Herein "f" is the Sun's spin frequency.
The radius R has to be taken around 0.94 R or 0.95 R according to different models.
The Dalsgaard model for the solar density shows a hyperbolic-like function, whereof the asymptotes intersect at about 0.98 R : at first, there occurs a very quick density increase from 10^-6g/cm³ at R, until 10^-2g/cm³ at nearly 0.95 R and next a slow, almost linear density increase until 1.5x10^2g/cm³ at the Sun’s center. On the other hand, S. Korzennik found that the highest value of the Sun’s rotation is located at about 0.94 R , where the corresponding density is 10^-2g/cm³.
What is your best guess to explain this relationship?
Taken literally (which no doubt is not what you mean by "explain the relationship"), it is telling us that spin frequency varies directly with solar mass and inversely with the square of radius. It is an interesting question inasmuch as variations in solar radius that have reportedly been observed from time to time suggest solar density variations, which in turn suggests an analogy between sun and the spinning ice skater who controls frequency by the extension or contraction of arms.
To clarify for considering the explanation: given differential rotation, what is the locus of reference for spin frequency and radius? Is radius R defined consistently in the Dalsgaard and Korzennik models and how; that is, down to the center from what latitude and from where within the outer layers (e.g. top of photosphere? of chromosphere? of corona?).
Dear Albert, thank you for your input.
The location is the sun's equator.
I was merely focusing on the presence of c in the equation.
Dear Thierry, this interaction does at least provide your respondent(s) a clue about focus on "the presence of c". Well aware that I may be on the wrong track with this line of inquiry into the assumptions on which the question rests, here goes anyway.
Ever since Galileo and Scheiner studied sunspots and Galileo mapped out their rotation, there has been debate or uncertainty about how the sunspot rotation rate relates to that of the solar sphere at other altitudes from the center. Given that we are considering rotation not of a solid body but of a ball of plasma and that we expect differential rotation of its various layers, I am not sure it is meaningful to speak of the spin frequency of the sun, or of its radius, without further qualification. Indeed the moment of inertia, and in turn angular momentum, depend on perpendicular distance of each particle to the axis of rotation, hence on the surface boundary of a particular sphere.
For these reasons, I would be more comfortable in possibly hazarding the sought-after "best guess" if the circle of which R is the radius were well defined. Though that circle might be identified with the sun's equator, a "best guess" at how or why spin frequency relates to solar density may depend on the altitude (or layer boundary) at which the equator is reckoned. Then one can think of the spinning object as the material within that altitude.
Dear Albert, the choice of the equatorial radius R around 0.94 R or 0.95 R is precisely due to the fact that at that place, there is some "consistence" in the Sun's matter. The density between 0.95 R and R becomes suddenly very thin.
The compliance of the equation occurs where the density begins to be rather high.
Thierry, the difference in velocity between two bodies ∆v is related to the difference in potential ∆V by this very simple relation.
∆v = c(∆V/Ø)
Where c is the speed of light and Ø is absolute potential. Absolute potential according to GPT is the proton energy divided by its charge. i.e ~ 938 million volts.
You can find the surface potential of any body just by dividing it's mass by the number of nucleons (a measure of density).
When a body such as the Earth or the Sun also has rotation, it simply tells us the the potential in the core is not the same as the potential at the surface, easy to see with the equation above.
∆v at ground potential is obviously zero, everything else moves...
Thanks, Thierry, for that clarification. Of course it is easy to see that you are not literally asserting that R = 0.945R ± 0.005, since algebraically that would reduce to the infinite set of falsehoods 1 = 0.95 or 0.94 or any intermediary between these last two. Rather, if we denote equatorial radius by Re, then your question statement asserts Re = 0.945R ± 0.005. Then to be sure of your characterization of R, I ask: is the intersection of asymptotes for solar density in the Dalsgaard model at about 0.98R or is it at about 0.98Re? And mutatis mutandis for all the other cases of “R” appearing in the descriptions of the two models?
Still pondering your interesting question and responsive elaborations on it, I take the question to be focused on theory and its supporting evidence that holds c to be a constant, as first well measured by Michelson -- and not a variable c(t) over aeons.
Dear Albert, I understand.
I add here a paper about the solar velocities and a diagram from Dalsgaard about the densities. (I meant 0.98R there).
Depending from the exact choice of Req and the rotation frequency at that place one gets a quite correct equality in f = GM/(2cR²).
Indeed, since it is subject to interpretation, a good fit is obtained around 0.98 and 0.94 Req with a rotation frequency feq somewhere between 474 and 515 nHz.
The justification of the choice is indeed a matter of goodwill. However, I think that the equality cannot just be called a coincidence.
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