Federation Artillery
The
Federation simply has no artillery. Nothing. Nada. Zilch.
This wasn't always the case; in the TOS episode "Arena", Captain Kirk exchanged mortar fire with a Gorn soldier on a nearby ridge. In the accompanying scene, we can see both the shells and the mortar itself, which strongly resembles a real life mortar. However, this scene was filmed at a time when Gene Roddenberry still had to fight with the networks over the style and content of his show, and as a result, the show always had a much more hard-edged style to it than the politically correct TNG spin-off (the Klingons, for example, were the brainchild of Gene Coon, not Gene Roddenberry).
Gene Roddenberry always had a problem with the military style of TOS (as muted as it was), and we didn't get a chance to see what he really wanted until TNG came out, with its "living room" bridge, effeminate dress uniforms, Braun Mixmaster hand phasers, and strident anti-military rhetoric (see "Peak Performance" for the most egregious example). The irony here is that Gene Roddenberry himself was a decorated war veteran, having flown the B-17 Flying Fortress on many sorties in the Pacific conflict of WW2. But he obviously had a problem with his own military past, as demonstrated most graphically in the pilot episode of TNG in which Captain Picard contemptuously referred to a 20th century American military uniform as the "costume" relic of an uncivilized, bygone era.
Of all the people involved with Star Trek gene over the past two decades, the only one who really impresses me is Nicholas Meyer. This is the man who directed both Star Trek II: The Wrath of Khan and Star Trek VI: The Undiscovered Country (he also co-wrote the screenplay for ST6). These films stand head and shoulders above the rest of the movies, some of which descended into outright drivel, and they both show an Enterprise whose operation is that of a military vessel, not a damned interstellar luxury liner. It is probably not a coincidence that both of Nicholas Meyer's films were made in times where Gene Roddenberry's influence was on the wane: ST2 was made after the box office failure of the self-indulgent, slow-moving Star Trek TMP, and ST6 was made after Mr. Roddenberry had already relinquished much of his control to Paramount (it was actually released the year of his death).
In any case, leaving aside all the behind-the-scenes stuff, the fact is that we would not see any sort of mortar again until the Klingons used one, in the DS9 episode "Nor the Battle to the Strong" (which, not surprisingly, was filmed long after Gene Roddenberry's death). During that battle, it was notable that no one in the entire Federation group had any mortars of their own with which to return fire. The same weakness befell the garrison in "The Siege of AR-588", which had no mortars or even sustained-fire automatic weapons with which to ward off Jem'Hadar attacks on their position.
It is tempting to argue that the Federation could simply replicate mortars, towed guns, and any other sort of artillery necessary if the need arises, but that's merely a knee-jerk reflex, and an oversimplistic analysis, for the following reasons:
It assumes that replicator patterns even exist, and there is no guarantee of that. The fact that they keep historical records of bygone eras doesn't mean they also have complete replicator patterns for every technological device invented over the past 400 years. If they don't, then they would have to re-invent those devices, and that process involves bureaucratic consensus, government approval, design, prototyping, testing, improvements and revisions, etc. The act of going back over old ground isn't as simple as it would seem; some experts within NASA have estimated that it would take well over a decade to restore our ability to land a man on the Moon, even though we did it way back in 1969.
It assumes that all ground units have replicators, and that they have all the necessary raw materials on hand, since replicators perform molecular rearrangement rather than elemental transmutation. The metallurgical requirements of artillery are probably quite unlike anything they normally replicate, so even if we presume that every ground unit has its own replicator and power source, and even if we presume that their replicators are capable of making the chemicals required for explosives and the metallic structures required for the artillery pieces themselves, it is still highly unlikely that they will have the raw materials. If they don't, then the weapons would have to be manufactured and then distributed through conventional means, or the appropriate raw materials would have to be manufactured and then distributed through conventional means. Either way, there are logistical concerns which would interfere with the simplistic Trekkie "push a button and everyone has mortars" mentality.
It ignores the issue of replicator limitations, whether they relate to size or complexity. We generally don't see large devices being replicated, with an extreme example being the fact that they must painstakingly assemble starships in shipyards rather than simply replicating them at the push of a button. We also see that starships must routinely stop at starbases in order to receive critical parts, which are manufactured using conventional techniques and which may include such seemingly mundane objects as a warp core hatch, which is a simple hunk of metal. Even if every ground unit has replicators, and raw materials, and pre-existing designs which have been fully tested, approved, converted into replicator patterns, and transmitted to all ground units, we still have no guarantee that the replicators can actually produce these weapons.
It assumes that there is no training whatsoever required for these weapons. Real-life soldiers expend a great deal of time and effort on the use of each type of weapon so that they're more likely to kill the enemy than their own allies, but if these weapons are not normally used by the Federation armed forces, then this means they haven't received any training on their use. In other words, the Federation would be asking soldiers to start using an entirely new type of weapon in the heat of combat, with no training whatsoever. What are they going to do? Hand a soldier a PADD with some pretty diagrams and videotaped instructions, pat him on the back, and expect him to instantly become proficient? E-mail him the Sally Struthers ICI correspondence course on mortar use and tactics? Hire Billy Blanks to make a videotape entitled "Tae-Mortar-Bo: How to incorporate mortars into your workout?"
The time-honoured Trekkie defense of arguing that they can simply replicate everything they don't already have is an empty evasion tactic. There is no guarantee that they have pre-existing designs, or that their replicators will be up to the job, or that they will have the necessary raw materials on hand at every base where the weapons are needed. Worse yet, their troops will have no training whatsoever, and with a mortar, a lack of training strikes me as distinctly dangerous. I may not be an experienced military instructor but I don't think that trial by fire is the best way to introduce a soldier to an entirely unfamiliar type of weapon.
Mortars
To the best of my knowledge, no Federation mortar has been seen or even mentioned in the entire series runs of Star Trek TNG, DS9, or Voyager. It's been nearly a century since a mortar was seen in the hands of the Federation, and every single piece of ground equipment has undergone major redesign since then. There is no reason to assume that mortars still exist in their inventory, particularly since we've seen several situations in which they would have been appropriate, yet they still didn't appear. This means that Federation soldiers are incapable of engaging ground forces without a line of sight and a short range to target.
Even if they were able to call up old mortar designs, rapidly produce them, and distribute them to ground units, it is highly questionable that they would be able to use them effectively. One does not simply guess how to use a mortar, as these weapons are much more idiosyncratic than something like a handgun. For example, most people are unaware that a mortar will "bed" itself into the ground when it is fired, so that its aim point will change with each of the first 2-4 rounds fired. After protracted firing, it can actually take a lot of effort to remove them from the ground, and all of this would be totally unfamiliar territory for the Federation soldier with a new toy. Furthermore, most people are unaware that this means there are rules as to where you can mount a mortar; for example, it is unadvisable to place one on a rooftop because it probably won't have the necessary strength, and that's exactly the sort of lesson that a Federation soldier would be learning the hard way.
Although mortars are not applicable to the Federation in its current state, it may be useful to note the following about the mortars that have been used by others:
The Klingon mortars in "Nor the Battle to the Strong" scored hits 2-3 metres from Jake Sisko, yet he wasn't wounded or even knocked down. A blast less than a metre from Dr. Bashir knocked him down without causing even the most superficial injury, thus suggesting that he was actually knocked down by some sort of inner-ear problem resulting from the shockwaves of multiple near-misses.
The Gorn mortar in "Arena" scored a hit less than 2 metres from Captain Kirk and Commander Spock. In each case, the blast caused a knock-down but not a kill or even a wound. By way of comparison, a fragmentation shell from the lightest modern mortar can kill at 15 metres.
The events of "Arena" merit further examination. The first casualty among Kirk's men vanished into thin air, as if hit by a phaser (and of course, there was no damage at all to the rocks he was touching). However, his disappearance coincided with the onset of what sounded like mortar fire, not phaser fire. Moreover, Spock complained that they had no hope of survival with their hand phasers against the enemy's "disruptors"; a strange thing to say when the enemy is clearly using mortars, not disruptors. And finally, Spock located the enemy with his tricorder, and they were more than 1300 yards away, on the other side of intervening high ground. The intervening high ground eliminates the possibility that the Gorns were using line-of-sight disruptors, and it also proves once again that they were using mortars, since that distance is too great for small-arms fire. It would therefore appear that the Gorn troops were indeed using a mortar, and that their mortar shells are not conventional fragmentation shells; instead, they seem to be some sort of disruptor-based projectile, which explodes like a tiny high-explosive charge and has a phaser-like effect on any organics in its immediate vicinity.
Spock's conduct during this encounter was grossly incompetent. He used his tricorder to locate the Gorns, but it didn't occur to him that they would be able to track his active tricorder signals! The shelling began within seconds of Spock using his tricorder to locate them, but he still didn't make the connection! He kept using the tricorder, and he kept drawing fire onto their position, until his tricorder suddenly exploded from some sort of signal feedback (probably an inadvertent side-effect of the tracking technology they were using). At that moment, the Gorn mortar fire immediately became much more random, with the explosions sounds clearly being more distant. Amazingly, even after this sequence of events, Spock made no comment about the obvious fact that the Gorns were using his tricorder signals to aim their mortar fire.
When they were finally able to locate a mortar of their own in order to return fire, they fired an interesting billiard-ball shell. This shell had a very large wide-area effect; so wide, in fact, that they fired it without a forward observer or even a tricorder reading, since Spock's tricorder had already been destroyed. Instead, they fired it based on a location estimate, roughly based on the Gorns' last known position. It detonated with a high airburst, and it silenced the Gorn mortar with just one shot! On the surface, this would suggest an extremely powerful weapon such as a low-yield tactical nuclear device. However, it produced none of the effects of a nuclear explosion. There was no shockwave. No deafening roar. No fireball. No prompt ionizing radiation. There was only a bright flash of light, from which the men briefly put their arms up in order to shield their eyes. Given the lack of nuclear or even high-yield chemical explosive effects, it is clear that this shell was not a high explosive or nuclear weapon. Moreover, there is no evidence that any Gorns were actually killed by the blast; the Gorn vessel took the risk of lowering its shields in order to beam its troops back up, which would be illogical if its troops were all dead. It is most likely that the projectile in question was actually some sort of electromagnetic pulse device, designed not to cause physical damage or radiation burns but to disable electronics (such as those in the Gorn projectile launchers, if any, or in the disruptor projectiles themselves). It might have even been capable of "shorting out" biochemical nervous systems, thus causing disorientation or perhaps even unconsciousness (although there is no way of ascertaining the validity of this speculation). In short, it is most likely that the weapon was an EMP grenade.
Towed Guns
No Federation towed artillery piece has been seen or even mentioned in the entire series runs of Star Trek TOS, TNG, DS9, or Voyager. It seems that an over-reliance on orbital fire support has probably been a fixture of Federation military strategy since Kirk's time, although that over-reliance has grown even worse in the TNG era.
Self-propelled Guns
No Federation self-propelled artillery piece has been seen or even mentioned in the entire series runs of Star Trek TOS, TNG, DS9, or Voyager.
Rocket Artillery
No Federation rocket launchers have been seen or even mentioned in the entire series runs of Star Trek TOS, TNG, DS9, or Voyager.