External HDD works on old MacBook but not new MacBook?

I have an external hard drive that works when connected to an old MacBook, but I bought a new MacBook, and the external hard drive appears briefly on my desktop and then disappears. After it disappears, I get the notification about improperly disconnecting the device. The external drive has a small light on the corner that remains on after the error message appears.

There is no external power supply for the external hard drive.

How do I make the drive stay on my desktop? Why does it disappear if the HDD is still powered on?