the event horizon (schwarzschild radius) of the sun is 3km.
i don't think planethood should be defined based on whether the barycenter of the system is within the orbited object, but I do think that's a pretty good definition for determining whether two objects are a planet moon pair or dual planets.
but in case you con't want to follow, here's a summary:
dual planet system (eg pluto charon): barycenter lies outside of either object
planet with moon (eg earth and moon): barycenter lies within the radius of the planet
planet: "(a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit[1]." [2]
as for why planethood shouldn't be determined by barycenter location alone:
"If the definition of a double- or binary-star system is used as a comparison, and it depended only on the location of the barycenter, then any revolving body with a barycenter beneath a star's surface would be a planet, and any body with a barycenter lying outside the surface of the star would be another star. In the Solar System, all of the major planets would be planets under this definition except one. The Sun–Jupiter barycenter is the only center of mass that lies outside the surface of the Sun. Therefore, since Jupiter is not a star, the difficulty faced by astronomers to derive a reality-based definition of double planet begins to become clear." [3]
[1]: meaning it has become gravitationally dominant, and there are no other bodies of comparable size other than its own satellites or those otherwise under its gravitational influence.
the event horizon (schwarzschild radius) of the sun is 3km.
i don't think planethood should be defined based on whether the barycenter of the system is within the orbited object, but I do think that's a pretty good definition for determining whether two objects are a planet moon pair or dual planets.
this wikipedia entry hasa fairly good description: http://en.wikipedia.org/wiki/Dual_planet#Definition_of_a_dou...
but in case you con't want to follow, here's a summary: dual planet system (eg pluto charon): barycenter lies outside of either object planet with moon (eg earth and moon): barycenter lies within the radius of the planet planet: "(a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit[1]." [2]
as for why planethood shouldn't be determined by barycenter location alone: "If the definition of a double- or binary-star system is used as a comparison, and it depended only on the location of the barycenter, then any revolving body with a barycenter beneath a star's surface would be a planet, and any body with a barycenter lying outside the surface of the star would be another star. In the Solar System, all of the major planets would be planets under this definition except one. The Sun–Jupiter barycenter is the only center of mass that lies outside the surface of the Sun. Therefore, since Jupiter is not a star, the difficulty faced by astronomers to derive a reality-based definition of double planet begins to become clear." [3]
[1]: meaning it has become gravitationally dominant, and there are no other bodies of comparable size other than its own satellites or those otherwise under its gravitational influence.
[2]: IAU
[3]: http://en.wikipedia.org/wiki/Dual_planet#Definition_of_a_dou...