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TL;DR: Last for couple of seconds except for Jupiter where it lasts for minutes. Mars Phobos and Deimos are too small to totally eclipse sun but there are annular eclipses. See this video which Curiosity Rover took on Mars September 13, 2012 where Phobos is eclipsing Mars. Jupiter Only four moons are large enough to create totality: Io, Ganymede, Europa, ...

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Jupiter appears to be 41 minutes http://adsabs.harvard.edu/full/1896JBAA....6..424W#:~:text=With%20us%2C%20the%20maximum%20duration,the%20third%20satellite%20of%20Jupiter. Saturn https://solarsystem.nasa.gov/news/13101/spectacular-eclipses-in-the-saturn-system/ Uranus has total solar eclipses once every 42 years https://en.wikipedia.org/wiki/...

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If you know somebody who still owns floppy disks, you might consider recycling their inside as filter. Please test before relying on them for protecting your eyes. I did use them during the last solar eclipse as filter for my analog camera with reasonable results. References https://www.instructables.com/Floppy-Disk-Sun-Filter/

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In your formula for lat: $$lat = Atan\bigg(\frac{y_i}{(dist_{axis} \times (1 - f) ^ 2)} \bigg) \times \frac{180 }{ \pi}$$ I think $(1-f)^2$ should be just $(1-f)$, because $\sqrt{1-\epsilon^2}=1-f$, since $1-\epsilon^2=\frac{b^2}{a^2}=(1-f)^2$. See for instance Chauvenet, A Manual of Spherical and Practical Astronomy, formula 539, p. 491.

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