On Dec 22nd I said I'd do some sums to estimate the reaction rates in the corona, so here we go. There are many simplifying assumptions (eg I only consider PP I chain, not PP II, PP III or CN), however the orders of magnitude will not be too far off.
The first step of the pp chain is mediated by the weak force and so does not occur readily but once it occurs the deuterium product of the first step will readily react with a proton to form 3He.
Looking at the core of the Sun, say, within ~0.3 solar radius:
Radius of core = 1.75 x 10^8 m
Volume of core = 2.24 x 10^25 m^3
Power output of core 3.9 x 10^26 W
Average density of power output = 17 W.m^-3
But the power output is greatest at the centre and falls to near zero on the edge of the core, so in the centre we can estimate a rate at least 10x the average: 170 W.m^-3
Each pp reaction to 4He releases ~25MeV of energy = 4 x 10^-12 J
Therefore the reaction rate for the first step in the centre is 2 x 4.25 x 10^13 m^-3.s^-1 = 8.5 x 10^13 m^-3.s^-1 (It takes two first step reactions to end in one 4He)
The reaction rate goes as the fourth power of temperature (at least in the temperature region we are interested i), and as the particle density.
The temperature in the centre = 1.5 x 10^7 K
The temperature in the quiet corona = 1 x 10^6 K and in the active corona 3 x 10^6 K. Let's take the latter.
The coronal reaction rate is thus 625 times less than the core on the basis of temperature.
The density in the centre is 1.5 x 10^5 kg.m^-3 of which one in ten nucleons is a unbound proton. So the density of protons is 1.5 x 10^4 kg.m^-3 or 9.4 x 10^30 protons m^-3.
The density in the corona is 10^15 protons m^-3.
The coronal reaction rate goes as (9.4 x 10^30/10^15) = 9.4 x 10^15 times less than in the core on the basis of density.
Therefore, the total rate in the corona is 5.9 x 10^18 less than in the core.
Thus the rate of pp reactions in the corona is 8.5 x 10^13/5.9 x 10^18 = 1.4 x 10^-5 s^-1.m^-3.
If we estimate the depth of the corona to be 50,000km (density drops by almost an order of magnitude by 70,000km) or 5 x 10^7m then its volume is 3.2 x 10^26 m^3 (this is generous as the corona becomes less dense, resulting in lower reaction rate with distance above the surface).
The total pp reactions in the entire corona would therefore be 3.2 x 10^26 x 1.4 x 10^-5 = 4.5 x 10^21 s^-1.
A complete pp chain through all three steps to 4He releases ~25MeV, but in the corona, the last reaction in the pp chain (3He + 3He > 4He + 1p + 1p + 12.8MeV) is unlikely because of the low density of 3He, so each pp chain stops after step 2 (the proton deuterium fusion) and results in a total 6.9MeV or 1.1 x 10^-12J.
The power from the pp chain in the entire corona is therefore ~4.5 x 10^21 x 1.1 x10^-12 = 5 x 10^9W (number of reactions per second times energy per reaction). So fusion in the corona is a completely insignificant part of the Sun's power output.
The power density in the corona is 1.4 x 10^-5 x 1.1 x 10^-12 = 1.5 x 10^-17 W.m^-3 (reactions per second per cubic meter times energy per reaction).
There is an interesting paper which proposes fusion in coronal flares: Ying-Zhi Zhang 2020 Res. Astron. Astrophys. 20 026,
https://arxiv.org/pdf/1912.00193.pdf. He models a coronal flare using MHD and then calculates the fusion rate and the contribution of fusion to the heating of the flare. He comes up with a energy production of 3 x 10^-7 J.m^-3 and a total production of 27,000 m^-3 reactions in an event that lasts 9,900s. That is equivalent to 10^-11W.m^-3, which is 6.7 x 10^5 times greater than the number I came up with above as an average for the entire corona. His modelling indicates that the centre of the flare will average over time about 2 x 10^7K and have a density of 1.25 x 10^16 particles m^-3 (cf my assumptions for the average across the corona are 3 x 10^6K and 10^15 particles m^-3 which taken together give a factor in reaction rate of 2.5 x 10^4, so we agree within an order of magnitude or so).
This woud be an explanation for the gamma production, the abundance of 3He and the positron annihilation signature in the corona, particukarly at flares.
On Dec 22nd I said I'd do some sums to estimate the reaction rates in the corona, so here we go. There are many simplifying assumptions (eg I only consider PP I chain, not PP II, PP III or CN), however the orders of magnitude will not be too far off.
The first step of the pp chain is mediated by the weak force and so does not occur readily but once it occurs the deuterium product of the first step will readily react with a proton to form 3He.
Looking at the core of the Sun, say, within ~0.3 solar radius:
Radius of core = 1.75 x 10^8 m
Volume of core = 2.24 x 10^25 m^3
Power output of core 3.9 x 10^26 W
Average density of power output = 17 W.m^-3
But the power output is greatest at the centre and falls to near zero on the edge of the core, so in the centre we can estimate a rate at least 10x the average: 170 W.m^-3
Each pp reaction to 4He releases ~25MeV of energy = 4 x 10^-12 J
Therefore the reaction rate for the first step in the centre is 2 x 4.25 x 10^13 m^-3.s^-1 = 8.5 x 10^13 m^-3.s^-1 (It takes two first step reactions to end in one 4He)
The reaction rate goes as the fourth power of temperature (at least in the temperature region we are interested i), and as the particle density.
The temperature in the centre = 1.5 x 10^7 K
The temperature in the quiet corona = 1 x 10^6 K and in the active corona 3 x 10^6 K. Let's take the latter.
The coronal reaction rate is thus 625 times less than the core on the basis of temperature.
The density in the centre is 1.5 x 10^5 kg.m^-3 of which one in ten nucleons is a unbound proton. So the density of protons is 1.5 x 10^4 kg.m^-3 or 9.4 x 10^30 protons m^-3.
The density in the corona is 10^15 protons m^-3.
The coronal reaction rate goes as (9.4 x 10^30/10^15) = 9.4 x 10^15 times less than in the core on the basis of density.
Therefore, the total rate in the corona is 5.9 x 10^18 less than in the core.
Thus the rate of pp reactions in the corona is 8.5 x 10^13/5.9 x 10^18 = 1.4 x 10^-5 s^-1.m^-3.
If we estimate the depth of the corona to be 50,000km (density drops by almost an order of magnitude by 70,000km) or 5 x 10^7m then its volume is 3.2 x 10^26 m^3 (this is generous as the corona becomes less dense, resulting in lower reaction rate with distance above the surface).
The total pp reactions in the entire corona would therefore be 3.2 x 10^26 x 1.4 x 10^-5 = 4.5 x 10^21 s^-1.
A complete pp chain through all three steps to 4He releases ~25MeV, but in the corona, the last reaction in the pp chain (3He + 3He > 4He + 1p + 1p + 12.8MeV) is unlikely because of the low density of 3He, so each pp chain stops after step 2 (the proton deuterium fusion) and results in a total 6.9MeV or 1.1 x 10^-12J.
The power from the pp chain in the entire corona is therefore ~4.5 x 10^21 x 1.1 x10^-12 = 5 x 10^9W (number of reactions per second times energy per reaction). So fusion in the corona is a completely insignificant part of the Sun's power output.
The power density in the corona is 1.4 x 10^-5 x 1.1 x 10^-12 = 1.5 x 10^-17 W.m^-3 (reactions per second per cubic meter times energy per reaction).
There is an interesting paper which proposes fusion in coronal flares: Ying-Zhi Zhang 2020 Res. Astron. Astrophys. 20 026, https://arxiv.org/pdf/1912.00193.pdf. He models a coronal flare using MHD and then calculates the fusion rate and the contribution of fusion to the heating of the flare. He comes up with a energy production of 3 x 10^-7 J.m^-3 and a total production of 27,000 m^-3 reactions in an event that lasts 9,900s. That is equivalent to 10^-11W.m^-3, which is 6.7 x 10^5 times greater than the number I came up with above as an average for the entire corona. His modelling indicates that the centre of the flare will average over time about 2 x 10^7K and have a density of 1.25 x 10^16 particles m^-3 (cf my assumptions for the average across the corona are 3 x 10^6K and 10^15 particles m^-3 which taken together give a factor in reaction rate of 2.5 x 10^4, so we agree within an order of magnitude or so).
This woud be an explanation for the gamma production, the abundance of 3He and the positron annihilation signature in the corona, particukarly at flares.