# Calculating mass of object by its orbit

Can you calculate the mass of an object by its own orbit around another object? I found many ways to calculate the mass of an object by measuring the trajectories of other objects orbiting it, but does it work the other way round?

E.g. you can calculate the mass of Jupiter by observing the orbit of Io, but can you calculate the mass of Jupiter by its own orbit around the sun? I tried using Keplers first law, since it relates the object mass to its trajectory, but to solve for the orbiting mass I would need to know its total energy (kinetic + potential). The kinetic energy is easily derived from it's orbit, but I can't find a way to determine the potential energy or the total energy without knowing its mass beforehand. Using Newtons Gravity equation doesn't work either since the orbiting mass cancels out if you compare gravitational and centripetal force. Does the orbit of an object even depend on its mass as long as it has some or would an object, that only weighs 1kg have the same orbit as the earth around the sun as long as its starting position and velocity is the same as earths?

• Jan 15 at 18:46

In theory you can. If you know the orbit, you know the barycenter, so you know the ratio of the two masses, and then the period fixes them.

The problem is whether you can know the orbit to that level of precision. Then your question becomes one of observation/technique.

The Jovian-Solar Barycenter is over the Sun's surface, for instance. I suspect with spacecraft at Jupiter, the radioscience group at JPL could figure it out.

No. This is due to the fact that (as you note) the acceleration of a body of a gravitational field is independent of its mass, since the gravitational mass (the "$$m$$" in $$F = mg$$, determining the force an object feels) and the inertial mass (the "$$m$$" in $$F = ma$$, determining its acceleration) are the same.

This is actually a surprisingly deep observation; it led Einstein to what's called the equivalence principle and from there to develop the theory of general relativity.

• No--the orbit is determined by the combined masses of the orbiting objects. It's just the central object normally dominates. Jupiter has .1% of the mass of the sun--going from the combined mass to Jupiter's mass thus loses 3 digits of precision--and you probably didn't have that to start with. Jan 16 at 1:36
• True. My answer is premised on the idea of a fixed gravitational field, which is pretty close to true (but not exactly true) in the case of Jupiter and the Sun. And note that in any event you can't extract Jupiter's mass from knowledge of the orbit, which is what the OP was asking about. Jan 16 at 2:37
• wait, isn't m the mass in F = ma ? @MichaelSeifert Jan 16 at 11:11
• the a determines the acceleration in F = ma right? Jan 16 at 11:11
• @WilliamMartens We can't calculate the force of gravity operating on the planet/object though. Given just the orbital parameters all we can work out is the primary body's mass. Jan 16 at 12:19