Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 53 / 2

Use the notation $I_{\nu }(\mu ,\tau )$ throughout, with optical depth increasing inward and $0<\mu \le 1$ for an outward ray. This removes the change in argument order used in the printed surface-intensity notation. Multiplying the radiative transfer equation by its integrating factor givesIntegrating from the surface to a deeper boundary $b$ yieldsFor a semi-infinite atmosphere with no exponentially growing homogeneous term, the boundary term vanishes as $b\to \infty$. The formal solution of the radiative transfer equation is thereforeThe usual deep-atmosphere boundary condition is necessary; a first-order differential equation alone does not fix the emerging specific intensity.

For a linear source function, the zeroth and first exponential moments give $I_{\nu }(\mu ,0)=a_0+a_1\mu$. Equivalently $I_{\nu }(\mu ,0)=S_{\nu }(\tau =\mu )$, the Eddington-Barbier relation. If the source rises inward, rays near the limb sample cooler, shallower layers, producing limb darkening.

For the finite polynomial, put $u=t/\mu$ and integrate each term. Repeated integration by parts gives ${\int }_0^{\infty }{u}^j{e}^{-u}du=j!$, so the polynomial source function moments for emergent intensity giveThe limiting grazing-ray specific intensity is $a_0$. If the displayed polynomial is only a local Taylor approximation to a more general source, this formula applies to that approximation and a source-function remainder also contributes; a finite local expansion is not automatically an exact description at every optical depth.

For a constant source in an overlying layer of optical thickness $\tau$, the radial ray has the exact transfer relationExpanding the exponentials gives the constant source transfer through a thin layer:This finite-layer result does not require the specific intensity below the layer to equal its source function. The sign of $S_{\nu }-I_{\nu }(1,\tau )$ already distinguishes emission from absorption.

An absorption line forms when a bound-bound transition increases opacity at selected frequencies. In a photosphere whose temperature decreases outward, optical depth of order one at the line frequency occurs higher than at a nearby continuum frequency. In an absorption-dominated region in local thermodynamic equilibrium, the source is the Planck function; the cooler line-forming layer then emits less specific intensity than the deeper continuum-forming layer. The result is a dark spectral feature. Scattering and departures from LTE modify the source, so this temperature-gradient picture is a useful mechanism rather than a universal identity for every line.

Two distinct situations can produce emission lines. First, a heated chromosphere has an outward temperature rise, so an optically thick line can sample gas hotter than the continuum photosphere beneath it. Secondly, a hot optically thin circumstellar envelope or stellar wind emits line photons through collisional excitation or recombination, with no comparably bright background along all rays; its added line flux can exceed the underlying continuum. Winds of hot luminous stars can also give a mixture of emission and absorption across a Doppler-broadened line profile. These examples explain the change in specific intensity through either an enhanced source function or additional emitting material.

The high luminosity-to-mass ratio makes radiative acceleration and atmospheric departures from the simple approximations important. For an illustrative fully ionized hydrogen-rich composition with electron-scattering opacity ${\kappa }_e\simeq 0.34\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$, the stellar electron-scattering Eddington factor isThus electron scattering alone removes roughly half of the effective surface gravity. It is below the Eddington luminosity, so electron scattering alone does not prove that a hydrostatic atmosphere is impossible. Line opacity can provide substantially more radiative acceleration and can drive a stellar wind.

For a plane-parallel atmosphere, the geometrical condition is that the line-forming region be thin compared with the radius. In a simple isothermal photospheric estimate, the atmospheric scale height isThis is plane-parallel validity from pressure scale height. The mass and luminosity alone do not specify $R$ or the atmospheric temperature, so they cannot decide the geometry uniquely. For example, if $T_e=3\times 10^4\mathrm{K}$ and $\mu \simeq 0.62$, the luminosity relation gives $R\simeq 37R_{\odot }$ and $H/R\simeq 3\times 10^{-3}$. Plane-parallel geometry could then be adequate for deep weak photospheric lines. It would still fail for lines formed over an extended wind, and a cooler, more extended supergiant needs a separate geometrical assessment. This numerical example is conditional, not an additional datum about the stated star.

LTE requires collisions and local thermal processes to establish the relevant excitation and ionization populations at the local gas temperature. In the dilute outer layers of a luminous hot star, radiative transition rates can dominate collisions, and the radiation field arriving from other depths is not the local Planck function. A non-LTE stellar atmosphere should therefore use statistical equilibrium with radiative and collisional rates coupled to transfer. Deep layers may approach LTE, but surface ionization balances and abundance-sensitive spectral lines are particularly susceptible to departures. An LTE abundance can be systematically wrong even when a plane-parallel approximation is geometrically good.

A static atmosphere requires velocities to be negligible where the diagnostic lines form. Massive luminous stars commonly have line-driven outflows, so static models cannot represent wind-formed profiles or their mass density and velocity gradients. A hydrostatic approximation can still be useful below the sonic region if the chosen lines form there, provided radiative acceleration is included in the effective gravity. Large outflows call for spherical moving-atmosphere models rather than merely a changed abundance in a static model.

The three assumptions cannot be accepted on the quoted mass and luminosity alone. LTE is especially doubtful in the outer layers; wind-sensitive abundance diagnostics require non-LTE moving models, and geometry must be checked through atmospheric extension and line-formation depth. Reliable stellar chemical abundances should be supported by observed wind signatures, temperature and gravity constraints, and agreement between several suitable ionization stages or weak photospheric lines.