Talk:Limb darkening

It occyurs to me that another cause of limb darkening might be because faculae - particularly bright features on the solar photosphere - occur in the "canyons" between solar granules, and so when looked at from an angle they'd be partially obscured by the dimmer granule plasma. I have no idea how much of an effect this would have, though, so I'm reluctant to just toss this into the article itself. Bryan 04:05, 23 Jan 2005 (UTC)

I am more an instrument person, but I work with some of the best solar physicists, and I will ask them about that. Paul Reiser 05:58, 23 Jan 2005 (UTC)

What is a limb, anyways?
Hi. I came here trying to find out what Newton meant by superior limb of the sun as quoted in Webster (definition 1). The dab for limb has a picture but doesn't link here, so I linked it here. Should there be a Limb (astronomy) as suggested by the dab page, or should it be part of this page, perhaps with the picture of the moon? But the moon isn't really a star… so, it doesn't experience limb darkening (oops). Or does it? But I'm not here to ask an astronomy question, I'm here to discuss how to improve the articles. I'll try to fix the dab so it doesn't imply that the moon is a star. 「ѕʀʟ·✎」 10:08, 31 December 2007 (UTC)
 * Phases of Venus mentions 'apparent disk' and moon mentions the 'limbs' which links to wiktionary (??). 「ѕʀʟ·✎」 10:17, 31 December 2007 (UTC)
 * The papers mentioned in Mariner 2 also use limb darkening for the planet Venus, so I guess the term limb darkening is not only used for stars. Van der Hoorn (talk) 00:41, 15 February 2009 (UTC)

Something seems incorrect
Can't seem to figure out what the problem is, but, assuming the simplest case of the first formula and the coefficients for the case of N=2 gives I(psi) = 0.3+0.93Cos(psi)-0.23Cos(psi)^2 (assuming I(psi=0) == 1).

Since the half angle subtended by the Sun as viewed from Earth is approximately the solar radius divided by the distance to the Sun = 7E8/1.5E11 = 0.00465 this means that the Cos(psi) and Cos(psi)^2 terms at the limb of the Sun will be Cos(0.00465) and Cos(0.00465)^2 or 0.999989 and 0.999978, respectively so that the limb darkening by this equation only works out to a very small fraction of a percent at the disk edge relative to the disk center, but we know that the observed darkening is on the order of 25%-50% for most visible wavelengths (this assumes theta ~ psi which is a very good approximation for these very small angles of psi and theta).

Unfortunately, none of the references appear to be open literature and there appear to be no other decent references I can find at the moment online, so I can't attempt to address this now. But I'll see what I can find. In the meantime, does anyone else have any ideas what the problem is? Thanks...

65.202.226.2 (talk) 20:54, 3 October 2011 (UTC)mjd


 * Looking at the diagram, you have calculated $$\theta$$, not $$\psi\,$$. At large distances, the edge of the solar disc corresponds to $$\psi\approx$$ 90 degrees. PAR (talk) 04:51, 4 October 2011 (UTC)


 * That actually gets back to my earlier comment that the lack of publicly available references is unfortunate. There certainly is a relation there for Cos[psi] that provides a solution corresponding to what you are saying, but how does one derive that equation? Alternatively, by the Law of Sines, Sin[theta]/r == Sin[psi']/R, where psi' is the supplimentary angle of psi (pi - psi). Sin[180-a] == Sin[a], so the equation then becomes Sin[theta]/r == Sin[psi]/R and therefore Sin[psi] == R Sin[theta]/r. So, finally, psi = ArcSin[R Sin[theta]/r]. I understand the reluctance to resort to inverse functions, but I can at least derive that. Now, ironically, I must have misassigned some constants when I first tried this, because I was getting a very different answer, but now I am getting very much the same result with both approaches. Thanks!

65.202.226.2 (talk) 15:20, 4 October 2011 (UTC)mjd


 * OK - sorry to be pedantic, but I'm bothered by the underived relationship between ψ and θ given on the main page:

From the diagram:

1. $$ \textrm{sin}(\Omega) = \frac{R}{r} $$

Also from the diagram, Law of Sines (and where ψ' is the supplimentary angle of ψ):

2. $$ \frac{\textrm{sin}(\theta)}{R} = \frac{\textrm{sin}(\psi')}{r} $$

2.a. $$ \textrm{sin}(\psi) = \frac{r}{R}\textrm{sin}(\theta) $$

2.b. $$ \textrm{cos}^2(\psi) = 1 - \frac{r^2}{R^2}\textrm{sin}^2(\theta) $$

The equation from the main page:

3. $$ \cos(\psi) = \frac{\sqrt{\cos^2(\theta)-\cos^2(\Omega)}}{\sin(\Omega)} $$

3.a. $$ \cos^2(\psi) = \frac{\cos^2(\theta)-\cos^2(\Omega)}{\sin^2(\Omega)} $$

So, the question now at hand is: Is(are) the relationship(s) in equation(s) 2 derived from the diagram consistent with the relationship(s) in equation(s) 3? Start by equating the expressions for $$\cos^2(\psi)$$ in 2.b. and 3.a.:

4. $$ \frac{\cos^2(\theta)-\cos^2(\Omega)}{\sin^2(\Omega)} = 1 - \frac{r^2}{R^2}\textrm{sin}^2(\theta) $$

4.a. $$ \cos^2(\theta)-\cos^2(\Omega) = \sin^2(\Omega) - \frac{r^2}{R^2}\textrm{sin}^2(\theta)\sin^2(\Omega) $$

4.b. $$ \sin^2(\Omega)+\cos^2(\Omega) = \cos^2(\theta) + \frac{r^2}{R^2}\textrm{sin}^2(\theta)\sin^2(\Omega) $$

4.c. $$ 1 -\cos^2(\theta) = \frac{r^2}{R^2}\textrm{sin}^2(\theta)\sin^2(\Omega) $$

4.d. $$ \sin^2(\theta) = \frac{r^2}{R^2}\textrm{sin}^2(\theta)\sin^2(\Omega) $$

4.e. $$ \sin^2(\Omega) = \frac{R^2}{r^2} $$

4.f. $$ \sin(\Omega) = \frac{R}{r} $$

This final result is consistent with what we know from the diagram (equation 1 above), so it is in fact true that expression 2 derived from the diagram and expression 3 given on the main page are consistent. Which is nice because they are numerically equivalent, but I wish I knew how the equation on the main page was derived. Thanks again.

65.202.226.2 (talk) 18:56, 4 October 2011 (UTC)mjd


 * Using your equations 1 and 2, solve 1 for R and substitute into 2. The r's cancel yielding;
 * $$\sin\Omega=\frac{\sin\theta}{\sin\psi '}=\frac{\sin\theta}{\sin\psi}$$
 * Solve for $$\sin\psi\,$$, squaring both sides and using $$\sin^2 x=1-\cos^2 x$$ gives:
 * $$1-\cos^2\psi=\frac{1-\cos^2\theta}{\sin^2\Omega}$$
 * Solving for $$\cos^2\psi\,$$ gives:
 * $$\cos^2\psi=1-\frac{1-\cos^2\theta}{\sin^2\Omega}=\frac{1-\cos^2\Omega}{\sin^2\Omega}-\frac{1-\cos^2\theta}{\sin^2\Omega}=\frac{\cos^2\theta-\cos^2\Omega}{\sin^2\Omega}$$
 * For the restricted range of angles considered in the diagram, we may take the positive square root of both sides. PAR (talk) 02:52, 5 October 2011 (UTC)

Cause of Limb Darkening
Something is badly missing in this article, namely a mention of what physical process is responsible for the limb darkening. After all, in the optical region of the spectrum there is no continuous absorption or scattering for hydrogen in its ground state.

Thomas — Preceding unsigned comment added by 86.25.189.56 (talk) 12:27, 21 December 2012 (UTC)
 * How does 6000 kelvin hydrogen behave? I am fairly certain that it behaves as a black body emitter. Rockphed (talk) 17:37, 1 August 2019 (UTC)

alternative equation
The "more conveniently written" equation could also be written in terms of the original coefficients. Deriving it from the earlier equation, it would be

\frac{I(\psi)}{I(0)} = 1+\sum_{k=1}^N a_k \, (\cos^k(\psi)-1) $$

Shawn — Preceding unsigned comment added by Sdivitt (talk • contribs) 08:44, 9 May 2013 (UTC)

Limb darkening of planets
My Cambridge Encyclopaedia of Astronomy (they used to make encyclopaedias out of trees, once) mentions the limb darkening of some planets and moons. Presumably this is a different effect, but why doesn't the article mention it? Stub Mandrel (talk) 21:14, 3 October 2015 (UTC)
 * I will have to look into that, but perhaps it is related to uneven lighting similar to dawn and dusk on earthRockphed (talk) 18:12, 1 August 2019 (UTC)
 * I agree. I came here directly from the article Atmosphere of Titan, wondering what limb darkening could mean (as English isn't my first language).  I got not much answer from this article. Fomalhaut76 (talk) 10:54, 18 April 2023 (UTC)
 * I have added mention in the lead that the effect is also observed on planets, and I have provided a new ref. CoronalMassAffection (talk) 14:23, 19 April 2023 (UTC)