Spock says it is impossible for such a young planet to have an atmosphere, but that is carless writing by non Vulcan scriptwriters. Even a new formed planet should have at least a very thin atmosphere.
No doubt what Spock actually said it that is impossible for a planet to form an atmosphere similar to Earth's, wiht lots of oxygen to breathe, in just a few thousand years. It actually took Earth several million times a thousand years tor acquire a lot of oxygen in the air, produced by photosynthetic plants.
Would a world approximately the size of Eerth's moon be able to retain an atmosphere? The ability of a world to retain its atmosphere depends mostly on it's escape velocity, which is calculated from its mass and radius.
According to
Habitable Planets for Man, Stephen H. Dole, 1964, the ability of a world to retain a gas in its atmosphere is roughly proportional to the ratio of the escape velocity divided by the root-mean-square speed of that gas in the exosphere, the outermost layer of the atmsphere,
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
According to table 5 on page 35 on page 35, the amount of time it will take the atmosphere to decrease to 0.3679 of its original amount will be a fewweeks if the ratio is 3, several thousand years if the ratio is 4, about 100 million years if the ration is 5, and infinite if the ratio is 6.
The exospheres of planets tend to be much hotter than the air at the surface. According to page 54, the temperatues in Earth's exosphere is about 1000 to 2000 degreess K. Dole said that if the maximum exopshere temperature with liquid water temperatures on the surface could be as low as 1000 degress K, an escape velocity of 6.25 kilometers per second would be 5 times the speed of atomic oxygen molecules in the exosphere, which would thus be 1.25 kilometers per second.
So if the maximum temperature in the exosphere of the Kalandan outpost planet, which I will call K.O.P. for short, is 1000K degrees, the planet could retain 0.3679 of its original atmosphere for as long as a few weeks if the escape velocity is 3.75 kilometers, per second, for several thousand years if hte escape velocity is 5.0 kllometers pr second, for about 100 million years if the escape velocity is 6.25 kilometers per second, and for infinity if the escape velocity is 7.50 kilometers per second.
And if the exosphere temperature of K.O.P.. is as high as 2000d degrees K, the escape velocity of K.O.P. would probably have to go up by a few kilometers per second to retain 0.3679 of its original atmosphere for the same lengths of time. Such higher escape velocities would probably still be lower than Earth's escape velocity of 11.186 kilometers per second.
So what escape velocity would K.O.P. probably have?
Spock didn't say that K.O.P. had Earth's mass, he said it had a similar mass, which allows for some uncertain degree of difference. Kirk, who tends to be less precise than Spock, said that the size (linear dimensions) of K.O.P. was approximately that of Earth's moon, which also allows for some uncertain degree of variation.
If K.O.P. has exactly the mass of Earth, and exactly the radius of the Moon, 1,737.4 kilometers, it will have 1 Earth mass within 0.0203017 of Earth's volume, and thus will have a much greater average density than Earth, 49.256958 times Earth's density of 5.414 grams per cubic centimeter, and thus 271.602549 grams per cubic centimeter (or gm/cm3 for short).
I used this online calculator for K.O.P.'s surface gravity,
https://philip-p-ide.uk/doku.php/blog/articles/software/surface_gravity_calc and this one for K.O.P.'s escape velocity.
https://www.omnicalculator.com/physics/escape-velocity Feel free to use other calculators to check my results.
So K.O.P. should have a surface gravity which is 13.47 g (the surface gravity of Earth being 1 g), and escape velocity 1.9146254 of Earth's 11.186 kilometers per second - 21.417 kilometers per second (km/sec for short).
The esacpe velocity is fine for keeping an atmosphere, but the surface gravity is murder! According to other episodes their anti gravity techology is too bulky to hide inside standard uniforms, and nobody mentioned anti gravity anyway. This could be simply a 20th century production oversight, or a serious flaw in the episode.
So I tried reducing the mass of K.O.P. by 10 percent of Earth's mass, and increasing the radius of by 10 percent of the Moon's radius, several times, to get a more survivable surface gravity for K.O.P..
If K.O.P. has 0.6 the mass of Earth, and 1.4 times the radius of the Moon, iw will have 0.5556494 the volume of Earth and will have 10.797183 the density of Earth, and so 49.541181 gms/cm3. The escape velocity will be a good 14.023 kilometers per second, but the surface gravity will still be a high 4.12 g.
The more I decresed the mass of K.O.P. and increased its radius to get an acceptable surface gravity, theless it resembeled its description by Kirk and Spock. I suppose that different people will have different tolerances for how much variation they will accept.
By making K.O.P. have a mass 0.4 Earth mass, and a radius 1.6 lunar radius, I gave it a density of 26.602185 gms/cm3, a surface gravity of 2.1 g, and an escape velocity of 10.71 km/s.
By making K.O.P. have a mass 0.3 Earth mass, and a radius 1.7 lunar radius, I gave it a density of 26.602185 gms/cm3, a surface gravity of 1.49 g, and an escape velocity of 8.998 km/s.
And some people might think that is going way too far in interpreting Kirk and Spock's words.
So I tried a different idea. Maybe Kirk made a mistake interpreting theddtata about the planet as it came in. Kirk might have erroniously compared the radius of the planet with the diameter of the Moon, and so K.O.P. could have twice the radius and diameter of the Moon. It is believed that the sinking of HMS
Victoria on 22 June 1893, with the loss of 358 lives, was probably due to an admiral mistaking a radius for a diameter while doing mental calcualtions.
https://en.wikipedia.org/wiki/HMS_Victoria_(1887)
So if Kirk heard that the radius of K.O.P. was about 3,474.8 kilometers, the diameter of the Moon, Kirk might have misremembered the radius of the Moon as 3,474.8 kilometers, and so said that K.O.P. was approximately the sixe of Earth's Moon. Out of friendship for Kirk, Spock might have resisted his scientific urge to correct him, and would have compensated for such a scientific sin during the rest of the episode by being even more picky with other people than he otherwise would have been.
So if K.O.P. has half the mass of Earth, and twice the radius of the Moon, it would have a density of 16.993018 gm/cm3, a surface gravity of 1.689 g, and an escape velocity of 10.71 km/s.
If K.O.P. has 0.6 the mass of Earth, and twice the radius of the Moon, it would have a density of 20.391621 gm/cm3, a surface gravity of 2.02 g, and an escape velocity of 10.732 km/s.
If K.O.P. has 0.7 the mass of Earth, and twice the radius of the Moon, it would have a density of 23.790225 gm/cm3, a surface gravity of 2.36 g, and an escape velocity of 12.673 km/s.
If K.O.P. has 0.8 the mass of Earth, and twice the radius of the Moon, it would have a density of 27.188829 gm/cm3, a surface gravity of 2.69 g, and an escape velocity of 13.547 km/s.
And I tend to think that the best and least objectionable figures for K.O.P. would probably be somewhere near there.
The problems in designing a planet K.O.P. to fit the episode are:
One) The planet has to be physically possible, since it existsi n the story. But it has to be very, very, very, very improbable to form naturally, so that every scientist would think that it was impossible, but there. And it should also be very improbable to build such a planet, since nobody would expect to see an artificial planet like that one, no matter how many other artificial planets they might know of. If building a planet like K.O.P. was easy, they would probably have said: "Hey, it must be another one of those type three artificial planets" or something.
Two) The planet has to satisfy the descriptions by Kirk and Spock, meaning it has to have no more than an acceptable variation from what they say. So it is a good thing that they said "approximately" and "similar to", allowing for more variation. And my theory that that the diameter of the Moon was similar to the radius of K.O.P., and Kirk mistook the diameter of the Moon for its radius, also helps a lot with varying the size of K.O.P.
Three) The Planet has to have a high enough escape velocity to retain its atmosphere for thousands of years since the Kalandans made it and died, and for as far in the future as they desired and planned for it to keep the atmospherre. And all of the designs I tried have sufficient escape velocity, and I don't see how any design fitting the description could have much lower escape velocity.
Four) The planet has to have a low enough surface gravity for Kirk's group to not suffer ill effects from it during a period of several hours, nor to expect and say that the gravity would kill them before they died of thirst and starvation. And as you can see, it took trying out a lot of planetary diameters and masses to get a few combinations with acceptably low surface gravities.
or:
Five) The planet can have a deadly, unbearably, high surface gravity if the Enterpise keeps a tractor beam pulling on the landing party to reduce the pull of gravity on them and make them feel light enough. But the Enterprise was thrown over 900 lgiht years away soon after they arrived, and was unable to reduce their surface gravity for hours until it returnd. So maybe they carried tiny antigravity devices hidden in their uniforms or swallowed a potion containing negative gravity matter before they beamed down, or something.
Unless someone believes TOS era technology can achieve some form of Five), problem Four) is in full force and K.O.P. must have a mass and radius that gives it a natural surface gravity which is low enough to be no problem for human explorers. Which is a severe restriction on the possible combinations of mass and radius.
Or maybe the Kalandans built a planet with a very high natural surface gravity but used an anti gravity field to reduce the gravity felt in the area near their underground base, the area which Kirk's Kirk's landing party would naturally selct to beam down to for reasons of their conformt. And that would explain the incredible coincidence that KIrk beamed down within walking distance of the entrance to the undergound Kalandan base.
Or maybe the Kalandans built a planet with a highly oblate shape, so that its equatorial radius was several times it s polar radius. And they spun the planet very fast, so that the speed of the rotating equartor would canccel out much of the surface gravity at the equator and make it low enough to be comfortable for the human-looking Kalandans, while the gravity at the poles would be high enough to suit the hypothetical high gravity allies of the Kalandans.
It is possible for worlds that resemble some of my suggested designs for the K.O.P. to physically esist. So how probable would such worlds be to form naturally, and how much more difficult would constructing such a world be for a society capable of building more normal planets?
I guess I will duscuss some of the problems constructing the K.O.P. to some of the suggested designs in a later post.
) rotating (day/night cycles) in a stable orbit around a star. It maintains a one g gravity field just like a space ship. It maintains a breathable atmosphere either by regenerating what escapes to space, or better yet, it uses some sort of tractor beam technology tuned to atmospheric gases to keep it from escaping to space. Again, our heroes do not mention these technologies even though it should be detected by their tricorders. Maybe the form of the technology was so foreign or advanced that it escaped the detection by the tricorders. 
