Why is sun's photosphere a million times less dense than air at the surface of the Earth?

Question

Sun's photosphere is about 19 times as hot as Earth's surface. (Source: http://solar-center.stanford.edu/vitalstats.html)

But Solar surface gravity is about 28 times terrestrial surface gravity.

So, shouldn't the density of gases at the photosphere of the sun be comparable to that on the surface of the Earth?

Why is the density of Sun's photosphere a million times lesser? (Source: http://solar-center.stanford.edu/vitalstats.html)

Essentially, what I am asking has been aptly worded in userLTK's answer, "how can the photosphere be so light and almost vacuum like under such high gravity (28 earth gravity)"?

Some of the answers have suggested that this may be because the Sun's photosphere is ionised. This doesn't seem to be accurate. The Hydrogen ionisation at optical depth = 1 is only 0.04%. (Source: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1968SoPh....3....5G&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf)

Answer 1

I think there is a misconception underlying your question. You write

But Solar surface gravity is about 28 times terrestrial surface gravity.

So, shouldn't the density of gases at the photosphere of the sun be comparable to that on the surface of the Earth?

It does sound natural that a stronger gravity would imply a thicker atmosphere, but no matter how strong the gravity is, there is a point where the atmosphere fades away almost to nothing. At the height of the ISS, earth's gravity is around 80% of the gravity at sea level. But yet the ISS is orbiting in the vacuum of space. Similarly, above Jupiter's atmosphere, the gravity is still more than twice that of earth, but there is still vacuum. With the sun, it just so happens that the part of the atmosphere that produces the light we see is close enough to the (imprecisely defined) edge of space that its density has gone down almost to nothing.

Certainly, if it were cooler, it would be denser. But some day the sun will be a white dwarf, about the size of the earth but with not that much less mass than it has now. Its gravity will be hugely strong, but there will still be a top of the atmosphere were it is less than a millionth of earth's atmosphere. As it cools, that atmosphere will cool so that it is no warmer than earth's atmosphere, but there will still be a point where the atmosphere is that thin. And even above that height, gravity will be much higher than even at the current surface of the sun.

Answer 2

In addition to the two answers above, I'll add that, first, your estimate of a million times seems wrong. These numbers are rough approximations of a fluid, not fixed volume, and I'd take them with a grain of salt too, but Wikipedia gives the density of the photosphere as about 2×10−4 kg/m^3. That's about 1/6,000 (not a millionth) of Earth's atmospheric density at the surface (1.2 kg/m^3).

It's not clear if that density figure is an average for the entire photosphere or closer to the surface where we can measure, but if we measure the entire Earth's atmosphere, the density of our atmosphere drops significantly (how much depends on where you determine the top of the atmosphere is), but that's a problem, there's no absolute boundary, so comparing density is a futile exercise, but the ratio drops to far less than 6,000 to 1, when you take the density of Earth's entire atmosphere. You could also compare the photosphere to Earth's Mesosphere and the photosphere probably becomes more dense, but no matter how you compare, it's always going to be apples to oranges and rather pointless.

The photosphere is about 500 km thick. I don't trust the numbers enough to calculate the pressure at the bottom of the photosphere with any accuracy. If you take a 500,000 meter column at the above density, .0002 kg/m^3, that's 100 kg or 220 lbs per column, worked out to square inches (PSI), .14 psi - 1/100th the pressure on the surface of Earth - but these numbers are terrible and prone to high inaccuracy. I just put this out there to show that the pressure and density at the bottom of the photosphere is still low, but not as low as 1 part in 6000 of Earth's surface.

The gist of your question is, how can the photosphere be so light and almost vacuum like over 500 km thickness under such high gravity (28 earth gravity), and that's a fair question. The answer, as others have pointed out is due to the high temperature and content that's mostly ionized hydrogen in a plasma state. Plasma is a different state of matter than gas and it tends to be much more spread out. The outward pressure of photons may be a key factor too (I'm not 100% sure on that point).

Answer 3

The Sun's photosphere is roughly a 400 km thick layer below which almost photons do not escape and above which almost all photons with an outward direction do escape the Sun. Even a fairly diffuse plasma (compared to Earth surface pressure) does a fairly good job of absorbing thermal photons. A somewhat diffuse plasma (e.g., the bottom of the photosphere) does an exceptionally good job of absorbing thermal photons.

The somewhat high temperatures (4400 kelvins or more) in the Sun's photosphere means that a good portion of the gas (mostly hydrogen and helium, plus some trace elements) is ionized. The much lower temperatures (~300 kelvins) in Earth's troposphere means that essentially none of the gas (mostly molecular nitrogen and oxygen, plus some trace compounds) is ionized. The very different temperatures and very different makeup mean that the Sun's photosphere and the Earth's troposphere are incomparable.

Answer 4

Where the photosphere lies is not only a function of density $\rho$, but also of opacities $\kappa$. One will find it at a particular wavelength wherever the integral optical depth integral $\tau = \int \rho \; \kappa \; dz$ along the line of sight $z$ becomes one.
As the opacities are a strong function of the composition of the atmosphere, it is insufficient to just take gravity and temperature and try to predict the location of the photosphere.

Answer 5

The density of the solar atmosphere decreases with radius. The position of the solar photosphere is defined by where the optical depth (measured inwards) reaches unity. There is no direct connection between pressure/density and gravity; only between the pressure gradient and gravity. $$\frac{dP}{dr}= -\rho g$$ A larger gravity merely increases the pressure and density gradient over what it would be in a lower gravitational field at the same density.

The photosphere occurs at the density and temperature it does, because that is where the optical depth reaches unity. The species that dominates the opacity at these temperatures is the H$^{-}$ ion (hydrogen with an extra electron).

The Earth's atmosphere is largely neutral and even at higher densities has a much lower opacity at visible wavelengths.