Number timestamped rows

Dear all,
maybe a stupid question, but: I have a table that is ordered like this:

user_id|timestamp|event
1      |0:1      |event_a
1      |0:2      |event_b
2      |0:1      |event_b
2      |0:3      |event_c
2      |0:4      |event_b
3      |0:1      |event_a

and I would like to number them according to their timestamps like this:

user_id|timestamp|event   |order
1      |0:1      |event_a |1
1      |0:2      |event_b |2
2      |0:1      |event_b |1
2      |0:3      |event_c |2
2      |0:4      |event_b |3
3      |0:1      |event_a |1

How would I do this with an UPDATE statement (e.g.) in pgsql?

Thanks in advance for any hint
Jan


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