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Asparagus is where you have the fuel flowing through the ring, peeling away pairs of tanks in that layer.
This is how I understand it. Whiped up a mspaint for ya.
http://imgur.com/2STLQvP
Generally Asparagus is seen as a litle bit more efficient (you shed "useless" weight earlier and, if your available TWR is high enough, it provides you with a good match to the rising TWR in your ascent).
But often you might want Onion staging instead ... it provides the full thrust of the whole ring of rocket motors for a longer time (which might be useful to get through the "soup" in the first 20km of ascent ) and it is faster/easier to setup if you use symmetry.
I for my part usually use more onion staging than Asparagus, for both of the reasons listed above ... and for roleplaying purposes (in order to avoid dropping stages/boosters onto my launchpad I force myself to do a small gravity turn halfway through the burn of the first stage [to get a few m/s sideways movement] [somethign which is also done for real rockets] ... the short burn duration of rockets with Asparagus would force me to have this gravity turn very early [which in itself would be ineficient and also mean that the boosters drop too close to launchpad for my tastes])
It still turned out that asparagus was more efficient in nearly every aspect but, as you may already know, more difficult to setup.
http://steamcommunity.com/app/220200/discussions/0/864975632445155984/
I've used setups like that. It's simple to do if the outer stage ONLY uses its own engines but then it's not a proper onion stage, is it? I'll reference the numbers from Kunzite's drawing: http://imgur.com/2STLQvP
Let's say that we have the asparagus layer setup like in the left picture and we add the onion layer, 6 tanks, which feed into the tanks labelled 1-3. What happens is that the outer tank connected to 3 will get drained by two engines (3 and 4), the one connected to 2 by three (2, 3 and 4) and the one connected to 1 by four (1, 2, 3, and 4). This means that the onion layer tanks are depleted at varying times and if you wait until the entire layer is empty before dropping it, part of the number 1 tanks will also have partially depleted.
There are a few ways to get around this. First, you can make all of the fuel lines of the onion layer feed into the number 1 tanks, which essentially turns it into a asparagus layer. Even if you drop it all at once you're doing the effort of building asparagus, and unless you drop it in pairs it's inefficient too! Second, you can hook the onion layer directly to the number 4 tank, and only rely on the middle engine and the ones in the onion layer. This might also not be super efficient but at least it's one way to do it.
In short, it's not simple.
Sure, easy enough. Nothing stopping that! A separate stage is easy, whether you actually place it as an outer layer or simply under the asparagus layer.
I'm sure you remember this beauty?
http://steamcommunity.com/sharedfiles/filedetails/?id=179184312
100% stock, 55 tons, enough wing components to lift a 747-800. It's not the heaviest thing I've launched, but it was the most challenging. Asparagus staging didn't help as it was dropping engines too quickly, and I needed the thrust vectoring from all those mainsails early on. Someone suggested I try using onion staging to keep my thrust and vectoring longer, and their suggestion worked.
So, if you need as much thrust as you can get during the initial ascent, then onion is the better choice. If you can afford to sacrifice some thrust for increased efficiency (which most of the time you can), then by all means go with asparagus.
And if you want to launch stuff reallistically without exploiting either staging system, then get KWRockery.
That, all boils down to TWR. If your first asparagus staging drops tanks/engines and your TWR goes from .40 to .20, then onion staging is clearly superior for that stage.
But, with smart design (ie making sure each asparagus staging leaves your design with high TWR), I'd go with asparagus every time. I use onion when I've already built a design, and don't feel like starting from scratch again, when I realize asparagus TWR isn't cutting it.
Look at the numbers in my thread. That is what I expected but the numbers don't bear this out. Remember that most of your thrust is required early and a higher TWR seems to be a better option on heavy. It's the simplicity when testing that wins hands down.