A 'Sun' lamp is so called because it simulates the light of the sun as perceived on Earth.
Its rays are parallel, from infinity. It has no position, and so no falloff with distance. (Good for you, in that respect.) The position of the lamp's gizmo has no bearing on its effect; only its direction makes any difference. All objects will cast shadows on other objects 'behind' them in that direction.
A quick solution to your problem, either in Cycles or EEVEE, would be to use one Sun lamp, anywhere you like, (inside the Sun object if that seems logical) pointing parallel to the Sun > Planet direction, and with 'Cast Shadow' or 'Shadow' switched off in the lamp's Properties panel > Data tab.
This would also mean planets do not cast shadows on one another.
If you want more particular effects, the answer will vary, depending on your renderer. For example, if you're in Cycles, you may want to prevent planets bouncing light on to one another, using a Light Path node. Or if you wanted the planets to go round the sun, you could use a shadowless point-lamp at the sun's position, with its falloff adjusted if necessary.