Talk:Roche limit

Oblateness term

I think that Weisstein got it wrong at Mathworld --- Let's say that the celestial body is perfectly spherical: then the semi-minor axis radius would equal the semi-major axis radius, so c=R, right? So, (1-c/R) would be zero, and the output of the formula would diverge :-(. I think Weisstein meant to say that c is the difference between the semi-major and semi-minor radii. This makes the formula work. We could define it that way, but it is quite a mouthful --- it turns out that the oblateness of a body is the ratio of the difference to the semi-major radius, so that seems more compact.

I tried to double check this against a non-Mathworld source, but could not. Can someone come up with a double check? Thanks! -- hike395 17:50, 15 Nov 2004 (UTC)

No, R is defined by both Weisstein and here as the radius of the primary, while c is the length of the semi-major axis of the satellite. So, while it's true that this is indirectly related to oblateness, the statement that c/R equals the oblateness of the satellite is incorrect- a perfectly spherical primary with a radius of 10 with a perfectly spherical satellite with radius of 1 would have c/R equal to .1, for example. We could reiterate the definition of R here if that would clarify the issue. --Noren 04:03, 16 Nov 2004 (UTC)

I'm sorry, that isn't clear to me at all. Weisstein defines R in the main body of the article [1]. c is not defined in the main article. The main article references an article about the oblate spheroidal gravitational potential[2]. There, c is defined as the semi-minor axis, not the semi-major axis of an oblate spheroid. It isn't clear to me (from the text description, at least) whether he is referring to the oblate potential of the primary body or of the satellite. It's quite confusing. Is there a different source that we can double check this with? It seems suspiciously mushy. -- hike395 07:42, 16 Nov 2004 (UTC)

I'd certainly like to read an alternative source, but I'm not very skeptical of the one we have, so my motivation to seek another one out is low. An important problem with your 'oblateness' definition of the $c/R$ term, as you pointed out, is that it becomes very messy in the limits, for example the 'spherical' limit when $c=R$. on the other hand, the obvious limiting case if the term relates to the relative sizes of the two bodies is that $R>>c$ and $M>>m$ ... in this case the $\left({\frac {(1+{\frac {m}{3M}})+{\frac {c}{3R}}(1+{\frac {m}{M}})}{1-{\frac {c}{R}}}}\right)^{\frac {1}{3}}$ term goes to 1, which is the behavior we expect given that this term is ignored entirely in the cruder first approximation.

Throughout the Roche limit discussion, capital letter variable definitions refer to the rigid primary, while lowercase letters refer to the satellite. Weisstein makes the assumption for purposes of this derivation that the primary is spherical- that's why the letter used is $R$ rather than an $A,B,C,a,b,$ or $c$- it refers to an assumed spherical body which has a Radius. Finally, you're right that $c$ refers to the semi-minor rather than the semi-major axis of the satellite, that was my mistake. --Noren 16:06, 16 Nov 2004 (UTC)

I'm confident one of the two following statements is true: 1) c refers to the difference in equatorial and polar radii of the primary, or 2) Weisstein's derivation is wrong. I believe this because a satellite deformed by tides is a prolate spheroid, while a fast rotating primary is an oblate spheroid. Weisstein specifically uses the oblate spheroidal potential: he must either be referring to the gravitational potential of the primary, or he is confused.

If c is the difference between the equatorial and polar radii, it goes to zero for a perfect sphere, so there is no problem with divergence at all. If c refers to the semi-minor axis of the primary (as stated by Weisstein), then the formula is easily shown to be incorrect. -- hike395 04:46, 17 Nov 2004 (UTC)

I found this paper [3], but is on only available as an on-line abstract, and I don't have a subscription to Icarus. Can anyone help out? -- hike395

This Roche limit stuff is really complicated. Check out [4][5]. There are many possible factors, including modeling the internal friction of the satellite and/or modeling the hydrodynamics of the satellite. The Weisstein derivation looks overly simple compared to what real astrophysicists use. -- hike395 06:58, 18 Nov 2004 (UTC)

Why clockwise?

Why are the illustrations (well, one of them anyway) of the disintegration process showing clockwise movement? The natural direction is counterclockwise, as we tend to look at things from the "north" pole...

Urhixidur 17:53, 2004 Nov 15 (UTC)

Okay, I fixed the image now.

Urhixidur 17:57, 2004 Nov 15 (UTC)

Help needed on wikijunior solar system book

I'm working on a book for children over at wikibooks and someone has written the following:

Mercury does not have a moon. Mercury's rotation is so slow that if Mercury had a moon, it would crash into Mercury or get broken up. This would happen because the moon's gravity would cause tidal effects on Mercury. There would be two bulges called tidal bulges on Mercury. One would bulge toward the moon, with the other bulge being on the opposite side of Mercury. The moon's motion in its orbit would be faster than Mercury's rotation because Mercury's rotation is very slow. That would cause the moon to be ahead of the tidal bulge all the time. The gravity from the bulge would pull back on the moon. This would cause the moon to become closer to Mercury and Mercury's rotation to speed up. This would continue to happen over millions of years until the moon got broken up by Mercury's gravity or crashed onto Mercury. Mercury had existed for billions of years, so if it had any moon, it is long gone.

[6]

Now aside from the fact that this explanation is confusing to me (let alone a kid) I'm not at all convinced that the science is actually correct. So I was wondering if any of you fine people (flattery will get me everywhere I hope) could help me rewrite? Theresa Knott (a tenth stroke) 2 July 2005 19:41 (UTC)

Sounds confusing but basically right. Tidal forces make satellites tend toward the planet's rotation period, given enough time. If the planet's rotation is faster (like with the Earth/Moon system), the planet donates angular momentum to the moon, causing it to move slowly away from the planet and (paradoxically) slow down. If the moon's revolution is faster (like with the Mars/Phobos system), the planet takes angular momentum from the moon, causing it to move slowly toward the planet, and to speed up. The good news is that all orbits eventually tend toward a stable, tidally locked orbit over time. The bad news is that they often get inside the Roche limit or outside the Hill sphere before they reach stability, and therefore stop being a moon altogether. --Doradus July 3, 2005 00:26 (UTC)

Having done the calculations, I think the remark is right. Mercury's Hill sphere radius is about 220 Mm, while its geosynchronous orbit radius is 244 Mm. That means a moon with enough angular momentum to sustain a geosynchronous orbit would not be within Mercury's sphere of influence, so it would be in orbit around the Sun. Likewise, a moon with an orbital radius within Mercury's Hill sphere would orbit faster than geosynchronous, and would therefore bleed angular momentum into Mercury until it found itself inside Mercury's Roche limit, and either broke apart or collided with Mercury. Perhaps someone can confirm my calculations? --Doradus July 3, 2005 03:59 (UTC)