Deriving a Body's Mass from its Orbit

[Kerbin Dallas Multipass]

I watched some documentaries and lectures about astronomy.

I understand how they find out the orbits of binary stars and exoplanets. So cool.

Question: If I know the orbital path of 2 bodies, how do I find out their mass?

My common sense tells me that a pot of petunias on the orbital path of.. lets say jupiter... could orbit the sun at the exact same speed and distance in a stable orbit as jupiter. How do we define the mass of the pot of petunias if it acts like jupiter?

Am I mixing up fundamental things here? I'd just like to understand

[UmbralRaptor]

For two objects that grossly differ in mass, you usually only find the mass of one. Mainly by some variant of Kepler's third law: T² == (4À²/µ)*A³

For two objects with vaguely similar masses, you want to consider that they rotate around a common barycenter, and use the difference in distances from that to find the mass distribution.

[Capt. Hunt]

Are the petunias at the Jovian L5 point?

[Extemporary]

You don't measure the orbit of the object in question, but rather the orbit of the other body around it.

Even with the mass difference of an exoplanet orbiting a star, the planet's mass causes the star to move a detectable amount. The petunias are so small that their effect on the sun is negligible, and we wouldn't be able to even detect them. Jupiter, on the other hand, does have a noticeable impact. The barycentre of the system is just above the sun's surface; i.e. the Sun and Jupiter both orbit a point very close but just outside the sun. This motion is detectable. Since we know the period and the radius of the sun's movement, we can calculate the mass and distance of the other object.

[Kerbin Dallas Multipass]

Are the petunias at the Jovian L5 point?

They are not, but they are "falling" at the same speed as the whale (jupiter)

Edited March 8, 2014 by Kerbin Dallas Multipass
s=t

[Kerbin Dallas Multipass]

You don't measure the orbit of the object in question, but rather the orbit of the other body around it.

Even with the mass difference of an exoplanet orbiting a star, the planet's mass causes the star to move a detectable amount. The petunias are so small that their effect on the sun is negligible, and we wouldn't be able to even detect them. Jupiter, on the other hand, does have a noticeable impact. The barycentre of the system is just above the sun's surface; i.e. the Sun and Jupiter both orbit a point very close but just outside the sun. This motion is detectable. Since we know the period and the radius of the sun's movement, we can calculate the mass and distance of the other object.

I understand the pot of petunias does not make the sun wobble. My question is relatively strict: what can we tell about an orbital path and how do we define masses based on those orbits

[TheShadow1138]

For a star, one could use the Mass-Luminosity Relationship to at least estimate the mass of the star, or use Kepler's Third Law as long as there is another body orbiting the star. As for the object orbiting the star, once the orbit has been defined I believe one could use the specific orbital energy to determine the mass of the orbiting body.

[stupid_chris]

Stritcly speaking in a perfect world two body system, you can't find the mass of the object orbiting the other one. Essentially, you could place an object of any mass you want on the same orbit of Jupiter without any clear difference. Basically, the orbit of an object is not defined by it's own mass, but by the mass of the object it's orbiting.

However, since reality is always a little more complex, you /could/, and the above statement isn't true when the orbiting body starts having a significant amount of mass compared to the parent body. In reality, neither does Jupiter orbit the Sun, neither does the Sun orbit Jupiter. They're both orbiting around their common mass barycenter. And unless I'm confused, I believe that Jupiter is the only body of the solar system for which the Sun/planet barycenter isn't located inside the Sun.

So if you want to know the mass of your pot of petunias, you could start by find the mass of the Sun with the usual formulas given the pot's perigee, apogee, and speed at those respective places. Then, if you had instruments precise enough, you could look at the oscillations of the sun as the pot of petunias orbits it. That would be insanely small, but hey it's a thing. Then from the Sun's displacement and from the distance of the pot of petunias at that time you could figure out the mass the pot of petunia must have to move the Sun in that way.

Of course this gets near impossible to do when you translate from a two body system to a system like the solar system that has a /lot/ of bodies involved, but it's as good as it gets I guess.

[Brotoro]

You can find the SUM of the masses of the two objects using Kepler's Third Law (as modified by Newton). If you can plot the two orbits, you can find the ratios of the masses of the two bodies from how far each is located from the center of mass of the system (and therefore get the individual masses). This can work fine with two stars because they are similar enough in mass…but will not be doable for a very low mass object orbiting a much more massive object.

[Alchemist]

Finding heavier object's mass by trajectory of something orbiting it - easy. Finding the smaller object mass - only if you can detect larger object's movement due to its gravity.

We are able to estimate mass of gas giants by observing the oscillations of the star, but it's not so easy to measure with enough precision.

[Ravenchant]

And unless I'm confused, I believe that Jupiter is the only body of the solar system for which the Sun/planet barycenter isn't located inside the Sun.

That's correct. The barycenter is located slightly outside the Sun- about 700 000 km from its center, I think.

[MrZayas1]

Basically if you want to find the mass of a planet orbiting a star, you can use Kepler's third law which is expressed in this equation: T² == (4À²/µ)*A³, so you could use that or, if you had a spectrometer you could measure the amount (or color ) of light coming from an object to observe whether or not it has an atmosphere, from which you could find it's density to determine the mass of the planet. (I think that is how it works) Hope this helps!