How close to the Sun can Earth still hold a moon? Out to its Hill sphere — the radius where a body’s own gravity beats the tidal pull of the bigger mass it orbits: rₕ ≈ a·(m/3M)⅓. Earth’s is about 1.5 million km (the Moon, at 384,000 km, sits comfortably inside). Outside it, the Sun takes what you tried to keep — the void, again, reclaiming.
THE TECHNIQUE rₕ ≈ a (m/3M)¹⁄³
The demo computes Earth’s Hill radius via a Newton cube root: live demo
HISTORY & CREDIT George William Hill
“A planet can hold a moon at any distance below its Hill radius.” — stable prograde moons only reach ~½ the Hill radius; the edge is not a hard wall. cited
the contest · your gravity vs the tidal pull of the mass you orbit. the reach · rₕ ≈ a(m/3M)⅓ — Earth’s is ~1.5 million km. Hill · George William Hill, from his lunar-theory work (1870s).
The edge of what your gravity still owns. physics
RECOMMEND FOR I-13 1.5 million km, on the compiler
On i-13 (Newton cbrt), Earth’s Hill radius is 1.496 million km:
$ i13 run gw_hillsphere.i13 # r_H = a / cbrt(3M/m)
RUN OK · 1075 step(s) · call depth 47
r_hill = 1496499000 -- metres
r_hill_Mkm = 1.4965 -- million km (Moon at 0.384 is inside)
Recommend as a NULL — a theorem (B39) + a resource cbrt (B40). The Hill radius is pinned by a, m, M; the cube root is a Newton workaround. NULL — how far your gravity still wins.