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In the most general case, there are three (spatial) degrees of freedom for each body, for a total of 9 degrees of freedom.
The circular restricted three-body problem forces the two larger masses to be in perfectly circular orbits defined by their masses and the chosen orbital radii (with the third body having negligible mass and thus no influence on their ...
6
There are two types of angular momentum of each planet: orbital angular momentum of the planet around the Sun, and rotational angular momentum of the planet around its rotational axis.
Orbital angular momentum $L_{orb}$ is typically calculated at perihelion or aphelion as $L_{orb}=mvr$, where $m$ is the mass of the planet, $v$ is the instantaneous orbital ...
5
Yes, the Solar System barycentre (SSB) is usually outside of the Sun. That is, over the long term, the mean distance from the centre of the Sun to the SSB is greater than the Sun's radius of 695,700 km. (That's the IAU's nominal solar radius).
As ProfRob commented, we don't really know the exact location of the SSB, since we can only calculate it based on ...
5
The Sun's motion relative to the Solar System barycenter is dominated by the four planets Jupiter, Saturn, Uranus and Neptune. Neptune is a lot lighter than Jupiter but it's also a lot farther away, and contrary to intuition the farther out it is the larger the induced motion in the Sun around the barycenter.
*To first order we can treat the Sun's response ...
4
There will be major limitations much worse than quantum effects.
There is always uncertainty in input variables; we only know the standard gravitational masses, positions and velocities to some amount of accuracy.
The chaotic nature of n-body dynamics as discussed in Wikipedia's N-body problem#; Planetary problem may cause the seemingly small uncertainty in ...
3
The general relativistic contribution to the precession of Mercury's orbit is 43 arc seconds per century. Since a complete circle has $360 \times 60 \times 60 = 1296000$ arc seconds, this means it will take approximately 30140 centuries (about 3 million years) for there to be one extra full precession cycle due to GR effects.
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Not a direct answer to your question (which others have given already), but maybe still worth considering here:
The dynamics of the solar system can to a high degree of accuracy be calculated by assuming the sun and planets as point masses, so whether the barycenter lies inside or outside the defined surface of the Sun is physically not really relevant for ...
1
Kepler's laws and the associated orbit only hold for a two-body problem, so the 'barycenter' in the question can only be understood as the center of mass of the sun and a particular planet (ignoring the other ones), not as the solar system barycenter. And in this sense the convention is to take the sun at the focus, which means the semi-major axis is the (...
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