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The Earth-Moon barycentre is said to be inside Earth. But Moon-Earth mass ratio (1.2%) is larger than Jupiter-Sun mass ratio (0.09%). Then how come the Jupiter-Sun barycentre is outside the surface of Sun?

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There are three factors that all play into this:

  • The Jupiter/Sun mass ratio
  • The Jupiter/Sun distance
  • The Sun's radius

The barycenter of any pair of orbiting masses lies on the line connecting their centers of mass, and its position depends on the masses of the two objects.

If two objects have the same mass, their barycenter will lie halfway between them. If one is twice as massive as the other, the center of mass will be 1/3 of the way from the larger mass to the smaller. The formula is:

DSb=DSP*MP/(MS+MP)

MS and MP are the masses of the Sun and planet, respectively, DSP is the distance between their centers of mass, and DSb is the distance from the center of mass of the Sun to the barycenter.

Finally, whether or not the barycenter is inside the Sun depends on the radius of the Sun. How far it is from the center of mass of the Sun is unaffected by the Sun's radius.

For the Earth-Sun pair, MP/(MS+MP) is 3.0×10-6

For the Jupiter-Sun pair, MP/(MS+MP) is 9.5×10-4

So the Earth-Sun barycenter is about 3.0×10-6 of the way from the Center of the Sun to the Earth, or about 279 miles. The Jupiter-Sun barycenter is about 9.5×10-4 (a number 300 times bigger) times Jupiter's distance (a number 5.2 times bigger) and comes out at 459,000 miles, which is just slightly larger than the Sun's radius of 430,000 miles.

It's outside the Sun because Jupiter is more massive than the Earth and further away from the Sun. It's just how the numbers work out -- nothing deeper than that.

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