<p>You might find this paper helpful:</p>

<p><a href="http://www.physics.hmc.edu/faculty/esin/a101/limbdarkening.pdf" rel="nofollow">http://www.physics.hmc.edu/faculty/esin/a101/limbdarkening.pdf</a></p>

<p>Look specifically at Eq. 5 for the wavelength dependence.  To get the intensity, use Equation 1.  The value of $u$ can be found in the caption to Figure 2.</p>

CC BY-SA 3.0
2016-06-10T01:10:38.843Z
Thanks. In section 2 1st paragraph: "Putting $u$ = 1 in Eq. 1, a single-parameter power law fit is performed..."  and since [Petro et al. (1984)](http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1984ApJ...283..426P&defaultprint=YES&filetype=.pdf) is only narrow-band continuum at 4451Å Its not clear to me yet if the paper really says $u=0.85$ is a good fit throughout the visible wavelength range. Can you elaborate on this?
uhoh
CC BY-SA 3.0
2016-06-10T01:11:28.750Z
also found **PS** ([Pierce and Slaughter 1977](http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1977SoPh...51...25P&defaultprint=YES&filetype=.pdf)) and **NL** ([Neckel and Labs 1994](http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994SoPh..153...91N&defaultprint=YES&filetype=.pdf)).
uhoh
CC BY-SA 3.0
2016-06-09T21:18:22.260Z
2016-06-09T21:18:22.260Z
J. O'Brien Antognini
<p>I've slightly modified the title to try to attract some attention. If we call sunlight "white" and limb darkening is a result of seeing deeper at normal incidence and shallower at oblique incidence, then <em>the center end edge of the solar disk viewed from earth should appear to have slightly different effective color temperatures, since the temperature is varying rapidly with depth</em>.</p>

<p>I'd like to have an approximate expression for the wavelength-dependent limb darkening of the sun in the visible spectrum, either a relatively simple analytical expression that I can understand (with the appropriate coefficients for the sun) or just some linear images of the sun in various bands in the visible wavelengths so I can try to make one of my own.</p>

<p>This <a href="https://astronomy.stackexchange.com/q/12331/7982">question</a> received this helpful <a href="https://astronomy.stackexchange.com/questions/12331/linear-limb-darkening-coefficient-u#comment17382_12331">comment</a> which links <a href="http://www.astro.keele.ac.uk/jkt/codes/jktld.html" rel="nofollow noreferrer">to here</a> but honestly I can't make my way through that to a practical expression I can use. The <a href="https://i.sstatic.net/MMyje.jpg" rel="nofollow noreferrer">Wikipedia</a> article is not helping me much either, except for the <a href="https://en.wikipedia.org/wiki/File:2012_Transit_of_Venus_from_SF.jpg" rel="nofollow noreferrer">image there</a>. I plotted scans of RGB but by the time an image gets into the internet, things like sRGB and gamma mean it may not be linear.</p>

<p><strong>update:</strong> At the <a href="http://sdo.gsfc.nasa.gov/data/" rel="nofollow noreferrer">Solar Dynamics Observatory (SDO)</a> website, I just found the image <a href="http://sdo.gsfc.nasa.gov/assets/img/latest/latest_1024_HMIIC.jpg" rel="nofollow noreferrer">sdo.gsfc.nasa.gov/assets/img/latest/latest_1024_HMIIC.jpg</a>. The color gradient of the limb darkening seems very similar to the Wikimedia image below. I've discovered that it is called a "colorized intensitygram" and the color gradient is purely artificial - the data is single channel intensity. The limb darkening is certainly real (compare to the artificially "flattented" display!)  </p>

<p>From <a href="http://www.solarham.net/latest_imagery/hmi1.htm" rel="nofollow noreferrer">http://www.solarham.net/latest_imagery/hmi1.htm</a></p>

<p><a href="https://i.sstatic.net/a8KdC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/a8KdC.png" alt="enter image description here"></a></p>

<p>I appreciate (the existence of) the complexities and intricacies of photon transport, instrumental effects, and color perception, but I am <em>just starting</em> to do astronomically correct animations, so please for right now, something imperfect, or not absolutely correct is good enough for me.</p>

<p><em>20 pixel wide averages of horizontal (-) and vertical (--) "scans" across the center of this 600x600 pixel <a href="https://en.wikipedia.org/wiki/File:2012_Transit_of_Venus_from_SF.jpg" rel="nofollow noreferrer">image from here</a></em></p>

<p><strong>note:</strong> It seems this is a somewhat futile example, as the image is likely to be a monochrome continuum image with false color!</p>

<p><a href="https://i.sstatic.net/28nHI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/28nHI.png" alt="enter image description here"></a></p>

<stellar-astrophysics><the-sun><photography>
Is the Sun slightly blue in the center? - Wavelength-dependent limb darkening of the Sun
uhoh
# Is the Sun slightly blue in the center? - Wavelength-dependent limb darkening of the Sun

(Asked by: username_0 on 2016-05-20)

I've slightly modified the title to try to attract some attention. If we call sunlight "white" and limb darkening is a result of seeing deeper at normal incidence and shallower at oblique incidence, then *the center end edge of the solar disk viewed from earth should appear to have slightly different effective color temperatures, since the temperature is varying rapidly with depth*.

I'd like to have an approximate expression for the wavelength-dependent limb darkening of the sun in the visible spectrum, either a relatively simple analytical expression that I can understand (with the appropriate coefficients for the sun) or just some linear images of the sun in various bands in the visible wavelengths so I can try to make one of my own.

This [question](https://astronomy.stackexchange.com/q/12331/7982) received this helpful [comment](https://astronomy.stackexchange.com/questions/12331/linear-limb-darkening-coefficient-u#comment17382_12331) which links [to here](http://www.astro.keele.ac.uk/jkt/codes/jktld.html) but honestly I can't make my way through that to a practical expression I can use. The [Wikipedia](https://i.sstatic.net/MMyje.jpg) article is not helping me much either, except for the [image there](https://en.wikipedia.org/wiki/File:2012_Transit_of_Venus_from_SF.jpg). I plotted scans of RGB but by the time an image gets into the internet, things like sRGB and gamma mean it may not be linear.

**update:** At the [Solar Dynamics Observatory (SDO)](http://sdo.gsfc.nasa.gov/data/) website, I just found the image [sdo.gsfc.nasa.gov/assets/img/latest/latest\_1024\_HMIIC.jpg](http://sdo.gsfc.nasa.gov/assets/img/latest/latest_1024_HMIIC.jpg). The color gradient of the limb darkening seems very similar to the Wikimedia image below. I've discovered that it is called a "colorized intensitygram" and the color gradient is purely artificial - the data is single channel intensity. The limb darkening is certainly real (compare to the artificially "flattented" display!)

From <http://www.solarham.net/latest_imagery/hmi1.htm>

[![enter image description here](https://i.sstatic.net/a8KdC.png)](https://i.sstatic.net/a8KdC.png)

I appreciate (the existence of) the complexities and intricacies of photon transport, instrumental effects, and color perception, but I am *just starting* to do astronomically correct animations, so please for right now, something imperfect, or not absolutely correct is good enough for me.

*20 pixel wide averages of horizontal (-) and vertical (--) "scans" across the center of this 600x600 pixel [image from here](https://en.wikipedia.org/wiki/File:2012_Transit_of_Venus_from_SF.jpg)*

**note:** It seems this is a somewhat futile example, as the image is likely to be a monochrome continuum image with false color!

[![enter image description here](https://i.sstatic.net/28nHI.png)](https://i.sstatic.net/28nHI.png)

**Tags:** stellar-astrophysics, the-sun, photography


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## 1 answers

### Answer by username_1 on 2016-06-09. Score: 2 | Accepted answer

You might find this paper helpful:

<http://www.physics.hmc.edu/faculty/esin/a101/limbdarkening.pdf>

Look specifically at Eq. 5 for the wavelength dependence. To get the intensity, use Equation 1. The value of $u$ can be found in the caption to Figure 2.

> **Comments:**
> **username_0**: Thanks. In section 2 1st paragraph: "Putting $u$ = 1 in Eq. 1, a single-parameter power law fit is performed..."  and since [Petro et al. (1984)](http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1984ApJ...283..426P&defaultprint=YES&filetype=.pdf) is only narrow-band continuum at 4451Å Its not clear to me yet if the paper really says $u=0.85$ is a good fit throughout the visible wavelength range. Can you elaborate on this?  
> **username_0**: also found **PS** ([Pierce and Slaughter 1977](http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1977SoPh...51...25P&defaultprint=YES&filetype=.pdf)) and **NL** ([Neckel and Labs 1994](http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1994SoPh..153...91N&defaultprint=YES&filetype=.pdf)).  

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