# What is the density profile within the Sun's photosphere? Which one of these is wrong?

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 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 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 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?

I usually don't answer my own questions, but sometimes when the question itself is called into question 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 there is a spread in values here.

The 1E-06 g/cm^3 density value (plot) is more consistent with 𝜏 = 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 of Timo Nieminen's thesis Solar Line Asymmetries: Modelling the Effect of Granulation on the Solar Spectrum

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.

8The 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. 𝜏 5000Å = 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. – ProfRob Jul 20 '19 at 7:32
• The quote is open to misinterpretation, but it isn't incorrect if the photosphere is defined by that range of temperature. – ProfRob Jul 20 '19 at 8:13
• @RobJeffries I'd meant to delete that; I went off and made an edit, how does it look now? – uhoh Jul 20 '19 at 8:16

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."

• 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 Jul 20 '19 at 1:23
• 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 Jul 20 '19 at 1:25
• have a look at this answer where all data is referenced to the Sun's surface. – uhoh Jul 20 '19 at 1:34
• 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 Jul 20 '19 at 4:38
• 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 Jul 20 '19 at 4:54