Talk:Equivalent radius

Non-spherical vs. non-spheroidal

The article claimed that: "For a non-spherical object, the mean radius (denoted R or r) is defined as the radius of the sphere that would enclose the same volume as the object." It then went on to give formulae for oblate spheroid and ellipsoids, which are not spherical in most cases. The two definitions appear to contradict each other. Praemonitus (talk) 14:26, 18 June 2024 (UTC)

@Praemonitus: There's clearly some misunderstanding, because there is only one definition. In the first paragraph, it distinguishes between the case of a sphere (which is trivial, see the last sentence), and every other shape. The formulae below apply to the special case of ellipsoids (either oblate spheroids or triaxial ellipsoids), for which the mean radius as defined before can actually be easily calculated. The dimensions (2 or 3 axes) for irregularly shaped objects are usually those of the ellipsoid with the same volume (because that's actually useful), motivating this definition of mean radius. Renerpho (talk) 16:13, 18 June 2024 (UTC)

By the way, the wording reflects the reference. Ref.1 ("Distorted, nonspherical transiting planets...") singles out non-spherical objects, and so did I. As I said, the case of the sphere is really trivial (its radius is defined as the radius). Renerpho (talk) 17:30, 18 June 2024 (UTC)

@Praemonitus: I edited the article, to hopefully clarify what is the definition, what motivates it, and how it is used in practice. I'm sure this can be improved much further, but hopefully it's less confusing now. Renerpho (talk) 16:32, 18 June 2024 (UTC)

Hat size

@Headbomb: Thanks for your expansion of the article. There's a problem with the example of hat size. The formula $R_{\text{mean}}={\sqrt {\frac {1}{\pi }}}L$ doesn't apply to irregularly shaped objects (the derivation uses the example of a square, where this is true, but it doesn't work for an ellipse, among most other shapes). Unless you have a rectangular or circular hat, it won't be applicable. Does this example assume that the hat is circular?

I could only check the 9th edition of Bello's book, where I think the referenced section is on p.524-525 (chapter 8.3 Perimeter and Circumference, section "Hats, Rings, and Circumferences"). It says nothing about mean diameter, so I think we need a new source for the example. Renerpho (talk) 10:31, 28 August 2024 (UTC)

The shape doesn't matter. It's taking the "perimeter/circumference" of the head, rather than the cross-sectional area, and divides by pi, and that's how mean radius is defined in that case. Headbomb {t · c · p · b} 12:47, 28 August 2024 (UTC)

Added the missing 1D case though, and rejiggled examples to match their 1d/2d/3d cases. Headbomb {t · c · p · b} 13:43, 28 August 2024 (UTC)

@Headbomb: The shape did matter, but generalizing the article has solved that. Before this, the article was about mean radius in astronomy, in which it has a very specific definition that was incompatible with the hat example. Now that we're treating astronomy as a special case, that's no longer an issue.

That said, please note that this article was created a few weeks ago for the sole purpose of having something to link to from the widely used Template:Infobox Planet. The previously undefined term "mean radius", used in many minor planet articles, had led to confusion. I think we really should keep that special application in the lead, clearly stating the connection to the moments of inertia (why has that been removed?). "Downgrading" the astronomy application to a mere example among many defeats the original purpose of the article.

I suggest to keep the article's main focus on astronomy, and move anything else to a section titled "other uses". Alternatively, the article could be renamed Mean radius (astronomy), to disambiguate it from other possible uses. There was no need for this previously, because there was no other article that talked about those. Renerpho (talk) 14:19, 28 August 2024 (UTC)

The infobox can link to the 3D version. I've commented out the moment of inertia thing for now simply because I didn't know how to integrate it in the current version of the article. It could simply be another example added to the 3D section. Headbomb {t · c · p · b} 17:57, 28 August 2024 (UTC)

@Headbomb: I'll try to reincorporate it. Could you please add citations to the examples you've added, to establish that the terms "mean radius" or "mean diameter" are actually used in those contexts? I cannot find any, and without them, the additions are WP:OR. Thank you. Renerpho (talk) 21:13, 28 August 2024 (UTC)

All the examples have references, and clearly are mean radii/diameter. If there's something specific, point it out. Headbomb {t · c · p · b} 21:18, 28 August 2024 (UTC)

@Headbomb: Nothing is "clear" about that. You've added all of these as examples of the term "mean radius/diameter" being used in applied sciences. You have provided zero evidence that this is true beyond your own interpretation of those terms. That's literally WP:OR. Maybe they can be interpreted as mean radii/diameters, but that's irrelevant. The question is if they are. If anything, there is a conflicting definition of "mean diameter" in the source "Tree and Forest Measurement", p.72, which applies to a specific way to calculate the average diameter of all trees within a stand (rather than that of a single tree). And as noted before, the book by Bello et al. doesn't seem to give any definition of mean radius/diameter, making it (and the hat example) irrelevant. Similar for the other sources I've checked. Renerpho (talk) 21:24, 28 August 2024 (UTC)

Yes, they are. And obviously so. And what's the conflicting definition? P. 72 is about stand basal area. You measure diameter at breast height of all the tree, convert to cross-sectional area with A = π (D/2)², add them all up and divide by the total plot area. This gives you a fraction, which you can express in many ways, that represents how much of an area is trees. If all your tree cross sections add up to 23 m² in a 500m² plot, then you have a stand basal area of 0.046. Headbomb {t · c · p · b} 21:43, 28 August 2024 (UTC)

WP:PROVEIT! If the term is obviously used then finding a citation that actually does so should be trivial. Sun (2016) doesn't even contain the word "mean". If you could provide a relevant quote from Wei (2023), which I have no access to, that would be very helpful. I am going to remove these examples tag these examples with [citation needed] unless you can provide sources that directly support your additions, and I may consider removing them if no citations are added (by you or someone else).

Also, if West defines a "quadratic mean diameter" then why isn't that the example provided in the article, rather than one that has nothing to do with the subject? I'd have no problem with its addition as an example, since it is clearly an alternative use of the term (this is clear and obvious because the term "mean diameter" is included in the text of the source). Renerpho (talk) 21:54, 28 August 2024 (UTC)

This is a broad concept article. It is known by several names, mean radius, effective radius, (or mean diameter, effective diameter, etc...) and so on. Sun, for example, does not have the word "mean" but it doesn't need to have the word "mean" in it to refer to this concept. A direct quote is "diameter of a circle with an equal aggregate sectional area". Sectional area of circle $A=\pi (D_{\text{magical-unnamed-concept}}/2)^{2}$, leading to $D_{\text{magical-unnamed-concept}}={\sqrt {\frac {A}{4\pi }}}$, which is clearly the same thing as $R_{\text{mean}}={\sqrt {\frac {A}{\pi }}}$. This stuff (effective quantities) is covered in any basic undergrad applied science programs (forestry, physics, engineering etc...). For Wei, if you don't have access to it, just look at our article on the hydraulic diameter/hydraulic radius. They're again (or really any source on hydraulic radius, Wei isn't unique here) very explicitly defining an effective diameter through an equivalent circle (hydraulic diameter D = 4A/P, which for a circle is $D={\frac {4\pi R^{2}}{2\pi R}}=2R$), the hydraulic radius is defined differently for historical reasons. Headbomb {t · c · p · b} 00:43, 29 August 2024 (UTC)

I'll think about spinning off mean radius (astronomy), so we can have both your BCA, and an article dedicated to the special application. I no longer think both can be covered well in the same article. As you said, there's no nice way to include the closely related concept of "dimensions" (based on the moments of inertia). I'm not a template editor so will need someone to fix the template (but we'll need that anyway, either to point the infobox to the 3D section or a spin-off). More later... Renerpho (talk) 07:04, 29 August 2024 (UTC)

The spin off has been created, and changes to the infobox template have been requested. As far as I am concerned, provided the proposed changes are accepted, this issue can be considered "closed". Renerpho (talk) 07:38, 29 August 2024 (UTC)