◄ WORLD V · SONNY 5DART 556 · a helldive down the well

THE ROCHE LIMIT where the tide tears a moon apart

Come too close and gravity itself shreds you: inside the Roche limit, the tidal difference across a moon exceeds its own self-gravity, and it disintegrates into a ring. For a rigid body d = R(2ρM/ρm)⅓ — about 1.26 planetary radii for equal densities (a fluid moon, which the tide can stretch, goes out to ~2.44R). Saturn’s rings sit inside its Roche limit: a moon that never could form, or one that did and died.

THE TECHNIQUE d = R (2ρM/ρm)¹⁄³

The demo computes the rigid Roche limit for equal densities via a Newton cube root (i13 has no cbrt): live demo







  
HISTORY & CREDIT Édouard Roche · 1848

  
“The Roche limit is one number.” — rigid vs fluid bodies differ (fluid ~2.44R, rigid ~1.26R for the classic form); density ratio matters. cited

  
the tide wins · tidal stretch > self-gravity ⇒ break up.
the radius · d = R(2ρM/ρm)⅓ — ~1.26R equal-density rigid; ~2.44R fluid.
1848 · Édouard Roche; Saturn’s rings lie inside it.

  
Get close enough and gravity un-builds you. physics






  
RECOMMEND FOR I-13 the shred line, on the compiler

  
On i-13 (Newton cbrt), the rigid Roche limit for equal densities is 1.26 R:

  
$ i13 run gw_roche.i13   # d = R*(2)^(1/3), equal densities, Newton cbrt
RUN OK · 1067 step(s) · call depth 47
d_rigid      = 125.99   -- for R=100
limit_over_R = 1.2599   -- ~1.26 planetary radii (rigid, equal density)

  
Recommend as a NULL — a theorem (B39) + a resource cbrt (B40). The Roche distance is pinned by the densities and radius; the cube root is a Newton workaround. NULL — where the tide tears a moon apart.