Short story about a moon of Mars whose orbit is only a few feet up

Question

A spaceship arrives on Mars and begins exploring. The crew notices a shallow semicircular trench that runs arrow-straight across the landscape, turning into a complete circle and boring right through any higher terrain. They continue to follow the feature until they come across a Martian village aligned along the trench and notice that there is a huge calendar in the center of town. It turns out that there is a 3rd moon of Mars and it orbits so low that it hits the surface. The Martians' calendar makes sure everyone is aware of its next pass (now why they didn't just move the village 100 yards to the left...).

Answer 1

This could be "The Holes Around Mars" by Jerome Bixby, as previously answered as part of this question.

The supporting quote from that answer:

That night in the ship, while we all sat around, still shaking our heads every once in a while, Allenby talked with Earth. He sat there, wearing the headphones, trying to make himself understood above the god-awful static.

". . . an exceedingly small body," he repeated wearily to his unbelieving audience, "about four inches in diameter. It travels at a mean distance of four feet above the surface of the planet, at a velocity yet to be calculated. Its unique nature results in many hitherto unobserved—I might even say unimagined—phenomena." He stared blankly in front of him for a moment, then delivered the understatement of his life. "The discovery may necessitate a reexamination of many of our basic postulates in the physical sciences."

[. . .]

"Inasmuch as Mars's outermost moon is called Deimos, and the next Phobos," he said, "I think I shall name the third moon of Mars—Bottomos."

The complete story may be read legally courtesy of Project Gutenberg.

Answer 2

Although not the story you're aiming for, a similar set of unlikely circumstances is found in Iain M. Bank's "The Hydrogen Sonata". A small moon has been hollowed out and dropped into a sub-surface orbit(!) around a slightly larger planet

So Fzan-Juym had been appropriately refitted, refurbished and improved, towed to Eshri, slung into a low orbit around it and then carefully lowered still further – kilometre by kilometre, metre by metre, eventually millimetre by millimetre, speeding up all the time – until its orbit now lay a kilometre beneath the planet’s surface, darting along one of the widest and deepest canyons of all in a blur of planet-girdling movement, its course held steady by a network of hermetically isolated AIs and multiply redundant thruster systems dedicated to doing nothing else.

Its own engines had done almost all the work at every stage, though various other craft had helped and been there to step in had anything started to go wrong, but a modest degree of seeming helplessness was deemed to be useful in providing a sort of camouflage of its own.

Fzan-Juym, headquarters of the Socialist-Republican People’s Liberation Regiment #14, had been in sub-surface equatorial orbit of Eshri ever since, zipping along like a super-fast bullet in a slab-sided groove open to the pitch-black sky, orbiting the planet in less than an hour and covering over two hundred million kilometres every year – nearly half a trillion altogether by now – while never coming closer than fifteen hundred metres to either the flat canyon floor or its sheer, polished sides.