What replaces Shimano Deore LX rear derailleur RD-M570? They stopped making my current derailleur in 2003. I need to replace it. How do I choose a replacement part?

A good answer will tell me a specific part (with part number) that will replace the RD-M570. A better answer will also teach me how I can look up this information for myself.

Thanks
Ignoring number of speed for a moment, derailleur rear specs are:


Total capacity
Max front ring difference
Max/min largest sprocket size
Min/max smallest sprocket size


All measured in numbers of teeth.

Total capacity is (diff in cassette smallest-largest sprocket teeth) + (diff in chainring smallest-largest sprocket teeth), it's basically a measure of how much chain slack a derailleur can take up. A longer cage means bigger total capacity, but also less chain tension and control. 

Shimano very kindly publishes all their specifications here. Note there is a link to archive PDF docs on that page which go back to 2004 so you can look up the RD-M570. Looks like it only came in an SGS (long) cage version.


Total cap: 43
Max front diff: 22
Large sprocket max: 34
Small sprocket min: 11


Putatively you need a Shimano 9 speed MTB derailleur, but in fact you could use an 8 or 9 speed derailleur that has the required capacity. The reason for this is that 8, 9 and 10 speed cassettes are all the same width and Shimano used the same derailleur actuation ratio for 7, 8 and 9 speed MTB systems (the ratio of length cable pulled to derailleur cage movement). 

If you want a new derailleur, you pretty much want to go for the highest spec 8 speed available, which at the moment is Alivio RD-M4000.


Total cap: 45
Max front diff: 22
Large sprocket: 36-32
Small sprocket: 11-12


That will actually enable you to run a slightly bigger cassette.

You could get an older but higher level model on Amazon or Ebay, say if you wanted to get something that has the same look at the rest of groupset components on the  bike. The Shimano Wikipedia page has a nice summary table of Shimano MTB groupsets where you can look up what groupsets were 8 or 9 speeds.