Orbital Ratios

[Jay-qu]

When one calculates the orbital ratios of our solar system, you tend to get a value around 1.8 - if you do this for the Jovian system one gets approximately 1.6 - and finally if you do this for the Saturnian system you get ~ 1.3

 

Is there any accepted idea for why these ratios form in such a way? I have been looking into the Modern Laplacian theory for planetary formation. This theory suggests that the planets formed through gas rings produced by equatorial mass shedding during the contraction of the protostellar cloud and not from a protostellar-disk.

 

Im guessing that somewhere it should be possible to predict this ratio given the initial mass of the protostellar gas cloud.

 

Has anyone seen anything like this done?

 

J

[Jay-qu]

I think I just found my own answer

 

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1984EM%26P...30..209P&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf

 

I couldnt be bothered latexing the whole thing so I just attached it below

 

Basically whats going on is in the first equation, thats the equation for the ratio between gas rings formed via equitorial mass shedding of the protostellar cloud. M is mass and f is the moment of inertia. The subscripts denote before and after any one gas ring shedding. When dealing with the entire solar system, the mass of all the planets makes up <1% the mass of the sun, so assuming that f and M stay relatively constant, you arrive at the second equation.

 

The little m in the second equation is the mass of the gas ring, while this varies largely between say jupiter and mars, when compared to the large values for f and M the ratios produced are approximately constant

 

J

 

Thanks go out to Dr. Prentice for the paper (who is also one of my lecturers for astrophysics)

[coldcreation]

When one calculates the orbital ratios of our solar system, you tend to get a value around 1.8 - if you do this for the Jovian system one gets approximately 1.6 - and finally if you do this for the Saturnian system you get ~ 1.3

 

Is there any accepted idea for why these ratios form in such a way? I have been looking into the Modern Laplacian theory for planetary formation. This theory suggests that the planets formed through gas rings produced by equatorial mass shedding during the contraction of the protostellar cloud and not from a protostellar-disk.

 

Im guessing that somewhere it should be possible to predict this ratio given the initial mass of the protostellar gas cloud.

 

Has anyone seen anything like this done?

 

J

 

I would look into Hamiltonian mechanics, Lagrangian-Euler dynamics and Laplace resonance. I think the answer can be found there, within the least action principle, rather than as a process of shedding equatorial mass during some initial condition.

 

Then equilibrium has to be explained. Shedding, even from a protostellar cloud or disk, would be insufficient to explain the subsequent stable orbit of object within the disk.

 

The large portion of the concept must be founded on multiple mean motion resonance patterns, by ensuring the repetition of specific geometrical configurations within the combined gravitational fields of bounded systems.

 

An extensive amount of literature exists on this topic, though definitive answers are still lacking.

 

Wether it will be one day possible to predict the ratios in advance given the initial mass of the protostellar gas cloud seems highly unlikely, especially in light of the fact that we still do not understand the mathematics of 3-body systems very well, let alone n-body systems (more that 3-bodies).

 

 

 

CC

[Jay-qu]

I dont think we need to know how to handle n-body systems to at least get a decent approximation, because the mass of the sun dominates all the rest.

 

The equitorial mass shedding is due to the fact that as the cloud contracts it spins faster to maintain angular momentum. At the equator of rotation of this gas cloud the gas is moving the fastest (radially) and hence when the contraction gets to a certain point the gas at the equator will be going to fast to continue contracting with the cloud ie its reached the orbital velocity required to maintain a stable orbit at that radius. And is thus shed.