Coronal radiative losses

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In

is about 1, and the radiation is thermal.

The

). The structure and dynamics of the .

The

. In active regions the energy flux is about 10⁷ erg cm⁻²sec⁻¹, in the quiet Sun it is roughly 8 10⁵ – 10⁶ erg cm⁻²sec⁻¹, and in coronal holes 5 10⁵ - 8 10⁵ erg cm⁻²sec⁻¹, including the losses due to the solar wind. [1] The required power is a small fraction of the total flux irradiated from the Sun, but this energy is enough to maintain the plasma at the temperature of million degrees, since the density is very low and the processes of radiation are different from those occurring in the photosphere, as it is shown in detail in the next section.

Processes of radiation of the solar corona

blackbody radiation

with the same radiative flux of the solar spectrum (yellow area).

The electromagnetic waves coming from the

).

The emitting plasma is almost completely ionized and very light, its density is about 10⁻¹⁶ - 10⁻¹⁴ g/cm³. Particles are so isolated that almost all the

.

In an optically-thin plasma the matter is not in thermodynamical equilibrium with the radiation, because collisions between particles and photons are very rare, and, as a matter of fact, the square root mean velocity of photons, electrons, protons and ions is not the same: we should define a temperature for each of these particle populations. The result is that the

blackbody radiation

, but it depends only on those collisional processes which occur in a very rarefied plasma.

While the

is sufficient to determine the temperature of the emitting plasma.

Some of these spectral lines can be forbidden on the Earth: in fact, collisions between particles can excite ions to metastable states; in a dense gas these ions immediately collide with other particles and so they de-excite with an allowed transition to an intermediate level, while in the corona it is more probable that this ion remains in its metastable state, until it encounters a photon of the same frequency of the forbidden transition to the lower state. This photon induces the ion to emit with the same frequency by stimulated emission. Forbidden transitions from metastable states are often called as satellite lines.

The

. The but not in the perpendicular plane. The of velocities at the temperature of line formation (thermal line broadening), while it is often larger than predicted. The widening can be due to : in this case the line width can be used to estimate the macroscopic velocity also on the Sun's surface, but with a great uncertainty. The magnetic field can be measured thanks to the line splitting due to the Zeeman effect.

Optically-thin plasma emission

The most important processes of radiation for an optically-thin plasma ^{[2] [3] [4]} are

• the emission in resonance lines of ionized metals (bound-bound emission);
• the radiative recombinations (free-bound radiation) due to the most abundant coronal ions;
• for very high temperatures above 10 MK, the bremsstrahlung (free-free emission).

Therefore, the radiative flux can be expressed as the sum of three terms:

${\displaystyle L_{r}=n_{e}\sum n_{l}C_{lk}h\nu _{lk}+L_{rec}+L_{brems}}$

where ${\displaystyle n_{e}}$ is the number of

per unit volume, ${\displaystyle n_{k}}$ the ion number density, ${\displaystyle h}$ the Planck constant, ${\displaystyle \nu }$ the frequency of the emitted radiation corresponding to the energy jump ${\displaystyle h\nu }$, ${\displaystyle C_{lk}}$ the coefficient of collisional de-excitation relative to the ion transition, ${\displaystyle L_{rec}}$ the radiative losses for plasma recombination and ${\displaystyle L_{brems}}$ the bremsstrahlung contribution.

The first term is due to the emission in every single spectral line. With a good approximation, the number of occupied states at the superior level ${\displaystyle n_{u}}$ and the number of states at the inferior energy level ${\displaystyle n_{l}}$ are given by the equilibrium between collisional excitation and spontaneous emission

${\displaystyle n_{l}n_{e}C_{lu}=n_{u}A_{ul}}$

where ${\displaystyle A_{lu}}$ is the transition probability of spontaneous emission.

The second term ${\displaystyle L_{rec}}$ is calculated as the energy emitted per unit volume and time when free electrons are captured from ions to recombine into neutral atoms (dielectronic capture).

The third term ${\displaystyle L_{brems}}$ is due to the electron scattering by protons and ions because of the

: every accelerated charge emits radiation according to classical electrodynamics. This effect gives an appreciable contribution to the continuum spectrum only at the highest temperatures, above 10 MK.

Taking into account all the dominant radiation processes, including satellite lines from metastable states, the emission of an optically-thin plasma can be expressed more simply as

${\displaystyle L_{r}=n_{e}n_{H}P(T)~~{W~m^{-3}}}$

where ${\displaystyle P(T)}$ depends only on the temperature. All the radiation mechanisms require collision processes and basically depend on the squared density (${\displaystyle n_{e}=n_{H}}$). The integral of the squared density along the line of sight is called the emission measure and is often used in X-ray astronomy. The function ${\displaystyle P(T)}$ has been modeled by many authors but with differences that depend strongly upon the assumed elemental abundances of the plasma, and of course on the atomic parameters and their estimation.

In order to calculate the radiative flux from an optically-thin plasma in a convenient analytic form, Rosner et al. (1978) ^[5] suggested a formula for P(T) (erg cm³ s⁻¹) as follows:

${\displaystyle P(T)\approx 10^{-21.85}~~~~~~~~~~~(10^{4.3}<T<10^{4.6}K)}$

${\displaystyle P(T)\approx 10^{-31}~T^{2}~~~~~~~~~(10^{4.6}<T<10^{4.9}K)}$

${\displaystyle P(T)\approx 10^{-21.2}~~~~~~~~~~~~(10^{4.9}<T<10^{5.4}K)}$

${\displaystyle P(T)\approx 10^{-10.4}~T^{-2}~~~~~~(10^{5.4}<T<10^{5.75}K)}$

${\displaystyle P(T)\approx 10^{-21.94}~~~~~~~~~~~(10^{5.75}<T<10^{6.3}K)}$

${\displaystyle P(T)\approx 10^{-17.73}~T^{-2/3}~~~(10^{6.3}<T<10^{7}K)}$

See also

References

Bibliography

• Güdel M (2004). "X-ray astronomy of stellar coronae" (PDF). Astron Astrophys Rev. 12 (2–3): 71–237.
  S2CID 119509015. Archived from the original
  (PDF) on 2011-08-11.
• Tucker W. H. (1977). Radiation processes in Astrophysics.
  ISBN 978-0262700108
  .