How Did Scientists Calculate the Orbital Radius of Jupiter's Moons?

[Forcino]

TL;DR Summary

How did the first scientists figure out the orbital radius of a moon of Jupiter? How can observations lead to a calculation of the orbital radius of a moon of another planet?

How did the first scientists figure out the orbital radius of a moon of Jupiter? How can observations lead to a calculation of the orbital radius of a moon of another planet?

 

[DaveC426913]

If we know the distance to Jupiter, it's pretty straightforward to work out the radius of its moon's orbits.

D = θ d / 206,265
where
D = linear size of an object
θ = angular size of the object, in arcsec
d = distance to the object

https://lco.global/spacebook/sky/using-angles-describe-positions-and-apparent-sizes-objects/

 

[Forcino]

Thank you.

So would you find the orbital radius of a Jovian moon by using a telescope to view Jupiter at opposition during a time when the moon is farthest from Jupiter? Then you could find the arcseconds of angle from the center of Jupiter to the center of the moon? Then you could use the equation D = θ d / 206,265? Then D would be the orbital radius of the moon?

 

[DaveC426913]

Yeah, although:
1. it doesn't really have to be at opposition
2. you'd need to measure the angular distance of the moon over several/many orbits to get a good estimate of the maximum orbital radius.

 

[snorkack]

How did the scientists figure out distance to Jupiter back in 17th century? Römer measured speed of light by 1676, but he got the speed about 30% too slow. How much of it was the error in travel time, how much the error in distance to Jupiter?
How much was Römer´s distance in 1676 better than Galilei´s distance, if any, in 1610?

 

[Vanadium 50]

Well, there's a good article in Wikipedia:
https://en.wikipedia.org/wiki/Rømer's_determination_of_the_speed_of_light