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Is the Oort cloud changing the position of the barycenter of the solar system? Does it affect in a significant way? How significant?

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The Oort cloud is a theoretical object. Its existence is more or less accepted on the basis of "comets have to come from somewhere", but no Oort cloud object has ever been observed (except for some comets that pass near us that presumably came from the cloud). It is theorized that it has two components, one disky and one spherical, both centered on the Solar System barycentre. Given the assumptions of symmetry and centering, the Oort as theorized cannot have any effect on the barycentre of the Solar System. Until such a time as it is actually observed - either directly via imaging, or indirectly via modeling of the orbits of comets - it is pure speculation to talk about an offset in the barycentre.

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If the Oort cloud is included to the Solar System and the calculation, it has a potential to influence the barycenter significantly. The distance of the Oort cloud from the Sun is some 100,000 AU (plus minus a factor of two) and its total mass is around 5 times the Earth's mass.

If there were a single orbiting object at the location of the Oort cloud, its effect on the barycenter would be equal to the effect of 500,000 Earths or so (because the influence increase both with the mass and with the distance from the Sun).

However, the Oort cloud is rather uniformly distributed along an orbit or orbits that are nearly circular, anyway. So the actual effect on the barycenter is much smaller than that of 500,000 Earths.

One must realize that with the existing technology, it is very difficult to measure the position of the barycenter too accurately, anyway. By the principles of relativity and equivalence, the Sun and the planets of the Solar System are basically freely falling in the gravitational fields created by the distant objects so no local measurement can easily reveal the location. To measure how the location changes with time, one would have to measure the distance from other stars etc. very accurately, say with the accuracy of thousands of kilometers, and it's not possible now (or a foreseeable future).

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