Talk:Earth radius

Untitled

I believe that in the comment where A is the authalic surface area of Earth, the word authalic is not needed. As the term means equal area, the "authalic surface area" of the Earth is identical to the surface area of the Earth! Making the distinction between the two may confuse people into thinking that they do not really understand what authalic means. If anything, make it clear A_r is the authalic radius.

Question: Why is this comment

Note: Earth radius is sometimes used as a unit of distance, especially in astronomy and geology. It is usually denoted by RE.

part of the "Volumetric radius" section?

Should it be perhaps "Earth volumetric radius is sometimes used...", or does this comment refers to the general case, situation in which it should be moved somewhere in the main text?

Radius vs. Sea Level

"The radius of Earth (or any other planet) is the distance from its center to a point on its surface at mean sea level." This doesn't seem right. Why is the mean radius of a planet exactly the sea level? The sea level of the earth is rising, does this mean the radius of the earth is increasing too? Why would the melting of the ice caps cause the land to rise? Anyone have a cite? -anon

Moreover many planets (most?) don't have a sea. Rich Farmbrough, 14:28, 28 February 2011 (UTC).

Mean sea level establishes the hydrostatic equilibrium more accurately than more rigid land can. If sea level rises, then mean radius increases. I am not sure what melting ice caps have to do with the remainder of your question, but land beneath and around ice caps rise after melting due to isostatic rebound. For bodies that have no fluid surface, a fictitious one is established, typically as an atmospheric pressure. See, for example Mars#Geography. Strebe (talk) 19:14, 28 February 2011 (UTC)

The Elevation of Mean Sea Level was lower in the Past when the climate was cooler, and there was more ice in all of the Glaciers, this includes Continental Glaciers, Glaciers that started on Continents, and ended in Oceans, and Glaciers that filled Mountain Valleys, and slowly moved down hill. Mean sea level was a very useful reference elevation when surveying equipment consisted of Theodolites, Chains, Wye Levels, Compass, Clocks, Transits, and similar devices to determine X, Y, and Z. Z is the vertical elevation above Mean Sea Level, set at a reference time period. Having a universally accepted reference elevation keeps things a lot simpler so that surveyors around the world can work together using the same system.

The Fact that sea level is rising is only an inconvenience if you happen to live too close to sea level, but it is not really an inconvenience for surveyors as long as every body agrees not to change the previously agreed upon reference elevation. Fortunately we now have the Global Positioning System ( GPS ), and the people who control that system take about 1700 continuously monitored reference points and so with the help of lots of big computers, they keep the reference elevation and the change in the reference such that the change sums up to zero. This is very good for Surveying as it makes one of the Three " Delta Z " = zero.

The other Two Delta X, and Delta Y are not so co-operative as Continents like to wander in various directions, and they even spin around about some axis. Fortunately, in most locations this motion is very small. The movement is around 1 mm per year in each of the two directions, Longitude, and Latitude. The actual movements only cause problems over very long periods of time, like millions of years when you are trying to find out where Dinosaurs actually lived, but the Location that was around the Equator 65 Million years ago is now over 20 degrees North of the Equator. That is Chicxulub, and The Deccan Traps were both near the Equator between 65 MYA and 66 MYA, and both are now around 20 degrees North of the Equator. The Impact location that is now at Chicxulub was Anti-podal to the Energy focal Point on the opposite side of the Globe at what is now the Deccan Traps in India. So millimeter per year movements really add up over millions of years to hundreds of Kilometers of movement. Mike Clark, Golden Colorado, 63.225.17.34 (talk) 04:05, 6 September 2016 (UTC)

Radius of curvature

I have changed the formula for Rc near the end of this section to reverse M and N, since when traversing along a meridian (with alpha = 0, i.e. facing North) one is moving along "M" the meridianal radius and when moving (e.g at the equator) eastward (for alpha = + 90 degrees) one is moving along "N" the radius of curvature in the prime vertical. JimC (C&B) 17:57, 7 January 2007 (UTC)

You just described it right——facing north-south the RoC = M, while facing east-west = N——so cos(0)M = M and sin(90°)N = N, which is how it was (your change went and inverted it!

Whoops, okay, I see the problem: Presenting it with MN as the numerator cancels and reverses the placement of M and N in the denominator. I've changed it to make it clearer. P=) ~Kaimbridge~ 15:21, 8 January 2007 (UTC)

"Neridional" radius of curvature???

The heading Neridonal appeared in a 1 April 2007 edit. Surely this should be Meridional. I don't think neridional is even a real word. Certainly it's not in any online dictionary. Also a Google search for it only gives 149 hits, and almost all of those hits are either this article or people who have copied this article verbatim (like answers.com). I'm going to edit this. If someone feels there is some really burning reason why it should be 'neridional' and wants to revert, please add a note here why such an obscure word that doesn't appear in dictionaries should be used. I'm looking at you, User:Kaimbridge, since you made that 1st April edit ... surely you wouldn't be playing an April Fool's joke on Wikipedia ;). Dr algorythm 03:19, 18 June 2007 (UTC)

LOL!!! Sorry about that——I'm blind in one eye and the other eye's no bargain, with an acutely advancing cataract further complicating things. So, needless to say, a few typos may escape me! P=) ...Hmmm, how come it took someone two and a half months to catch?!? P=/ ~Kaimbridge~14:27, 18 June 2007 (UTC)

ERROR?

On this page the polar radius is derived as being larger than the equatorial radius. Please check this derivation. —The preceding unsigned comment was added by 216.59.226.224 (talk) 04:48, 25 April 2007 (UTC).

What do you mean, a (equatorial radius) = 6378.135 and b (polar) = 6356.75? ~Kaimbridge~10:49, 25 April 2007 (UTC)

Regarding the above given values a and b, I think they should be corrected and precised as follows: a=6378.137 km (according to GRS80 and GRS84) and b~= 6356.752 km (as a derived value). Gil

The shape of the Earth is called an "OBLATE SPHEROID". This means the Radius of the Earth is greater at the Equator ( 6378.137 km ) versus the Polar Radius that is smaller, (6356.752 KM). The shape is caused by the balance of forces between Gravity ( that wants to make a perfect sphere ), and Centripetal forces ( That wants to turn the Earth into a thin flat disc ). Gravity mostly wins, as the Earth is not spinning on its axis at a very fast rate. If you simply average the two numbers, the Simple radius is 6367.74445. This radius is about 3.5555 Km below the reference Radius of 6371 KM. They actually cheat a little to get this number, but it is fairly close to the correct value as they take 1 times the Polar Radius, and 2 Times the Equatorial radius, and divide by 3 ( 6356.752 + 6378.137 + 6378.137 ) / 3 = 6371.008667. This is rounded to 6371.009, or more commonly to just 6371.0 Km Radius. The actual Value at the Tilt Angle of a Perfect Sphere is 6372.4567 at a Tilt angle of 23 degrees 19 minutes, and 39.3 seconds ( 23.327 583 333 degrees of tilt). Note that the actual tilt angle of the Earth is similar to this number. The Earth's Tilt angle oscillates between about 22.5 degrees, and 24.5 degrees. It is currently close to 23.4 degrees. The slightly higher Radius is really only useful if you want to determine the actual Mass, and actual Average Density of the Earth. For 6371 KM, the density is around 5514 kg/m^3. For 6372.4567 km Radius, the Density is lower at 5510.183 736 kg/m^3. The Best radius to choose is the one that gives you a surface Area of the Earth that is closest to the actual surface area of the Earth. For R = 6372.4567, the Surface Area ( Simple sphere ) is 5.102 977 464 E 14 m^2, the Volume is then 1.083 950 098 E 21 m^3, the density is about 5510.183 736 kg/m^3, and the mass is about 5.972 764 201 E 24 kg. These are all close to generally accepted values. Mike Clark, Golden, Colorado, 63.225.17.34 (talk) 04:55, 6 September 2016 (UTC)

Latitude dependent radius error?

When using the formula *RADII WITH LOCATION DEPENDENCE* I've had a not logical result, for a 43ºS. calculation. My carefully result was no logical because it was 5417.374347 Km., not between the equatorial and polar radius, can spmebody explain this abnormal fact, I will be very happy if somebody explains me this anomaly, Tks. Carlos J.J.Vial <carlosjjv@terra.cl>Carlos J. Jiménez Vial 21:02, 3 August 2007 (UTC)

Well, if its the actual radius at 43º, it should be about 6368 km (depending on the actual values of a and b); but if it involves cos(43º), 5417 is actually too big——it should be somewheres between 4633 and 4681 km (based on the arcradii extremes of 6335 and 6399 km)! What values does it give for 0 and 90°? ~Kaimbridge~17:37, 4 August 2007 (UTC)

I am interested on the equation of estimating earth radius depend on latitude, could you give me information the reference of this equation (from where this equation come from?) Marksteven2 04:45, 3 December 2008

Please sign your talk-page posts with four tildes.

These may help:

• http://www.mathworks.com/access/helpdesk/help/toolbox/aeroblks/geocentrictogeodeticlatitude.html
• http://lists.maptools.org/pipermail/proj/2004-April/001212.html
• http://mathforum.org/library/drmath/view/61528.html

-Ac44ck (talk) 07:47, 3 December 2008 (UTC)

SI Units

Why are we not using SI units here (radii in meters)? MeddlerOnTheRoof (talk) 02:21, 9 May 2008 (UTC)

Kilometres are SI. JIMp _talk·cont 18:11, 29 December 2009 (UTC)

Changes to section on authalic radius

I removed this text:

This number is derived by square rooting the average (latitudinally cosine corrected) geometric mean of the meridional and transverse equatorial, or "normal" (i.e., perpendicular), arcradii of all surface points on the spheroid

It may be that the number approximates the root of the average. I don't find evidence that this average is used to derive the exact expression. One derivation is here.

I also modified this text

the radius of a hypothetical perfect sphere which has the same, geometric mean oriented surface area as the spheroid.

The "mean oriented surface area" of a perfect sphere is zero, as each elemental vector area has an equal antipodal counterpart. The vector sum of the oriented area is zero. -Ac44ck (talk) 02:43, 1 November 2008 (UTC)

Approximation

The original Meridional definition of a meter was one ten-millionth the distance from the North Pole to the Equator, the circumference of the earth is then (about) 40 million meters. Treating the earth as a perfect sphere then gives a nice, and easy to remember approximation for the average radius of the earth: (2/pi)*10^7 meters. This works out to 6366.2km, which is within 0.1% of all of the various figures for the average radius. Pulu (talk) 06:14, 12 November 2008 (UTC)

Reference or monograph

A reference for radius of curvature section was recently expanded. It seems lengthy for a reference.

If the subject in this reference is to be expounded upon, I suggest moving the longer treatment to the great circle distance article, which doesn't seem to include any approximate formulae at the moment.

I found the reference written with:

A related application of M and N: if two nearby points have the difference in latitude of $d\phi \,\!$ and longitude of $d\lambda \,\!$ (in radians) with mean latitude $\phi \,\!$, then the distance D between them is

$D\approx {\sqrt {(Md\phi )^{2}+(N\cos \phi d\lambda )^{2}}}.\,\!$

I tweaked it because two-glyph symbols like $d\lambda \,\!$ may be difficult for some readers to parse and $\phi \,\!$ seemed to be used in different ways. So I changed it to:

A related application of M and N: if two nearby points have the difference in latitude of $d\phi \,\!$ and longitude of $d\lambda \,\!$ (in radians) with $M\,\!$ and $N\,\!$ calculated at mean latitude $\phi _{m}\,\!$, then the distance D between them is

$D\approx {\sqrt {(M\cdot d\phi )^{2}+(N\cdot \cos \phi _{m}\cdot d\lambda )^{2}}}.\,\!$

The current version contains undefined symbols, such as $dH\,\!$. Why the new symbol is multiplied by a trigonometric identity to obtain $dH={\sqrt {(dH\cos(\alpha ))^{2}+(dH\sin(\alpha ))^{2}}}\,\!$ is lost on me. And current indications are that the results are exact. The previous version gave a formula that was an approximation. Perhaps the current formula is exact for a differential distance $dD.\,\!$ But the introducing paragraph advertises that a way to calculate $D\,\!$ is to be presented, and the now-more-elaborate treatment doesn't seem to deliver it.

I wondered whether the formula for $D\approx ...\,\!$ belonged in a reference when I tweaked it. But it is similar to the procedure prescribed by the FCC for moderate distances in 47 CFR 73.208, an allusion to which was recently omitted from the external references in this or a related article, and I didn't want to find a place for it in the article. That it is an application of the Pythagorean theorem over short distances on a curved surface seemed more easily discernable when the formula was by itself. - Ac44ck (talk) 22:54, 3 December 2008 (UTC)

I'm the guilty party! P=)

Everything you said appears right (I, too, questioned whether it should be presented as a footnote reference, but no one moved it, and now you tweaked it, so I attempted to clarify it further). I added H ("hypotenuse") to isolate the angular distance.

First of all, the FCC formula (a better presentation of it is here) is the same formula——"KPD_lat" is M and "KPD_lon" is cos(lat)N, in binomial series expansion form, based on the Clarke 1866 spheroid (a = 6378.2064, b = 6356.5838). As for "(w)hy the new symbol is multiplied by a trigonometric identity to obtain $\scriptstyle {dH={\sqrt {(dH\cos(\alpha ))^{2}+(dH\sin(\alpha ))^{2}}}}\,\!$", that is important because $\scriptstyle {M\cos(\alpha )H=M\Delta \phi }\,\!$ and $\scriptstyle {N\sin(\alpha )H=N\cos(\phi _{m})\Delta \lambda }\,\!$, so if you can isolate and separate H, the left over $\scriptstyle {\sqrt {(M\cos(\alpha ))^{2}+(N\sin(\alpha ))^{2}}}\,\!$ is the transverse meridional radius of curvature, T, or arcradius, $\scriptstyle {\overset {{}_{\smile }}{R}}\,\!$, that can be applied to orthodromic calculations (by using the orthodromic azimuth), for any distance (as for infinitesimal distances, it equals the geodetic distance), though it is technically the "elliptical great-circle distance", as it follows the geographical delineation, rather than the geodetic!

Something that needs to be corrected, though, is a distinction between spherical/geographical and geodetic azimuth ("whole" and "local"):

${\tilde {\alpha }}({\widehat {\sigma }})=\lim _{H\to 0}{\tilde {\alpha }}=\arctan \left({\frac {N}{M}}\tan({\widehat {\alpha }})\right);\,\!$

$R_{c}={\frac {{}_{1}}{{\frac {\cos ^{2}({\tilde {\alpha }})}{M}}+{\frac {\sin ^{2}({\tilde {\alpha }})}{N}}}};\,\!$

${\begin{aligned}{\color {white}{\frac {\big |}{}}}{\overset {{}_{\smile }}{R}}=T&={\sqrt {(M\cos({\widehat {\alpha }}))^{2}+(N\sin({\widehat {\alpha }}))^{2}}},\\&={\frac {{}_{1}}{\sqrt {\left({\frac {\cos({\tilde {\alpha }}({\widehat {\sigma }}))}{M}}\right)^{2}+\left({\frac {\sin({\tilde {\alpha }}({\widehat {\sigma }}))}{N}}\right)^{2}}}};\end{aligned}}\,\!$

I give a more in-depth analysis here, though some of the notation has evolved since then (e.g., "O" is now "T", and the loxodromic azimuth notation has simplified).

As for adding it to the great circle distance article, I would suggest changing the section to "Approximate elliptical great-circle distance formula" (and after giving the FCC form, let $\scriptstyle {SA_{m}={\frac {\cos(LAT_{1})\cos(LAT_{2})\sin(\Delta \lambda )}{\cos(LAT_{m})\sin(\Delta {\widehat {\sigma }})}}}\,\!$, $\scriptstyle {{\color {white}{\frac {\big |}{}}}KPD_{m}={\sqrt {M^{2}(LAT_{m}){\big (}1-SA_{m}^{2}{\big )}+{\big (}N(LAT_{m})SA_{m}{\big )}^{2}}}}{\color {white}.}\,\!$, then $\scriptstyle {DIST=KPD_{m}\times \Delta {\widehat {\sigma }}}\,\!$) and moving it to after the "A worked example" section.

If you are really interested in geodetic formulation, in general, I would STRONGLY recommend Richard Rapp's Geometric Geodesy (both parts), now downloadable in PDF, free, from OSU, here: This is the bible of geodetic formulation! P=)  ~Kaimbridge~ (talk) 18:07, 4 December 2008 (UTC)

Thanks for this and for your reply at Talk:Great-circle_distance. I moved the section in the great circle distance article, as you suggested. I would hesitate to bring up "elliptical" in an article on "great-_circle_ distance". At the moment, the character string "ellip" appears only in two words near the bottom of the page.

I still don't understand the relationship between $\scriptstyle {dD}\,\!$ and $\scriptstyle {D}\,\!$ in the current note in this article. As this Earth radius article recognizes the earth is not spherical, it seems that the accuracy of any constant-radius formula, such as $\scriptstyle {dD={\overset {{}_{\smile }}{R}}\cdot {dH}}$ would decrease with range. It looks like a formula for a differential distance as opposed to one which is applicable for any distance.

Thanks for the pointer to Rapp's documents. Part 1 is 189 pages; Part 2 is 225 pages! I have only scanned them so far, but they look very interesting.

You have a much better handle on this topic than I do. I wouldn't feel comfortable copying things from the note here to the great-circle distance article. And on reflection, things that talk about the elliptical shape of the earth seem out of place there. Which makes the FCC formula out of place there, too. A short-distance approximation using the Pythagorean theorem would not be so complicated on a _sphere_.

There seems to be good stuff in the note here, but it makes for a longish note compared with its neighbors in the "Notes and references" section. I don't know how standard the term "great ellipse" is, but a section to address distances along such a curve seems to be missing. The article on Earth _Radius_ may not be the place for it. The article on geodesy doesn't seem to be so much about application. The Vincenty's formulae article seems to be the only treatment of non-spherical distance. Maybe a separate article is needed to discuss exact and approximate distances along the surface of a spheroid? One name for the new article might be "Geodetic formulae".-Ac44ck (talk) 21:03, 4 December 2008 (UTC)

Right, $\scriptstyle {dD={\overset {{}_{\smile }}{R}}\cdot {dH}}\,\!$ is the formula for a differential distance, just like $\scriptstyle {dD=M\cdot {d\phi }}\,\!$ is the formula for a differential meridional distance: So how do you find a distance of any length along a meridian? By finding the average value of M between the latitudes and multiply by the difference: $\scriptstyle {D=M_{avg}\Delta \phi }\,\!$. Likewise, to find the elliptical great-circle (technically, due to the way it is differentiated, I think this would be the proper term, rather than "great-ellipse") distance, you would find the average value of T (or $\scriptstyle {\overset {\smile }{R}}\,\!$) along the great circle between the points and multiply by the angular distance: $\scriptstyle {D=T_{avg}\Delta {\widehat {\sigma }}}\,\!$. Andoyer's Approximationapproximates this. I have a more direct/simpler form of his equation, but I'm shutting down now, so I'll give it here tomorrow! P=) ~Kaimbridge~ (talk) 20:39, 5 December 2008 (UTC)

Will (eventually) add to Geographic distance, instead.  ~Kaimbridge~ (talk) 02:14, 7 December 2008 (UTC)

Contrary to what the first paragraph in the note says about showing how "D between them can be found", the current version doesn't seem to present a way to find D. Instead, it provides something that needs to be averaged via a forthcoming mechanism, which I gather will be quite more complicated than the formula which existed in the note last week. I don't think we need to make an either-or choice here. Both versions could be helpful to have in an article.

Interesting stuff, but finding distances along the surface doesn't seem on-topic in the Earth _radius_ article.

Agreed, though another possibility would be to change it from a reference for R_c, to its own subsection underneath ("Transverse meridional or arcradius", $\scriptstyle {T{\mbox{ or }}{\overset {\smile }{R}}}\,\!$), or even its own section, "Radius of arc", $\scriptstyle {\overset {\smile }{R}}{\mbox{ or }}R_{a}\,\!$ to highlight

${\tilde {\alpha }}({\widehat {\sigma }})=\lim _{H\to 0}{\tilde {\alpha }}=\arctan \left({\frac {N}{M}}\tan({\widehat {\alpha }})\right);\,\!$

${\begin{aligned}{\color {white}{\frac {\big |}{}}}{\overset {{}_{\smile }}{R}}=T&={\sqrt {(M\cos({\widehat {\alpha }}))^{2}+(N\sin({\widehat {\alpha }}))^{2}}},\\&={\frac {{}_{1}}{\sqrt {\left({\frac {\cos({\tilde {\alpha }}({\widehat {\sigma }}))}{M}}\right)^{2}+\left({\frac {\sin({\tilde {\alpha }}({\widehat {\sigma }}))}{N}}\right)^{2}}}};\end{aligned}}\,\!$

and its relationship to

$R_{c}={\frac {{}_{1}}{{\frac {\cos ^{2}({\tilde {\alpha }})}{M}}+{\frac {\sin ^{2}({\tilde {\alpha }})}{N}}}};\,\!$

if you didn't think it would violate NOR!  ~Kaimbridge~ (talk) 02:14, 7 December 2008 (UTC)

What do you think of having a separate article on "Geodetic formulae" with its own content (including the FCC formula and the growing contents of the note under discussion here) plus links to the Vincenty's formulae and great-circle distance articles?

The term "elliptical great-circle" seems awkward to me. A circle is a special kind of ellipse. The term reads something like an "elliptical special ellipse". The term "elliptical great-perimeter" comes to mind as an alternative. -Ac44ck (talk) 21:25, 5 December 2008 (UTC)

The thing is, I think, where a great circle or great ellipse would be a straight arc ("----------"), the way this would slightly deviate from the whole geodetic delineation at each increment (as you are delineating it spherically, then squashing it down to an ellipse) would result in a broken arc, something like "-’¯‘-’¯‘-", thus it wouldn't be a smooth geodetic elliptic arc, but increments of spherical arc elliptically adjusted, independently (i.e., a polygonal arc?).  ~Kaimbridge~ (talk) 02:14, 7 December 2008 (UTC)

First note

The first note is not clear to me:

The center of the Earth is somewhat model dependent. Exceptions to the cited range occur near the South Pole and along the equator. Also, differences due to variation of mass density within the planet and tidal forces require data for the entire surface of the Earth and are not included here.

• How is the "center of the earth" model dependent? Wouldn't the center of the ellipsoid be at the same location as that of a spherical model? The center of curvature varies for the ellipsoid, but the center of the ellipsoid is a unique point. The geometric "center of the earth" may not be at the center of mass. I don't see a need to distinguish between "geometric center" and "center of mass" here. The "center of gravity" for an observer on the surface may be elsewhere, but that is a different concept.
• How are the limits "exceptions to what cited range"? Why only the South Pole and not the North Pole?
• How is an anomaly near the South Pole related to the location of the center of the earth or the stated value of the polar radius?
• How do tidal forces affect the location of the center of the earth? Or is this referring to deformation of the surface by tidal forces? The discussion so far in the article seems to assume that the surface is a static shape of some form (spherical, ellipsoidal) as opposed to a moving surface.
• Two of the three wikilinks are already in the "See also" section.

Do we need to keep this footnote? -Ac44ck (talk) 20:48, 26 December 2008 (UTC)

I rewrote the first paragraph. It had become incoherent. I think you still have questions, but let's start over, referring to the new first paragraph. Strebe (talk) 02:24, 27 December 2008 (UTC)

Every shape (solid or not) has a radius at every point. A straight line has an infinite radius at each point. I thought the phrase "a radius that is the same by all measures" was fairly elegant. The article covers both "radius to a geometric center" and "radius of curvature" (which generally varies with both latitude and azimuth). Those are different ways to measure the radius. They are the same for all points (in every direction) on a sphere.

The mention that the radius "varies from place to place" might say something about the role of topography in that variation. The radius is constant at a given latitude for the models discussed in the article.

The part about "Exceptions to this range occur near the South Pole" is confusing to me. The polar radius is one of the limits in the "6,356.750 km to 6,378.135 km" range, so it isn't an exception. The polar radius is equal for both poles, so I don't see a need to single out the South pole. Likewise, the equator is one of the limits, not an exception.

Other prior questions remain. - Ac44ck (talk) 03:28, 27 December 2008 (UTC)

I am not aware of any definition of radius that applies to anything but (hyper)spheres. If you know otherwise, please cite it? I reverted your changes to the lead paragraph for several reasons. For one, the problem with definition of radius. For another, it's too chatty. For yet another, the restriction of the discussion to a geometric model of the sphere is not made until later in the paragraph; hence it makes no sense to discuss variation by "latitude" when the discussion is about the geoid at that point. It is true that the south pole disclaimer is bogus; local topography there never exceeds the equatorial radius. But the equatorial disclaimer holds; local topography exceeds the model ellipsoid radius. Still, it is poorly written as is because the notion of a model ellipsoid has not even been introduced and yet it is comparing measurements against that. I'll make another attempt. It sure was a mess before! I think we're making progress. Strebe (talk) 09:50, 27 December 2008 (UTC)

An ellipsoid may be described using spherical coordinates, where every point on the surface is defined by a radius from a geometric center: http://mathworld.wolfram.com/Ellipsoid.html .

I shortened sentences; I thought that would make it less convoluted than it was. A complaint seems to be that things were done out of order: restricting discussion to a sphere too early and not introducing an ellipsoidal model early enough. But the first sentence was and is about a sphere. The second sentence did and does talk about various values for the radius. Mentioning variation with latitude doesn't seem a grievous offense.

I get the immediate impression from the talk about spheres in the first sentence that the geoid is not the main topic to be discussed. "The notion of a model ellipsoid" is at least implied by giving two orthogonal values for the radius. What shape should the reader have in mind if not a sphere or ellipsoid? I wouldn't expect an average reader to have the complex shape of the geoid in mind while scanning the lead paragraph.

"The distance from mean sea level at each point on the surface" seems to exclude topography, so an "equatorial disclaimer" wouldn't seem to be needed. The equatorial radius is stated to a resolution of less than one meter. Is that actually a "sea level" radius or some average of land and sea levels?

To say that the earth "approximates a sphere" sounds like it is an action taken by the earth, but humans are making the approximation. I would change the current text to say "the earth's radius deviates from that of a perfect sphere by only a third of a percent" because only one property of the earth is being compared with a sphere. "Earth" is not capitalized when it follows "the": http://grammar.ccc.commnet.edu/grammar/grammarlogs1/grammarlogs212.htm .

How does reversion of a "too chatty" edit that was "poorly written" qualify as "progress" as opposed to reversion? -Ac44ck (talk) 12:05, 27 December 2008 (UTC)

At the risk of touching off another harangue, I tried fixing the lead again.

I note that in this diff, I made two changes:

• "A true sphere has a unique radius" to "A true sphere has a constant radius"
• "Strictly speaking only a sphere has a true radius" to "Strictly speaking only a sphere has a radius that is the same by all measures"

And the lead was suddenly found to be "an incoherent mess."

In this diff, I made modifications to the text as left by a previous editor to:

• Attribute action to people rather than the inanimate earth
• Allow for other shapes being perceived as roundish
• Change "varies from place to place" to "varies with latitude"
• Shorten a confusing sentence
• Add one sentence
• Insert a paragraph break

And its previous editor declared it "chatty" and "poorly written".

The insistence that only a sphere may have a radius is contrary to the action of providing two values for the radius in the lead.

I think that "varies from place to place" is too general. The text specifically talks about "The distance from mean sea level at each point". I haven't investigated whether the stated radii are actually sea level values, but that's what it says. And those distances don't vary randomly from place to place. They vary comprehensibly with latitude.

To say that "the earth deviates" is also too general. It is the radius which deviates from the value for a sphere.

I think there are too many "to"s strung together in the second sentence for ease in reading, but I'm not going to change that at the moment.

The footnote still says "The center of the earth is somewhat model dependent." How so? For which two models is the center at a different location?

The capitalization of "Earth" seems contrary to usual practice, but it occurs throughout the article. -Ac44ck (talk) 19:35, 27 December 2008 (UTC)

This document http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350.2-a/Chapter%203.pdf suggests that the equatorial radius of the WGS 84 Ellipsoid is within 2 meters of a measured "mean sea level" at the equator. -Ac44ck (talk) 22:02, 27 December 2008 (UTC)

First of all, my comments about an incoherent mess do not apply to your recent edits; they apply to the state of the lead paragraph as it has been for a long time, so I will not respond to that portion of your essay.

Your citation for definition of "radius" is not satisfying. You cited a usage, not a definition. Meanwhile, that quote is simply a loose usage by that same site's dictionary of math terms: "Radius: The distance from the center of a circle to its perimeter, or from the center of a sphere to its surface. The radius is equal to half the diameter." (http://mathworld.wolfram.com/Radius.html) In particular, radius does NOT mean "the distance from a point on a solid's surface to its center". At least, I can find no such citation, and since all the citations I can find say otherwise, your inferred definition seems to be on unsupportable ground. In general "center" is not even defined for a (hyper)solid, and while it may be for an ellipsoid and a few other solids by virtue of symmetry, it is not well-defined even for the geoid. Therefore I strongly resist the verbiage of "constant radius"; it is self-redundant and implies more generality than accepted usage allows for.

Your lexicography of "the Earth" is some random site's statement of usage, which in fact says that any combination is in use. Meanwhile, Wikipedia's convention is found here:

http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style

Hence, "the Earth".

I will work more on the lead sentence to make plain when the real earth is being discussed, versus the geoid, versus a geometric model. Meanwhile I have reverted your edits for the reasons given. Strebe (talk) 01:16, 28 December 2008 (UTC)

Oh, and with regard to the Earth "approximating" a sphere: I disagree with your analysis. At the very least it seems uselessly pedantic, since no one would be confused by the usage as it stands, but I would also argue that your point is semantically incorrect. If you were to remove tidal effects, the Earth would approximate a sphere somewhat better, and if you were to remove centripetal forces, it would approximate a sphere very closely indeed. "Approximation" is reciprocal. If a Taylor polynomial approximates a sine wave, then the sine wave approximates the Taylor polynomial. There is nothing incorrect in stating the Earth approximates a sphere. Therefore in the interest of a pithy text and elimination of excessive qualifiers, which themselves open up ambiguities, a responsible editorial policy opts for terseness. Strebe (talk) 01:32, 28 December 2008 (UTC)

Thanks for your opening remark. Yet it seems that you are intent on reverting what I write. I tried to modify your text after you discarded mine; and you discarded that.

I see that the lead is now much longer than the "chatty" version I had tweaked. I think the current lead is overblown; it is practically a mini-article, if not longer than some articles in WP. But I'm done trying to fix it for now.

I find the insistence on such a narrow definition for the word "radius" to be flabbergasting. The radius article says:

More generally ... the radius of something ... refers to the distance from its center or axis of symmetry to its outermost points.

I am going to agree to disagree. I won't say that we will agree to disagree because that may be assuming too much. I don't understand the reason for what I perceive as hostility being directed my way. Did I do something that ticked you off? If so, what? -Ac44ck (talk) 04:56, 28 December 2008 (UTC)

Ac44ck, I apologize if you are feeling persecuted. Let me be plain: It is your edits and comments that led me to take a good look at that lead paragraph and see what a complete mess it really was. Hence I am grateful. If you will notice, it is not your edits in particular that I edited away. It's everyone's, including my earlier ones. The problems were too pervasive not to just start over.

"Chatty" does not mean "long"; it means "long-winded": too many words to convey the meaning. A 2,000 page book can be terse; a single sentence can be chatty. Passive voice, parenthetical asides, and strings of words like "may be approximated by a sphere" are chatty.

With respect to the definition of "radius", I do not dispute that the term is used more generically, particularly in various specialties — that is this article's raison d'être! But dictionaries — even mathematical dictionaries — define the term in a standard way, and other usages are field-specific. A typical reader will be equipped neither with the knowledge nor the references to understand it in any other way. In particular, since we are trying to explain what we mean by this specialized meaning of "radius", it is circular to depend upon that specialized meaning to explain it.

The article is about Earth radius, but the lead paragraph's explanation of what was meant by "Earth radius" was incoherent. It's what led to all your questions in the first place. Random facts were inserted between what should have been a continuous narrative, leading to confusion about what was being said and why. The narrative was missing information needed to understand what it was trying to say. Statements were made before dependent information was introduced. A lead paragraph(s) fails if it cannot explain cogently what the article is about. There was nothing cogent about the presentation, and therefore it was useless to make incremental edits, as we were both doing.

The lead paragraph should now contains enough information and sources to understand what all uses of "Earth radius" mean, and the rationale for using the term, without duplicating information in the article body and without redundancy or irrelevant material. If that is not so, then it needs more work. Strebe (talk) 06:25, 28 December 2008 (UTC)

Thanks, Strebe, for the reply above.

So what was lacking was a definition. This gets it done in short order:

• For the purposes of this article, the term "radius" means the distance from the "center" of the Earth to an idealized point representing the "surface" of the Earth.
• This article represents the "surface" of the Earth either as an ellipsoid or a sphere. For other shapes, please refer to the Figure of the Earth article.
• The "center" of the earth is taken to be the geometric center of that ellipsoid or a sphere.
• Local topography is ignored; distances from the center of the Earth to particular points on the actual surface of the Earth are not address in this article.
• This article also discusses radii of curvature, which may have other centers.

If an average reader arrives at an article entitled "Earth Radius", I don't know what they might be looking for other than a "center-to-surface" distance. By providing definitions of what are meant in the article by "center", "surface" and "radius", the reader now has enough information to decide whether they are in the right place.

I think the current lead is too detailed and covers too much ground. For me, a function of a lead is to decide whether I want to read more of the article. I think that I am now probably more familiar with the concepts mentioned in the lead than an "average" reader might be, but my eyes start to glaze over about half way though the lead because it is telling me way more than I usually want to know to answer the question: "How big is the Earth?" If I find an article that says it is about the radius of the Earth, I expect to answer that question by finding a number and multiplying it by two.

I think that the whys and wherefores about the decisions to use an ellipsoid, sphere, or whatever, might wait until the reader actually decides that they want to delve further into the article. Maybe parts of the current lead would be of more interest the introduction. - Ac44ck (talk) 08:26, 28 December 2008 (UTC)

Ac44ck, you make an alluring case, and I am nearly convinced. For example, perhaps the article should lead out with the range of values useful as an "Earth radius", since some major fraction of visitors are probably only looking for that. From there it can get more involved. My biggest concern is that the lead paragraph(s) needs to mention all the usages of the term "Earth radius" so that the visitor can tell early on whether this is even the right article. I see no way to avoid a discussion of the various models in that case, since "radius of the earth" is also a concept in geoidal models. It is easy enough to then shuffle the person off to some other page, but it's also true that this article does discuss the geoid in several places, rather contrary to your assertions. What you propose is simple, though. I like it a lot.

Please note that I simplified more before I found your most recent response. Those simplifications do not alter the basic structure, though, and I can certainly see that most people's needs would be met by nothing more than a couple of numbers they can use as a value for "the" radius. Strebe (talk) 10:48, 28 December 2008 (UTC)

Nice job on the new lead, Strebe. It doesn't exhaust my interest before I get to the table of contents anymore, and it tells me that I am likely to find news I can use here.

It seems to retain the problem that you complained about: a (shall we say) purist may have a narrow interpretation of the word "radius". They may stumble over the first use being "natural radius" when, to them, "radius" means only one thing. The term "natural radius" may seem redundant to them. If we are concerned about a misunderstanding of the word "radius" as it is used in this article, then clearing that up might be the first order of business. Whether the earth is spherical or a cube, our topic here is "radius"; and that means ...

Or maybe the "purist" needs to read with more flexibility.

There was good stuff in the longer version of the lead. It all seems to be gone now. Including some of it elsewhere in the article may be beneficial, as other sections are pretty lean on explanation. - Ac44ck (talk) 19:13, 28 December 2008 (UTC)

It's all thanks to your suggestion, Ac44ck. I am considering how to work in the material I removed from the lead; it just got too late last night! I think it should go into the Introduction in some manner; perhaps most of it as-is. Concerning "natural radius", I vacillated between "serves naturally as its radius" and "serves as its natural radius", the former being faintly more accurate by some interpretation of semantics, and the latter flowing better without any realistic confusion. I do not think it proper to simply leave out "natural" or at least some word conveying that we will be using the term "radius" with respect to the Earth even though we're not talking about a sphere. Of course I'd like others' opinions, and of course feel free to edit as you see fit. —Preceding unsigned comment added by Strebe (talk • contribs) 21:03, 28 December 2008 (UTC)

Polar radius

The current value in the article for polar radius is b = 6,356.7523 km, which would seem to more than accurate enough for an average visitor.

But I have a consistency problem: The value is attributed to "WGRS 80/84" in the "Fixed radii" section. It is attributed to WGS (E2008) in the "Mean radii" section.

In following links to here, it appears that WGS (E2008) is using the values of a and f from WGS 84.

The table here says those values make b ≈ 6,356,752.314,245 m. It uses the " ≈ " symbol, but reports the value to the nearest micron (1e-6 m)!

For comparision:

Item	Size
Grain of Sand	100 to 2000 microns

So the precision of b in WGS (E2008) is adequate to make a topographic map of surface features on grains of sand.

What I have read suggests that the value for a does well to match an observable value for MSL by +/- 2 meters. Applying a flatness factor to it wouldn't seem to justify six decimal places in the result. The article also notes that the geoid heights at the poles are 1) unequal, and 2) "off" by at least 13 meters. Stating the value of b to the nearest micron seems absurd.

But I am uncomfortable with rounding "for" the visitor without telling them, though I did it myself in this edit, where I rounded to the nearest millimeter because the values for both GRS 80 and WGS 84 were the same through that precision here.

The value of b was further rounded to the nearest 0.1 meter for unstated reasons in this diff.

So I have three questions:

1. When might one want the value of b to the nearest micron?
2. What reference "should" we cite for the value of b?
3. If we cite a source but round its value, do we need to tell the visitor that we rounded "for" them?

-Ac44ck (talk) 01:01, 30 December 2008 (UTC)

1. Surely the only point of that much precision would be for use in other calculations, such as recovering the flattening value or computing the eccentricity. Sub-meter precision does not say anything useful about the "actual" polar radius, of course.
2. http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350_2.html
3. I would think so, along with the reason.

Strebe (talk) 03:13, 30 December 2008 (UTC)

WGS (E2008)

The "Mean radii" section refers to WGS (E2008). Does it use values for a and f that differ from those in WGS-84?

I downloaded the latest PowerPoint viewer to look at the file in the reference, but it doesn't render most of the slides.

Slide 23 would seem to be the key: "Summary and Model Availability", but there is a box with a red 'x' in it where there is supposed to be an image.

This page suggests to me that they might use the same values.

http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/egm08_wgs84.html

It appears that the "improvements" in WGS (E2008) are in the values for GM and ω, as they are not noted as being adopted from WGS-84.

These changes don't seem to be relevant in an article on "Earth radius". WGS (E2008) may be the "latest and greatest" version, but the PPT file is mostly unintelligible unless one has the "right" way to read it. I haven't found an HTML version of it.

Why not refer to WGS-84, for which information seems to be more accessible? - Ac44ck (talk) 19:52, 30 December 2008 (UTC)

This may contain the same slides in a different order:

http://www.dgfi.badw.de/typo3_mt/fileadmin/2kolloquium_muc/2008-10-08/Bosch/EGM2008.pdf

The "Summary and Model Availability" is Slide 4 in the PDF file. It doesn't give values for a and f.

Shall we use http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350_2.html for the reference in both the "Fixed radii" and "Mean radii" sections? -Ac44ck (talk) 23:08, 30 December 2008 (UTC)

There is no point in using E2008 or being so specific as to note E2008. The datum is different but the ellipsoid is the same for all WGS84-based datums. We don't care where the ellipsoid is anchored. Feel free to improve accordingly. Strebe (talk) 02:31, 31 December 2008 (UTC)

Not sure what this means

• At the geographic coordinates 0°00′S 121°50′E the geoid height rises to 63.42 m above the reference ellipsoid (WGS-84), giving a total radius of 6,378.200 km.

This is listed as a notable radius. The text doesn't say why it is notable. At first glance it looks like a random location. I'm guessing that this is the maximum height above the reference ellipsoid. If so, the word "greatest" or "maximum" should be in the sentence. Also, if this is the maximum, shouldn't the minimum be listed as well? Is the minimum in the Indian Ocean? -- ☑ SamuelWantman 07:06, 9 March 2009 (UTC)

I do not understand the significance of the entry, either. Feel free to remove it. Strebe (talk) 07:35, 9 March 2009 (UTC)

It makes sense to me (though I didn't add it): It is showing that the actual maximum value of a is 6378.2 km (at 121°50'E), not 6378.137, as WGS-84 defines. And, right below that, the different, actual values of b for both poles are given. Perhaps it would be clearer to rename the subsection as "Radii extremes" (and maybe move the equatorial and polar extremes above the "Maximum:" and "Minimum:" values)? ~Kaimbridge~ (talk) 14:03, 9 March 2009 (UTC)

Probably best not to argue over what "maximum value of a" could mean, since presumably that's in the Andes, not the Indian Ocean. If the intent is to state that the point is the greatest distance of the geoid from the center of the earth, then it needs to be written more plainly... and preferably cited, since it's not readily calculable, the way Chimborazo is. But is it really the greatest distance of the geoid from the center of the earth? Why would that just happen to be on the equator? If the intent is to state that it is the greatest distance of the geoid from earth's center along the equator, then I start to wonder why that's notable. It's not actually any sort of contender for the semjimajor axis of the approximating ellipsoid. If it has anything at all to do with an ellipsoid then it would be something like, Defines the smallest ellipsoid that encloses the geoid as long as we require the earth's equator to lie on the plane of the ellipsoid's semimajor axis, even though the whole point of an approximating ellipsoid was to get a best fit, not a minimally-enclosing ellipsoid. Therefore defeating any significance of it. Strebe (talk) 01:41, 10 March 2009 (UTC)

I added the section in this diff. My initial purpose was to provide focus for a reader who was interested only in values for 'a' and 'b'. I moved the min/max geoid radii to a different section. The values for 'a' and 'b' and min/max geoid radii seem theoretical to me, so I added min/max radii for permanent landmarks which one might be able to visit.

The article is about "Earth radius". I would keep all the various min/max values in the article. - Ac44ck (talk) 04:21, 10 March 2009 (UTC)

But what IS the value given? What does it correspond to? Strebe (talk) 04:38, 10 March 2009 (UTC)

It seems to be the location of maximum distance from the center of the earth to a point on the geoid.

The image in the article labeled "Geoid Height" shows the difference between geoid radius and ellipsoid radius. The 0°S 121.83°E location is within a region of the image where geoid height is a maximum. The maximum ellipsoid radius is on the equator. Maximum geoid height on top of maximum ellipsoid radius yields maximum geoid radius. Another relative maximum is shown in the North Atlantic, but the ellipsoid radius is shorter there.

The minimum in the Indian Ocean is not a minimum radius. The greatest negative difference between radii occurs in the Indian Ocean. The ellipsoid radii have minimum values at the poles. The "Geoid Height" image indicates that relatively little adjustment to the ellipsoid radius is needed at the poles. Geoid radii at the poles are unequal; both are shorter than the geoid radius in the Indian Ocean.

The text giving the location of the maximum geoid radius was added in this diff.

As noted, the wording could use tweaking after we have agreement on what it was intended to communicate.

My original name for the section in this diff was "Extreme radii" but it didn't seem to fit with listing the geoid radii for both poles. If the section contains only "extreme" radii, then only one polar geoid radius should be listed. But I wanted to keep both values, so I changed the name of the section to "Notable radii". Maybe the section needs a better name. - Ac44ck (talk) 18:01, 10 March 2009 (UTC)

I went to http://www.oc.nps.edu/oc2902w/geodesy/geodexer/geoidmn.html to peruse the geoid data. The maximal value I found was +81.11m at 5°S/150°E, but the high latitude more than canceled out the rise above the ellipsoid. Walking around the entire region from 120°E to 155°E, ±5° of latitude, I find the point 0°N/133°E to have +72.81, greater than the +63.42 quoted in the article. The geodetic coordinate resolution of the map isn't very good, though, so it's hard to say with any precision where the real maximum is. Hence I looked at the underlying data, which can be found here: http://www.oc.nps.edu/oc2902w/geodesy/geodexer/geoidata22a.js.

The geodetic resolution of the data is only 1°. Meanwhile the radial "penalty" for a one degree change in latitude on the ellipsoid is -64.59m at the equator. Since the radial penalty is on the same order as the geoidal data's maximal deviations from the ellipsoid, and since the geoid changes only slowly at that scale, it's clear the maximum radius of the geoid must be very close to the equator. The smoothly varying geoid is amenable to polynomial interpolation, so a reasonable location could be inferred by finding the neighborhood along the equator with the highest cluster value. A casual look at the data suggests 140°E, with an equatorial rise of +74.21, flanked by +72.62 and +73.54 north and south. The problem with this methodology, however, is that we can go considerably west to 131°, finding a rise of +73.70 flanked by +75.03 and +70.99 north and south. There are several other candidates. With data of this resolution we cannot infer where in this region the real maximum is. The best we can do is state something like "about +74m along the equator in the region of 130°E–143°E." Strebe (talk) 22:00, 10 March 2009 (UTC)

The calculator here http://sps.unavco.org/geoid/ says:

Latitude = 0° N = 0° 0' 0" N

Longitude = 133° E = 133° 0' 0" E

GPS ellipsoidal height = 0 (meters)

Geoid height = 73.762 (meters)

Unhappily, the calculator says 73.762 instead of +72.81. The discrepancy may suggest that, as when seeking a value for _the_ polar radius, there may not be _one_ geoid which is "correct". But the calculator reinforces the notion that maximum geoid radius is not at 0°S 121.83°E. With enough persistence, one might this (or a similar) calculator to find a precise location where geoid radius is maximum. - Ac44ck (talk) 04:32, 11 March 2009 (UTC)

I perused a few papers dealing with accuracy of geoidal models. Accuracy seems to vary quite a bit from locale to locale and by method of measurement. It does not look like sub-meter precision means much in most places under most circumstances because the height of the geoid just isn't known that precisely. The data set here http://earth-info.nga.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html, brought into MS Excel, penalized for latitude, and sorted, shows clearly that the only candidates for maxima occur as I describe above. Given uncertainties, the best that could be said is that it occurs within 0°15' of the equator between 129° and 142° east. But I haven't found a single reference that seems to care about such geoidal extrema (except in terms of deviation from the ellipsoid), and I do not believe one will be found. Therefore I retrospectively support SamuelWantman's deletion. Any further comments? Strebe (talk) 09:29, 12 March 2009 (UTC)

If this is in fact notable or significant, it must have been written about somewhere, and mentioned in a third party source. If so, it should have a citation. I'm removing it for being uncited information, that might be original research. If someone can produce a cite, feel free to revert my change. -- ☑ SamuelWantman 03:57, 12 March 2009 (UTC)

This is under discussion. What purpose is served by this haste? Strebe (talk) 05:26, 12 March 2009 (UTC)

Your first comment was that I should feel free to remove it! I read through the discussion, and nobody has produced a citation for the information, and the discussion indicates that it is original research. I don't mind waiting a few days to delete it again, but I want to make it clear that without a citation I believe it should probably be deleted. I came to this article as a reader trying to understand how the Earth's radius varies because of gravitational anomolies. As a reader I find the article very confusing because there seems to be two ways of looking at radius, one is an idealized radius that ignores the topology of the earth and the other is from actual measurements. For the benefit of readers, the difference between the two needs to be clarified using non-technical English. -- ☑ SamuelWantman 18:42, 12 March 2009 (UTC)

Please see my note above. I agree the citation should be deleted. I am waiting for other participants in the discussion to weigh in. Concerning the clarity of the article, it would be good to know your thoughts on specific passages. The introduction should have made clear that there are several different models, and what "radius" means with respect to them. Any help would be appreciated. Strebe (talk) 21:00, 12 March 2009 (UTC)

I don't have an application for geoidal extrema in mind today. I just moved existing text rather than throwing it away. I can imagine that the values might be of interest to some; they apparently were of interest to the editor who added them. I have no objection to their deletion. - Ac44ck (talk) 05:35, 14 March 2009 (UTC)

Yet another source of confusion

The lede paragraph begins:

Because the Earth is not perfectly spherical, no single value serves as its natural radius. Instead, being nearly spherical, a range of values from 6,357 km to 6,378 km (≈3,950 – 3,963 mi) spans all proposed radii according to need,...

I cannot make sense of "spans all proposed radii according to need". Does this mean that these figures a are the range of numbers used in geologic and astrophysical computations? If so, it should say so. The numbers are clearly within the range of the actual maximum and minimus listed later in the article. How about this for the lede:

"Radius" is normally a characteristic of perfect spheres. Because of the equatorial bulge, the Earth is not perfectly spherical, so no single value serves as its natural radius. The radius is at its minimum at the floor of the Arctic Ocean at the North Pole. The average of the two poles is 6,357 km. The radius is at its maximum at the summit of Mount Chimborazo near the equator. The average radius along the equator is 6,378 km. As a result of this variation, Earth radius has come to mean the distance from some "center" of the Earth to a point on some idealized surface that models the Earth. Mathematical models are used to calculate approximate values for the variation in Earth's radius. Several different ways of modeling the Earth as a sphere all yield a convenient mean radius of 6371 km (≈3,959 mi). This article deals primarily with spherical and ellipsoidal models of the Earth. See Figure of the Earth for a more complete discussion of models.

Note: Earth radius is sometimes used as a unit of distance, especially in astronomy and geology. It is usually denoted by $R_{\oplus }$.

--☑ SamuelWantman 02:47, 19 March 2009 (UTC)

SamuelWantman, thanks for the comments. We discussed the problem of the lead paragraph extensively up above in "First note". We felt it best to supply immediately the kinds of numbers people might be looking for when they visit the article, but to do it in a way that they can understand (if they choose to think about it) what the numbers mean. Your proposed edit seems clear to me, and of good quality, but, as discussed above in "First note", we'd like to get to real numbers more quickly. If you're having trouble understanding the verbiage as it is, then it probably needs work, though it seems lucid to me. Rather than rewriting in a way that pushes needed numbers further down into the paragraph, I suggest we simply mend the sentence that gives you trouble. We don't want to explain "too much" in the lead paragraph, since the main article repeats all of it anyway.

How about replacing "Instead, being nearly spherical, a range of values from 6,357 km to 6,378 km (≈3,950 – 3,963 mi) spans all proposed radii according to need, and several different ways of modeling the Earth as a sphere all yield a convenient mean radius of 6371 km (≈3,959 mi)" with "The distance from its center to hard surface ranges between 6,353 km and 6,384 km (≈3,947 – 3,968 mi). Therefore the various quoted values for its radius fall within that range. Several ways of modeling the earth as a sphere converge on 6371 km (≈3,959 mi) as a convenient mean radius."? Strebe (talk) 21:11, 19 March 2009 (UTC)

http://en.wikipedia.org/wiki/Earth_radius

This gfx http://en.wikipedia.org/wiki/File:Lowresgeoidheight.jpg is kind of wrong, that Longitudes goes from 0 --> 180° (E) and from 0 --> -180° (W)

193.162.192.70 (talk) 08:22, 29 June 2009 (UTC)

-- steen bondo ---

It's a reasonable representation. Just interpret the x axis as longitudinal difference from the prime meridian. It doesn't claim to represent "absolute" longitude, and longitude isn't absolute in any case. 180°W – 180°E is just a convention, and not the only one in use. Strebe (talk) 03:53, 30 June 2009 (UTC)

Meridional mean radius

Wikipedia contains three values for the meridional mean radius in the spherical Earth article:

• rectifying radius: 6,367.449 km
• quadratic mean: 6,367.454 km
• simple mean: 6,367.445 km

This Earth radius contains another, based on a root mean semicubic calculation:

• 6367.4491 km

Presumably, the "rectifying radius" is an "exact" solution based on the elliptic integral. The approximation in this article is given to higher precision.

The approximations of the meridional mean radius probably belong here rather than in the spherical Earth article.

Whence the calculation of the root mean semicubic? I have not seen a mean calculated this way before or elsewhere. -Ac44ck (talk) 03:41, 29 August 2010 (UTC)

The entire section should be removed from the Spherical Earth article and any unique material integrated with the Earth radius article. The two just duplicate each other uselessly as it is.

The rectifying radius comes from the integral, as you surmise. I do not know the source of the “semicubic” formula. It may well be original research from the contributor (User:kaimbridge [1]). It is remarkably effective, though; I see it agrees to 12 decimal places on earth-like ellipsoids. Strebe (talk) 04:42, 29 August 2010 (UTC)

I wonder whether the mean radii formulae belong in the spherical Earth article instead. This Earth radius article is mostly about the "real" values. As various flat-Earth approximations are compiled in an article separate from great circle distance, the various radii for "equivalent" spherical Earths might belong in the spherical Earth article.

In looking further for the "semicubic" formula, it was found under the "Muir-1883" heading in the circumference article.- Ac44ck (talk) 15:09, 29 August 2010 (UTC)

The lede states, “…Several different ways of modeling the Earth as a sphere all yield…”, and “This article deals primarily with spherical and ellipsoidal models of the Earth.” The Introduction states, “It is also common to refer to any mean radius of a spherical model as "the radius of the earth".” There is supporting material throughout. It does seem as if mean radii is very much the purview of this article. Spherical Earth, on the other hand, is much more about the conception of Earth as a sphere. Certain historical values proposed for radii belong there (Eratosthenes &c.), but otherwise this article seems more apt. Does it not? Strebe (talk) 19:28, 29 August 2010 (UTC)

OK. I moved the unique material from the spherical earth article to here. -Ac44ck (talk) 01:53, 30 August 2010 (UTC)

The second sentence ("all proposed radii...")

The second sentence of the article needs an easy improvement:

"Because the Earth is not perfectly spherical, no single value serves as its natural radius. Instead, being nearly spherical, a range of values from 6,357 km to 6,378 km (≈3,950 – 3,963 mi) spans all proposed radii according to need..."

As everyone knows, the earth spheroid's radius of curvature varies between 6335 and 6400 km-- so apparently the author intends "all proposed radii" to not include radius of curvature. Which it does, unless the exclusion is made plain. Tim Zukas (talk) 18:29, 30 August 2010 (UTC)

Good point. Strebe (talk) 19:18, 30 August 2010 (UTC)

Much better now-- but might as well say "to points on the sea-level surface..." Dunno offhand how far from the center of the earth to the top of the highest equatorial mountain. Tim Zukas (talk) 19:51, 30 August 2010 (UTC)

It does not (or need not) refer to mean sea level specifically. The article lists actual minimum and maximum, so I have revised the lede accordingly. Strebe (talk) 20:28, 30 August 2010 (UTC)

Date of Moritz

Not sure if the two Moritz refs are the same or different, but the one I'm citing seems to be dated 2000 even though 1980 appears in the title. It's here: http://www.springerlink.com/content/0bgccvjj5bedgdfu/about/ and on the "about" page it gives a year 2000 date. Also it's cited as year 2000 in Method of calculating tsunami travel times in the Andaman Sea region by Monte Kietpawpan et al. ☺Coppertwig (talk) 23:24, 11 December 2010 (UTC)

According to this page, Volume 74, Number 1 of the Journal of Geodesy was published in March 2000, so I think you're right. Also, as corroborating evidence, I was able to export a BibTex-formatted citation, which also includes a date of 2000. Jakew (talk) 21:52, 12 December 2010 (UTC)

For just an average guy, this article is sheer CRAP

I don't want a mathematical treatise on how to measure the radius of the earth. This is an encyclopedia, I expect to see a number. The radius of the earth from the centre to the equator is X. The radius of the earth from the centre to the poles is Y. Why can't some of you freakin rocket scientists do something straight forward and simple to answer the average person's question for a change? And no it doesn't have to be this way. And yes, you can provide a simple bloody answer. Everything isn't a shade of flippin grey. This is the problem with Wikipedia, you egg heads make it worthless to look for information. This is not a math course. It is an encyclopedia. Please provide information that is understandable. For God's sake. — Preceding unsigned comment added by 75.119.254.110 (talk) 08:12, 25 December 2011 (UTC)

Fix it if is broken. Or whine. Your choice. I might as well whine about people who think the world should be arranged to suit their own personal, simplistic ways of thinking and, instead of helping, throw abusive tantrums when they don’t get exactly what the want… for free, even. And, Happy Holidays. Strebe (talk) 21:26, 25 December 2011 (UTC)

Ellipsoidal quadratic mean radius

This radius definition was added in these edits. “Ellipsoidal quadratic mean radius” doesn’t exist in the literature. It appears to be original research which has a lot of references on the Web, but they all go back to a single, unpublished source who appears to be promoting it. I will delete this edit in a week if there is no published source forthcoming. Strebe (talk) 06:28, 2 January 2012 (UTC)

Isnt that just some fancy name for a three dimentional quadratic mean? Wouldnt "triaxal quadratic mean" be the proper term? — Preceding unsigned comment added by 173.9.95.205 (talk) 18:22, 20 July 2012 (UTC)

Standard gravitational radius

Newton's law of universal gravitation#Gravitational field

With a few simplifications:

$g=G{M \over {r^{2}}}$

Gravitational constant

$G=6.67384\times 10^{-11}\ {\mbox{m}}^{3}\ {\mbox{kg}}^{-1}\ {\mbox{s}}^{-2}=6.67384\times 10^{-11}\ {\rm {N}}\,{\rm {(m/kg)^{2}}}$

Mass of Earth

M = 5.9722 × 10²⁴ kg

Earth's gravity

Standard gravity is, by definition, 9.80665 m/s²

Putting it all together:

9.80665 = (6.67384e-11 * 5.9722e24) / r^2

r = sqrt((6.67384e-11 * 5.9722e24) / 9.80665) = 6,375,213.43562 meters

Standard gravitational radius = 6,375.213 km -Ac44ck (talk) 03:22, 3 July 2012 (UTC)

Earth radius at N32.704220, W90.000000?

The location in radians spherical coordinates: (1,-π/2,1) = (1, 3τ/4,1). The first one is radius of Earth, the -π/2 or 3τ/4 is 90 degrees west, and the last 1 is the angle from N. Pole or (π/2-1)rad north = (τ/4 -1) rad north which is 32.7042204869 degrees N.

I know this location by Google maps is north of Canton, MS, USA just a few hundred feet from US 51. But, problem is the radius. I put one in for the radius as for a place holder, but I would like to know is the exact value of the Earth's radius at N32.7042204869, W90.0000000000 taking into account all of the different elevations (like changes caused by the moon and sun), deformations (like geoid height), and drifts (like continental drift). I am thinking of having something like the annual South Pole remarking ceremony marking this location. How do I find all of this together in a simple way?

I am also thinking about the fact that the length of the arc with an ellipse is different from an arc with a circle. How does this affect radians?

John W. Nicholson (talk) 12:55, 19 February 2013 (UTC)

This is not a trivial problem to solve. Note that the Talk page is for discussing improvements to the article. You should find a geodesy discussion board for this sort of question. Thanks. Strebe (talk) 00:44, 20 February 2013 (UTC)

External links modified

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Spherical

See Talk:Figure_of_the_Earth#Spherical. --Jack Upland (talk) 12:54, 3 March 2016 (UTC)

Let's centralize the discussion there. fgnievinski (talk) 15:56, 3 March 2016 (UTC)

Definitional value (or approximation)

There was a note (above) on an approximation for the radius of the Earth. It is perhaps better counted as a definitional value based on the definition of units, both SI kilometers and the still widely-used nautical miles. The radius of the Earth (in the sense of the simple mean (a+b)/2) is very nearly given by 20000/pi kilometers or 10800/pi nautical miles. Do the division, and you get approximately 6366 km or 3438 nautical miles. These values match the simple mean radius to 99.98%. In both cases, the numerator is the number of base units from pole to pole, which for both kilometers and nautical miles were originally defined to be an integral number from equator to pole. There are 10,000 km from pole to equator or 20,000 from pole to pole, and there are 90·60=5400 nautical miles from pole to equator or 10800 from pole to pole. Meanwhile pi here is just the angle from pole to pole expressed in radians. Another way to put it: the radius of the Earth in nautical miles is numerically equal to the number of minutes of angle in one radian: approximately 3438. 174.199.34.212 (talk) 16:39, 4 November 2016 (UTC)

Average height, including land

@Mindbuilder: Concerning this edit:

“The ellipsoid is sized to approximate global sea level (actually, more precisely, the geoid)[13] and therefore these values for average radius would be a little larger if the height of the land were included. However the average global height of the land is only about 835m[14], and the land is only about 29% of the surface of the earth, so the average radius would increase by less than 1km, and the average radius would be the same if rounded to the nearest km.

Thank you for the well-intentioned edit. It could be interesting to some, but it demonstrates the hazards of WP:OR and WP:SYNTH. The conclusion is false because the average radius might flip up or down one digit regardless of how small the average land height is when the unrounded value without land height is close to an integer.

Isn’t it sufficient to simply note that the average values do not include land height? As it stands, this edit isn’t really acceptable, and I don’t see any way to make it acceptable without replacing the multiple sources that have been synthesized into an erroneous conclusion with a single source that states the correct conclusion. Strebe (talk) 00:23, 14 September 2017 (UTC)