Apparently Japanese scientists have discovered a large lava tube below the surface of the moon which they say might be used as a moon base. Seeing how surface temperatures on the moon range from 107C (225F) during the day to -153C (245F) at night, what temperatures could we expect in the lava tube? And how much protection would the tube offer from radiation?
Anything worthy of the name lava tube would offer near total protection from radiation arriving at the moon’s surface.
From this site:
The linked lava tube is apparently at a latitude of about 12.6 degrees, which implies a subsurface temperature around 15 C (59 F).
Underground, you get more or less the average temperature. And it’s not surprising that the average temperature on the Moon would be about the same as on Earth, because they are after all both at the same distance from the Sun. The Moon is a bit cooler than the Earth, because without an atmosphere, it doesn’t have a greenhouse effect, which is why you need to be at the equator to get a comfortable temperature. It might still be a good idea to build your base at higher latitude, though, because people and all of our various machinery will produce waste heat, which will raise the temperature somewhat.
Another way to think about it is that if the habitat is a sealed box sitting inside a larger open cavity (ie minimal physical contact with the surrounding bulk material), then it only cares about radiative heat gain/loss anyway.
You might find Selenites raising Mooncalves in those tubes.
What is a lava tube, please? 
A tunnel in rock left behind from flowing lava.
I think of lava rock as being pretty porous, but I wonder if this tube could be sealed to make an artificial atmosphere underground.
The lava tubes on the moon are a bit different than those on earth, as they are all really old, left over from when the moon still had a molten core and liquid mantle.
Not all that different though, in that they are the same basic idea, magma forced its way up to the surface, leaving behind a tunnel.
lava rock on earth is often porous, as it cooled rapidly in water. All igneous rock is also “lava rock” that cooled over a longer period of time.
Actually, all natural stone is somewhat porous to some degree. The technology to seal stone and concrete is readily available, including polyurethanes, acrylics, siloxanes, and siliconates. The need for sealing the tube surfaces doesn’t come so much from containing an atmosphere, however, as it does to protect occupants from the fine dust produced from the friable lunar regolith that would cause both respiratory problems and act as an irritant and pollutant for any long term habitation. The Apollo astronauts reported significant problems with lunar dust even on the relatively short durations of the Apollo 15 through 17 J-class missions, and while regolith in a lava tube may not be subject to the same ionization and electrostatic charge buildup as that exposed to the solar wind, the fine and dry structure of the material will likely present the same inhalation and cementing action.
You’d want to locate a base underground not for the warmth (which, as Chronos points out will be generated as a waste product of power generation and various necessary equipment for atmospheric processing) but actually insulate from excessive heating, ready access to a thermal reservoir (the surrounding regolith), and protection from solar and galactic radiation and micrometeorites. Surface structures would require significant reinforcement to afford occupants the same degree of protection, and given the extreme costs and limitations of transporting mass from Earth to Luna it would be desirable to utilize indigenous materials to the maximum extent possible. Inflatable or sprayable materials within underground vaults and voids such as lava tubes or hollow volcanic tumuli.
However, this still fails to address one of the most significant problems with a long duration lunar habitat; the lunar surface gravity at only 16% of that of Earth is thought to likely pose significant musculoskeletal, cardiopulmonary, macular degeneration, and other physiological issues. While we have almost no experience with human physiological response to low fractional gravity for durations of more than a few days, what we have learned from long duration stays at the International Space Station is that gravity plays a significant role in the normal functioning of mammals, from the large scale anatomical down to the cellular level, and it is very likely that 0.16 g is not enough to provide an adequate environment for long term human health. There is, of course, little that we can do about that short of building a giant centrifuge to simulate the acceleration of a higher gravitational field.
It remains unclear what purpose a base on Earth’s moon would serve other than to provide a destination for a space program. Luna has few precious resources that have been found to date (and none that would be fiscally worth transporting back to Earth…no, not even [SUP]3[/SUP]He, which is not currently in any significant demand), it is not particularly scientifically interesting beyond some narrow areas of geology, it does not present a particularly useful depot or waypoint to interplanetary space versus geostationary orbit or the L4/L5 libration points, and is not otherwise particularly scenic or aesthetically appealing. For the costs of establishing a crewed lunar base we could send multiple probes to each of the outer planets to further explore their structure, behavior, and moon systems, including the investigation for potential life on liquid water and hydrocarbon rich moons such as Europa, Ganymede, Titan, and Enceladus, all of which present targets of interest rich in both scientific interest and natural resources that could ultimately sustain an eventual interplanetary human presence.
Stranger
Well, it is obvious: An amusement park!! (Blackjack and hookers optional).
More seriously, it might act as some kind of “dry dock” for interplanetary ships; facilities for docking and repair and stuff like that. Perhaps also as some sort of “entry point” to Earth, a “customs” of sorts? If in the far future there is asteroid mining, the raw materials might be brought to the Moon for processing and/or temporary storage before ferrying them to the Earth itself?
Perhaps in the far future the Moon would end up with industrial refineries or something…? If they are suitably automated, there might be only a minimal complement of flesh-and-blood humans keeping an eye on things that would be rotated out regularly.
Or something 
If you’re mining asteroids, you might as well just do all of the processing in zero-g. No sense bringing in an unnecessary extra gravity well, even one as shallow as Luna’s. For that matter, you won’t even necessarily bring it back to Earth at all: The big advantage to doing asteroid mining is for materials to use in space.
Hard to say without the actual experiments, but it seems to me that there’s a big difference between 1/6 g as on Luna and zero g as in orbit. On the Moon, you could wear a few hundred kilograms of lead during your daily routine, and get the same total weight you’d have on Earth, but nothing of the sort is possible in orbit. This wouldn’t perfectly simulate Earth’s gravity (for instance, the blood pressure difference between your head and feet would be much less), but it would at least put normal loading on your muscles and skeleton.
It makes for a lovely fantasy to imagine that we finally manage to get someone up there with the equipment to break into these tubes and make a real start at establishing humanity. Then, after monumental effort, they finally break into this space that has lain undisturbed for unthinkable amounts of time, only to find that it’s all fitted out and equipped as an alien rec room sort of affair.
With apologies, this evokes the sort “Age of Sail” notions of ports and docks that, while filling the tomes of science fiction and space opera, don’t really represent the technology of a spacefairing civilization. Luna, of course, isn’t any kind of point of entry; in fact, it would represent an additional energy barrier, having to interact with (come to its orbital velocity) it rather than entering into orbit of Earth through a low energy path through a libration point; the same is true for vehicles leaving the Earth-Luna system, where there would only be an advantage from the Moon’s orbital momentum a few times per year for most targets due to the inclination of the lunar orbit to the plane of the ecliptic. We currently crash spacecraft returning from the moon into Earth’s atmosphere in dramatic aerobraking maneuvers prior to descent and landing, but this isn’t practical for vessels intended to remain in orbit, which we’d rather bring in with as little impulse as possible.
The notion of lowering material down into the gravity well of Luna for processing, only to have to lift the products back up doesn’t make much sense unless such impulse is so cheap as to be irrelevant. Earth’s moon does not appear to have much in the way of natural resources to be worth investing facilities in. (Yes, there is water ice in shadowed areas of the higher latitudes and potentially locked below in regolith, but that just makes it more difficult to access notwithstanding that there are objects in near Earth orbit containing water ice and mineral resources that are much easier to extract and use without having to fly down into a hole and climb back up.) And given that most of the time and resources are in free orbit around the Sun, it makes far more sense to develop an infrastructure to extract and refine them in situ rather than trying to transport them elsewhere, especially given that they are most valuable to support a space-based exploration and manufacturing infrastructure rather than transporting materials to Earth. (Although we are expecting to hit limits on certain materials, such as copper and uranium, some time later this century or early in the next, it is unlikely to be fiscally viable to procure them from interplanetary space unless there is already a substantial infrastructure for processing and distribution in space, and that only makes sense in the context of supporting a space-faring industry; it’s a classic chicken-and-egg problem that is only resolved by some entity putting the focus on bootstrapping a space manufacturing infrastructure without clear fiscal viability.)
The biggest piece of practical infrastructure we need in the near future to support interplanetary exploration is actually a space-based, solar-orbiting communications and telemetry system consisting of several (3-6) satellites in an orbit between Venus and Earth. We are currently dependent upon the Earth-based Deep Space Network and a handful of auxiliary systems which are strained at supporting even the handful of current interplanetary missions, and Earth’s moon does not offer a significant improvement on that. The next thing we need is a way to manufacture propellant and other consumables from space resources. Again, Luna offers little of utility in that vein.
Space physiology and medicine observations from the last forty-odd years have found that while exercise in zero gravity does have some positive benefits for maintaining muscular capability and bone density, there are still dramatic losses regardless. And as previously noted, some of the most serious problems are those found at the cellular level, especially metabolic dysfunction. How much a problem these will pose for occupants in 0.16 g versus freefall is unknown, but the expectation of most space medicine experts is that there will be substantial problems that we do not currently have the means to protect against or rectify. Stays in such an environment may be limited to only a few months at a time as they are on the ISS, or they may be longer but with attendant degradation and increased mortality and morbidity currently seen in older former astronauts. We’re likely either going to have to more accurately reproduce terrestrial conditions or substantially modify the human form in order to have permanent habitation of space.
Stranger
The relevant factor here is permeability, not porosity - the interconnectedness of the spaces between crystals, not the spaces themselves. Basalt like the mare basalts is actually very low permeability *and *porosity. It only gets permeable when it’s highly fractured as a whole layer (like columnar basalt). Even the volcanic rock you’d think were permeable, like pumice, isn’t really - all the gas vesicles (bubbles) make it porous, but they don’t really interconnect very well (or pumice wouldn’t floatfor very long)
Fun fact - the unit for measuring permeability is the darcy. Sadly, not a Jane Austen reference