Editing Radiative zone (section)

==Stability against convection==
{{see also|Natural convection}}
The radiation zone is stable against formation of [[convection cell]]s if the density gradient is high enough, so that an element moving upwards has its density lowered (due to [[Adiabatic process|adiabatic expansion]]) less than the drop in density of its surrounding, so that it will experience a net [[buoyancy]] force downwards.

The criterion for this is:
::<math>\frac{\text{d}\,\log\,\rho}{\text{d}\,\log\, P} > \frac{1}{\gamma_{ad}}</math>
where ''P'' is the pressure, &rho; the density and <math>\gamma_{ad}</math> is the [[heat capacity ratio]].

For a homogenic [[ideal gas]], this is equivalent to:
::<math>\frac{\text{d}\,\log\,T}{\text{d}\,\log\, P} < 1-\frac{1}{\gamma_{ad}}</math>

We can calculate the left-hand side by dividing the equation for the temperature gradient by the equation relating the pressure gradient to the gravity acceleration ''g'':

: <math>\frac{\text{d}P(r)}{\text{d}r}\ =\ g\rho \ = \ \frac{G\,M(r)\,\rho(r)}{r^2}</math>
''M''(''r'') being the mass within the sphere of radius ''r'', and is approximately the whole star mass for large enough ''r''.

This gives the following form of the [[Schwarzschild criterion]] for stability against convection:<ref name="Pols2011"/>{{rp|64}}
::<math>\frac{3}{64\pi\sigma_B\,G} \frac{\kappa\,L}{M}\frac{P}{T^4} < 1-\frac{1}{\gamma_{ad}}</math>
Note that for non-homogenic gas this criterion should be replaced by the [[Paul Ledoux#Ledoux criterion|Ledoux criterion]], because the density gradient now also depends on concentration gradients.

For a [[polytrope]] solution with ''n''=3 (as in the Eddington stellar model for radiative zone), ''P'' is proportional to ''T''<sup>4</sup> and the left-hand side is constant and equals 1/4, smaller than the ideal [[monatomic gas]] approximation for the right-hand side giving <math>1-1/\gamma_{ad}=2/5</math>. This explains the stability of the radiative zone against convection.

However, at a large enough radius, the opacity &kappa; increases due to the decrease in temperature (by [[Kramers' opacity law]]), and possibly also due to a smaller degree of ionization in the lower shells of heavy elements ions.<ref>{{Cite journal |last1=Krief |first1=M. |last2=Feigel |first2=A. |last3=Gazit |first3=D. |date=2016-04-10 |title=Solar opacity calculations using the super-transition-array method |journal=[[The Astrophysical Journal]] |volume=821 |issue=1 |pages=45 |arxiv=1601.01930 |doi=10.3847/0004-637X/821/1/45 |doi-access=free |bibcode=2016ApJ...821...45K |issn=0004-637X}}</ref> This leads to a violation of the stability criterion and to the creation of the [[convection zone]]; in the sun, opacity increases by more than a tenfold across the radiative zone, before the transition to the convection zone happens.<ref>{{Cite journal |last1=Turck-Chièze |first1=Sylvaine |last2=Couvidat |first2=Sébastien |date=2011-08-01 |title=Solar neutrinos, helioseismology and the solar internal dynamics |journal=Reports on Progress in Physics |volume=74 |issue=8 |pages=086901 |arxiv=1009.0852 |doi=10.1088/0034-4885/74/8/086901 |pmid=34996296 |bibcode=2011RPPh...74h6901T |issn=0034-4885}}</ref>

Additional situations in which this stability criterion is not met are:
*Large values of <math>L(r)/M(r)</math>, which may happen towards the star core's center, where ''M''(''r'') is small, if nuclear energy production is strongly peaked at the center, as in relatively massive stars. Thus such stars have a convective core.
*A smaller value of <math>\gamma_{ad}</math>. For semi-ionized gas, where approximately half of the atoms are ionized, the effective value of <math>\gamma_{ad}</math> drops to 6/5,<ref name="Pols2011"/>{{rp|p=37}} giving <math>1-1/\gamma_{ad}=1/6</math>. Therefore, all stars have shallow convection zones near their surfaces, at low enough temperatures where ionization is only partial.