<p>I usually don't answer my own questions, but sometimes when the question itself is <a href="https://astronomy.stackexchange.com/a/32732/7982">called into question</a> I make an exception.</p>
<p>The density of the photosphere at <span class="math-container">$\tau_{5000}=1$</span> is predicted to be <span class="math-container">$3 \times 10^{-7} \text{g/cm}^3$</span> in the Holweger-Müller Model Atmosphere<sup>7</sup>.</p>
<p>As pointed out <a href="https://astronomy.stackexchange.com/questions/32727/what-is-the-density-profile-within-the-suns-photosphere-which-one-of-these-is/32737?noredirect=1#comment60449_32737">in comments</a> there is a spread in values here.</p>
<p>The 1E-06 g/cm^3 density value (plot) is more consistent with  = 1 or &quot;bottom&quot; of the photosphere, while the density in the quote is more consistent with the cooler &quot;top&quot; of the photosphere (circa 4300 K).</p>
<p>From <a href="https://people.smp.uq.edu.au/TimoNieminen/papers/thesis/chapter2_photosphere.pdf" rel="nofollow noreferrer">Chapter 2: The Photosphere</a> of Timo Nieminen's thesis <a href="https://people.smp.uq.edu.au/TimoNieminen/thesis.html" rel="nofollow noreferrer">Solar Line Asymmetries: Modelling the Effect of Granulation on the Solar Spectrum</a></p>
<p><a href="https://i.sstatic.net/JS8JR.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JS8JR.png" alt="density versus hight above the Sun's surface" /></a></p>
<p><a href="https://i.sstatic.net/s6zLZ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/s6zLZ.png" alt="density and other parameters versus hight above the Sun's surface" /></a></p>
<blockquote>
<p>Figure 2-3: The Holweger-Müller Model Atmosphere</p>
<p><sup>7</sup> Holweger, H. and Müller, E. A. “The Photospheric Barium Spectrum: Solar Abundance and Collision Broadening of Ba II Lines by Hydrogen”, Solar Physics 39, pg 19-30 (1974). Extra points have been cubic spline interpolated by J. E. Ross. The optical properties (such as the optical depth and the opacity) of a model atmosphere are, obviously, very important, and will be considered later. See table C-4 for complete details of the Holweger-Müller model atmosphere including all depth points used.</p>
<p><sup>8</sup><strong>The height scale is not arbitrary.</strong> The base of the photosphere (height = 0 km) is chosen to be at standard optical depth of one (i.e.  <sub>5000Å</sub> = 1 ).</p>
</blockquote>

CC BY-SA 4.0
2019-07-20T07:32:47.387Z
The HM atmosphere perfectly well shows that the "photosphere" covers a range of temperatures and densities. The density at the temperature minimum is around $10^{-9}$ g/cc and increases as you go inwards. The quote and picture are consistent.
ProfRob
CC BY-SA 4.0
2019-07-20T08:13:01.917Z
The quote is open to misinterpretation, but it isn't incorrect if the photosphere is defined by that range of temperature.
ProfRob
CC BY-SA 4.0
2019-07-20T08:16:40.603Z
@RobJeffries I'd meant to delete that; I went off and made an edit, how does it look now?
uhoh
CC BY-SA 4.0
2019-07-20T05:10:53.120Z
2019-11-14T05:13:54.977Z
2020-06-17T09:47:40.653Z
uhoh
<p>The density is what you read from the graph, correctly.  Don't worry about what it says in that quote, it's all just a matter of what is meant by the "photosphere", a term that is rather vaguely defined and used to mean different things in different places.  You can see the problem in the temperatures used in that quote-- they correspond to what the graph considers to be entirely above the photosphere.  The quote seems to think of the photosphere as the region from the tau ~ 2/3 point to the minimum in the temperature, whereas the graph seems to think of the photosphere as something quite noticeably hotter.  Other places regard the photosphere as a shell of zero width, right where tau ~ 2/3.  It's all just the different ways the word is used, there's nothing to worry about.  The graph matches up density with height and temperature, so you can just use that-- and note that even that is a kind of average situation, the reality is much more complicated.</p>

<p>As for positive and negative heights, why would you care what point is getting called x=0?  It's completely arbitrary where the zero height is set, every different source could likely use a different meaning for "the top of the photosphere."</p>

CC BY-SA 4.0
2019-07-20T01:23:46.247Z
So you are sure it's 1E-06 g/cm^3 and not 1E-09 g/cm^3? Can you provide an authoritative, independent source to support your conclusion?
uhoh
CC BY-SA 4.0
2019-07-20T01:25:49.227Z
Your "why would you care what point is getting called x=0?" is inappropriate. A plot of density versus depth needs a reference point. The gradient is so steep in this region that it would be absurd to measure from the center of the Sun. Instead, the "surface" or "x=0" is a much better point of reference in this particular case.
uhoh
CC BY-SA 4.0
2019-07-20T01:34:33.113Z
have a look at [this answer](https://astronomy.stackexchange.com/a/30233/7982) where all data is referenced to the Sun's surface.
uhoh
CC BY-SA 4.0
2019-07-20T04:38:57.720Z
The location ofthe center of the Sun is of no interest here.  As for support for my statement, the graph itself shows where those two densities appear.  So it is simply different language about what constitutes the "photosphere."  To wit, the quoted statement is talking about the density at the temperature minimum, so it is clear they regard the photosphere as extending up to the bottom of the "chromosphere", which starts at the temperature minimum.  Also, there is no clear meaning to "the Sun's surface", because the Sun is a gas.   So it's all pure semantics, the graph says all you need.
Ken G
CC BY-SA 4.0
2019-07-20T04:54:19.490Z
I'm primarily looking for an answer to the question "What is the density profile within the Sun's photosphere?" The "Which one of these is wrong?" is less important; it's okay if they are both wrong. The accepted answer to my question will likely show or link to a mass density profile of the Sun's photosphere as defined by some model. "You don't want to know the answer to your question" or "Astrophysicists can't agree where the photosphere begins and ends" probably won't be accepted.
uhoh
CC BY-SA 4.0
2019-07-20T13:39:51.163Z
The graph gives the information you seek.  If your issue is that it is contained in a rather small space, then you seek a similar graph that expands that portion, look at other plots like https://www.researchgate.net/figure/Background-model-Dependence-with-height-above-the-photosphere-of-a-temperature-and-b_fig2_316640073 .  If it is a mathematical expression you seek, it is given by d rho /dr = -mg rho / kT which is solved by an exponential function that drops on the scale of H = kT/mg, where m is the average particle mass.
Ken G
CC BY-SA 4.0
2019-07-20T13:46:06.690Z
Besides, it is a simple fact that the term "photosphere" is vague and used differently in different places.  I can't help if you don't want to hear that.
Ken G
CC BY-SA 4.0
2019-07-20T01:12:42.390Z
2019-07-20T01:12:42.390Z
Ken G
<p>The Sun's photosphere contains the Sun's surface as defined by opacity = 2/3 point. I'd like to see the profile of mass density from bottom to top of the photosphere. I did a quick search and got confused.</p>

<p>The image below is found on the Wikipedia <a href="https://en.wikipedia.org/wiki/Photosphere" rel="nofollow noreferrer">Photosphere</a> page. If I trace the dotted line labeled "Density" to the photosphere layer and read the density axis at the top, I read something like <span class="math-container">$8 \text{ to } 3 \times 10^{-7} \text{ g/cm}^{-3}$</span>, which you could call <span class="math-container">$1 \times 10^{-6} \text{ g/cm}^{-3}$</span>. However, the <a href="https://en.wikipedia.org/wiki/Photosphere#Sun" rel="nofollow noreferrer">Sun section</a> of Wikipedia page where this image is shown says:</p>

<blockquote>
  <p>The Sun's photosphere has a temperature between 4,500 and 6,000 K (4,230 and 5,730 °C) (with an effective temperature of 5,777 K (5,504 °C)) <strong>and a density of about 1×10−6 kg/m3;</strong> increasing with depth into the sun.</p>
</blockquote>

<p>and links to the solar-center.stanford.edu page <a href="http://solar-center.stanford.edu/vitalstats.html" rel="nofollow noreferrer">The Sun's Vital Statistics</a> for the <span class="math-container">$1 \times 10^{-6} \text{ kg/m}^{-3}.$</span> Converting the units, this is only <span class="math-container">$1 \times 10^{-9} \text{ g/cm}^{-3}.$</span></p>

<p><strong>Question:</strong> Is it possible to clear up this disparity, and to see a plot of the density versus depth from the bottom to the top of the Sun's photosphere, which would likely contain both positive and negative heights above the Sun's surface?</p>

<hr>

<ul>
<li>Source: <a href="https://commons.wikimedia.org/wiki/File:Sun_Atmosphere_Temperature_and_Density_SkyLab.jpg" rel="nofollow noreferrer">File:Sun Atmosphere Temperature and Density SkyLab.jpg</a></li>
<li>Original source: <a href="https://history.nasa.gov/SP-402/p2.htm" rel="nofollow noreferrer">SP-402 A New Sun: The Solar Results From Skylab</a></li>
</ul>

<p><a href="https://i.sstatic.net/S24zQ.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/S24zQ.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2019-07-19T23:39:35.900Z
companion question: [At what depth below the Sun's surface does the density reach that of water?](https://astronomy.stackexchange.com/q/32729/7982)
uhoh
<the-sun><stellar-atmospheres><photosphere>
What is the density profile within the Sun's photosphere? Which one of these is wrong?
uhoh
# What is the density profile within the Sun's photosphere? Which one of these is wrong?

(Asked by: username_0 on 2019-07-19)

The Sun's photosphere contains the Sun's surface as defined by opacity = 2/3 point. I'd like to see the profile of mass density from bottom to top of the photosphere. I did a quick search and got confused.

The image below is found on the Wikipedia [Photosphere](https://en.wikipedia.org/wiki/Photosphere) page. If I trace the dotted line labeled "Density" to the photosphere layer and read the density axis at the top, I read something like $8 \text{ to } 3 \times 10^{-7} \text{ g/cm}^{-3}$, which you could call $1 \times 10^{-6} \text{ g/cm}^{-3}$. However, the [Sun section](https://en.wikipedia.org/wiki/Photosphere#Sun) of Wikipedia page where this image is shown says:

> The Sun's photosphere has a temperature between 4,500 and 6,000 K (4,230 and 5,730 °C) (with an effective temperature of 5,777 K (5,504 °C)) **and a density of about 1×10−6 kg/m3;** increasing with depth into the sun.

and links to the solar-center.stanford.edu page [The Sun's Vital Statistics](http://solar-center.stanford.edu/vitalstats.html) for the $1 \times 10^{-6} \text{ kg/m}^{-3}.$ Converting the units, this is only $1 \times 10^{-9} \text{ g/cm}^{-3}.$

**Question:** Is it possible to clear up this disparity, and to see a plot of the density versus depth from the bottom to the top of the Sun's photosphere, which would likely contain both positive and negative heights above the Sun's surface?

---

* Source: [File:Sun Atmosphere Temperature and Density SkyLab.jpg](https://commons.wikimedia.org/wiki/File:Sun_Atmosphere_Temperature_and_Density_SkyLab.jpg)
* Original source: [SP-402 A New Sun: The Solar Results From Skylab](https://history.nasa.gov/SP-402/p2.htm)

[![enter image description here](https://i.sstatic.net/S24zQ.jpg)](https://i.sstatic.net/S24zQ.jpg)

**Tags:** the-sun, stellar-atmospheres, photosphere

> **Comments:**
> **username_0**: companion question: [At what depth below the Sun's surface does the density reach that of water?](https://astronomy.stackexchange.com/q/32729/7982)  

---

## 2 answers

### Answer by username_0 on 2019-07-20. Score: 3 | Accepted answer

I usually don't answer my own questions, but sometimes when the question itself is [called into question](https://astronomy.stackexchange.com/a/32732/7982) I make an exception.

The density of the photosphere at $\tau\_{5000}=1$ is predicted to be $3 \times 10^{-7} \text{g/cm}^3$ in the Holweger-Müller Model Atmosphere7.

As pointed out [in comments](https://astronomy.stackexchange.com/questions/32727/what-is-the-density-profile-within-the-suns-photosphere-which-one-of-these-is/32737?noredirect=1#comment60449_32737) there is a spread in values here.

The 1E-06 g/cm^3 density value (plot) is more consistent with = 1 or "bottom" of the photosphere, while the density in the quote is more consistent with the cooler "top" of the photosphere (circa 4300 K).

From [Chapter 2: The Photosphere](https://people.smp.uq.edu.au/TimoNieminen/papers/thesis/chapter2_photosphere.pdf) of Timo Nieminen's thesis [Solar Line Asymmetries: Modelling the Effect of Granulation on the Solar Spectrum](https://people.smp.uq.edu.au/TimoNieminen/thesis.html)

[![density versus hight above the Sun's surface](https://i.sstatic.net/JS8JR.png)](https://i.sstatic.net/JS8JR.png)

[![density and other parameters versus hight above the Sun's surface](https://i.sstatic.net/s6zLZ.png)](https://i.sstatic.net/s6zLZ.png)

> Figure 2-3: The Holweger-Müller Model Atmosphere
>
> 7 Holweger, H. and Müller, E. A. “The Photospheric Barium Spectrum: Solar Abundance and Collision Broadening of Ba II Lines by Hydrogen”, Solar Physics 39, pg 19-30 (1974). Extra points have been cubic spline interpolated by J. E. Ross. The optical properties (such as the optical depth and the opacity) of a model atmosphere are, obviously, very important, and will be considered later. See table C-4 for complete details of the Holweger-Müller model atmosphere including all depth points used.
>
> 8**The height scale is not arbitrary.** The base of the photosphere (height = 0 km) is chosen to be at standard optical depth of one (i.e. 5000Å = 1 ).

> **Comments:**
> **username_1**: The HM atmosphere perfectly well shows that the "photosphere" covers a range of temperatures and densities. The density at the temperature minimum is around $10^{-9}$ g/cc and increases as you go inwards. The quote and picture are consistent.  
> **username_1**: The quote is open to misinterpretation, but it isn't incorrect if the photosphere is defined by that range of temperature.  
> **username_0**: @RobJeffries I'd meant to delete that; I went off and made an edit, how does it look now?  

---

### Answer by username_2 on 2019-07-20. Score: 0

The density is what you read from the graph, correctly. Don't worry about what it says in that quote, it's all just a matter of what is meant by the "photosphere", a term that is rather vaguely defined and used to mean different things in different places. You can see the problem in the temperatures used in that quote-- they correspond to what the graph considers to be entirely above the photosphere. The quote seems to think of the photosphere as the region from the tau ~ 2/3 point to the minimum in the temperature, whereas the graph seems to think of the photosphere as something quite noticeably hotter. Other places regard the photosphere as a shell of zero width, right where tau ~ 2/3. It's all just the different ways the word is used, there's nothing to worry about. The graph matches up density with height and temperature, so you can just use that-- and note that even that is a kind of average situation, the reality is much more complicated.

As for positive and negative heights, why would you care what point is getting called x=0? It's completely arbitrary where the zero height is set, every different source could likely use a different meaning for "the top of the photosphere."

> **Comments:**
> **username_0**: So you are sure it's 1E-06 g/cm^3 and not 1E-09 g/cm^3? Can you provide an authoritative, independent source to support your conclusion?  
> **username_0**: Your "why would you care what point is getting called x=0?" is inappropriate. A plot of density versus depth needs a reference point. The gradient is so steep in this region that it would be absurd to measure from the center of the Sun. Instead, the "surface" or "x=0" is a much better point of reference in this particular case.  
> **username_0**: have a look at [this answer](https://astronomy.stackexchange.com/a/30233/7982) where all data is referenced to the Sun's surface.  
> **username_2**: The location ofthe center of the Sun is of no interest here.  As for support for my statement, the graph itself shows where those two densities appear.  So it is simply different language about what constitutes the "photosphere."  To wit, the quoted statement is talking about the density at the temperature minimum, so it is clear they regard the photosphere as extending up to the bottom of the "chromosphere", which starts at the temperature minimum.  Also, there is no clear meaning to "the Sun's surface", because the Sun is a gas.   So it's all pure semantics, the graph says all you need.  
> **username_0**: I'm primarily looking for an answer to the question "What is the density profile within the Sun's photosphere?" The "Which one of these is wrong?" is less important; it's okay if they are both wrong. The accepted answer to my question will likely show or link to a mass density profile of the Sun's photosphere as defined by some model. "You don't want to know the answer to your question" or "Astrophysicists can't agree where the photosphere begins and ends" probably won't be accepted.  
> **username_2**: The graph gives the information you seek.  If your issue is that it is contained in a rather small space, then you seek a similar graph that expands that portion, look at other plots like https://www.researchgate.net/figure/Background-model-Dependence-with-height-above-the-photosphere-of-a-temperature-and-b_fig2_316640073 .  If it is a mathematical expression you seek, it is given by d rho /dr = -mg rho / kT which is solved by an exponential function that drops on the scale of H = kT/mg, where m is the average particle mass.  
> **username_2**: Besides, it is a simple fact that the term "photosphere" is vague and used differently in different places.  I can't help if you don't want to hear that.  

---

