I'm not sure whether this paper is a great overlooked observation or a trivial result that one can prove with a couple diagrams and no calculation at all. Maybe it's both :)
If you assume all orbits are spherical, then the distance from your planet to the sun is 1 AU.
Now draw a graph, with the sun at the origin, our planet P at 1 AU 'below' the sun, and the other planet P' going around in a circle at distance R around the sun. Now draw a right triangle, with the sun at a near corner, the other planet P' at the far corner, and one of the short sides pointed straight at our planet P.
__x__P' (t2)
| /
y /
| /
|/
sun
|\
y| \
| \
|_x_\
| P' (t1)
P
You can see that the y-distance of the planet from the sun exactly averages out to zero - half the time it's on the far side, half the time it's near. Therefore the y-distance from our planet to the other planet always averages out to exactly 1 AU.
However, the x-component is sometimes to the 'left' and sometimes to the 'right' of P. We know from the triangle inequality that |y + x| >= |x|, so the x component always makes the distance between P and P' larger than it would otherwise be, therefore the average must be larger than 1 AU, and this discrepancy grows with the radius of the other planet P'.
That's only if P' is closer to the sun than P. In other words, the average distance between the two planets will be approximately the radius of the outermost planet's orbit.
If you assume all orbits are spherical, then the distance from your planet to the sun is 1 AU.
Now draw a graph, with the sun at the origin, our planet P at 1 AU 'below' the sun, and the other planet P' going around in a circle at distance R around the sun. Now draw a right triangle, with the sun at a near corner, the other planet P' at the far corner, and one of the short sides pointed straight at our planet P.
You can see that the y-distance of the planet from the sun exactly averages out to zero - half the time it's on the far side, half the time it's near. Therefore the y-distance from our planet to the other planet always averages out to exactly 1 AU.However, the x-component is sometimes to the 'left' and sometimes to the 'right' of P. We know from the triangle inequality that |y + x| >= |x|, so the x component always makes the distance between P and P' larger than it would otherwise be, therefore the average must be larger than 1 AU, and this discrepancy grows with the radius of the other planet P'.