====== Heat transfer in stars ======

The energy produced in the core of a star by nuclear reactions must be transported outward through its layers until it escapes as radiation from the surface.  
This **heat transfer** occurs mainly by two mechanisms — **radiative transfer** and **convective transfer** — each operating in different regions depending on the temperature, density, and opacity of stellar material.

===== Radiative transfer =====

Consider two spherical layers inside a star separated by one **mean free path**,

$$
\Delta r = (\kappa\rho)^{-1},
$$

where \(\kappa\) is the [[opacity]] in m\(^2\) kg\(^{-1}\), the effective cross-section for absorption and scattering per unit mass.  
Opacity represents the resistance of the medium to the flow of radiation — the larger the opacity, the shorter the mean free path.

{{:courses:ast301:radiative.png?nolink&550|Radiative energy transfer between two thin spherical layers.}}

Each layer emits radiation as a blackbody with flux \(\mathscr{F} = \sigma T^4\), where \(\sigma\) is the **Stefan–Boltzmann constant**.  
The inner layer, at radius \(r_1\), is hotter (\(T_1 > T_2\)) and therefore radiates more intensely than the outer one.  
The **net outward flux** is

$$
\mathscr{F} = \sigma T_1^4 - \sigma T_2^4,
$$

in units of W m\(^{-2}\).  
Multiplying this flux by the surface area \(4\pi r^2\) gives the **luminosity** through that spherical surface:

$$
L = -4\pi r^2 (\sigma T_1^4 - \sigma T_2^4)
      \approx -4\pi r^2 \Delta r \frac{d}{dr}(\sigma T^4),
$$

where the minus sign indicates that energy flows outward as temperature decreases with radius.  
Replacing \(\Delta r = (\kappa\rho)^{-1}\) and differentiating gives

$$
L = -16\pi\sigma r^2 \frac{T^3}{\kappa\rho} \frac{dT}{dr}.
$$

Rearranging,

$$
\frac{dT}{dr} = -\frac{1}{16\pi\sigma} \frac{\kappa\rho L}{T^3 r^2}.
$$

This simplified relation underestimates the true gradient by about 30%.  
A more accurate treatment yields the **radiative temperature gradient**:

----
\begin{equation}\label{4}
\dot{T}_r = \left.\frac{dT}{dr}\right\rvert_{\text{rad}}
           = -\frac{3}{64\pi\sigma} \frac{\rho\,L\,\kappa}{T^3 r^2}.
\end{equation}
----

This equation shows that high luminosity \(L(r)\) or large opacity \(\kappa\) requires a **steeper temperature gradient** to maintain the same energy flow.  
The dependence of \(\dot{T}_r\) on \(L\) and \(\kappa\) is analogous to **Ohm’s law**:

  * Luminosity \(L\) behaves like electric current \(I\),  
  * Opacity \(\kappa\) acts like resistance \(R\),  
  * Temperature difference \(\Delta T\) corresponds to potential difference \(\Delta V\).

Thus, radiative heat transport can be viewed as a kind of “thermal diffusion,” where higher resistance (opacity) requires a larger temperature drop to maintain the same energy flux.

===== Convective transfer =====

{{:courses:ast301:convective.png?nolink&500|Convective heat transport by rising and sinking gas bubbles.}}

Where radiation becomes inefficient—due to high opacity or steep gradients—energy is instead carried by **convection**.  
Hot gas bubbles rise from deeper layers, cool near the surface, and sink again, creating circular convection currents.  
On the Sun’s photosphere these appear as bright **granules**, each about \(1000\) km across, as shown below.

{{youtube>CCzl0quTDHw?large}}

This is a high-resolution video of the Sun’s surface taken by the Daniel K. Inouye Solar Telescope ([[wp>Daniel_K._Inouye_Solar_Telescope|DKIST]]).  
The granular pattern arises from the alternating hot upflows and cool downflows of convective motion.

{{:courses:ast301:bubbles.png?nolink&600|Rising hot bubbles and sinking cool bubbles in a convective layer.}}

To understand when convection occurs, consider two neighboring spherical layers at radii \(r_1\) and \(r_2\), and a small bubble of gas initially identical to its surroundings at \(r_1\).  
As it rises, the ambient pressure decreases, causing the bubble to **expand and cool adiabatically** (\(\delta Q = 0\)).  
If the bubble becomes cooler (and hence denser) than its surroundings at \(r_2\), it sinks and convection ceases.  
If instead it remains hotter and less dense, it continues to rise, sustaining convection.

The **criterion for convection** is therefore:

$$
\dot{T}_r = \left.\frac{dT}{dr}\right\rvert_{\text{rad}}
            > \left.\frac{dT}{dr}\right\rvert_{\text{ad}} = \dot{T}_a.
$$

In words:  
the **radiative temperature gradient** of the surroundings must be **steeper** than the **adiabatic gradient** of the bubbles.

Because \(\dot{T}_r \propto L\) while \(\dot{T}_a\) is independent of luminosity, **higher luminosity promotes convection**.

===== Adiabatic temperature gradient =====

Let us derive the expression for \(\dot{T}_a\).  
Starting from the thermodynamic identity

$$
\left.\frac{dT}{dr}\right\rvert_{\text{ad}}
 = \frac{\partial T}{\partial P}\frac{dP}{dr}
 + \frac{\partial T}{\partial S}\frac{dS}{dr}
 = \frac{dT}{dP}\frac{dP}{dr},
$$

since \(dS/dr = 0\) for an adiabatic process.  
For an ideal gas, \(PV^\gamma = \text{constant}\) where \(\gamma = C_P / C_V\).  
Differentiating \(PV = RT\) for one mole gives

$$
\frac{dT}{dP} = \left(1-\frac{1}{\gamma}\right)\frac{T}{P}.
$$

Using the hydrostatic equilibrium equation \(dP/dr = -\rho g\), we find

$$
\dot{T}_a = \left(1-\frac{1}{\gamma}\right)\frac{T}{P}(-\rho g).
$$

For a non-degenerate ideal gas \(P = \rho kT/m_{av}\), giving

----
\begin{equation}\label{5}
\dot{T}_a = \left.\frac{dT}{dr}\right\rvert_{\text{ad}}
           = -\left(1-\frac{1}{\gamma}\right)\frac{m_{av}}{k}\,g.
\end{equation}
----

This **adiabatic temperature gradient** is independent of luminosity and depends only on the local gravity \(g\) and gas composition through \(m_{av}\).

{{:courses:ast301:sun.webp?nolink&550|The Sun’s interior showing the radiative core and outer convective envelope.}}

In the Sun, **radiative transfer** dominates up to about \(0.7\,R_\odot\), while the outer \(0.3\,R_\odot\) is **convective**.  
In the outer layers, the luminosity \(L(r)\) remains nearly constant (energy generation occurs only in the core), so the gradient \(\dot{T}_r \propto T^{-3}\) becomes steep as temperature falls toward the surface.  
When \(|\dot{T}_r| > |\dot{T}_a|\), convection begins.

===== Radiative and convective zones in stars =====

  - **Low-mass stars** (\(<0.3\,M_\odot\)) are almost completely convective.  
  - **Solar-type stars** (~1 \(M_\odot\)) have a **radiative core** and **convective envelope**.  
  - **Massive stars** (>1.5 \(M_\odot\)) develop **convective cores** surrounded by **radiative envelopes**.

These internal zones strongly affect nuclear burning, element mixing, and stellar evolution.

===== Insights =====
  - Radiative diffusion follows \(\dot{T}_r \propto L\kappa / (T^3 r^2)\); greater opacity or luminosity steepens the temperature gradient.  
  - Convection starts when the radiative gradient exceeds the adiabatic one.  
  - The adiabatic gradient depends only on gravity and composition, not on luminosity.  
  - Radiative and convective zones determine where a star is mixed or stratified.  
  - The Sun’s outer layers provide a clear example of convective heat transfer.

===== Inquiries =====
  - Derive the radiative temperature gradient starting from the blackbody flux \(\mathscr{F}=\sigma T^4\).  
  - Explain physically why high opacity or luminosity promotes convection.  
  - How does the adiabatic gradient depend on composition and gravity?  
  - Describe the energy-transport structure of stars of different masses.  
  - What observational features of the Sun demonstrate convective motion?