has to do with the excitation frequency of your plasma and its plasma frequency.
there is a cut-off density where the electromagnetic waves exciting the plasma get reflected. then you are overdense.
in the case of corona (DC, no frequency) I do not know.. there, it is more a temperature condition.
i fully agree with Dr. Lukas. regarding corona u can excite different types of corona , which will have a different density profile. Please be specific what u want to ask.
Regarding astrophysics and Solar corona:
Book of E. Priest "Solar Magnetohydrodynamic" contains description of the layers of solar atmosphere and you can get numbers from there.
It is not clear if you ask about astrophys. plasma or about plasma from other field (gas discharge, etc). Common meaning of the term corona may differ, I guess.
I attached typical scales for ideal plasma of the solar convective zone four you to get the idea (r and other units are cgs: cm, sec, gr). It shows that for spatial scale a i have ideal MHD case.
The NRL Plasma Formulary has a useful plot of plasma density and temperature showing different physical regimes.
Nornally, the plasma characteristics (EOS, collective properties, transport properties, etc.) depend on two dimensionless parameters: 1. The coupling parameter which is the Landau length divided by the Wigner-Seitz radius, 2. degeneracy parameter, which is the plasma temperatute divided by the Fermi energy. In astrophysics usually there are no quantum-mechanical effects, no degeneracy of the electron subsytem and the coupling parameter is small. This is why no correlations matter. But when the electron density is about that of solid state and/or the temperature is relatively low (but still the ionization level is high), both parameters are of the order of 1 or higher and correlations begin to influence the plasma properties: strongly coupled plasmas.
The first question when you wantto characterize something is "For what purpose?" If for example you wish to consider the "strength" of binary interactions between plasma particles, then the relevant quantity is the action integral, or, simply put average interaction*collision duration and the collision duration depends on velocities, i.e. temperature. If you want to access whether a laser pulse will enter the plasma then basically what you want is the plasma frequency. So you can have supersolid density plasma that is weakly coupled or more moderate electron densities with very low temperatures. So you might get more helpful answers if you stated your purpose.
On different plasma electromagnetic scales and dimensionless parameters in hot kinetics of electrons for moving (solar wind/laser) plasmas you can find formulas at http://www.vniitf.ru/images/zst/2012/s3/3-13.pdf
The characterization over-dense vs. under-dense is more or less subjective as, already, pointed out by the answers above. On the other hand, you might put into good use the characteristic parameters of your particular problem: For example a plasma is "dense and hot enough" to behave like a plasma if the Debye Length is much smaller than the characteristic length of your problem (the size of the container for example) in order to assure quasi-neutrality; in the opposite case you don't have a plasma but a set of individual particles.Now for a wave entering a plasma region, the plasma might be under-dense if its effect on the wave propagation is negligible or over-dense if it hinders propagation (Langmuir Frequency being the criterion). Last but not least, it might be more important to see if you need to use MHD or Two-Fluid or Kinetic approach. My favorite discussion on these is the old but good "A Physicist's ABC of Plasma Physics" by the late LA. ARTSIMOVICH which, now, is free on the Internet.(https://archive.org/details/APhysicistsAbcOfPlasmaPhysics)
In the ionosphere/magnetosphere ionization ducts exist which extend thousands of kilometers from one hemisphere to the other hemisphere along the the earth's magnetic field lines. The electron density in the duct is usually a few percent above (overdense) or a few percent below (underdense) the ambient density outside the duct. Overdense ducts trap Whistler waves. Underdense ducts trap high-frequency waves up to about 7 MHz.
Also, if the electron density is above the density at which a high-frequency wave gets reflected it can be referred to as overdense and if the electron density is below the density at which a high-frequency wave gets reflected it can be referred to as underdense.
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