You don't need a second battery for that. A battery is a storage bank, and an alternator recharges. If you want to be able to play your music longer with the vehicle off, then you need more battery bank. If you want your battery to not be drained while you are driving then an alternator recharges it.
There's a lot more to it than a straight answer, it's relative to you.
Lets say you have 1500w going to your sub.
1500w / (yeah 80% amp efficient is easy, but you can usually find what yours is) .8 = 1875w draw
1875w draw / 14.4v (which is standard running voltage, some alts go higher), = 130 amp draw.
So when this is at full volume (set properly for 1500w), and hitting hard, it's drawing 130amps. Between those bass hits, its drawing less. If you're listening to bass heavy rap, or generated bass tracks, it could be drawing constantly. If you listen to rock or country, you've got a few seconds between bass hits, and are mostly charging. You could have a larger amperage draw if there is more recovery time, and you won't notice the draw.
Then you have to add that on top of what your vehicle uses. Your stock alt is sized so that if everything is on: AC, electric fans, efi, taillights, head lights, turns, horns, cig lighter, map lights, etc.. are all drawing, you have a little head room.
What you should really do is start with an alternator. When you want to put in a sub amp, you need to see what larger alt options you have. You need to see what is in your budget and what are your goals. Then you can work backwards if you want:
Lets say you can find a 300amp alt. If your stock is 130, and you use the AC, and don't want any trouble, you could take 300- lets say 100-120 out, and you would have 180-200 amps to work with.
200 x 14.4v = 2880 watts
2880w x amp efficiency = 2304 amp watt output
Then you have to figure out what you listen to, and about how often it's going to have subs hit. With rock and country, you could easily bump that up to 3,000 watts and not have an issue. With generated bass tracks, you are already right there. You just have to figure out your budget and goals, and most of the time meet somewhere in the middle.