The Photosphere

5.1 Limb Darkening

Derivation 1: Eddington Approximation for Limb Darkening

The Sun appears brighter at its center than at its limb because we see deeper (hotter) layers when looking straight down than when looking at a tangent. The Eddington approximation provides an elegant analytical solution.

Step 1. The formal solution of the radiative transfer equation along a ray at angle \(\theta\) to the normal (\(\mu = \cos\theta\)) is:

where \(S(\tau)\) is the source function and \(\tau\) is the optical depth.

Step 2. In the Eddington approximation, the source function varies linearly with optical depth: \(S(\tau) = a + b\tau\). For LTE (local thermodynamic equilibrium),\(S = B_\nu(T)\), and \(T\) increases linearly with \(\tau\) to first order.

Step 3. Evaluating the integral:

This is the Eddington-Barbier relation: the emergent intensity at angle \(\theta\)equals the source function at optical depth \(\tau = \mu = \cos\theta\).

Step 4. The limb-darkening law is commonly written as:

where \(u\) is the limb-darkening coefficient. For the Eddington model,\(u = 3/5 = 0.6\). Observations give \(u \approx 0.56\) in the visible (wavelength-dependent), confirming the basic radiative transfer picture.

5.2 Granulation

Derivation 2: Rayleigh-Benard Convection and Granule Size

Solar granulation is the surface manifestation of convection. Bright granules (hot upflows) are surrounded by dark intergranular lanes (cool downflows). Typical granule diameter is ~1000 km with lifetimes of 5-10 minutes.

Step 1. Convective instability occurs when the Rayleigh number exceeds a critical value:

where \(\alpha\) is the thermal expansion coefficient, \(d\) is the layer depth,\(\nu\) is kinematic viscosity, and \(\kappa\) is thermal diffusivity.

Step 2. The preferred horizontal wavelength at onset is \(\lambda \approx 2\sqrt{2} \, d\). In the solar context, the pressure scale height at the photosphere sets the relevant depth:

Step 3. The granule diameter scales with a few pressure scale heights:

The convective velocity can be estimated from mixing-length theory:\(v_{\text{conv}} \sim (F / \rho)^{1/3} \approx 1\text{--}2\) km/s, consistent with spectroscopic measurements of Doppler shifts in granules. Supergranulation (~30,000 km) and mesogranulation (~5,000 km) represent larger-scale convective patterns.

5.3 Spectral Lines and the Curve of Growth

Derivation 3: Equivalent Width and Curve of Growth

The solar spectrum contains thousands of absorption lines (Fraunhofer lines). The curve of growth relates the equivalent width of a line to the column density of absorbers.

Step 1. The equivalent width is defined as the width of a rectangular perfect absorber that removes the same flux:

Step 2. The line optical depth for a Voigt profile (Gaussian core + Lorentzian wings):

where \(f_{\ell u}\) is the oscillator strength and \(\Delta\nu_D\) is the Doppler width.

Step 3. The curve of growth has three regimes:

The curve of growth is the principal tool for determining elemental abundances in the solar photosphere. Asplund et al. (2009) used 3D hydrodynamic model atmospheres to derive the current standard solar composition.

5.4 Fraunhofer Lines

Derivation 4: Line Formation in a Schuster-Schwarzschild Atmosphere

Joseph von Fraunhofer catalogued hundreds of dark lines in the solar spectrum in 1814. We can understand their formation using simple atmospheric models.

Step 1. In the Schuster-Schwarzschild model, a cooler absorbing layer sits above a blackbody continuum source. The emergent specific intensity is:

Step 2. In the line center, if \(\tau_\nu \gg 1\), \(I_\nu \to S_\nu\). If \(S_\nu < B_\nu(T_{\text{cont}})\) (cooler layer), we see an absorption line. The line depth relative to the continuum is:

Step 3. Notable Fraunhofer lines include:

5.5 Photospheric Temperature Structure

Derivation 5: Grey Atmosphere Temperature Profile

Step 1. For a grey (frequency-independent opacity) atmosphere in radiative equilibrium, the temperature is related to optical depth by:

Step 2. This is the Milne-Eddington relation. At \(\tau = 0\) (surface):\(T_{\text{surface}} = (1/2)^{1/4} T_{\text{eff}} \approx 0.841 T_{\text{eff}} \approx 4860\) K. At \(\tau = 2/3\): \(T = T_{\text{eff}} = 5778\) K.

Step 3. The photosphere spans from \(\tau \approx 10^{-4}\) to \(\tau \approx 1\), corresponding to a geometric thickness of only ~500 km (less than 0.1% of \(R_\odot\)). The temperature minimum of ~4400 K occurs about 500 km above \(\tau = 1\), marking the boundary with the chromosphere.

Numerical Simulation

5.6 Eddington Approximation for Limb Darkening — Full Derivation

We now derive the classic result \(I(\theta)/I(0) = \tfrac{2}{5}(1 + \tfrac{3}{2}\cos\theta)\) from first principles, starting with the radiative transfer equation.

The Transfer Equation

Step 1. Along a ray at angle \(\theta\) to the outward normal (\(\mu = \cos\theta\)), the monochromatic transfer equation reads:

where \(\tau\) is the vertical optical depth increasing inward, and\(S(\tau)\) is the source function.

Step 2. In LTE the source function equals the Planck function,\(S = B_\nu(T)\). The Eddington approximation assumes \(S\) is linear in \(\tau\):

Step 3. The formal solution of the transfer equation for outgoing radiation (\(\mu > 0\)) emerging from a semi-infinite atmosphere is:

Step 4. Substituting \(S = a + b\tau\) and evaluating term by term:

since \(\int_0^\infty e^{-\tau/\mu}\,d\tau = \mu\) and\(\int_0^\infty \tau\,e^{-\tau/\mu}\,d\tau = \mu^2\).

Connecting to the Grey Atmosphere

Step 5. For a grey atmosphere in radiative equilibrium, we showed\(T^4(\tau) = \tfrac{3}{4}T_{\text{eff}}^4(\tau + \tfrac{2}{3})\). Therefore:

So \(a = F/(2\pi)\) and \(b = 3F/(4\pi)\), giving:

Step 6. The disk-center intensity is \(I(0) = I(\mu=1) = 5F/(4\pi)\), so the normalised limb-darkening profile is:

At the limb (\(\theta = 90°\), \(\mu = 0\)) the intensity drops to\(2/5 = 40\%\) of disk center. This corresponds to a linear darkening coefficient\(u = 3/5 = 0.6\), close to the observed visible-light value of 0.56.

5.7 Granulation and Mixing-Length Theory

Convective Velocity from MLT

Mixing-length theory (MLT) provides an estimate of the convective velocity by assuming that a fluid parcel travels one mixing length \(\ell\) before thermalising with its surroundings.

Step 1. A parcel rising adiabatically through a superadiabatic background acquires a temperature excess:

where \(\delta = \nabla - \nabla_{\text{ad}}\) is the superadiabatic gradient and\(H_P\) is the pressure scale height.

Step 2. The buoyancy force on the parcel produces an acceleration\(a = g\,\delta T / T\). After travelling distance \(\ell\) the kinetic energy gained is \(\tfrac{1}{2}\rho v_{\text{conv}}^2 \sim \rho\,g\,(\delta T / T)\,\ell\), giving:

Step 3. The convective energy flux is \(F = \rho c_p\,\delta T\,v_{\text{conv}}\). Eliminating \(\delta T\) in favour of \(F\) gives the MLT convective velocity:

Step 4: Numerical estimate. At the photosphere:\(g = 274\) m/s\(^2\),\(\delta/T \sim 10^{-6}\) K\(^{-1}\),\(\ell \sim H_P \sim 150\) km,\(F \sim 6.3 \times 10^7\) W/m\(^2\),\(\rho \sim 3 \times 10^{-4}\) kg/m\(^3\),\(c_p \sim 1.3 \times 10^4\) J/(kg K):

Granulation Observables

Diagram: Photospheric Granulation and Spectral Line Formation

Advanced Simulation: Limb Darkening Profile and Granulation Power Spectrum