I would imagine planetary orbits are measured from the Sun's centre and not its surface. Is that true?
I can't find anywhere that actually states this.
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From the Sun's center, always. When you deduce the equations of motion of planets, you're always calculating from the center. Plus, the results don't change when the Sun blows up as a red giant, or collapses as a dwarf.
But even if you measure from the surface, in most cases it won't make a huge difference. In Earth's case, it's a 0.5% error. It would be a larger error for internal planets, smaller error for external planets.
EDIT: Incorrect. Planets (and the Sun, too) orbit the common barycenter of the Solar System.
It's how the maths/physics works - barring miniscule alterations due to non-uniform distribution of mass, the centre of mass of a planet orbits around the centre of mass of the Sun.
And the centre of mass of the Solar_System (which is very near the centre of the Sun) orbits the centre of mass of the galaxy, etc.
Positions and distances are calculated with respect to the center of mass of a body, not with respect to the geometric center, or surface. The center of the Earth (and therefore its surface) moves with respect to its center of mass by about a centimeter. Mercury's center of mass is offset from its center of figure (geometric center) by 640 meters.
Radar ranging directly measures the surface to surface distance, but these distances are converted to center-of-mass distances in order to calculate orbits.