Can USB 2.0 power a USB 3.0 external HDD?

I have an USB 3.0 2.5'' Drive (without external power source) from verbatim and when I plug it into my old laptop which only has USB 2.0 it is incredibly slow (much slower than my USB 2.0 2.5'' hard drives of comparable size). My guess is that this has something to do with the power, but I haven't ruled out that it might be some other issue, as I'm not sure how to check that. Needless to say, it runs just fine and much faster than my USB 2.0 drive on a USB 3.0 port.

So, my (statistically not significant) experimental result is: It works, but it might be slow, and I'm not sure if it is good for the hard drive to run on too low power.

And about how to find out: I tried to find out about that by looking at the technical details and comparing them, but there was nothing in there that seemed helpful. If I remember correctly, there were other USB drives that explicitly stated that they work on USB 2.0 ports, and mine didn't, so maybe that is something to look out for, although it also might be a case of asbestos free cereal.

My personal experience with a 2.5-inch Toshiba Cavino 1TB drive and it works fine in a USB 2 slot. It only has a single USB cable.

The only difference is when it is plugged into a USB 2 port, the activity light is white while in USB 3 port the activity light is blue.