1. Convection
John Crooke
3/26/2019

2. Stars We know stars emit energy
We know stars create that energy in their core via fusion
So how does that energy reach the surface?
Credit: Universe Today

3. Methods of Energy Transfer

4. In Stars Conduction Radiation Convection When particles collide
“Generally insignificant in most stars throughout the majority of their lifetimes” (Carroll & Ostlie 2017)
Radiation
Photons
Primary component in “radiative zone”
Convection
Moving fluids
Primary component in “convection zone”

5. Radiative/Convection Zones
M < ~0.3 solar masses:
Star is solely convective
~0.3 – ~1.2 solar masses:
Core  radiative zone  convection zone (Behrend & Maeder 2001)
Larger mass = larger radiative zone (Padmanabhan 2001)
Notice: similar to when p-p chain is fusion method
> ~1.2 solar masses:
Core  convection zone  radiative zone (Behrend & Maeder 2001)
Larger mass = larger convection zone (Martins et al. 2013)

6. Credit: Sun.org - www.sun.org, released under CC-BY-SA 3.0`
So why the difference?

7. Radiation
Radiative diffusion (Carroll & Ostlie , Prialnik 2005, Ryan & Norton )
High pressure and temperature creates an environment in which photons have an exceptionally short mean free path before scattering off of electrons
Energy travels through the radiative zone in a random walk
Mean free path decreases as opacity increases
When does this occur?

8. Radiation Pressure Gradient
To see how opacity affects energy transfer, we start with the radiation pressure gradient:
𝑑 𝑃 𝑟𝑎𝑑 𝑑𝑟 =− 𝜅 𝜌 𝑐 𝐿 𝑟 4𝜋 𝑟 2
Recall,

9. Radiation Pressure Gradient
Radiation pressure can be expressed as:
𝑃 𝑟𝑎𝑑 = 1 3 𝑎 𝑇 4
Taking the derivative gives:
𝑑 𝑃 𝑟𝑎𝑑 𝑑𝑟 = 4 3 𝑎 𝑇 3 𝑑𝑇 𝑑𝑟
So,
𝑑𝑇 𝑑𝑟 =− 3 4𝑎𝑐 𝜅 𝜌 𝑇 3 𝐿 𝑟 4𝜋 𝑟 2

10. Temperature Gradient 𝑑𝑇 𝑑𝑟 =− 3 4𝑎𝑐 𝜅 𝜌 𝑇 3 𝐿 𝑟 4𝜋 𝑟 2
So now we have a temperature gradient in the radiative zone:
𝑑𝑇 𝑑𝑟 =− 3 4𝑎𝑐 𝜅 𝜌 𝑇 3 𝐿 𝑟 4𝜋 𝑟 2
Notice that as the opacity increases, the steeper the temperature gradient is. This makes the radiative diffusion process take longer (to the point that it becomes ineffective) because of the decreased mean free path (Carroll & Ostlie 2017).
We need a different, more efficient, energy transfer method in areas with steep temperature gradients.

11. Convection
In all sorts of fluids, so it must apply to stars as well

12. Convection (a recap) Caused by temperature gradients
In a fluid, an increase in T leads to an increase in V
Increased V means decreased ρ
Less dense objects are more buoyant compared to more dense ones, so a less dense parcel of a fluid will rise in the medium until it loses enough energy to return to normal and sink
This is a cyclic process
Constantly moving energy throughout the fluid

13. Schwarzschild Criterion
Prialnik 2005
Suppose the blob at point 1 is slightly perturbed and rises to point 2
P2 < P1 in stars
Lower surrounding pressure at point 2 lets blob expand (assume adiabatically)
Blob then has new density, ρ*
If ρ* > ρ2 the blob sinks and the star is “stable”
If ρ* < ρ2 the blob rises and the star is “unstable”
Conditions of star

14. Conditions for Convection
Because of ideal gas laws, if the blob’s T is higher than the new surrounding T, it will continue to rise and the star is unstable (Bohm-Vitense 1993).
So again, the most efficient energy transfer method is determined by T
Convective instability (Bohm-Vitense 1993):
𝑑𝑇 𝑑 𝑃 𝑔, 𝑏𝑙𝑜𝑏 < 𝑑𝑇 𝑑 𝑃 𝑔, 𝑠𝑡𝑎𝑟
Convective stability (Prialnik 2005):
𝑑𝑃 𝑑𝜌 𝑠𝑡𝑎𝑟 < 𝑑𝑃 𝑑𝜌 𝑏𝑙𝑜𝑏

15. Changing Terms ∇= 𝑑 ln 𝑇 𝑑 ln 𝑃 𝑔 = 𝛾−1 𝛾
Convective instability becomes (Bohm-Vitense 1993, Pasetto et al ):
∇ 𝑏𝑙𝑜𝑏 < ∇ 𝑠𝑡𝑎𝑟
∇ 𝑠𝑡𝑎𝑟 = 3𝜋 𝐹 𝑟 𝜅 𝑔𝑟 𝑃 𝑔 16𝜎 𝑇 4 𝑔
We can then say that convection occurs when either:
𝐹 𝑟 increases (happens in hotter stars)
Increase 𝜅 𝑔𝑟 while keeping 𝑃 𝑔 large
Reaffirms what we arrived at earlier as to why radiation is not the energy transfer method

16. Mixing Length Theory Method of describing superadiabatic convection
𝑑𝑇 𝑑𝑟 𝑎𝑐𝑡𝑢𝑎𝑙 > 𝑑𝑇 𝑑𝑟 𝑏𝑙𝑜𝑏
𝑑𝑇 𝑑𝑟 𝑏𝑙𝑜𝑏 − 𝑑𝑇 𝑑𝑟 𝑎𝑐𝑡𝑢𝑎𝑙 >0
Difference in T between blob and surrounding gas as the blob rises a distance dr
𝛿𝑇= 𝑑𝑇 𝑑𝑟 𝑏𝑙𝑜𝑏 − 𝑑𝑇 𝑑𝑟 𝑎𝑐𝑡𝑢𝑎𝑙 𝑑𝑟=𝛿 𝑑𝑇 𝑑𝑟 𝑑𝑟
Mixing length l is just the distance the blob travels before it reaches equilibrium with surrounding environment(Bohm-Vitense 1993, Carroll & Ostlie 2017, Pasetto et al. 2016)

17. Mixing Length Theory 𝑙=𝛼 𝐻 𝑝 𝐻 𝑝 is the pressure scale height
𝐻 𝑝 = 𝑃 𝜌𝑔
𝛼 is a free parameter. Typically has values between 0.5 and 3 (Carroll & Ostlie 2017, Pasetto et al. 2016)
Substituting l as dr in the equation for 𝛿𝑇 we can calculate the heat flow per unit volume from the blob to the surrounding material:
𝛿𝑞= 𝐶 𝑃 𝛿𝑇 𝜌
By Nedtheprotist - Own work, Public Domain,

18. 𝑣 = 2𝛽 𝑓 𝑛𝑒𝑡 𝑙 𝜌 = 𝛽 𝑇 𝑔 𝑘 𝜇 𝑚 𝐻 𝛿 𝑑𝑇 𝑑𝑟 𝛼
Mixing Length Theory
With 𝛿𝑞 we can find the convective flux
𝐹 𝑐 =𝛿𝑞 𝑣 = 𝐶 𝑃 𝛿𝑇 𝜌 𝑣
𝑣 is the average velocity. To find it we must know the net force per unit volume acting on the bubble
𝑓 𝑛𝑒𝑡 = 1 2 𝜌𝑔 𝑇 𝛿𝑇 𝑓𝑖𝑛𝑎𝑙
With that,
𝑣 = 2𝛽 𝑓 𝑛𝑒𝑡 𝑙 𝜌 = 𝛽 𝑇 𝑔 𝑘 𝜇 𝑚 𝐻 𝛿 𝑑𝑇 𝑑𝑟 𝛼
𝛽 is another free parameter. It is used in finding the average value of 𝑣 2 and has a value between 0 and 1.

19. 𝐹 𝑐 =𝜌 𝐶 𝑃 𝑘 𝜇 𝑚 𝐻 2 𝑇 𝑔 3/2 𝛽 𝛿 𝑑𝑇 𝑑𝑟 3/2 𝛼
Mixing Length Theory
In its final form, the convective flux is
𝐹 𝑐 =𝜌 𝐶 𝑃 𝑘 𝜇 𝑚 𝐻 𝑇 𝑔 3/2 𝛽 𝛿 𝑑𝑇 𝑑𝑟 3/2 𝛼
Assuming you knew all the values and free parameters, you could use it in the equation for total flux
𝐹= 𝐹 𝑟𝑎𝑑 + 𝐹 𝑐 = 𝜎 𝑇 4 𝜋
Mixing length theory in this form is used in stellar modeling.

20. Comparison between mixing length theory and numerical simulations
Not perfect, as it involves two free parameters, but a good analytic approximation of a complex phenomenon

21. Conclusion
Radiation and convection are both important mechanisms for transporting energy from the core of a star to the surface
They operate in two different regimes dependent on thermal gradients and opacities
Relative placement of regions depend on stellar mass
Conditions for convection rise from simple assumptions of adiabatic “blobs” and gas laws
Mixing length theory is a decent analytic approximation of convection in stars and is an efficient way of creating stellar models
However, the process is inherently inaccurate due to the two free parameters
New theories, such as scale-free convection theory aim to fix this but do not currently fit numerical models as well (Pasetto et al. 2016)

22. References Behrend, R. & Maeder, A. 2001, A&A, 373, 190
Bohm-Vitense, E. 1993, Introduction to Stella Astrophysics, Vol. 2 (New York, NY: Cambridge University Press)
Carroll, B. & Ostlie, D. 2017, An Introduction to Modern Astrophysics (2nd ed.; New York, NY: Cambridge University Press)
LeBlanc, F. 2010, An Introduction to Stellar Astrophysics (West Sussex, United Kingdom: John Wiley and Sons, Ltd.)
Martins, F. et al. 2013, A&A, 554, A23
Ryan, S.G. & Norton, A.J. 2010, Stellar Evolution and Nucleosynthesis (New York, NY: Cambridge University Press)
Padmanabhan, T. 2001, Theoretical Astrophysics, Vol. 2 (New York, NY: Cambridge University Press
Pasetto, S. et al. 2016, MNRAS, 459, 3182
Prialnik, D. 2005, An Introduction to the Theory of Stellar Structure and Evolution (New York, NY: Cambridge University Press)