At first, let me quote some dialogues from Pirates of the Caribbean $5$: Dead Men Tell No Tales.
Also in another scene when she accused to be a witch, she defends herself in this way "I'm an astronomer."
"This chronometer keeps the exact time in London.
I'm using it to make an altitude measurement to determine longitude.
Only then will we find our exact spot at sea."
Following this scene, she says: "I'm not only an astronomer, I'm also a horologist."
so we can conclude that:
she is an astronomer and a horologist with facilities and possibilities of only a chronometer and a pen and a book(or notebook), and an astronomical table in that book or in her mind, with no other additional equipments.
But Captain Jack's crew look at her like fools! It seems that I am on the same side as Jack's fool crews! Also, the film emphasizes that Carina is a woman of science and an astronomer, and I believe that there is a scientific support behind this dialogue.
How did she measure the altitude and determine the longitude by use of the chronometer?