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But the Jews went back into Israel.. and the Jews of today can trace their lineages all the way back to the exodus. And they don't look like the people of Iraq.
So just because they spent a short time period in Iraq doesn't mean they looked like the people who live there now… and we know what they look like. Because they live in Israel now and other places. It's not a mystery.
Short time being 2,600 years?
Yes, the Persians did eventually conquer Babylon and in doing so allowed the Jewish Diaspora to return but most historical studies point out that a sizeable contingent never went home and stayed in what is modern day Iraq, hence that community being the purest distillation of what the tribe of Judah looked like.
There is a while cottage industry dedicated to this topic but most people note that the expansion of Judaism into colder northern climates involved a lot of mixing, marriage, and reproduction with non-Jews and also the eventual genetic transformation from living in colder climates with less sunlight, hence the much more diverse range of appearances in Jews of today.
The oldest imagery of Jews from that era is from the Dura Europos Synagogue which was located just across the Iraqi border with Syria and right in the middle of old time Babylon. Here is their depiction of Samuel anointing David
Here is there depiction of Mordechai and Esther