NavList:

I have never tried to find longitude by timing eclipses of Jupiter's satellites but it might be worth a try at some point. At present Jupiter is near opposition meaning that eclipses occur close to the planet's limb and are difficult to observe.

Predictions for eclipses (and other phenomena) can be obtained from https://ssp.imcce.fr/forms/satellites-events. Attached is the geocentric data eclipses and occultations for 2024. There are a few things to note:

1)There are no events involving Io (501) in which both the immersion and emersion are visible. One of these occurs behind Jupiter.
2) There are no events involving Callisto (504)
3) The average duration of eclipses is for Io (501) 3.78 min; Europa (502) 4.54 min; Ganymede (503) 16.20 min. This is the time from the start of the event to when the satellite fully disappears or reappears.

Looking at the documents provided in other posts, the timing involves determining when the last vestiges of light disappear from the eclipsed satellite and when it first reappears. This seems quite tricky. In Raper from 1840 that Lars Bergman provided it states

...in which observers have been found to differ 40s or 50s.

The observation may be considered complete only when the immersion and emersion of the same satellite are observed on the same evening.

Elsewhere others advise using the first satellite because its moves the fastest but because of item 1) above you can't have it both ways!

Robin Stuart