Stellar Plasmas

Derivation: Hydrostatic Scale Height

Step 1. In hydrostatic equilibrium along a radial magnetic field:

Step 2. Using the ideal gas law p = nk_BT for an isothermal corona:

Step 3. The solution is exponential with pressure scale height:

For T = 1.5 MK: H ≈ 75 Mm ≈ 0.1 R_⊙. This explains why the corona is geometrically extended — the hot plasma has a large scale height.

Derivation: Conductive Temperature Profile

Step 1. Spitzer thermal conductivity parallel to B is dominated by electrons:

Step 2. In a static, radiatively negligible corona, the energy equation reduces to:

Step 3. In spherical symmetry: d/dr(r²κ₀T^5/2 dT/dr) = 0, so r²T^5/2 dT/dr = const. Integrating:

This very gradual decline (T ∝ r^−2/7) explains why the corona remains hot out to several solar radii. The conductive heat flux at the base is:

Energy Loss Estimates

Radiative loss: The optically thin radiative loss function Λ(T) peaks near T ~ 10⁵ K (transition region lines) and has a broad minimum near coronal temperatures:

Required heating rates:

• Active regions: ~10⁴ W/m² (at the coronal base)
• Quiet Sun corona: ~300 W/m²
• Coronal holes: ~100 W/m² (mostly conducted back or carried away by wind)

Alfvén Wave Heating

Alfvén waves propagating from the photosphere carry an energy flux:

For photospheric values (δv ~ 1 km/s, B ~ 100 G, ρ ~ 10⁻⁴ kg/m³): F_wave ~ 10⁷ W/m² — more than sufficient, but the wave must be damped efficiently. In the corona, phase mixing provides dissipation on a length scale:

Nanoflare Heating (Parker Model)

Parker (1988) proposed that photospheric shuffling of magnetic footpoints braids coronal field lines, building up current sheets that dissipate via reconnection in many small "nanoflares" (E ~ 10²⁴ erg each). The energy input rate is:

where v_ph ~ 1 km/s is the photospheric shuffling velocity and L is the loop length. This gives a heating rate ~10³ W/m², consistent with quiet Sun requirements. The statistical signature would be a power-law distribution of flare energies dN/dE ∝ E^−α with α > 2, so that nanoflares dominate the total energy budget.

RTV Scaling Law

For a static coronal loop in radiative and conductive equilibrium, Rosner, Tucker & Vaiana (1978) derived the scaling:

where T_max is in K, p is pressure in dyn/cm², and L is the half-length in cm. For an active region loop (L ~ 50 Mm, n ~ 10¹⁵ m⁻³, T ~ 3 MK), this successfully predicts the observed scaling between loop length and temperature.

Derivation: Radiative Temperature Gradient

Step 1. The radiative diffusion equation relates the energy flux to the temperature gradient. In an optically thick medium, photons random-walk with mean free path l_mfp = 1/(κρ):

where u = aT⁴ is the radiation energy density (a = 4σ_SB/c).

Step 2. Differentiating u with respect to r:

Step 3. Setting F = L/(4πr²) and solving for dT/dr:

Derivation: Central Temperature (Virial Theorem)

Step 1. The virial theorem for a self-gravitating gas states:

where U is the thermal energy and Ω is the gravitational potential energy.

Step 2. For a uniform-density sphere: Ω = −3GM²/(5R). The average temperature is related to U by U = (3/2)Nk_BT̄ = (3/2)(M/m_p)k_BT̄:

Step 3. Solving for the mean temperature and using T_c ≈ 2T̄ (density weighting):

for solar values. This is close to the actual central temperature of the Sun (1.57 × 10⁷ K), confirming the virial estimate.

Reimers Mass-Loss Formula

For cool giant stars, Reimers (1975) proposed an empirical scaling:

where L, R, M are in solar units and η_R ~ 0.5 is an efficiency parameter. This gives Ṁ ~ 10⁻¹⁴ M_⊙/yr for the present Sun, increasing to ~10⁻⁸ M_⊙/yr on the red giant branch.

Derivation: CAK Radiation-Driven Wind

Step 1. Hot star winds are accelerated by absorption of photospheric UV photons in spectral lines. The line radiation force per unit mass is:

where σ_e is the electron scattering opacity and M(t) is the force multiplier — a dimensionless factor representing the enhancement due to line absorption. Castor, Abbott & Klein (CAK, 1975) parameterized it as:

where t is the Sobolev optical depth parameter, k ~ 0.1–0.4 and α ~ 0.5–0.7.

Step 2. The equation of motion becomes:

where Γ_e = σ_eL/(4πGMc) is the Eddington parameter for electron scattering.

Step 3. The CAK critical solution gives the mass-loss rate:

and the terminal velocity:

For an O5 V star: Ṁ ~ 10⁻⁶ M_⊙/yr, v_∞ ~ 2500 km/s.

Wind Momentum–Luminosity Relation

The modified wind momentum Ṁv_∞√R is observed to correlate tightly with luminosity:

with x ≈ 1/α ≈ 1.8 and D depends on spectral type. This provides a powerful extragalactic distance indicator.

Force-Free Field Classifications

Potential (α = 0): ∇ × B = 0, so B = −∇Ψ with ∇²Ψ = 0. Minimum-energy state for given boundary flux distribution.

Linear force-free (α = const): ∇²B + α²B = 0 (Helmholtz equation). The constant-α field has minimum energy for given helicity.

Nonlinear force-free (α = α(r)): Most general. B · ∇α = 0 implies α is constant along each field line. Must be solved numerically.

Derivation: Flux Tube Pressure Balance

Consider a magnetic flux tube of radius a embedded in an external pressure p_e. Radial pressure balance:

For a straight tube (B_φ = 0), integrating across the boundary:

A flux tube with stronger internal field must have lower internal gas pressure — the basis for sunspot darkening (Wilson depression).

CME: Loss of Equilibrium & Torus Instability

Coronal mass ejections (CMEs) can be understood as a loss of MHD equilibrium. A current-carrying flux rope (modeled as a torus of major radius R and minor radius a) experiences an outward "hoop force":

and is confined by the strapping external field B_ex(R). The torus instability occurs when the external field decreases faster than a critical rate:

This decay index criterion n ≥ 3/2 successfully predicts CME onset in both simulations and observations. CME speeds range from 200–3000 km/s with kinetic energies of 10²³–10²⁵ J.