid and for each of their subsequent generational fragments. We show that objects with radii up to 100 kilometers in size are efficiently destroyed by the YORP effect doing so in a timescale that is much younger than the age of the Solar System...

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