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Because since he didn't c-bet and the 9 probably didn't helped him I assumed he was weak and would fold to a bigger bet.
Its hard to say with certainty but lets go through this scenario with assuming villain is either a TAG or a fish.
A TAG will likely c-bet all TP hands, and air hands. Although I don't agree with checking back to pot control with a pocket pair below K, I've seen 'TAGs' deathly afraid to bet non top pair type hands on dry-ish 1-overcard flops. If this is a TAG, his check back range is most likely going to be a weak-ish underpair type hand he hope to pot control and see a cheap showdown. If he does have TT, while the turn card doesn't help him, it certainly won't scare him. Since a lot of regs auto-bet turn if flop is checked back by default, I expect him to peel a lot of a 1-pair type hands to reevaluate the river. You can probably bet closer to $3 and a TAG will still call with what he has.
Since I don't claim to know what fish is thinking about, then think about how you can extract max value. Fish doesn't take much convincing to see another card. I mean, the point of poker is to see showdowns right? So is a fish anymore likely going to fold a draw or a gutshot or 2nd pair if you bet $3 instead of $2?