# Are sunspots vertically displaced from the surrounding photospheric plasma?

While reading a paper on the helioseismology of sunspots (Cameron, et al.), very brief reference is made to the "vertical displacement of a blob of plasma." This caused me to wonder if the emissive surface of the sunspot itself were perhaps depressed from the remainder of the photosphere. If this does occur, is there any literature on mechanisms? I would be very interested to learn more. There was one paper I came across around formation of sunspots from a toroidal field (Parker); but, it seemed to focus more on the vertical displacement of magnetic field lines rather than the plasma itself.

Thanks.

• You sure that's referring to the sunspot itself, rather than a "side-effect" ? IIRC, temperature, magnetic, etc. steep gradients lead to a 'burp' of plasma being ejected at high speed. – Carl Witthoft Nov 9 '16 at 15:09

# Yes.

The idea that sunspots are depressed slightly came as a possible explanation for the Wilson effect. The Wilson effect was discovered as the shape of sunspots as viewed from Earth changed as the Sun rotates, in a way consistent with the change in perspective looking onto a slightly depressed region. While this isn't the only explanation for the effect, it's certainly the most prevalent.

More specifically, as Solanki (2003) writes, the depressions indicate a lowering of the layer where the optical depth $$\tau=1$$ (keep in mind that the bottom of the photosphere is the layer where $$\tau=2/3$$). There are two causes mentioned: lower temperature and magnetic effects.

Sunspots are cooler than the surrounding areas, as is well known, and thus appear darker. We see the same thing apparent at the boundaries of solar granules: Cooler, darker gas sinks and lets the hotter gas (which is less dense) move upward. Additionally, the opacity $$\kappa$$ is temperature-dependent, which may impact how far one can see into the star.

Not only is temperature a factor, but so is the magnetic field. Sunspots are, at heart, a magnetic phenomenon, and thus the radial force equation is substantially different. Normally, in a star, the equation of hydrostatic equilibrium is $$\frac{\mathrm{d}P}{\mathrm{d}r}=-\rho g$$ for pressure $$P$$, density $$\rho$$ and gravitational acceleration $$g$$. However, when the magnetic field becomes important in a sunspot on the solar surface, the force balance is $$\frac{\mathrm{d}P}{\mathrm{d}r}=\frac{B_z}{4\pi}\left(\frac{\mathrm{d}B_r}{\mathrm{d}z}-\frac{\mathrm{d}B_z}{\mathrm{d}r}\right)$$ where $$r$$ and $$z$$ are the radial and vertical coordinates (note the change of coordinate system - $$r$$ is along the surface, and $$z$$ is perpendicular to it!). The force from the magnetic field implies a lower gas pressure and a greater depression.

• Astounding knowledge! – Fattie Nov 9 '16 at 18:49
• Thanks for this! The paper from Solanki is exactly what I was looking for. – user14781 Nov 9 '16 at 20:57
• Is Kramer's law relevant to the photosphere? – Rob Jeffries Nov 9 '16 at 21:47
• It was a genuine question. I actually don't know how bulk opacity varies with temperature at a few thousand K, but molecules and atmosphere are more opaque than ions, so I'm sure opacity sharply increases with decreasing temperature. – Rob Jeffries Nov 9 '16 at 21:59
• @RobJeffries (I deleted my previous comment; $H_{-}$ ionization might not be as dominant in relatively cool sunspots as I thought.) I couldn't tell you. My knowledge of stellar atmospheric models is minimal, especially at such low temperatures. I've always just naively assumed Kramer's law is valid roughly up to the photosphere or beyond, which I now don't think it is. – HDE 226868 Nov 9 '16 at 22:08