And Mary Goes Round and Round and Round

All these calculations and more depended on surface gravity of the planet and the mass of the planet.  Surface gravity can be determined from a low orbiting moon, like Phobos of Mars.  That quantity is then used to determine the mass of the planet using data from another orbiting moon or the planet orbit around the Sun. Together with the measured radius, the mean density of the planet is found.  Only with the blessing of a closer moon...

So consider Mertis of Jupiter,

Orbital radius = 128000 km
                       
Orbital period = 0.29478*24*60*60 = 25468.992 s.

Orbital angular velocity = 2*pi/ 25468.992 = 0.000247 rads^-1.

Centripetal force, \({F}_{J}\) = \(w^2 r\) = (0.000247)^2*128000 = 0.007809 kms^-2

The surface gravity of Jupiter is then estimated to be, 48.724 kms^-2.  The gravity of Jupiter is given by
g = 48.72*e^(-x/69911) kms^-2.

A check with Jupiter second closest moon, Adrastea orbiting at 129000 km with a period of 0.29826 days confirms the near value of 0.007669 kms^-2 as the centripetal force as given by the expression for Jupiter gravity,  g = 48.72*e^(-x/69911) at x = 129000.

The we consider Jupiter orbit about the Sun,

Orbital radius = ( 816520800 + 740573600 ) / 2 = 778547200 km.

Orbital speed =  13.07 kms^-1.

Centripetal force, \({F}_{J}\) = \(\frac{v^2}{r}\) = (13.07)^2/778547200 = 2.194e-7 kms^-2

But, centripetal can also be calculated from,

\({F}_{J}= \frac{ {M}_{J}}{{M}_{J} + {M}_{s}}{g}_{J}\),   so

\({M}_{J} = {M}_{s}. \frac{{F}_{J} }{({g}_{J}-{F}_{J})}\),

where \({M}_{J}\) is the mass of Jupiter and \({M}_{s}\) is the mass of the Sun.

\({M}_{J}\) = ( 9.942779860e27)*2.194e-7/(48.724-2.194e-7 ) =8.9567e+21 kg

A value that needs to be checked, as does the new Sun mass estimate made previously.  Ridiculously small!

And here we are stuck for more data, on surface gravity, or planet mass, or the lack of a moon about another moon.  If there's another way to estimate surface gravity...