I'm writing a novel, and need the orbit of a body that would fly past Earth every 50 years years.

Question

A highly reflective inert satellite is launched from Earth, which will orbit the Sun and fly past Earth every 50 years, close enough to be visible in the night sky, for 200 million years. It has no thrusters for course correction. It'll be massive enough to not be knocked off course (much) by micrometeorites. How big will it have to be? What's its velocity and orbit? If this is an impossible scenario, what can be changed in my "givens" to make it work?

Edit: Rather than 200 million years, the body would have to survive for only 300,000 years.

Answer 1

The scenario as written contains two elements that are in direct conflict.

A close flyby requires the body to interact with Earth's gravity.

A 300,000-year stable orbit, which requires the body to avoid Earth's gravity.

A body with no thrusters cannot have both. Its orbit is ballistic. A close flyby is a gravitational perturbation. This "slingshot" effect will alter the body's velocity and orbital period. The 50-year return schedule would be destroyed after the very first encounter.

If the body's orbit were only slightly different from Earth's (e.g., a period of 1.02 years, giving a 50-year synodic "lap"), it would be violently unstable. It would be deep inside Earth's gravitational "sphere of influence." It would eventually collide with Earth, Venus, or be ejected by Jupiter.

Therefore, the premise must be changed to make it work.

The Plausible Solution: The 50:1 Resonance

The only way to achieve a stable 50-year return is to remove the "close flyby" and replace it with a "distant encounter." The body must be in a 50:1 orbital resonance with Earth.

This means the body completes one orbit of the Sun in the exact time it takes Earth to complete fifty orbits.

This solution is stable because the body spends more than 49 years far away in the outer solar system. It only approaches Earth's orbit (not necessarily Earth itself) once every 50 years at its perihelion (closest point to the Sun).

This orbit is stable in the long term only if it does not pass close to Earth or, crucially, Jupiter. Over 300,000 years (6,000 orbits), even tiny gravitational kicks from the outer planets will accumulate. This will cause the orbit to drift. For the sake of the novel, we must assume a "perfect" orbital lane that minimizes these perturbations.