A Moon Orbiting Nothing

The configuration sounds impossible: a moon tracing circles around empty space. The point of this page is that, dynamically, nothing about it is exotic — replace Earth with an equal point mass and the Moon’s orbit does not care. The only question is whether the orbit fits inside the region where the star lets the pair stay bound: the Hill sphere.

Code

from lastmoon.physics import constants as c
from lastmoon.physics.orbits import (
    HILL_STABILITY_FRACTION,
    hill_radius,
    is_moon_orbit_stable,
    kepler_period,
)

r_hill_m = hill_radius(c.AU, c.M_EARTH, c.M_SUN)
stable_limit_m = HILL_STABILITY_FRACTION * r_hill_m
moon_ok = is_moon_orbit_stable(c.LUNAR_SEMI_MAJOR_AXIS, c.AU, c.M_EARTH, c.M_SUN, c.LUNAR_MASS)
period_days = kepler_period(c.LUNAR_SEMI_MAJOR_AXIS, c.M_EARTH, c.LUNAR_MASS) / 86400.0
print(f"Hill radius of an Earth-mass primary at 1 AU: {r_hill_m/1e9:.2f} million km")
print(f"Stable zone (1/3 Hill): {stable_limit_m/1e9:.2f} million km")
print(f"Lunar orbit (0.38 million km) stable? {moon_ok}; period {period_days:.1f} days")

Hill radius of an Earth-mass primary at 1 AU: 1.50 million km
Stable zone (1/3 Hill): 0.50 million km
Lunar orbit (0.38 million km) stable? True; period 27.3 days

The Moon’s actual orbit sits comfortably inside the stable third of the Hill sphere, with its familiar 27-day period intact.

Code

import matplotlib.pyplot as plt
import numpy as np

from lastmoon.figures.style import apply_style

apply_style()
a_outer = np.linspace(0.5, 1.7, 100) * c.AU
limit = [HILL_STABILITY_FRACTION * hill_radius(a, c.M_EARTH, c.M_SUN) for a in a_outer]
fig, ax = plt.subplots()
ax.plot(a_outer / c.AU, np.array(limit) / 1e9, label="stability limit (1/3 Hill)")
ax.axhline(c.LUNAR_SEMI_MAJOR_AXIS / 1e9, ls="--", label="the Moon's orbit")
ax.set_xlabel("orbital distance from a Sun-like star (AU)")
ax.set_ylabel("moon orbital radius (million km)")
ax.legend()
fig

Stable moon orbits (below the line) for an Earth-mass primary across the habitable zone.

What this means for the paper

A mass-conserving conversion leaves the moon’s orbit untouched, and that orbit is comfortably Hill-stable everywhere in the habitable zone. The “moon orbiting nothing” is dynamically ordinary — which is exactly what makes it a clean tracer of the invisible primary.

Idealizations

Two-body + Hill-criterion treatment; no long-term resonant effects. The paper uses the same conservative 1/3-Hill stability bound.

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