What excactly govern the rules for bombing planets in Master of Orion?
Early in my current game the Alkari had a fleet around one of my planets and bombed it several turns in a row.
Now I have fleet parked around one of their planets but the bombardment screen does not come up after the second turn at the planets. There were two Alkari ships there the second turn, and it took several turns for my two regular battleships to destroy one of them. In the meantime my two bombers rained nukes on the planet for several turns, but at the start of the next turn the population had not decreased (it was halved in the initial attack, though).
So what am I missing here?

Planetary shield in place? Trying to remember, but I think if the rating of the shield is high enough it'll pretty much block entirely the lower lever bombardment weapons, now matter how many you throw at the planet.

I think it works like this (I hope I'm not screwing this up): if a bomb does 3-12 damage and you have 20 of them, it does not equal a single attack at 60-240 damage. Instead, it's 20 individual attacks at 3-12 damage, and there's a hidden die roll for each bomb. That means a shield that will block 10 (Class X) damage will block entirely any individual bomb hit that does 10 or less damage, and then those that hit with 11 or 12 explodey power will only do 1 or 2 damage. And if the planet has a Class XV shield then those nuclear bombs will do no damage at all. In that case, you're better off getting just a few bombs with higher damage potential since you'll be better able to overcome the shield with each dropped bomb.

Someone will correct all my mistakes above. ; ) Page 53 talks a little bit about it, both from the shield side and the planetary attack side.

Edited: "10 or less damage"

Also, I believe the missile bases themselves are protected by the best shield tech available in addition to the Planetary Shield. If someone has Class V Planetary Shields and Class IV Deflector Shields, the missile bases have 9 points of protection.

There must be some bug that sometimes prevents you from bombing planets.
In my recent game I'm paying the Sakkras and my neighbours the Bulrathi declared war. I sent a Bomber to one of their recently colonized planet. I only get the "new planet explored" message, but no option of bombing it, despite it having 0 bases.
My bombers have Heavy Lasers in weapon slots 1-3, and 35 Fusiom Bombs in slot 4.
On a hunch I designed a Bomber with 35 Fusion Bombs in slot 1, and the lasers in slots 2-4. And now I could bomb the planet! Un-fucking-believably stupid!
It's possible it is mentioned in the manual, but in any case it's not exactly logical...

You don't bombard the planet in the fighting screen. That is just to kill enemy fighters and missile bases on the planet.

Once you have cleared out the enemy fleet around the planet and destroyed all the missile bases it will give you an option later if you want to bombard the planet.

The part in the fight scene where your fleet attacks the planet even when there are no bases doesn't actually do anything as far as I can see.

Actually depending on a number of factors it is actually possible to do damage to the actual population of the planet as well during a battle, with a population of 1 on a planet and sufficient ground facilities you can actually cause the game to crash (depopulating the planet at this point with under some versions cause it to crash). I think the GOG version doesn't have this crash problem but with sufficient bombs you can still cause the population count to go down (it takes a lot of bombardment damage with high level weaponry to do this however, nuclear bombs won't cut it).